by Vahid Sarhangian - University of Toronto T-Space · 2016-02-19 · Acknowledgements Iwould liketoexpress my sincere gratitudetomy advisors, Professors OdedBerman, Opher Baron,

Managing Perishability in Service Operations

by

Vahid Sarhangian

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Rotman School of ManagementUniversity of Toronto

c© Copyright 2015 by Vahid Sarhangian

Abstract

Managing Perishability in Service Operations

Vahid Sarhangian

Doctor of Philosophy

Graduate Department of Rotman School of Management

University of Toronto

2015

We study three problems in service operations where either the supply or demand is per-

ishable. In the first chapter, we study a perishable inventory system for items whose quality

deteriorates in time, e.g., blood products. Specifically, we assume that supply and demand

are driven by independent Poisson processes, units have a constant shelf-life, and unsatisfied

demand is lost. We consider a threshold-based allocation policy that trades off the age and

availability of allocated units. We characterize the sojourn time distribution of units in inven-

tory and evaluate the performance of the threshold policy in terms of the distribution of the

age of allocated units and the proportion of outdates and lost demand. Our numerical results

demonstrate the importance of system parameters on the performance of the policy and identify

important properties of the distribution of the age of allocated units.

In the second chapter, we study the performance of certain practical ordering and allocation

policies in reducing the age of transfused blood in hospitals while keeping the outdate and

shortage rates low. We develop a data-driven (evidence-based) simulation model based on the

operations of the blood bank of an acute care hospital in Hamilton, Ontario. We use empirical

data to validate our model and estimate its inputs. The results suggest that by properly

adjusting the ordering and allocation policies at the hospital level, a significant reduction of

issue age could be achieved, without compromising availability or resulting in excessive outdates.

In the third chapter, we study the rational abandonment behavior of utility-maximizing

customers in the context of an observable priority queue, and identify novel pricing implications.

We first characterize the equilibrium abandonment strategy of low-priority customers. We then

consider pricing as a means to control the abandonment behavior and investigate its implications

on system welfare and firm revenue. A distinguishing feature of our model is that in the presence

ii

of abandonment the timing of payment matters. We show that the welfare can be maximized

using only a single fee charged upon service completion. In contrast, revenue maximization

generally requires a combination of both an entrance and a service fee.

iii

Dedication

To my parents.

iv

Acknowledgements

I would like to express my sincere gratitude to my advisors, Professors Oded Berman, Opher

Baron, and Philipp Afeche for their guidance, help and support throughout the course of my

PhD studies. I have been extremely fortunate to have the opportunity to learn from and

collaborate with them.

I would also like to thank my committee members, Professors Gonzalo Romero and Dmitry

Krass, and the external appraiser, Professor Ramandeep Randhawa.

I am indebted to Professor Jeffery S. Rosenthal whom I had the privilege of having as the

instructor in four graduate courses that greatly impacted my interest in and understanding of

probability theory and stochastic processes.

Finally, I thank all my friends and fellow PhD students (past and present) at Rotman

School of Management. Especially, I am grateful to my friend and colleague, Professor Hossein

Abouee-Mehrizi for helping me getting access to the data used in Chapter 3, his support, and

always being available to discuss research problems.

v

Contents

1 Introduction 1

2 Threshold–Based Allocation Policies for Perishable Inventory Systems 4

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Review of the FIFO and LIFO Policies . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 The LIFO Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 The FIFO Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Additional Results for the FIFO Policy . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 The Threshold Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6.1 Obtaining the Performance Measures . . . . . . . . . . . . . . . . . . . . . 20

2.6.2 Sojourn Time of Units in Stage 2 . . . . . . . . . . . . . . . . . . . . . . . 21

2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Concluding Remarks and Future Research . . . . . . . . . . . . . . . . . . . . . . 32

2.9 Appendix A: Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.10 Appendix B: MDP formulation for finding the optimal policy . . . . . . . . . . . 47

3 Reducing the Age of Transfused Red Blood Cells in Hospitals 48

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 A data-driven simulation model of RBC inventory . . . . . . . . . . . . . . . . . 50

3.3.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.2 Estimating the model inputs using data . . . . . . . . . . . . . . . . . . . 50

3.3.3 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

vi

3.4 Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Results of the simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5.1 Trade–off between the average issue age and the outdate rate . . . . . . . 56

3.5.2 Distribution of the issue age . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.3 Order size variability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5.4 Type distribution among outdated units . . . . . . . . . . . . . . . . . . . 61

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Rational Abandonment from Priority Queues 65

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 The Equilibrium Strategy of Low-priority Customers . . . . . . . . . . . . . . . 71

4.4.1 Structure of the Equilibrium Strategy . . . . . . . . . . . . . . . . . . . . 71

4.4.2 Explicit Characterization of the Equilibrium Thresholds and Customer

Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.5 Pricing under Rational Abandonment: Preliminaries . . . . . . . . . . . . . . . . 76

4.5.1 Equilibrium Behavior of Low-Priority Customers in the Presence of Pricing 77

4.5.2 Steady-State System Performance Measures . . . . . . . . . . . . . . . . . 78

4.6 Pricing for Welfare Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.1 Welfare Maximization Requires Equal Balking and Abandonment Thresh-

olds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6.2 Welfare-Maximizing Pricing Requires a Service Fee and No Entrance Fee 81

4.7 Pricing for Revenue Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.7.2 Revenue-Maximizing Pricing Requires Service Fee and Entrance Fee . . . 83

4.7.3 Maximizing Revenues with a Single Fee: Charge for Entrance or Service

Completion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.9 Appendix A: Proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.10 Appendix B: Additional Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Bibliography 114

vii

List of Tables

3.1 ABO/Rh substitution rule inferred from data. . . . . . . . . . . . . . . . . . . . . 53

3.2 The output of simulation (baseline ) versus empirical data. . . . . . . . . . . . . 54

3.3 ABO/Rh compatibility in order of preference used in simulation scenarios when

an exact match is not available. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

viii

List of Figures

2.1 A typical sample path of the process W . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 A two–stage representation of the system under the threshold policy . . . . . . . 19

2.3 An illustration of random variables R, Ii, Zi, Uk on the time line . . . . . . . . . 21

2.4 The trade–off curves under the threshold policy for different supply-to-demand

ratios and demand rates. From left to right the tick marks on each curve corre-

spond to threshold values of 40 (FIFO), 28, 24, 20, 16, 12, 8, 4, and 0 (LIFO). . 27

2.5 The proportion of products allocated from Stage 2. . . . . . . . . . . . . . . . . . 28

2.6 The conditional distribution of the age of products given that they are allocated

from Stage 2 for a system with λ = μ = 1 and threshold value (a) T = 4 and (b)

T = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Tail probabilities under different threshold values for a system with λ = μ = 1. . 30

2.8 Age-availability tradeoffs under the optimal and the threshold policy for the ap-

proximating discrete systems, and under the threshold policy for the continuous

system; left: λ = μ = 0.5, and right: λ = 0.5, μ = 0.48. . . . . . . . . . . . . . . . 31

3.1 Distribution of the age of units at the time of receipt by the hospital. . . . . . . . . . . 51

3.2 A time series plot of daily demand in 2011. The dotted lines mark the first,

second and third quartiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Issue age distribution obtained from simulation compared to the empirical distribution

obtained from data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.4 Tradeoff between average issue age and proportion of outdates under different ordering and

allocation policies. Each line corresponds to the specified ordering policy. The tick marks on

each line (from left to right) correspond to allocation policies with T = 42 (FIFO), 28, 21, 14,

7, and 0 (LIFO). The symbol (×) marks the performance of the hospital in 2011 obtained from

the data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

ix

3.5 Issue age distribution for selected allocation policies when keeping (a) current

levels of inventory, (b) 5 days of inventory and (c) 3 days of inventory. . . . . . 58

3.6 Fraction of units issued with age below or above the threshold under selected

policies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Daily demand and orders observed (a) in empirical data; (b) when keeping 5 days

of inventory and using a FIFO policy; and (c) when keeping 5 days of inventory

and using a threshold policy with T = 21. . . . . . . . . . . . . . . . . . . . . . . 60

3.8 Type distribution of outdated units for selected scenarios. . . . . . . . . . . . . . 61

4.1 An illustration of the equilibrium abandonment strategy of low-priority cus-

tomers in Example 1 with Case (i) on the left and Case (ii) on the right. A

customer joins/stays in positions marked with � and � and abandons if her po-

sition reaches a position marked with ×. Customers do not abandon once they

reach the positions marked with �. . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 Optimal service and entrance fees as a function of λh. System parameters: R =

60, μ = 1, λl = 0.4, cl = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Regions where a service/entrance fee is optimal for Rl = 40 (left) and Rl = 20

(right) and cl = μ = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Comparison of service fee vs. entrance fee. System parameters: R = 60, μ =

1, λl = 0.4, cl = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

x

Chapter 1

Introduction

In this thesis, we study three problems in service operations where either the supply or demand

is perishable.

In Chapter 2, we study a perishable inventory system for products whose quality deteriorates

in time. The problem is motivated by applications in the healthcare and retail industries,

where fresher products are associated with higher quality. Hence, when allocating products of

different ages to demand, one faces a tradeoff between the age and availability of the product.

An important example can be found in inventory management of Red Blood Cells (RBCs).

RBC transfusion is one of the most frequently used medical interventions. While currently

RBC units can be transfused up to 42 days after donation, recent medical studies suggest an

association between the age of RBCs and the risk of adverse medical outcomes for transfused

patients (Flegel et al. 2014). Therefore, there is an interest in inventory management policies

that could reduce the age of transfused RBCs in hospitals without compromising the availability

or resulting in excessive outdates.

More specifically, we consider a perishable inventory model with stochastic supply and

demand. We assume that supply and demand are driven by independent Poisson processes,

units have a constant finite shelf-life, and unsatisfied demand is lost. The model captures the

main operational features of a hospital which locally collects its required blood from donors. We

consider a threshold-based allocation policy that trades off the age and availability of allocated

units. The threshold policy was first suggested in the literature by Haijema et al. (2007) for

inventory management of blood platelets, and further investigated in a simulation study by

Atkinson et al. (2012) for reducing the age of transfused RBCs. Under the threshold policy,

1

Chapter 1. Introduction 2

demand is satisfied using the oldest unit that is younger than a given threshold. If all units

are older than the threshold, the freshest unit available is allocated. We evaluate the exact

performance of the threshold policy in terms of the distribution of the age of allocated units

and the proportion of outdates and lost demand. Our key technical contribution is to provide an

exact characterization of the sojourn time distribution of units in inventory for a given threshold

value, from which various performance measures of interest can be obtained. Our numerical

results demonstrate the effects of system size and supply-to-demand ratio on the performance

of the policy and identify important properties of the distribution of the age of allocated units.

Furthermore, for small problem instances, we numerically compare the performance of the

threshold policy with the “optimal policy” that minimizes the weighted sum of loss probability

and average age of allocated units. Our results provide preliminary evidence that although

more complicated state-dependent allocation policies can outperform the threshold policy, the

relative improvement in performance is likely to be insignificant in practice.

In Chapter 3, we further investigate the problem of reducing the age of transfused RBCs in

hospitals. We develop a data-driven (evidence-based) simulation model based on the operations

of the blood bank of an acute-care hospital in Hamilton, Ontario, which orders its required blood

from a central supplier. We use empirical data to validate our model and estimate its inputs.

We then evaluate and compare the outcome of certain practical ordering and allocation policies

in satisfying the same historical demand as observed in the data. During the 1 year period

for which we analyzed the data, 10,349 units were transfused with an average issue age (age of

transfused units) of 20.66 days and 6 units were outdated (outdate rate: 0.06%). By adopting a

strict first in, first out (FIFO) allocation policy and an order-up-to ordering policy with target

levels set to 5 times the estimated daily demand for each blood type, the average issue age is

reduced by 29.4% (to 14.59 days), without an increase in the outdate rate (0.05%) or resulting

in any unmet demand. Further reduction of issue age without a significant increase in outdate

rate is observed when adopting non-FIFO threshold-based allocation policies and appropriately

adjusting the order-up-to levels. Our results suggest that a significant reduction of issue age

could be achieved, without compromising availability or resulting in excessive outdates, by

properly adjusting the ordering and allocation policies at the hospital level. We also discuss the

impact of using a systematic ordering policy on demand variability at the supplier level and its

potential benefits in reducing the age of transfused units in hospitals.

In Chapter 4, we study the rational abandonment behavior of utility-maximizing customers

Chapter 1. Introduction 3

in the context of an observable priority queue, and identify novel pricing implications. Although

there is a large literature on equilibrium customer behavior in queues (Hassin and Haviv 2003),

the focus is predominantly on the queue-joining decisions of customers. This ignores subse-

quent customer abandonment which is an important aspect of customer behavior in queueing

systems. Such abandonment behavior is particularly important in priority queues, which are

quite prevalent in practice.

We consider an observable M/M/1 priority queue with two classes and a preemptive priority

discipline. Assuming a linear waiting cost and a constant service reward, we characterize the

equilibrium abandonment strategy of low-priority customers and show that it has a simple

threshold structure. We then consider pricing as a means to control the abandonment behavior

and investigate its implications on system welfare and firm revenue. A distinguishing feature

of our model is that in the presence of abandonment the timing of payment matters. We show

that the welfare can be maximized using only a single fee charged upon service completion.

In contrast, revenue maximization generally requires a combination of both an entrance and a

service fee. Moreover, charging only an entrance fee may generate more or less revenue than

charging only a service fee. To the best of our knowledge, this is the first work that (i) gives

an analytical characterization of equilibrium abandonment behavior in priority queues, and (ii)

studies pricing for a queueing system in presence of rational customer abandonment.

Chapter 2

Threshold–Based Allocation Policies

for Perishable Inventory Systems

2.1 Introduction

An important problem in inventory management of perishable products is to determine how

units of different ages should be allocated to demand. In many applications, fresher products

are associated with higher quality. Thus, when choosing an allocation policy, one typically has

to balance opposing objectives: allocating fresher products to demand leads to higher customer

utility, but at the same time may increase the number of outdates and hence increase costs

and/or decrease the availability of the product.

Important examples can be found in inventory management of blood products. Under

current regulations, the shelf-life of refrigerated Red Blood Cell (RBC) units is 42 days. How-

ever, an extensive body of recent cohort studies (e.g., Koch et al. 2008, Eikelboom et al. 2010

and Offner et al. 2002), suggests a range of moderate to strong correlation between receiving

“older” RBCs and increased risk of adverse medical outcomes. Similar issues exist in transfu-

sion of blood platelets (shelf-life of 5–7 days), where fresher platelets are preferred, at least for

certain treatments. As a result, there is an interest in inventory management policies which

could reduce the age of transfused blood without compromising its availability.

Other examples can be found in the retail industry. Freshness is an important quality factor

for many perishable products (e.g., seafood, fruits, bagels, or cut flowers) and can affect the

demand, competitiveness, and customers’ perception of the store (see, e.g., Li et al. 2012 and

4

Chapter 2. Threshold Policies for Perishable Inventory Systems 5

the references therein). In many cases, retailers can directly choose how products are allocated

to customers. Even in grocery stores, where customers pick the products off the shelves, the

store can affect the allocation through displaying products of certain age on the shelf (Chen

et al. 2014). In the case of online grocery stores such as AmazonFresh, the retailer has full

control over the allocation of products. Therefore, a fundamental question is whether firms can

better manage their perishable inventory using different allocation policies.

Motivated by the above question, we study a practical family of threshold-based allocation

policies (hereinafter referred to as the threshold policy). The threshold policy, which was first

suggested by Haijema et al. (2007) for inventory management of platelets, allocates products

according to a mixture of two extreme policies: (i) the policy that allocates the oldest unit

available, or First In, First Out (FIFO), and (ii) the policy that allocates the youngest unit

available, or Last In, First Out (LIFO). The FIFO policy is effective in reducing the outdates

as it always allocates the next unit to be outdated, but may lead to allocation of old units

and hence reduce the quality of the allocated products. In contrast, the LIFO policy keeps

the allocated products as fresh as possible, but may result in unacceptable wastage and/or

compromise the availability of the product. The threshold policy aims to balance these two

extremes. Specifically, under the threshold policy demand is satisfied using the oldest unit that

is younger than a given threshold age; if however all units are older than the threshold, demand

is satisfied using the youngest unit. By changing the value of the threshold, a range of policies

between FIFO and LIFO is obtained. In particular, FIFO and LIFO policies are retrieved by

setting the threshold value to the shelf-life of the product and zero, respectively.

The threshold policy is appealing from a practical point of view since it does not require

full information on the state of the inventory (number and age of all units available) and can

be implemented by dividing the inventory into two categories depending on whether units are

younger or older than the threshold. While the threshold policy is practical and intuitive, its

performance in terms of the resulting availability and age of allocated products is not clear, es-

pecially in presence of uncertainty: What is the shape of the age–availability trade–off achieved

under the threshold policy and how does it depend on system parameters? What is the resulting

distribution of the age of allocated units under the threshold policy? How does the threshold

policy compare to disposal of units before passing the shelf-life?

We address such questions in the context of an inventory model with stochastic supply and

demand. In particular, the model assumes that supply and demand are driven by independent


Poisson processes, products have a constant shelf-life, and unmet demand is lost. The model is

well-studied in the literature (see Section 4.2) and its simple structure together with exogenous

supply and demand allows us to capture the effect of allocation policies on the performance of

the system.

Our key technical contribution is to provide an exact characterization of the sojourn time

distribution of products in inventory for a given threshold value, from which the performance

measures of interest can be obtained. Our analysis are based on a two–stage model of the

system operating under the threshold policy. In this model, fresh units arrive at Stage 1, but

move to Stage 2 if their sojourn time exceeds the threshold value. Demand is satisfied using

the oldest unit from Stage 1, but if Stage 1 is empty, the freshest unit in Stage 2 is allocated.

The main complexity in the analysis of the threshold policy is that the supply and demand

processes of Stage 2 depend on the state of Stage 1, i.e., empty (idle) or with units available

(busy). Consequently, direct analysis of Stage 2 inventory without considering the dynamics

of Stage 1 is not possible. The main idea behind our approach is to track a tagged unit in

Stage 2 and decompose its sojourn time into idle and busy periods of Stage 1. We first assume

that units have infinite shelf-life in Stage 2 and then retrieve the sojourn time distribution of

units in the actual system from the analysis of the system with infinite shelf-life. To analyze

the latter, we construct a sequence of modified systems in which the sojourn time of units is

limited to a finite number of Stage 1 idle and busy periods. We show that as the number of idle

and busy periods considered tends to infinity, the sojourn time of units in the modified systems

converges (in distribution) to that of the units in the system with infinite shelf-life. Finally,

we employ the asymptotic analysis of the modified systems to characterize the sojourn time of

units in Stage 2.

Equipped with our exact results, we numerically examine and bring new insights into the

performance of the threshold policy in terms of the resulting distribution of the age of allocated

units and the proportion of outdates and lost demand. Our main findings are summarized as

follows: (i) We find that the shape of the trade–off curve significantly depends on the system

parameters. When the system is larger (i.e., has higher demand and supply rate) and supply

is greater than demand, the threshold policy performs better, in the sense that reduction of

average issue age is obtained in return for a smaller increase in outdate and loss probabilities.

We also observe that as the size of the system increases the proportion of units allocated from

Stage 2 inventory (i.e., with age above the threshold) decreases, regardless of the supply-to-


demand ratio. For a sufficiently large system, the threshold policy effectively leads to disposal

of all units that pass the threshold age. (ii) We observe that the distribution of the age of

allocated units from Stage 2 inventory highly depends on the size of the system, but is less

sensitive to the supply-to-demand ratio. Specifically, for larger systems where the utilization

of Stage 2 inventory is low, the age of units allocated from this stage tends to be closer to

the shelf-life. Further, we find that the age distributions under different threshold policies are

not stochastically ordered. That is, while decreasing the threshold value reduces the average

age of allocated units, it may increase the proportion of allocated units that are older than a

certain value. (iii) By comparing the performance of the threshold policy with the “optimal

policy” that minimizes the weighted sum of loss probability and average age of allocated units,

we provide preliminary evidence that, although more complicated state-dependent allocation

policies can outperform the threshold policy, the relative improvement in performance is likely

to be minimal in practice.

In summary, our main contributions are to (i) provide exact analysis of the threshold policy

for a stochastic perishable inventory system; (ii) generate insights into the performance of the

threshold policy and the shape of the resulting age-availability trade-off in different parameter

regimes; and (iii) identify important properties of the distribution of the age of allocated units

under the threshold policy.

The rest of the chapter is organized as follows. In the next section we review the related

literature and position our work. In Section 3 we describe the model and performance measures

of interest. In Section 4, we study the FIFO and LIFO policies. We provide some additional

theoretical results for the FIFO policy in Section 5. These results are required for our analysis

of the threshold policy provided in Section 6. Section 7 contains the results of our numerical

study. Finally, in Section 8 we present our concluding remarks and discuss future research. All

proofs are given in Appendix A.

2.2 Literature Review

There is a vast literature on managing perishable inventory; see Nahmias (1982) and Karaes-

men et al. (2011) for comprehensive literature reviews, and Baron (2011) and Nahmias (2011)

for tutorials on models and techniques used in their analysis. The majority of this literature

assumes that the products are issued according to the FIFO policy. This assumption is rea-


sonable when the quality of the product remains constant during its lifetime, as it minimizes

the outdates. However, even under this assumption, there are examples where FIFO policy is

not optimal (see, e.g., Pierskalla and Roach 1972 and Remark 1 in Chen et al. 2014). Clearly,

when the quality of the product deteriorates in time and the objective function is not indepen-

dent of the age of allocated units, FIFO is no longer necessarily optimal. Nevertheless, while

a few studies (e.g., Li et al. 2012 and Xue et al. 2012) consider disposal policies, i.e., depleting

old inventory before expiration in return for a salvage value, little attention has been paid to

allocation policies.

The threshold policy considered in this study was first introduced by Haijema et al. (2007)

in the context of inventory management of blood platelets. They consider a discrete-time model

and study optimal/near optimal replenishment policies assuming fixed threshold policies. Re-

cently Atkinson et al. (2012) investigate the performance of the threshold policy for reducing the

age of transfused RBCs at hospitals using a simulation model. They calibrate their simulation

model using data from the blood bank of Stanford University Medical Center. They investi-

gate the trade-off between the expected age of transfused units and fraction of imported units

(lost demand) under the threshold policy. They conclude that the threshold policy can reduce

the average age of transfused blood without significantly affecting availability in many U.S.

hospitals. In contrast our study provides both a contribution in analyzing the threshold policy

within a rigorous analytical framework, and new insights (as discussed in the introduction) that

are hard to gain from a simulation study. For example, we investigate the distribution of the

age of allocated units. The age distribution is particularly important for the blood banking

application since the actual relation between the risk of adverse outcomes and age of blood

is currently under investigation and may not be linear (Pereira 2013). Of course, when exact

analysis are not possible, e.g., when generalizing the model to include multiple blood types with

substitution, there may be a need for simulation or approximations.

The inventory model considered in this study was first introduced by Graves (1978) and

further studied in Kaspi and Perry (1983) who obtained the performance measures of the system

under the FIFO policy. Many papers have since considered variations and extensions of the

so-called stochastic perishable inventory problem. Examples include the problem with renewal

supply (Kaspi and Perry 1984), quality inspections (Perry 1999), batch demand and donations

(Goh et al. 1993) and controlled supply and demand (Perry and Posner 1990). The model

is specifically suited for the study of blood banks (Nahmias 2011b, Chapter 9) where supply


corresponds to donations and demand to transfusions. While the model stylizes the operations

of real-world blood banks (see the discussion in Section 2.8), it is still useful in generating

insights into the performance or comparison of practical policies (see, e.g., Kopach et al. 2008,

and Goh et al. 1993). A common assumption among almost all these papers is however that the

system operates under the FIFO policy. Keilson and Seidmann (1990) study the LIFO policy

and compare its performance with FIFO. Parlar et al. (2011) compare FIFO and LIFO policies

in a profit maximization setting. To the best of our knowledge, our work is the first to present

exact performance analysis of the system under a general family of allocation policies.

Allocation policies have also been studied for systems with separate demand streams for

products of different ages. In this setting, the decision is whether to satisfy demand for a

specific age category using products of other categories in case of a shortage. Goh et al. (1993)

consider the stochastic perishable inventory problem with two stages of inventory. The first

stage contains units with age below a threshold and the second stage holds units older than

the threshold. Each stage has an independent demand stream, which is satisfied according

to FIFO. The authors develop approximations and compare the performance of two policies:

restricted and unrestricted. Under the restricted policy demand for each stage can only be

satisfied using the inventory of that stage. Under the unrestricted policy, demand for the

second stage can be satisfied using the first stage inventory in case of a shortage. In contrast,

our system has a single demand stream that is satisfied using the oldest unit in the first stage

as long as there are units available, and otherwise, using the freshest unit in the second stage.

More examples can be found in discrete-time settings. For instance, Pierskalla and Roach

(1972) show the optimality of the FIFO policy for certain objective functions assuming known

replenishment quantities at each period, and Deniz et al. (2010) analytically compare practical

joint allocation–replenishment policies for a product with a life-time of two periods.

2.3 Model Description

In this section we describe the model and formally introduce the performance measures of

interest. Units arrive at the inventory according to a Poisson process with intensity λ. It is

assumed that units are “fresh”, i.e., have age zero upon arrival to the inventory. However, the

analysis can be easily extended to the case where all units are delayed for some constant time,

e.g. for tests or preparation, before arriving at the inventory. Demand occurs according to


an independent Poisson process with intensity μ. Shelf-life of units in the inventory is equal

to a constant γ. Demand occurring while the inventory level is at zero is lost. Otherwise, if

there are available units in the inventory then one is allocated to the demand according to the

allocation policy in effect.

It is convenient to view the system as an M/M/1+Ds queue in which units are the arrivals

and each service completion corresponds to a demand allocation. The queue has arrival rate λ,

service rate μ, and its service discipline coincides with the allocation policy of the system. In

addition, units have a deterministic patience until the end of service (the +Ds in the Kendall

notation) equal to γ, i.e., abandon the system if their sojourn time exceeds γ.

Consider the system operating under some allocation policy denoted by π, and let Sπ

denote the random variable associated with the steady-state sojourn time of units in inventory

(or equivalently in the corresponding queueing system). Observe that Sπ has a probability

mass at γ and a continuous density on (0, γ). For outdated units we have Sπ = γ, while for

each transfused unit Sπ ∈ (0, γ) is equal to the age of the unit at the time of transfusion.

We study the outdate probability qπ, and the cumulative distribution function (cdf) of the

random variable associated with the age of transfused units Aπ. Observe that both performance

measures can be expressed in terms of the random variable Sπ. The outdate probability is

qπ = P (Sπ = γ), (2.1)

and the cdf of Aπ is given by

Aπ(x) =

⎧⎪⎨⎪⎩P (Sπ ≤ x|Sπ < γ) = P (Sπ ≤ x)/(1− qπ), 0 ≤ x < γ,

1, x ≥ γ.

(2.2)

Another performance measure is the probability of a demand being lost �π. As discussed in

Parlar et al. (2011), a simple conservation law relates this measure to the outdate probability.

Under any policy π, as long as the steady-state limits exist, we have

λ (1− qπ) = μ (1− �π) . (2.3)

We close this section by mentioning that, throughout the chapter, whenever we refer to the

Laplace transform (LT) of a random variable we mean the LT of its probability density function


(pdf) or equivalently the Laplace-Stieltjes transform (LST) of its cdf.

2.4 Review of the FIFO and LIFO Policies

The above model has been previously studied in the literature under FIFO and LIFO policies.

Some of the methods used in the analysis, however, will be used in our evaluation of the

threshold policy. Furthermore, as previously mentioned, FIFO and LIFO belong to the family

of threshold policies. Therefore, we next review the relevant results and use them to obtain the

performance measures of interest under these policies.

2.4.1 The LIFO Policy

The analysis of the LIFO policy is due to Keilson and Seidmann (1990) and Parlar et al. (2011),

who study the distribution of the sojourn time of units in the inventory, SL (where “L” is the

shorthand notation for LIFO). The analysis is based on the following observations, valid under

the LIFO policy. First, the sojourn time of a new unit arriving in inventory only depends on

future demand and unit arrivals. Second, for any unit with sojourn time less than γ, we know

that all units which arrived during the sojourn time of the unit also had sojourn times less than

γ, and hence were not outdated. It follows that SL = min(SL, γ), where SL is the random

variable associated with the sojourn time of units in inventory if they had infinite shelf-life. As

discussed in Parlar et al. (2011) since both the demand and supply are Poisson, SL has the

same distribution as the length of a busy period in an M/M/1 queue with arrival rate λ and

service rate μ.

Given the above, the outdate probability is P (SL = γ) = P (SL ≥ γ). Parlar et al. (2011)

also give an expression for the LT of the truncated busy period SL, from which the LT of AL can

be obtained. The formula is however not computationally useful. In the following proposition

we present all performance measures of interest directly in terms of the cdf of the busy period

B(x) ≡ P (SL ≤ x), which can be computed efficiently by numerically inverting its LT,

∫ ∞

0e−θxB(x)dx =

2μ

θ(λ+ μ+ θ +

√(λ+ μ+ θ)2 − 4λμ

) ,

(see, e.g., Gross et al. 2008, page 102). Note that when λ > μ, P (SL < ∞) = (μ/λ) < 1, that

is B(x) is improper. However, since SL is bounded, for any positive λ and μ, P (SL < ∞) = 1.


γ

W (t)

t

Busy Period

IdlePeriodResidual Busy Period

Figure 2.1: A typical sample path of the process W

Proposition 2.4.1 Under the LIFO policy the outdate probability is

qL = 1−B(γ). (2.4)

Furthermore, the cdf of the age of transfused units is

AL(x) =

⎧⎪⎨⎪⎩B(x)/(1− qL), 0 ≤ x < γ,

1, x ≥ γ,

(2.5)

and the expected age of transfused units is given by

E[AL] = γ −∫ γ0 B(y)dy

1− qL. (2.6)

2.4.2 The FIFO Policy

The stochastic perishable inventory problem under the FIFO policy was first studied in Graves

(1978) and Kaspi and Perry (1983) and recently revisited in Parlar et al. (2011). The analysis

is based on the so called Virtual Outdating Process (VOP) W ≡ {W (t); t ≥ 0}, which returns,

as a function of time, the remaining time until the next outdate if no new demands were to

arrive. VOP is useful since it is a (strong) Markov process and contains important information

about the state of the system. In particular, the age of the oldest unit in inventory at any time

t ≥ 0 is γ − W (t), the event {W (s) > γ} implies no inventory at time s, and {W (s−) = 0}


indicates that a unit was outdated at time s.

W has cadlag (right-continuous with left limits) sample paths with upward jumps. Jumps

occur either when the oldest unit is allocated to a demand (when 0 < W < γ, at Poisson rate μ)

or is outdated (when W hits zero). In both cases, jump sizes are equal to the inter-arrival time

of units to the inventory and hence exponentially distributed with rate λ. Figure 2.1 illustrates

a sample path of process W .

Kaspi and Perry (1983) show that W has the same distribution as the virtual waiting time

process of an M/M/1 + D queue with arrival rate μ and service rate λ, in which the idle

periods are deleted and customers do not join the system if they have to wait more than γ

before starting service (the +D in the Kendall notation). Using this observation they obtain

the steady-state distribution of W . Let f denote the steady-state pdf of W . We have (Parlar

et al. 2011)

f(x) =

⎧⎪⎨⎪⎩f(0)e−(λ−μ)x, 0 < x < γ,

f(0)eμγ−λx, x ≥ γ,

(2.7)

where

f(0) =

⎧⎪⎨⎪⎩

λ(λ−μ)

λ−μe−(λ−μ)γ , λ �= μ,

λ1+λγ , λ = μ.

Having the steady-state distribution of W , we can obtain the required performance measures.

Let “F” be the shorthand notation for FIFO policy. The following proposition summarizes the

results.

Proposition 2.4.2 Under the FIFO policy the outdate probability is

qF =

⎧⎪⎨⎪⎩

λ−μλ−μe−(λ−μ)γ , λ �= μ,

11+λγ , λ = μ.

(2.8)

Furthermore, the cdf of the age of transfused units is

AF (x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

e−(λ−μ)(γ−x)−e−(λ−μ)γ

1−e−(λ−μ)γ , λ �= μ, 0 ≤ x < γ,

x/γ, λ = μ, 0 ≤ x < γ,

1, x ≥ γ,

(2.9)


and the expected age of transfused units is given by

E[AF ] =

⎧⎪⎨⎪⎩

(λ−μ)γ−(1−e−(λ−μ)γ)

(λ−μ)(1−e−(λ−μ)γ), λ �= μ,

γ2 , λ = μ.

(2.10)

The tractable form of the performance measures under the FIFO policy allows us to obtain

simple yet useful structural results presented in the following corollary without a proof.

Corollary 2.4.3 Under the FIFO policy:

1. For a fixed supply-to-demand ratio (λ/μ), both the outdate probability qF and loss proba-

bility �F are strictly decreasing in μ.

2. For a fixed supply-to-demand ratio (λ/μ), as μ → ∞, if λ < μ then qF → 0 and �F →(μ − λ)/μ; if λ > μ then qF → (λ − μ)/λ and �F → 0; and if λ = μ then qF → 0 and

�F → 0.

3. The distribution of the age of transfused units AF (·) is strictly convex for λ > μ, strictly

concave for λ < μ and uniform for λ = μ.

4. For a fixed supply-to-demand ratio (λ/μ), as μ increases, the expected age of transfused

units E[AF ] increases for λ > μ, decreases for λ < μ and is constant for λ = μ.

The first two points demonstrate the impact of the size of the system on the outdate and

loss probabilities. The results are intuitive noting that increasing the demand and supply rates

by a certain factor has the same effect as multiplying the shelf-life by that factor. To see this,

consider a system with parameters λ, μ, and γ. Then, increasing the supply and demand rates

by a factor of n, we have a system with parameters nλ, nμ, and γ. By decreasing the unit

of time by a factor of n, however, this system has the same performance as that of a system

with parameters λ, μ, and nγ. The third point highlights the impact of supply-to-demand

ratio on the age of allocated products. It implies that by increasing the supply-to-demand

ratio, allocated units become older. Finally, the last point states that increasing the size of the

system results in an increase in the average age of allocated units when the supply is greater

than demand, and a decrease when the supply is less than demand. We shall demonstrate and

discuss some of these results further in the numerical study of Section 2.7.


2.5 Additional Results for the FIFO Policy

Before turning to the analysis of the threshold policy we need some additional results for a

FIFO system in which units have a shelf-life T . Consider the queueing counterpart of the

system. The queue alternates between busy and idle periods. During a busy period, there are

units available in inventory while during an idle period, the inventory is empty. We also define

the residual busy period as the time interval between the epoch when a unit is outdated until

the start of the next idle period. In what follows, we present the distribution of the number

of units that are outdated during a (residual) busy period as well as the LT of the length of

the (residual) busy period given the number of outdates. We also obtain the distribution of

lost demand during an idle period as well as the LT of the length of the idle period given the

number of lost demand.

We prove the results using sample path analysis of the process W . First, following Kaspi

and Perry (1983), we define the stopping time

τ = inf{t ≥ 0;W (t) = 0 or W (t) > T},

on {W (0) > 0}. We shall use the notation Px and Ex to denote conditional probability and

expectation given the initial value W (0) = x > 0. Let us define

gx(θ) ≡ Ex[e−θτ1{W (τ)=0}], (2.11)

hx(θ) ≡ Ex[e−θτ1{W (τ)>T}]. (2.12)

Then for x > T , gx(θ) = 0 and hx(θ) = 1. For each x ∈ (0, T ] from Kaspi and Perry (1983)

(see also Cohen 1982, page 548), we have

gx(θ) =e−α1(θ)x(λ+ α1(θ))e

−α2(θ)T − e−α2(θ)x(λ+ α2(θ))e−α1(θ)T

(λ+ α1(θ))e−α2(θ)T − (λ+ α2(θ))e−α1(θ)T, (2.13)

hx(θ) =(e−α2(θ)x − e−α1(θ)x)(λ+ α1(θ))(λ+ α2(θ))

λ((λ+ α1(θ))e−α2(θ)T − (λ+ α1(θ))e−α1(θ)T ), (2.14)

where

α1(θ) = (θ + μ− λ+ ((λ+ μ+ θ)2 − 4λμ)1/2)/2,

α2(θ) = (θ + μ− λ− ((λ+ μ+ θ)2 − 4λμ)1/2)/2.


From this one can obtain, given the starting point x ∈ (0, T ], the probability that W hits zero

before upcrossing T , that is

Px(W (τ) = 0) = gx(0) =e−(μ−λ)x − (λ/μ)e−(μ−λ)T

1− (λ/μ)e−(μ−λ)T, (2.15)

and the probability of its complimentary event, i.e., that W upcrosses T before hitting zero,

Px(W (τ) > T ) = hx(0) = 1− gx(0) =1− e−(μ−λ)x

1− (λ/μ)e−(μ−λ)T. (2.16)

Next, consider W right after a jump caused by hitting zero. Let t = 0 be the time of the

jump and note that the starting point W (0) ∈ (0,∞) is exponentially distributed with rate λ.

Let

p ≡ P (W (τ) > T ), g(θ) ≡ E[e−θτ |W (τ) = 0], h(θ) ≡ E[e−θτ |W (τ) > T ].

Observe that p is the probability that W upcrosses T before hitting zero, g(θ) is the LT of the

time it takes for W to hit zero given that it happens before upcrossing T , and h(θ) is the LT

of the time it takes for W to upcross T given that it occurs before hitting zero.

Lemma 2.5.1 For λ �= μ,

g(θ) =

(λe−α2(θ)T (1− e−(λ+α1(θ))T )− λe−α1(θ)T (1− e−(λ+α2(θ))T )

(λ+ α1(θ))e−α2(θ)T − (λ+ α2(θ))e−α1(θ)T

)/(1− p),

h(θ) =

(e−λT +

(λ+ α1(θ))(1− e−(λ+α2(θ))T )− (λ+ α2(θ))(1− e−(λ+α1(θ))T )

(λ+ α1(θ))e−α2(θ)T − (λ+ α2(θ))e−α1(θ)T

)/p,

and for λ = μ,

g(θ) =

(2λ(eTδ(θ) − 1)

θ(eTδ(θ) − 1) + 2λ(eTδ(θ) − 1) + (eTδ(θ) + 1)δ(θ)

)/(1− p),

h(θ) =

(2e(T/2)(θ+δ(θ))δ(θ)

θ(eTδ(θ) − 1) + 2λ(eTδ(θ) − 1) + (eTδ(θ) + 1)δ(θ)

)/p,

where δ(θ) ≡√θ(θ + 4λ) and

p =

⎧⎪⎨⎪⎩

μ−λμ−λe−(μ−λ)T , λ �= μ,

1/(1 + λT ), λ = μ.

(2.17)


We proceed with the analysis of the residual busy period R. Let M be the number of

outdates during the residual busy period. Also, let r(θ) denote the LT of R, and let rm(θ)

denote the LT of R given M = m.

Proposition 2.5.2 The number of units that are outdated during the residual busy period is

geometrically distributed with parameter p as given in (2.17), i.e.,

P (M = m) = (1− p)mp, (2.18)

for m ≥ 0. Moreover, the LT of the length of the residual busy period given M = m is

rm(θ) = h(θ)(g(θ)

)m, (2.19)

and the LT of the length of the residual busy period is given by

r(θ) =ph(θ)

1− (1− p)g(θ). (2.20)

We next consider full busy periods (see Figure 2.1 for a realization). Note that the length

of the busy periods are i.i.d. random variables. Let N be the number outdates during a generic

busy period denoted by Z. Also, let z(θ) denote the LT of Z, and let zn(θ) denote the LT of Z

given N = n.

Proposition 2.5.3 The distribution of the number of units that are outdated during a busy

period is given by

P (N = n) =

⎧⎪⎨⎪⎩hT (0), n = 0,

gT (0)p(1− p)n−1, n > 0.

(2.21)

Moreover, the LT of the length of the busy period given N = n is

zn(θ) =

⎧⎪⎨⎪⎩

hT (θ)

hT (0), n = 0,

gT (θ)gT (0) h(θ) (g(θ))

n−1 , n > 0,

(2.22)

and the length of the busy period has LT given by

z(θ) = hT (s) +ph(θ)gT (θ)

1− (1− p)g(θ). (2.23)


Finally, we consider the idle periods which are independent and exponentially distributed.

Let L be the number of lost demand during a generic idle period I. Also let i(θ) and il(θ)

denote the LT of I and the LT of I given L = l, respectively.

Proposition 2.5.4 Idle periods are exponentially distributed with parameter λ, that is

i(θ) =λ

λ+ θ. (2.24)

Moreover, the number of lost demand during an idle period is Geometrically distributed with

parameter λ/(μ+ λ), i.e.,

P (L = l) =

(μ

μ+ λ

)l ( λ

μ+ λ

), (2.25)

and the length of the idle period given the number of lost demand L = l has an Erlang(l+1, λ+μ)

distribution, that is,

il(θ) =

(λ+ μ

λ+ μ+ θ

)l+1

. (2.26)

2.6 The Threshold Policy

We analyze the threshold policy by considering a two-stage representation of the system op-

erating under a threshold policy with parameter T ∈ (0, γ). Figure 2.2 depicts this two-stage

representation of the system. Fresh units arrive at Stage 1 according to a Poisson process with

rate λ and stay there for a maximum of T time units after which they are transferred to Stage

2. Units remain in Stage 2 for up to γ − T additional time units and are eventually outdated

if their age exceeds the shelf-life γ before they are allocated. Demand for Stage 1 inventory

occurs according to a Poisson process with rate μ and is satisfied according to a FIFO policy.

If there are no units available in Stage 1, a unit from Stage 2 is allocated according to a LIFO

policy. Demand occurring while there are no units available is lost.

Let “T” be the shorthand notation for a threshold policy with parameter T , so that ST is

the random variable representing the steady-state sojourn time of units under a threshold policy

with parameter T . Let ST1 ∈ (0, T ] denote the random variable representing the steady-state

sojourn time of units in Stage 1. Also, for units which transfer to Stage 2, let ST2 ∈ (0, γ − T ]

denote the random variable representing their steady-state sojourn time in Stage 2. To simplify


μ

••

••T

λμ•

Stage 1(FIFO)

•

Stage 2(LIFO)

Figure 2.2: A two–stage representation of the system under the threshold policy

the notation we omit the superscript T from ST1 and ST

2 . Then,

ST = S11{S1<T} + (T + S2)1{S1=T}. (2.27)

It is easy to see that Stage 1 is an independent system operating under the FIFO policy,

in which units have shelf-life T . Hence, given that a unit is allocated from Stage 1, its sojourn

time distribution is known through the analysis of Subsection 3.2. In particular, let q1, �1,

A1(·) and E[A1], respectively, denote the outdate probability, loss probability, cdf of the age of

transfused units, and the expected age of transfused units in a FIFO system where units have

shelf-life T . Then all these measures can be obtained from Proposition 2 by setting γ = T .

However, both demand and arrival processes of Stage 2 depend on the state of Stage 1.

During a busy period in Stage 1, demand is only satisfied from Stage 1 inventory but units may

pass the threshold age T and hence move to Stage 2. During an idle period in Stage 1, demand

is satisfied from Stage 2 inventory but there are no arrivals at Stage 2. To characterize the

distribution of S2 we first consider a system in which units have infinite shelf-life in Stage 2.

Let S2 denote the random variable representing the steady-state sojourn time of units in this

system. Then since the allocation policy in Stage 2 is LIFO, using similar arguments to those

in Section 3.1, we have S2 = min(S2, γ − T ).

Note that a unit can arrive at Stage 2 during a busy period in Stage 1 and can be allocated

to a demand during one of the idle periods of Stage 1. In general, however, since the shelf-life

is infinite, to find the sojourn time of allocated units one needs to consider infinitely many such

idle periods. Note that S2 has an improper distribution whenever λq1 > μ�1, in which case

P (S2 < ∞) = (μ�1)/(λq1). Our approach is based on analyzing a sequence of modified Stage

2 systems in which each unit can only be allocated during a finite number k of Stage 1 idle

periods, after its arrival at Stage 2. Specifically, in the kth modified system a unit which is not


allocated by the end of the kth busy period after its arrival at Stage 2 is discarded. We denote

the random variable associated with the steady-state sojourn time of units in the kth modified

system by S2,k and show that as k tends to infinity, the distribution of S2,k converges to that

of S2. By analyzing the more tractable random variable S2,k, we are then able to obtain the

LT of S2.

In the next subsection, we explain how the required performance measures under the thresh-

old policy can be obtained from the distribution of S2. In Subsection 2.6.2 we analyze the

modified systems and use them to obtain the LT of S2.

2.6.1 Obtaining the Performance Measures

We first examine the outdate probability. For a unit to be outdated it must first move to Stage

2 and then, assuming it has an infinite shelf life, spend more than γ − T in Stage 2. Note that

q1 is the probability that a unit moves to Stage 2. Hence, the outdate probability of a policy

with threshold T is given by

qT = q1P (S2 ≥ γ − T ). (2.28)

To obtain the distribution of the age of transfused units AT , we condition on whether the

unit is allocated while in Stage 1 or 2. Denote these events by S1 and S2, respectively, and note

that P (S1) + P (S2) + qT = 1. First, clearly P (AT ≤ x|S1) = A1(x). Second, given that a unit

is allocated from Stage 2, we know that its age is greater than T , and hence for x ≤ T we have

P (AT ≤ x|S2) = 0. For T < x < γ,

P (AT ≤ x|S2) = P (S2 ≤ x− T )/P (S2 < γ − T ).

Noting that

P (S1) =1− q11− qT

, P (S2) =q1P (S2 < γ − T )

1− qT,

and combining the two cases we have

AT (x) ≡ P (AT ≤ x) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

P (S1)A1(x), x < T,

P (S1) +q1

1−qTP (S2 ≤ x− T ), T ≤ x < γ,

1, x ≥ γ.

(2.29)


R I2

...

... Zk

Uk0 U1

IkZ1I1

t

...

...

Tagged unit moves toStage 2

Z0

Figure 2.3: An illustration of random variables R, Ii, Zi, Uk on the time line

Finally, the expected age of transfused units E[AT ], can be computed using

E[AT ] = E[A1]P (S1) +(T + E[S2|S2 < γ − T ]

)P (S2),

where

E[S2|S2 < γ − T ] = γ − T −∫ γ−T0 P (S2 ≤ x)dx

P (S2 < γ − T ). (2.30)

For all the performance measures, P (S2 ≤ x) for x ≤ γ − T can be computed by numerically

inverting its LT, i.e., E[e−θS2 ]/θ, where E[e−θS2 ] will be given in Theorem 4 of Subsection

2.6.2. We note that evaluating the integral in (2.30) could be computationally demanding since

P (S2 ≤ x) needs to be computed using numerical inversion of its LT. A simple approximation

is to discretize the age of products according to the unit of measurement. For example, in the

case of RBC units, where the age of units is measured in days, one can compute the expected

age of allocated units using

E[AT ] =

γ∑n=1

n · P (n− 1 < AT < n). (2.31)

2.6.2 Sojourn Time of Units in Stage 2

In this subsection, we obtain the LT of S2 by considering a sequence of modified systems. Before

formalizing the approach we introduce some notation. Consider a tagged unit which has just

arrived at Stage 2. Let Zi, i ≥ 0 and Ii, i ≥ 1 denote, respectively, the length of the ith busy

and idle period in Stage 1 following the arrival of the tagged unit to Stage 2. Accordingly, Z0

corresponds to the length of the busy period during which the unit arrives at Stage 2. Note

that {Ii; i ≥ 1} and {Zi; i ≥ 0} are sequences of i.i.d. random variables having the same


distribution as I and Z, respectively. Thus, we have their LTs from Propositions 2.5.3 and

2.5.4, respectively. Furthermore, the time interval between the epoch when the tagged unit

moves to Stage 2 until the start of the first idle period is a residual busy period R, the LT of

which is given by Proposition 2.5.2. Let t = 0 be the instance the tagged unit moves to Stage

2 and let Uk, k ≥ 1, denote the time when the kth busy period ends. That is, for k ≥ 1,

Uk = R+

k∑i=1

Ii +

k∑i=1

Zi. (2.32)

Letting X0 = R and Xi = I +Z for i = 1, 2, ..., and noting that {Xi; i ≥ 1} is an i.i.d sequence

with Xi > 0 for all i ≥ 1, {Uk, k ≥ 1} can be viewed as the arrival epochs of a delayed renewal

process. Figure 2.3 presents an illustration of the corresponding renewal process on the time

line.

Now recall that S2,k is the sojourn time of units in the kth modified system in which a unit is

discarded if it is not allocated by the end of the kth busy period (or equivalently the beginning

of the (k + 1)st idle period). Therefore, for a unit that is allocated by the end of the kth busy

period we have S2,k = S2, while for a unit that is still in the system by the end of the kth busy

period we have S2,k = Uk. Formally, for k ≥ 1,

S2,k = S21{S2<Uk} + Uk1{S2≥Uk}. (2.33)

Theorem 2.6.1 Consider the sojourn time of units in Stage 2 assuming infinite shelf-life S2,

and the sojourn time of units in the kth modified system S2,k. We have

limk→∞

P (S2,k ≤ x) = P (S2 ≤ x), x ∈ [0,∞),

limk→∞

E[e−θS2,k ] = E[e−θS2 ], θ > 0.

The theorem implies that for sufficiently large number of idle periods, the sojourn time

distribution of the units in the modified system becomes arbitrary close to that of units in

the system with infinite shelf-life. It also indicates that the LT of S2 can be obtained by first

obtaining the LT of S2,k and then letting k → ∞. While the result is sufficient for our analysis,

in the following theorem we state a stronger convergence result for S2, i.e., the actual sojourn


time of units in Stage 2. Recall that S2 = min(S2, γ − T ). Hence, similarly if we let

S2,k ≡ min(S2,k, γ − T ), k ≥ 1, (2.34)

denote the truncated sojourn time of units in the kth modified system, one would expect S2,k

to converge to S2 as k tends to infinity. Indeed, the following theorem establishes their almost

sure convergence.

Theorem 2.6.2 Consider S2 the actual sojourn time of units in Stage 2, and S2,k as defined in

(2.34). We have P (S2,k → S2) = 1, that is the sequence of random variables {S2,k} converges

to S2 with probability 1.

We now turn to the analysis of the modified systems. Consider the kth modified system.

Note that given the number of units that are in front of the tagged unit at the beginning of any

idle period, its remaining sojourn time is independent of the past. For the kth modified system,

let ϕkν,i(θ), 1 ≤ i ≤ k+1 denote the LT of the remaining sojourn time of the tagged unit at the

beginning of the ith idle period, given that it has ν units in front of it. Then, ϕkν,1(θ) is the LT

of the remaining sojourn time of the unit at the beginning of the first idle period, given that

the number of units moving to Stage 2 during the residual busy period is ν ≥ 0. Recall that

M denotes the number of outdates during the residual busy period and rν(θ) denotes the LT

of the length of the residual busy period given M = ν. Thus, we have

E[e−θS2,k ] =

∞∑ν=0

P (M = ν)rν(θ)ϕkν,1(θ). (2.35)

We first express ϕkν,1(θ) for k ≥ 1 in Theorem 2.6.4, then we use (2.35) to find the LT of S2,k.

Finally, in Theorem 4 we apply Theorem 1 to obtain the LT of S2.

The following lemma presents a recursive relation for ϕkν,i(θ), which can be used to obtain

ϕkν,1(θ). We consider the tagged unit at the beginning of the ith idle period given that it has

ν ≥ 0 units in front of it. We then condition on the number of demand arrivals during the idle

period. By considering two cases depending on whether the unit is allocated during the idle

period or not, we are able to relate the LT of the remaining sojourn time of the tagged unit

at the beginning of the ith idle period to that of the unit at the beginning of the (i+ 1)st idle

period.


Lemma 2.6.3 For 1 ≤ i ≤ k and ν ≥ 0 we have

ϕkν,i(θ) =

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)ϕkν+n−l,i+1(θ) +

(μ

μ+ λ+ θ

)ν+1

. (2.36)

Note that by definition of ϕkν,i(θ) we have ϕk

ν,k+1(θ) = 1 for all ν ≥ 0. Thus, for a given

k and starting from i = k one can use Lemma 2.6.3 to recursively solve for ϕkν,1(θ). The next

theorem expresses ϕkν,1(θ) as a function of k. First, define

c1(θ) ≡ h(θ)p

(μ

λ+ μ+ θ

), c2(θ) ≡ (1− p)g(θ)

(μ

λ+ μ+ θ

), (2.37)

with g(θ), h(θ) and p given in Lemma 2.5.1, and let

ξi(θ) ≡

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

hT (θ) + gT (θ)c1(θ)/ (1− c2(θ)) , i = 0,

ξ0(θ) + gT (θ)c1(θ)/ (1− c2(θ))2 , i = 1,

gT (θ)c1(θ) (c2(θ))i−2 / (1− c2(θ))

i+1 , i ≥ 2,

(2.38)

βi(θ) ≡

⎧⎪⎨⎪⎩c1(θ)/ (1− c2(θ)) , i = 0,

c1(θ) (c2(θ))i−1 / (1− c2(θ))

i+1 , i ≥ 1,

(2.39)

with hT (θ) and gT (θ) given in (2.15) and (2.16) respectively. Next, define the nested sum Yν(d)

for non-negative integers ν and d as

Yν(d) ≡0∑

j0=0

ξj0(θ)1∑

j1=0

ξj1(θ)

2−j1∑j2=0

ξj2(θ)

3−j2−j1∑j3=0

ξj3(θ) · · ·

(d−1)−jd−2−···−j1∑jd−1=0

ξjd−1(θ)

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

d− jd−1 − · · · − j1

⎞⎠ , (2.40)

where we adopt the convention that any empty sum is equal to 0 and any empty product equal

to 1. Note that d is the number of sums in Yν(d). For d = 0, Yν(d) has no sums, that is

Yν(0) =

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

0

⎞⎠ =

(μ

λ+ μ+ θ

)ν+1

,

and for d = 1 the expression only contains the first sum. Noting that the first sum simplifies


to ξ0(θ), we have

Yν(1) = ξ0(θ)

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

1

⎞⎠ .

Similarly, for d ∈ {2, 3, ...}, (2.40) includes the first d sums.

Theorem 2.6.4 The LT of the remaining sojourn of a unit at the beginning of the first idle

period in the kth modified system, given it has ν ≥ 0 units in front of it, is given by

ϕkν,1(θ) =

k−1∑i=0

Yν(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

+(i(θ)z(θ)

)k. (2.41)

Using (2.35) we obtain E[e−θS2,k ] and let k → ∞ to obtain the LT of S2. Define the nested

sum X(d, w) for positive integers d and w as

X(d, w) ≡w∑

j1=0

ξj1(θ)

w+1−j1∑j2=0

ξj2(θ)

w+2−j2−j1∑j3=0

ξj3(θ) · · ·w+(d−2)−jd−2−···−j1∑

jd−1=0

ξjd−1(θ)βw+(d−1)−jd−1−···−j1(θ).

(2.42)

Note that X(d, w) contains the first d−1 sums, such that X(1, w) = βw(θ) for all w ∈ {1, 2, . . .}.Moreover, X(d, w) satisfies the recursive relation given by

X(d, w) =w∑i=0

ξi(θ)X(d− 1, w + 1− i),

for d ∈ {2, 3, ...}, which can be used to calculate X(i, 1) for i ∈ {1, 2, ...}, as needed in Theorem

4 below.

Theorem 2.6.5 The LT of S2 is given by

E[e−θS2 ] = β0(θ) + ξ0(θ)

∞∑i=1

X(i, 1)

(λ

λ+ μ+ θ

)i

.

2.7 Numerical Results

In this section we discuss the results of our numerical study. Our numerical study has four

parts. In the first part, we investigate the trade-off between the expected age of allocated units,

and the proportion of lost demand and outdates under the threshold policy. In the second

part, we study the distribution of the age of allocated units. In the third part, we compare


the performance of the threshold policy with the “optimal policy” by considering a discrete

approximation of the system. Finally, we discuss the implications of our observations for the

blood bank of a hospital which locally collects blood from donors and aims to use the threshold

policy to reduce the age of transfused RBCs without compromising the availability. In this

case, the supply process corresponds to the arrival of donated blood to the blood bank and the

demand process to transfusion of RBC units to patients. We note that the stylized setting we

consider in this chapter has been previously used to study the operations of blood banks (e.g.,

Goh et al. 1993, Kopach et al. 2008).

To streamline our discussions, we fix the shelf-life at γ = 40 days (the effective shelf-life

of RBC units in inventory) in our examples and vary the demand rate and the supply-to-

demand ratio. We note that the effect of changing the shelf-life is captured in our examples,

as through scaling, it has the same effect as varying the system size. For example the system

with λ = μ = 1, γ = 40 and T = 16 has the same performance as that of the system with

λ = μ = 8, γ = 5 and T = 2. We consider three supply-to-demand ratios: 0.98, 1, and 1.02.

We focus on supply-to-demand ratios close to 1, since otherwise the loss or outdate probability

is very high even under the FIFO policy. We capture the effect of the system size by setting the

demand rate to 1, 3, and 8. For each scenario, we consider nine threshold values and compute

the distribution of the age of allocated units, as well as the outdate and loss probabilities. We

used the Fixed Talbot method (Abate and Valko 2004) to numerically invert the LT of S2. The

expected age of allocated units are computed using (2.31).

The trade-off between expected age, availability and outdates. We first investigate

the trade–off between the expected age of allocated units, and the loss and outdate probabilities.

The trade-off curves are depicted in Figure 2.4. From left to right, the threshold value is

decreasing from 40 (FIFO) to 0 (LIFO). As a result, the mean age of allocated units is decreasing

while both the outdate and loss probabilities are increasing. For each demand rate, increasing

the supply-to-demand ratio leads to higher availability under all threshold values (shifts the

age–availability curves to the left), and increases the proportion of outdates (shifts the age–

outdates curves to the right).

We observe that the shape of the trade–off curves depend on both the supply-to-demand

ratio and demand rate. When the demand rate is high (i.e., μ = 8) and supply is greater

than demand (i.e., λ/μ = 1.02) the threshold policy performs well: the average issue age is

significantly reduced by lowering the threshold value while only slightly increasing the outdate


0 2 4 6 8 100

5

10

15

20

25

30

35

Proportion of lost demand %

Meanageofallocatedunits λ/μ= 1.02

λ/μ= 1

λ/μ= 0.98

μ = 8

0 2 4 6 8 100

5

10

15

20

25

30

35


Meanageofallocatedunits λ/μ = 1.02

λ/μ = 1

λ/μ = 0.98

μ = 3

0 2 4 6 8 100

5

10

15

20

25

30

35



λ/μ = 1

λ/μ = 0.98

μ = 1

0 2 4 6 8 100

5

10

15

20

25

30

35

Proportion of outdated units %

Meanageofallocatedunits λ/μ= 1.02

λ/μ= 1

λ/μ= 0.98

μ = 8

0 2 4 6 8 100

5

10

15

20

25

30

35



λ/μ = 1

λ/μ = 0.98

μ = 3

0 2 4 6 8 100

5

10

15

20

25

30

35



λ/μ = 1

λ/μ = 0.98

μ = 1

Figure 2.4: The trade–off curves under the threshold policy for different supply-to-demand ratiosand demand rates. From left to right the tick marks on each curve correspond to thresholdvalues of 40 (FIFO), 28, 24, 20, 16, 12, 8, 4, and 0 (LIFO).

and loss probabilities in comparison to FIFO. As the supply-to-demand ratio decreases, however,

the range of average age of allocated units between FIFO and LIFO decreases. The reason is

that by lowering the supply rate units spend less time on the shelf, hence by changing the way

they are allocated to demand less difference in average age is observed. Furthermore, we observe

that for smaller demand rates the tradeoff curves are flatter. That is, reducing the mean age

of allocated units leads to a higher increase in outdate and loss probabilities. The effect of the

supply-to-demand ratio is however less significant, as for smaller system sizes the difference in

supply and demand rates becomes smaller.

To gain more insight into the performance of the threshold policy, we next look at the

proportion of units allocated from each stage. Figure 2.5 presents the proportion of units

allocated from Stage 2 inventory under different scenarios. We observe that for sufficiently

large demand and threshold value, almost no units are allocated from Stage 2. When the

demand rate is μ = 8, the proportion of units allocated from Stage 2 is very small (less than

1%). In contrast, when the demand rate is small, e.g., μ = 1, a much higher utilization of Stage

2 inventory is observed, especially for lower threshold values. The observation is robust with

respect to the supply-to-demand ratio.


4 8 12 16 20 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Threshold value

Prop.of

unitsallocatedfrom

Stage

2(%

)

λ/μ= 1.02

λ/μ= 1

λ/μ= 0.98

μ = 8

4 8 12 16 20 240

0.5

1

1.5

2

2.5

3

3.5

4

Threshold value

Prop.of

unitsallocatedfrom

Stage

2(%

)

λ/μ= 1.02

λ/μ= 1

λ/μ= 0.98

μ = 3

4 8 12 16 20 240

5

10

15

Threshold value

Prop.of

unitsallocatedfrom

Stage

2(%

)

λ/μ= 1.02

λ/μ= 1

λ/μ= 0.98

μ = 1

Figure 2.5: The proportion of products allocated from Stage 2.

Recalling that Stage 1 inventory is a FIFO system with shelf-life T , the above observation

can be explained using the results of Corollary 1. When the system size is large and/or the

threshold value is high, the availability of units in Stage 1 is high. That is, a low proportion of

units move to Stage 2 inventory and a small fraction of demand is directed to Stage 2. As a

result, almost all units in Stage 2 are eventually outdated and never allocated. Consequently, the

performance of the system under the threshold policy with parameter T is almost equivalent to

that of FIFO with the shelf-life reduced to T . Put another way, in this case the policy results

in disposal of almost all units that pass the threshold age T . Conversely, when the system

and/or the threshold value is small, the availability of units in Stage 1 is low and hence a higher

utilization of Stage 2 inventory is observed. In this case, by allocating only a small proportion

of units from Stage 2, the threshold policy keeps the average issue age close to that of a FIFO

system with shelf-life T , yet achieves a higher availability and a lower outdate rate compared

to it.

Distribution of the age of allocated units. As we saw in Section 2.6, the age dis-

tribution of units that are allocated from Stage 1 is the same as that of a FIFO system with

shelf-life T . As a result, its properties are known through Proposition 2.4.2 and Corollary 1.

To investigate the age of units allocated from Stage 2, we compute the conditional distribution

of the age of units given that they are allocated from Stage 2. Since units in this stage are

allocated in a LIFO order, one might expect the units to be mostly fresh, i.e., have an age

close to the threshold value. We find however that the shape of the distribution, while not very

sensitive to the supply-to-demand ratio, highly depends on the system size and the threshold

value. Specifically, the smaller the system size and threshold value, the fresher the allocated

units. The reason is that, as we observed earlier, in such cases there is a higher utilization of

Stage 2 inventory and hence units do not spend a long time on the shelf. This is illustrated in


4 10 16 22 28 34 400

0.2

0.4

0.6

0.8

1

Age

Cumilitiveprobab

ility

μ = 1

μ = 3

μ = 8

(a)

8 16 24 32 400

0.2

0.4

0.6

0.8

1

Age

Cumilitiveprobab

ility

μ = 1

μ = 3

μ = 8

(b)

Figure 2.6: The conditional distribution of the age of products given that they are allocatedfrom Stage 2 for a system with λ = μ = 1 and threshold value (a) T = 4 and (b) T = 8.

Figure 2.6 for a system with supply-to-demand ratio equal to 1 and threshold values 4 and 8 (for

brevity we preclude the graphs for other supply-to-demand ratios and thresholds). Observe, for

example, that for the system with μ = 8 and T = 8, the distribution is clearly skewed to the

left. That is, the small proportion of demand satisfied from Stage 2, is mostly satisfied from

old units with age close to the shelf-life.

We next consider the age distribution for different threshold values. We observed earlier

that the expected age of allocated units under the threshold policy reduces as we lower the

threshold value. A natural question regarding the distribution of the age of allocated units is

whether the age of allocated units under different threshold values are stochastically ordered,

that is, whether P (AT1 > x) < P (AT2 > x) for all x and T1 < T2. We find that this is not

the case. We demonstrate this result via an example in Figure 2.7, where we plot the tail

probabilities for x ∈ {20, 24, 28, 32} under thresholds T ∈ {8, 12, 16} and for a system with

λ = μ = 1. Observe that the policy with the lowest threshold value (T = 8) and hence the

lowest expected age of allocated units results in the highest proportion of units older than x,

for all 4 values considered.

How does the threshold policy perform in comparison with the “optimal pol-

icy”? Here, we compare the performance of the threshold policy with that of the policy that

minimizes the weighted sum of loss probability and average age of allocated units. Computing

the optimal allocation policy is complicated since due to the constant shelf-life assumption one

needs to keep track of the age of all units in the inventory. To alleviate this difficulty, we

consider a discrete approximation of the system. We discretize the time unit into n slots of

length δ = 1/n. Note that since the supply and demand are Poisson processes, the probability


20 24 28 320

0.005

0.01

0.015

0.02

0.025

Age x

Tailprobab

ilityP(A

T>

x)

T = 16

T = 12

T = 8

Figure 2.7: Tail probabilities under different threshold values for a system with λ = μ = 1.

of having more than one event during a slot (supply or demand arrival) is o(δ). Therefore, we

can approximate the system in discrete-time by assuming that the shelf-life is nγ periods, and

at each period we either have a unit arrival with probability λ/n, a demand with probability

μ/n, or neither with probability 1−λ/n−μ/n. The Markov Decision Process (MDP) formula-

tion for the discrete system is presented in Appendix B. We numerically compute the optimal

allocation policy for different cost parameters and values of n using value iteration and evaluate

its performance in terms of the average age of allocated units and proportion of lost demand

using simulation. Due to the large size of the state-space, we only obtain the optimal policy for

small problem instances. However, recall that the threshold policy performs worse for smaller

systems (i.e., for smaller systems, the reduction of average age is obtained in return for a larger

increase in loss probability). Therefore, it is interesting to see how much improvement can be

obtained over the threshold policy for small systems.

In Figure 2.8 we present results for two systems with shelf-life γ = 5, one with λ = μ =

0.5 (λ/μ = 1) and the other with λ = 0.5, μ = 0.48 (λ/μ = 1.04). For each number of slots,

n, we plot the tradeoff curves under both the threshold and the optimal policy. The tradeoff

curve for the continuous system under the threshold policy is also depicted. Note that the

curves under the discrete systems are approaching that of the continuous system. For discrete

systems, we observe a relatively small improvement (lower curves) under the optimal policy

comparing to the threshold policy. To measure this improvement, for each discrete system,

we fitted a linear regression to the tradeoff curve under the threshold policy and a polynomial

regression to the tradeoff curve under the optimal policy (R2 > 0.99). For different average age


●●

●

●

●

●

●

0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38

1.5

2.0

2.5

3.0

3.5

4.0

Proportion of lost demand

Aver

age

age

of a

lloca

ted

units

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

n = 1n = 2n = 3Continuous system

●●

●

●

●

●

●

0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38

1.5

2.0

2.5

3.0

3.5

4.0

Proportion of lost demand

Aver

age

age

of a

lloca

ted

units

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

n = 1n = 2n = 3Continuous system

Figure 2.8: Age-availability tradeoffs under the optimal and the threshold policy for the ap-proximating discrete systems, and under the threshold policy for the continuous system; left:λ = μ = 0.5, and right: λ = 0.5, μ = 0.48.

(loss probability) values in the range between FIFO and LIFO, we then computed the relative

reduction in loss probability (average age) achieved under the optimal policy in comparison

with the threshold policy. The maximum relative improvements in average age were 8.1%,

8.8%, and 8.8% for n = 1, 2, and 3, respectively, for the system with λ/μ = 1, and 7.5%, 8.9%,

and 9.4% for n = 1, 2, and 3, respectively, for the system with λ/μ = 1.04. Noting that the

marginal increase in the relative improvement is decreasing, one would expect the maximum

relative improvement to be around 10% for both systems. The maximum relative improvements

in loss probability were 7.1%, 4.7%, and 4.3% for n = 1, 2, and 3, respectively, for the system

with λ/μ = 1, and 6.8%, 5.0%, and 4.6% for n = 1, 2, and 3, respectively, for the system

with λ/μ = 1.04. Thus, we expect the maximum relative reduction in loss probability under

the optimal policy to be below 4.3% and 4.6% for the continuous systems with λ/μ = 1 and

λ/μ = 1.04, respectively.

In a broader sense, our results suggest that while more complicated state-dependent policies

can outperform the threshold policy, at least for small systems, the improvement is likely

insignificant, especially in view of the lower practicality of state-dependent policies. We note

that characterizing the structure of the optimal allocation policy or a thorough computational

investigation of it is out of the scope of this work and is left for future research.

Implications for the hospital blood bank application. In the following, we summarize

the implications of our findings for the blood banking application.


• Our results imply that when the supply and demand rates are high (e.g., for a large

hospital or a blood type with high demand), the benefits of the threshold policy may be minimal

compared to adopting a FIFO policy and reducing the shelf-life to the threshold value. It is

important to mention that in such cases, shelf-life reduction is expected to have a minimal

effect on the outdate rate and availability, as empirically observed in Blake et al. (2012). For

smaller hospitals or blood types with lower demand, however, the threshold policy is expected

to maintain some utilization of Stage 2 inventory and hence differentiate itself from shelf-life

reduction. The increase in proportion of lost demand or outdates in this case may however be

higher than suggested in the simulation study of Atkinson et al. (2012).

• Our results on the distribution of the age of allocated units under the threshold policy

has important implications. Although the threshold policy allocates the freshest unit available

when there are no units younger than the threshold, we find that the allocated units in this

case are more likely to be close to expiry (unless the threshold is very small). Moreover,

although lowering the threshold value leads to fresher transfusions on average, we find that it

may increase the proportion of “very old” transfusions. This is particularly important if the

probability of adverse medical outcomes (currently under investigation in clinical trials) turns

out to be nonlinear in the age of transfused RBCs (see Pereira 2013 for examples of potential

nonlinear relationship functions).

• We observe that the supply-to-demand ratio has a significant effect on the age of allocated

units, especially when the demand rate is high. This implies a potential in reducing the age of

transfused units in hospitals by reducing the amount of inventory held.

• Finally, our results suggest that the threshold policy performs well in comparison with

the optimal allocation policy. Therefore, given its practicality, it should be considered as

a viable policy for managing the age-availability of RBCs units, and more generally, other

perishable goods.

2.8 Concluding Remarks and Future Research

Our study is a step toward understanding the performance of more general allocation policies

(other than FIFO and LIFO) for perishable products. We present an exact analysis of a

stochastic perishable inventory system operating under a family of threshold-based allocation

policies. We quantify the age–availability trade–off achieved under the threshold policy, by


computing the distribution of the age of transfused units as well as the proportion of lost

demand and outdates. Our results highlight the importance of system parameters on the

performance of the threshold policy. More specifically, we observe that the threshold policy

performs better for larger system sizes and when the supply is greater than demand. We also

find that for a sufficiently large system the threshold policy effectively leads to disposal of units

that pass the threshold age. Moreover, we identify important properties of the distribution of

the age of allocated units. For instance, we show that the age distributions under different

threshold values are not stochastically ordered.

The practicality of the threshold policy makes it a natural candidate to be used in practice,

especially in applications where complicated state-dependent policies are often not considered,

such as the blood banking application we discussed. Our preliminary numerical results sug-

gest that although more complicated policies can outperform the threshold policy, the relative

improvement is likely not very significant.

We also discuss the implications of our findings for an application in inventory management

of RBCs. Our model captures the main operational features of a hospital blood bank that locally

collects its required blood from donors. However, it should be noted that most hospitals order

their required blood from an external source (e.g., American Red Cross in the U.S. or Canadian

Blood Services in Canada) rather than facing an exogenous supply. The age of units at the

time of receipt by the hospital may also vary. Nevertheless, we expect our qualitative insights

to hold under more general assumptions. We verify some of our results and investigated the

effect of more general assumptions in a simulation study based on data from the RBC inventory

of a Canadian acute-care hospital in the next chapter.

We suggest two different venues for future research. One is to study the performance of

the threshold policy, or other practical allocation policies, in a more general setting. Important

examples are systems with batch supply and demand, and inhomogeneous initial age of units.

Since an exact analysis of the system under such assumptions is less likely to be possible, it

would be useful to explore approximations. A different direction would be to further investigate

the optimal allocation policy. This is currently being investigated by Sabouri et al. (2015) in a

discrete-time setting.


2.9 Appendix A: Proofs.

Proof of Proposition 2.4.1. The outdate probability qL is equal to the probability that the

busy period lasts longer than γ, which gives (2.4). By definition P (SL ≤ x) = P (SL ≤ x) for

all x < γ, yielding (2.5) using (2.2). To obtain the expected value we use (2.5) to write

E[AL] =

∫ ∞

0(1−AL(y))dy =

∫ γ

0(1−AL(y))dy = γ −

∫ γ0 B(y)dy

1− qL,

as given in (2.6). �

Proof of Proposition 2.4.2. The outdates occur at instances when W hits zero, the long-run

average rate of which is f(0), and hence qF = f(0)/λ which yields (2.8). To obtain AF (·), notethat the remaining age of units at the time of transfusion is embedded at epochs right before

jumps on a W sample path when 0 < W < γ. Thus, from (2.7) and using the Poisson Arrival

Sees Time Average (PASTA) property of the demand process, we have for x < γ,

AF (x) = P (SF ≤ x|SF < γ) = limt→∞P (γ −W (t) ≤ x|0 < W (t) < γ)

= limt→∞P (γ − x ≤ W (t) < γ)/P (0 < W (t) < γ) =

∫ γγ−x f(y)dy∫ γ0 f(y)dy

which after simplification gives (2.9). Using (2.9) the derivation of (2.10) is straightforward.

For the case where λ = μ, the results are obtained by letting λ → μ and applying L’Hopital’s

rule. �

Proof of Lemma 2.5.1. We first derive h(θ). Note that with probability e−λT , we have

W (0) > T , i.e., the first jump is greater than T and hence τ = 0. Conditioning on W (0) we

have

h(θ) ≡ E[e−θτ |W (τ) > T ] =E[e−θτ1{W (τ)>T}]P (W (τ) > T )

=

∫ T0 hx(θ)λe

−λxdx+ e−λT∫ T0 hx(0)λe−λxdx+ e−λT

.

Next, we consider g(θ). Note that given W (τ) = 0, the first jump must be some x ≤ T . Again

by conditioning on W (0) we get

g(θ) ≡ E[e−θτ |W (τ) = 0] =E[e−θτ1{W (τ)=0}]P (W (τ) = 0)

=

∫ T0 gx(θ)λe

−λxdx∫ T0 gx(0)λe−λxdx

.


Computing the integrals and noting that P (W (τ) = 0) + P (W (τ) > T ) = 1, gives the results.

�

Proof of Proposition 2.5.2. Let t = 0 be the time the outdate occurs. Then W has just

hit zero, causing a jump with size W (0) ∈ (0,∞) independent of the history of the process.

If W (0) > T , then the residual busy period is over with R = 0. Otherwise, the residual busy

period ends the first time W upcrosses T (see Figure 1 for a realization of R). However, before

this happens, W may hit zero as new outdates may occur. Therefore, the probability that there

are M = m ≥ 0 outdates during the residual busy period is equal to the probability that W

hits zero m times before upcrossing T . Note that from the strong Markov property of W , every

time it hits zero the process regenerates and a new i.i.d. cycle starts. Also, the probability of

hitting zero before crossing over T for each cycle is p ≡ P (W (τ) > T ). Therefore, we have

P (M = m) = (1− p)mp. From Lemma 2.5.1, it then follows for the length of the residual busy

period given the number of outdates, that

rm(θ) ≡ E[e−sR|M = m] = h(θ) (g(θ))m .

Removing the condition on M , we get the LT of the length of the busy period:

r(θ) ≡ E[e−θR] =

∞∑m=0

P (M = m)E[e−θR|M = m] =

∞∑m=0

(1− p)mph(θ) (g(θ))m

=ph(θ)

1− (1− p)g(θ).

�

Proof of Proposition 2.5.3. Each busy period starts with a fresh unit arriving at an empty

system. Let t = 0 be the start of the busy period, then W (0) = T . The busy period ends

the first time W upcrosses T . Each time W hits zero before this happens, a unit is outdated.

Note that starting from level T two cases can occur: either W upcrosses T before hitting zero,

with probability PT (W (τ) > T ) = hT (0), or it first hits zero, with probability PT (W (τ) = 0) =

gT (0) = 1 − hT (0). In the first case, the busy period ends with no outdates occurring during

it, so P (N = 0) = hT (0), and the LT of the conditional length of the busy period is

z0(θ) ≡ E[e−θZ |N = 0] = ET [e−θτ |W (τ) > T ] = hT (θ)/hT (0).


In the second case after W hits zero, by the strong Markov property of W , a new i.i.d. cycle

independent of the history of the process starts and the time until W upcrosses T has the same

distribution as a residual busy period. Thus, for n > 0 we have

P (N = n) = gT (0)P (M = n− 1) = gT (0)p(1− p)n−1.

That is, to have n outdates during the busy period, starting from W (0) = T , W must first hit

zero and then before crossing level T it must hit zero n− 1 (n ≥ 1) additional times. It follows

that

zn(θ) ≡ E[e−θZ |N = n] = ET [e−sτ |W (τ) = 0]E[e−θR|M = n− 1] =

gT (θ)

gT (0)rn−1(θ).

Substituting from (2.19) we get the result for the n > 0 case in (2.22). Finally, by the same

argument and again due to the strong Markov property of the process W , given W (0) = T we

have Z = τ + 1{W (τ)=0}R. It follows that

z(θ) ≡ E[e−θZ ] = ET [e−θτ1{W (τ)>T}] + ET [e

−θ(τ+R)1{W (τ)=0}]

= ET [e−θτ1{W (τ)>T}] + E[e−θR]ET [e

−θτ1{W (τ)=0}]

= hT (θ) + r(θ)gT (θ). (2.43)

Substituting r(θ) from (2.20) we get (2.23) which completes the proof. �

Proof of Proposition 2.5.4. An idle period starts whenever W upcrosses T , with the length

of the idle period being equal to the size of the over-shoot and hence exponentially distributed

with parameter λ (see e.g., Kaspi and Perry 1983). To see (2.25), note that the demand and

unit arrivals can be viewed as two competing Poisson processes, and hence P (L = l) is the

probability that the demand process with intensity μ wins l times before the arrival process

does. Finally, (2.26) follows from the fact that the time between subsequent demand arrivals,

given that the idle period has not ended, is exponentially distributed with rate λ+ μ. �

Proof of Theorem 2.6.1. We first show that the cdf of S2,k converges to that of S2 for all

x ∈ [0,∞). Then, the second part follows from the continuity theorem for Laplace transforms

(see Feller 1971, page 431). Let Aj denote the event that “the unit is allocated during the jth

idle period” and let Ak denote the event that “the unit is not allocated during any of the first


k idle periods”. Note that ∪kj=1Aj = {S2 < Uk} and Ak = {S2 ≥ Uk}. Now consider the cdf of

S2,k. Using (2.33) we can write

P (S2,k ≤ x) =

k∑j=1

P (S2,k ≤ x|Aj)P (Aj) + P (S2,k ≤ x|Ak)P (Ak)

=

k∑j=1

P (S2 ≤ x|Aj)P (Aj) + P (Uk ≤ x|Ak)P (Ak). (2.44)

Letting k → ∞ in (2.44), the last term on the RHS vanishes. To see this note that

limk→∞

P (Uk ≤ x|Ak)P (Ak) = limk→∞

P (Uk ≤ x, Ak) ≤ limk→∞

P (Uk ≤ x) = 0,

where the last equality holds because Uk is the kth renewal epoch of a delayed renewal process.

Thus, for any x ∈ [0,∞) we have

limk→∞

P (S2,k ≤ x) =

∞∑j=1

P (S2 ≤ x|Aj)P (Aj) = P (S2 ≤ x).

Note that since x is finite the last equality follows even if the cdf of S2 is improper. Hence, the

proof is complete. �

Proof of Theorem 2.6.2. To prove the theorem it is sufficient to show that for each ε > 0,∑∞k=1 P (|S2,k − S2| ≥ ε) < ∞. Then from the Borel-Cantelli Lemma we have P (|S2,k − S2| ≥

ε i.o.) = 0 (where i.o. stands for infinitely often) implying that P (S2,k → S2) = 1 (see Billingsley

1995, page 70) as claimed. To this end, we consider the random variable S2 defined on a

probability space (Ω,F , P ) and decompose the sample space Ω for k ≥ 1 as

C1,k ≡ {ω ∈ Ω; S2(ω) ≤ Uk(ω)},

C2,k ≡ {ω ∈ Ω; γ − T ≤ Uk(ω) < S2(ω)},

C3,k ≡ {ω ∈ Ω;Uk(ω) < γ − T ≤ S2(ω)},

C4,k ≡ {ω ∈ Ω;Uk(ω) < S2(ω) < γ − T},

such that ∪4i=1Ci,k = Ω for any k ≥ 1. Note that since S2,k = min(S2,k, γ − T ), for any

ω ∈ C1,k ∪ C2,k we have S2(ω) = S2,k(ω). Thus, for each ε > 0, {|S2,k − S2| ≥ ε} ⊆ C3,k ∪ C4,k,


implying that for all k ≥ 1, P (|S2,k − S2| ≥ ε) ≤ P (C3,k ∪ C4,k). Therefore,

∞∑k=1

P (|S2,k − S2| ≥ ε) ≤∞∑k=1

P (C3,k ∪ C4,k) ≤∞∑k=1

P (Uk ≤ γ − T ).

It remains for us to show that∑∞

k=1 P (Uk ≤ γ − T ) < ∞. Indeed, defining the stopping time

σ = inf{n ≥ 1;Un > γ − T} we have

∞∑k=1

P (Uk ≤ γ − T ) =

∞∑k=1

P (σ > k) = E[σ] < ∞,

where the inequality follows from the fact that Uk is the kth renewal epoch of a delayed renewal

process and hence the expected time for it to pass any constant threshold is finite. �

Proof of Lemma 2.6.3. Consider the tagged unit at the beginning of the ith idle period with

ν units in front of it. We condition on the number of demands arriving during the ith idle

period and consider two cases: (i) there are L = l ≤ ν and (ii) there are L ≥ ν + 1 demands

during the ith idle period. In case (i), the unit is not allocated during the ith idle period and

hence will be in the system at the beginning of the (i + 1)st idle period. Conditioning on the

number of outdates during the (i+ 1)st busy period N = n ≥ 0, the time interval between the

start of the ith and (i + 1)st idle periods has LT il(θ)zn(θ). Also, the unit will have ν + n − l

units in front of it at the beginning of the (i+1)st idle period and hence the LT of its remaining

sojourn time at the beginning of the (i + 1)st idle period is ϕkν+n−l,i+1(θ). It follows that the

remaining sojourn time of the unit at the beginning of the ith idle period given N = n and

L = l ≤ ν has LT il(θ)zn(θ)ϕkν+n−l,i+1(θ). Removing the conditions on L and N , the first

term on the right-hand side (RHS) follows. In case (ii), the unit is allocated during the ith idle

period. Thus, its remaining sojourn is equal to the time it takes until the (ν + 1)st demand

arrival. Note that given the idle period has not ended, the time between demand arrivals are

exponentially distributed with rate λ + μ. Thus, the time until the arrival of the (ν + 1)st

demand is Erlang distributed with parameter (λ + μ) and (ν + 1) phases, and hence its LT is

given by (λ+ μ

μ+ λ+ θ

)ν+1

. (2.45)


Also, from Proposition 5 the event {L ≥ ν + 1} has probability

∞∑l=ν+1

P (L = l) = 1−ν∑

l=0

(μ

μ+ λ

)l ( λ

μ+ λ

)=

(μ

μ+ λ

)ν+1

,

which after being multiplied by (2.45) gives the second term on the RHS. �

Proof of Theorems 2.6.4 and 2.6.5

We need the following lemmas before presenting the proofs.

Lemma 2.9.1 For ω ≥ 0 we have

v∑l=0

⎛⎝ v + n− l + 1

ω

⎞⎠ =

ω∑κ=0

⎛⎝ v + 1

ω + 1− κ

⎞⎠⎛⎝ n+ 1

κ

⎞⎠ . (2.46)

Proof. Starting from the left-hand side (LHS) we first claim that

v∑l=0

⎛⎝ v + n− l + 1

ω

⎞⎠ =

⎛⎝ v + n+ 2

ω + 1

⎞⎠−

⎛⎝ n+ 1

ω + 1

⎞⎠ , (2.47)

which can be proved by induction on v. For v = 0 (2.47) becomes

⎛⎝ n+ 1

ω

⎞⎠ =

⎛⎝ n+ 2

ω + 1

⎞⎠−

⎛⎝ n+ 1

ω + 1

⎞⎠ ,

which is the Pascal’s recurrence (see, e.g., Gross 2008, page 218). Now assume for some v ≥ 1

thatv−1∑l=0

⎛⎝ v + n− l

ω

⎞⎠ =

⎛⎝ v + n+ 1

ω + 1

⎞⎠−

⎛⎝ n+ 1

ω + 1

⎞⎠ ,

then

v∑l=0

⎛⎝ v + n− l + 1

ω

⎞⎠ =

⎛⎝ v + n+ 1

ω

⎞⎠+

v∑l=1

⎛⎝ v + n− l + 1

ω

⎞⎠ =

⎛⎝ v + n+ 1

ω

⎞⎠+

v−1∑l=0

⎛⎝ v + n− l

ω

⎞⎠

=

⎛⎝ v + n+ 1

ω

⎞⎠+

⎛⎝ v + n+ 1

ω + 1

⎞⎠−

⎛⎝ n+ 1

ω + 1

⎞⎠ (induction hypothesis)

=

⎛⎝ v + n+ 2

ω + 1

⎞⎠−

⎛⎝ n+ 1

ω + 1

⎞⎠ , (Pascal’s recurrence)


as claimed. Next, applying Vandermonde’s convolution (see, e.g., Gross 2008, page 226) to

the first term we get

⎛⎝ v + n+ 2

ω + 1

⎞⎠ =

ω+1∑κ=0

⎛⎝ v + 1

ω + 1− κ

⎞⎠⎛⎝ n+ 1

κ

⎞⎠

=ω∑

κ=0

⎛⎝ v + 1

ω + 1− κ

⎞⎠⎛⎝ n+ 1

κ

⎞⎠+

⎛⎝ n+ 1

ω + 1

⎞⎠ ,

which after substituting in (2.47) gives (2.46). �

Lemma 2.9.2 For i ≥ 0 we have

∞∑n=0

(μ

λ+ μ+ θ

)n

P (N = n)zn(θ)

⎛⎝ n+ 1

i

⎞⎠ = ξi(θ), (2.48)

and∞∑

m=0

(μ

λ+ μ+ θ

)m+1

P (M = m)rm(θ)

⎛⎝ m+ 1

i

⎞⎠ = βi(θ) (2.49)

with ξi(θ) and βi(θ) given in (4.4) and (2.39), respectively.

Proof. We give a proof for (2.49); (2.48) can be obtained similarly. Substituting for P (M = l)

and rm(θ) from (2.18) and (2.19) into (2.49), and using the definitions in (2.37) the LHS becomes

c1(θ)

∞∑m=0

(c2(θ))m

⎛⎝ m+ 1

i

⎞⎠ ,

establishing (2.49) for i = 0. To obtain the formula for i ≥ 1, note that

i!

(1− c2(θ))i+1=

di

d (c2(θ))i

∞∑m=0

(c2(θ))m =

∞∑m=0

di

d (c2(θ))i(c2(θ))

m

=∞∑

m=0

m(m− 1) · · · (m− i+ 1) (c2(θ))m−i

=∞∑

m=i

m(m− 1) · · · (m− i+ 1) (c2(θ))m−i .


Multiplying both sides by c2(θ)i−1/i! yields:

(c2(θ))i−1

(1− c2(θ))i+1=

∞∑m=i

(c2(θ))m−1

⎛⎝ m

i

⎞⎠ =

∞∑m=i−1

(c2(θ))m

⎛⎝ m+ 1

i

⎞⎠ =

∞∑m=0

(c2(θ))m

⎛⎝ m+ 1

i

⎞⎠ ,

from which (2.49) follows. �

Lemma 2.9.3 For ν, i ≥ 0 the following identity holds:

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)Yν+n−l(i) =

(λ

λ+ μ+ θ

)Yν(i+ 1). (2.50)

Proof. Substituting for P (L = l), il(θ) and Yν+n−l(i) into the LHS from (2.25),(2.26) and

(2.40), respectively and changing the order of sums, after some simplifications we can rewrite

the LHS as

(μ

λ+ μ+ θ

)ν+1( λ

λ+ μ+ θ

)ξ0(θ)

1∑j1=0

ξj1(θ)

2−j1∑j2=0

ξj2(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)

∞∑n=0

(μ

λ+ μ+ θ

)n

P (N = n)zn(θ)

ν∑l=0

⎛⎝ ν + n− l + 1

i− ji−1 − · · · − j1

⎞⎠ . (2.51)

Using Lemma 2.9.1 the last sum is

ν∑l=0

⎛⎝ ν + n− l + 1

i− ji−1 − · · · − j1

⎞⎠ =

i−ji−1−···−j1∑ji=0

⎛⎝ ν + 1

(i+ 1)− ji − ji−1 − · · · − j1

⎞⎠⎛⎝ n+ 1

ji

⎞⎠ ,

which allows us to rewrite (2.51) as

(μ

λ+ μ+ θ

)ν+1( λ

λ+ μ+ θ

)ξ0(θ)

1∑j1=0

ξj1(θ)

2−j1∑j2=0

ξj2(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)

i−ji−1−···−j1∑ji=0

∞∑n=0

(μ

λ+ μ+ θ

)n

P (N = n)zn(θ)

⎛⎝ n+ 1

ji

⎞⎠⎛⎝ ν + 1

(i+ 1)− ji − · · · − j1

⎞⎠ . (2.52)


Rearranging the terms in (2.52) and using (2.48) we arrive at the RHS:

(λ

λ+ μ+ θ

)ξ0(θ)

1∑j1=0

ξj1(θ)

2−j1∑j2=0

ξj2(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)

i−ji−1−···−j1∑ji=0

ξji(θ)

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

(i+ 1)− ji − · · · − j1

⎞⎠ =

(λ

λ+ μ+ θ

)Yν(i+ 1).

This completes the proof. �

Proof of Theorem 2.6.4. The proof is by induction on k. For k = 1, noting that ϕkν,k+1(θ) = 1

and using (2.36) we have

ϕ1ν,1(θ) =

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)ϕ1ν+n−l,2(θ) +

(μ

μ+ λ+ θ

)ν+1

=

ν∑l=0

P (L = l)il(θ)

∞∑n=0

P (N = n)zn(θ) +

(μ

μ+ λ+ θ

)ν+1

. (2.53)

Noting that

ν∑l=0

P (L = l)il(θ) =

ν∑l=0

(λ

μ+ λ

)(μ

μ+ λ

)l ( λ+ μ

λ+ μ+ θ

)l+1

=

(λ

λ+ μ+ θ

) ν∑l=0

(μ

λ+ μ+ θ

)l

=

(λ

μ+ λ

)(1−

(μ

μ+ λ+ θ

)ν+1), (2.54)

and using (2.24), we can simplify (2.53) to obtain

ϕ1ν,1(θ) =

(1−

(μ

μ+ λ+ θ

)ν+1 )i(θ)z(θ) +

(μ

μ+ λ+ θ

)ν+1

=

(μ

μ+ λ+ θ

)ν+1

(1− i(θ)z(θ)) + i(θ)z(θ)

= Yν(0)(1− i(θ)z(θ)) + i(θ)z(θ),


which establishes (2.41) for k = 1. Now assume that (2.41) holds for some k ≥ 1. From (2.36)

we have

ϕk+1ν,1 (θ) =

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)ϕk+1ν+n−l,2(θ) +

(μ

μ+ λ+ θ

)ν+1. (2.55)

Note that by construction of the kth modified system we have ϕk+1ν+n−l,2(θ) = ϕk

ν+n−l,1(θ).

Substituting ϕkν+n−l,1(θ) for ϕ

k+1ν+n−l,2(θ) in (2.55) and using (2.41) we get

ϕk+1ν,1 (θ) =

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)

×(

k−1∑i=0

Yν+n−l(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

+ (i(θ)z(θ))k

)+

(μ

μ+ λ+ θ

)ν+1

=

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)

k−1∑i=0


)( λ

λ+ μ+ θ

)i

+ν∑

l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)(i(θ)z(θ))k +

(μ

μ+ λ+ θ

)ν+1. (2.56)

We start with the first term in (2.56). Rearranging the terms and applying Lemma 2.9.3 we

have

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)k−1∑i=0


)( λ

λ+ μ+ θ

)i

=

k−1∑i=0

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)Yν+n−l(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

=

k−1∑i=0

(λ

λ+ μ+ θ

)Yν(i+ 1)

(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

=

(k+1)−1∑i=1

Yν(i)(1− (i(θ)z(θ))(k+1)−i

)( λ

λ+ μ+ θ

)i

. (2.57)


The second term in (2.56) can be evaluated by rearranging the sums and using (2.54):

ν∑l=0

∞∑n=0

P (L = l)P (N = n)il(θ)zn(θ)(i(θ)z(θ))k

=(i(θ)z(θ)

)k ν∑l=0

P (L = l)il(θ)

∞∑n=0

P (N = n)zn(θ)

=(i(θ)z(θ)

)k(1−

(μ

μ+ λ+ θ

)ν+1 )i(θ)z(θ)

=(1−

(μ

μ+ λ+ θ

)ν+1 )(i(θ)z(θ)

)k+1. (2.58)

Finally, substituting (2.57) and (2.58) into (2.56) we get

ϕk+1ν,1 (θ) =

(μ

μ+ λ+ θ

)ν+1 +

(k+1)−1∑i=1

Yν(i)(1− (i(θ)z(θ))(k+1)−i

)( λ

λ+ μ+ θ

)i

+(1−

(μ

μ+ λ+ θ

)ν+1 )(i(θ)z(θ)

)k+1

=

(μ

μ+ λ+ θ

)ν+1

(1− (i(θ)z(θ))k+1

)

+

(k+1)−1∑i=1

Yν(i)(1− (i(θ)z(θ))(k+1)−i

)( λ

λ+ μ+ θ

)i

+(i(θ)z(θ)

)k+1

=

(k+1)−1∑i=0

Yν(i)(1− (i(θ)z(θ))(k+1)−i

)( λ

λ+ μ+ θ

)i

+(i(θ)z(θ)

)k+1.

This completes the proof. �

Proof of Theorem 2.6.5. Substituting ϕkν,1(θ) from (2.41) into (2.35) we have

E[e−θS2,k ] =

∞∑ν=0

P (M = ν)rν(θ)

(k−1∑i=0

Yν(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

+ (i(θ)z(θ))k

)

=k−1∑i=0

∞∑ν=0

P (M = ν)rν(θ)Yν(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

+

∞∑ν=0

P (M = ν)rν(θ)(i(θ)z(θ))k. (2.59)

We deal with the two terms separately. First, substituting for Yν(i) in the first term and using


Lemma 2.9.2, we have

∞∑ν=0

k−1∑i=0

P (M = ν)rν(θ)Yν(i)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

=

k−1∑i=0

⎡⎣ξ0(θ) 1∑

j1=0

ξj1(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)

∞∑ν=0

P (M = ν)rν(θ)

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

i− ji−1 − · · · − j1

⎞⎠

×(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i]

=

∞∑ν=0

P (M = ν)rν(θ)

(μ

λ+ μ+ θ

)ν+1 (1− (i(θ)z(θ))k

)

+

k−1∑i=1

⎡⎣ξ0(θ) 1∑

j1=0

ξj1(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)

∞∑ν=0

P (M = ν)rν(θ)

(μ

λ+ μ+ θ

)ν+1⎛⎝ ν + 1

i− ji−1 − · · · − j1

⎞⎠

×(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i]

=(1− (i(θ)z(θ))k

)β0(θ)

+k−1∑i=1

⎡⎣ξ0(θ) 1∑

j1=0

ξj1(θ) · · ·(i−1)−ji−2−···−j1∑

ji−1=0

ξji−1(θ)βi−ji−1−···−j1(θ)

(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i⎤⎦ . (2.60)

Using the definition of X(d, w) in (2.42) for w = 1, (2.60) becomes

(1− (i(θ)z(θ))k

)β0(θ) + ξ0(θ)

k−1∑i=1

X(i, 1)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

. (2.61)

Next, the second term in (2.59) is simply

∞∑ν=0

P (M = ν)rν(θ)(i(θ)z(θ))k = (i(θ)z(θ))kr(θ). (2.62)

Substituting (2.61) and (2.62) back into (2.59) we have

E[e−θS2,k ] = (i(θ)z(θ))kr(θ) +(1− (i(θ)z(θ))k

)β0(θ)

+ξ0(θ)

k−1∑i=1

X(i, 1)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

(2.63)

= (i(θ)z(θ))kr(θ) +(1− (i(θ)z(θ))k

)β0(θ)

+ξ0(θ)

k−1∑i=1

X(i, 1)

(λ

λ+ μ+ θ

)i

− ξ0(θ)

k−1∑i=1

X(i, 1)(i(θ)z(θ))k−i

(λ

λ+ μ+ θ

)i

.

(2.64)


Letting k → ∞ in (2.64), the first term goes to 0 and the second term converges to β0(θ).

Therefore, to get the final result it remains to show that for all θ > 0 the last term converges

to 0 as k → ∞. To this end, consider E[e−θS2,k ] in (2.63) and note that it converges if and only

if the sequence {Fk}k≥2 with

Fk ≡k−1∑i=1

X(i, 1)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

,

is convergent. However, from Theorem 1 we know that E[e−θS2,k ] converges and therefore {Fk}is indeed convergent. Thus, we have |Fk+1 − Fk| → 0 as k → ∞. Now observe that

Fk+1 − Fk =

k∑i=1

X(i, 1)(1− (i(θ)z(θ))k+1−i

)( λ

λ+ μ+ θ

)i

−k−1∑i=1

X(i, 1)(1− (i(θ)z(θ))k−i

)( λ

λ+ μ+ θ

)i

= X(k, 1)(1− i(θ)z(θ)

)( λ

λ+ μ+ θ

)k

+(1− i(θ)z(θ)

) k−1∑i=1


(λ

λ+ μ+ θ

)i

=(1− i(θ)z(θ)

) k∑i=1


(λ

λ+ μ+ θ

)i

,

and hence,

limk→∞

k∑i=1


(λ

λ+ μ+ θ

)i

= 0,

for all θ > 0. Finally, noting that for k ≥ 2

k∑i=1


(λ

λ+ μ+ θ

)i

≥k−1∑i=1


(λ

λ+ μ+ θ

)i

,

we can conclude that the last term in (2.64) vanishes as k → ∞, which completes the proof. �


2.10 Appendix B: MDP formulation for finding the optimal

policy

Let s = [s1, s2, . . . , sN ]T (T denotes transpose) denote the system state where si is the number

of units of age i in the inventory and N is the shelf-life. Since at each period there is at

most one unit arrival or demand, we have si ∈ {0, 1}. Denote by ps and pd the probability

of having a supply or demand at each period, respectively. We assume that each unsatisfied

demand incurs a cost of cs, and allocating a unit with age i incurs a cost of cai. Note that since

both the demand and supply processes are exogenous, minimizing the shortages is equivalent

to minimizing outdates.

Let V denote the minimum expected long-run α-discounted total cost starting in state s.

Then V satisfies the optimality equation:

V (s) = (1− ps − pd)αV (A(s)) + psαV (A(s+ e1))

+pd

[1{s �= 0} min

{i;si=1}{αV (A(s− ei)) + cai}+ 1{s = 0}(cs + αV (s))

], (2.65)

where ei denotes the vector with a 1 in the ith coordinate and 0’s elsewhere, and A(s) = As

with A = [aij ]N×N such that aij = 1 if i − j = 1, and aij = 0 otherwise. The three terms in

(2.65) correspond to having no supply or demand, a unit arrival, and a demand arrival during

the period, respectively. Note that A is a function that “ages” the inventory, i.e., increases the

age of all available units by 1. Since the state and action spaces are bounded in our problem,

the long-run average optimal policy can be obtained by letting α → 1 (see Putterman 1994).

Chapter 3

Reducing the Age of Transfused Red

Blood Cells in Hospitals: Ordering

and Allocation Policies

3.1 Introduction

Red Blood Cell (RBC) transfusion is an integral part of many medical treatments and surgeries;

in 2011, over 13 million units of voluntarily donated RBCs were transfused to patients across

the United States (NBCUS 2011). While advances in storage solutions led to an increase of

RBCs shelf-life from 35 to 42 days in the late 1970s, an extensive body of recent cohort studies

(e.g., Koch et al. 2008, Eikelboom et al. 2010, Offner et al. 2002, Pettila et al. 2011) suggests a

range of moderate to strong correlation between receiving “older” blood and increased risk of

adverse medical outcomes such as infection, morbidity, and mortality. The results are however

still inconclusive (Lelubre et al. 2009). It remains for Randomized Controlled Trials (RCTs) to

clarify the relationship between the age of blood and health outcomes for transfused patients.

Recently completed RCTs could not show an increased risk of harm for premature infants

(Fergusson et al. 2012), cardiac surgery patients (Steiner et al. 2015) and critically ill adults

(Lacroix et al. 2015). However, the results of four ongoing RCTs (e.g., Heddle 2011) enrolling a

total of 40,835 patients are yet to be published and collectively they will have the sufficient power

to detect even small differences in mortality (1-1.5%). Even these small differences would be of

clinical relevance given that transfusion is one of the most frequently used medical interventions.

48

Chapter 3. Reducing the Age of Transfused Red Blood Cells in Hospitals 49

Therefore, in anticipation of the results of RCTs confirming the adverse outcomes of “older”

RBCs, it is imperative to be equipped with inventory policies that could reduce the age of

transfused RBCs without compromising availability or resulting in excessive outdates. A recent

commentary (Sayers and Centilli 2012) in the journal of Transfusion, specifically emphasizes

the need for inventory and supply chain management policies that could contribute to reducing

the age of transfused RBC units without compromising the adequacy of supply. Nevertheless,

although there are a few papers investigating the impact of shortening the shelf-life of blood,

limited attention has been given to alternative inventory policies.

Previous studies have investigated the impact of reducing the shelf-life of RBCs (currently 42

days) at the level of hospital (Fontaine et al. 2011) and on regional blood supply chains (Blake

et al. 2012, Grasas et al. 2014). In a simulation study Atkinson et al. (2012) investigated the

potential of a family of threshold–based allocation policies in reducing the age of transfused

RBCs without jeopardizing its availability. In the previous chapter, we further investigated

the threshold policy using an analytical approach. In both studies, however, the supply of

blood to the hospital was treated as exogenous and hence ordering policies were not considered.

However, RBC units in hospitals are typically ordered from a central supplier. By changing the

ordering policy, the hospital can control the amount of RBC inventory on hand, which in turn

affects the age of units at the time of issue.

The aim of this study was to asses the performance of practical ordering and allocation

policies in reducing the age of transfused RBCs while keeping the outdate and shortage rates

low. We developed a data-driven (evidence-based) simulation model based on the operations of

the RBC inventory of an acute care teaching hospital in Hamilton, Ontario. We used historical

data for 2011 to validate the model and estimate its inputs. For a given allocation and ordering

policy, the model estimates the annual outdate rate, shortage rate, and the distribution of the

age of issued RBC units. Using the model, we evaluated and compared the outcome of several

ordering and allocation policies when satisfying the same historical demand as observed in 2011.

The outputs were compared to the performance of the hospital as a benchmark.

3.2 Data

We used transaction-level records for RBC inventory of the hospital to validate our model

and estimate its required inputs. The data were extracted from the Transfusion Registry for


Utilization, Surveillance, and Tracking (TRUST) database, a prospective hospital-based registry

of all patients admitted to acute care hospitals in Hamilton, Ontario. We used records for all

RBC units received by the hospital in 2011, which amounted to a total of 10,411 records. Each

record corresponded to a RBC unit and included the (i) ABO/Rh type of the unit; (ii) the

date it was collected from the donor by Canadian Blood Services (CBS); (iii) the date it was

received by the hospital; (iv) the final disposition of the unit (transfused, expired, etc.) and

the date it occurred; and (v) the patient ABO/Rh type for transfused units.

3.3 A data-driven simulation model of RBC inventory

3.3.1 Simulation procedure

We developed and implemented our simulation model in R (R Development Core Team 2013).

The simulation model has 4 main parts. Below we briefly describe each part.

1. Initialize. At the beginning of the simulation, the inventory is initialized by setting the

number of available inventory for each blood type and the initial age of each unit.

2. Replenish inventory. At the beginning of each simulation cycle (corresponding to a

day) the inventory is replenished according to the ordering policy.We assume that the

orders are fully satisfied.

3. Satisfy demand. The demand for the day is realized one-by-one and satisfied (if possi-

ble) according to the allocation policy and the ABO/Rh rule. Any demand that cannot

be satisfied using the inventory on hand is assumed to be satisfied using an emergency

order and is counted as unmet demand.

4. Update inventory. At the end of the simulation cycle, the age of all units in inventory

are increased by 1. All units exceeding the shelf-life are removed from the inventory and

counted as outdated.

3.3.2 Estimating the model inputs using data

The model has two types of inputs; fixed and controlled. Fixed inputs were estimated or in-

ferred using the data and kept identical in all scenarios. Controlled inputs however varied

among different scenarios. Since the objective of our study was to evaluate the outcome of


1−7 8−14 15−21 22−28 29−35 36−42

Receipt age distribution

Days

% o

f RB

C u

nits

010

2030

4050

60

Figure 3.1: Distribution of the age of units at the time of receipt by the hospital.

alternative ordering and allocation policies, they were set as the controlled inputs. To validate

the simulation model we also modeled the current ordering and allocation policies of the hos-

pital. The resulting model is called the “baseline scenario” and its performance is compared to

the empirical estimates in the following subsection.

In order to estimate some of the model inputs we needed to observe the state of the inventory.

To this end, we implemented a trace-driven simulation by setting all inputs of the simulation

model according to the data. That is, for each day i, the inventory was replenished with units

that had receipt date equal to i and with the same receipt age observed in the data, demand

was set to all units which were transfused on day i, and the units were allocated to the demand

exactly as observed in the data. This way, we were able to observe the state of the inventory

(count and types) each time a unit was transfused. The trace-driven simulation was utilized in

estimating some of the required inputs as explained below.

Initial inventory. The initial inventory was obtained using data for 2010. Specifically, the

inventory was initiated by identifying and calculating the age of all units which were received

by the hospital in 2010 but their final disposition occurred in 2011.

Initial age of units. We used the data for 2011 to obtain the empirical distribution of

the age of units at the time of receipt by the hospital for each blood type. When replenishing

the inventory, we simulated the receipt age of units by randomly drawing from these empirical

distributions. The receipt age distribution is illustrated in Figure 3.1.

Demand. Daily demand data for 2011 was directly used as the demand input in the

simulation model. A time series plot of the demand is presented in Figure 3.2. The average


daily demand was 28.6 units with standard deviation of 12.2. The daily demand fluctuated

considerably during the year with a maximum of 76 units observed in November, and a minimum

of 6 units in August.

Other final dispositions. Beside transfused and expired units, there were also a few units

that were contaminated, destroyed or transferred to another hospital. These events were taken

into account directly using the data. For example, if a unit was contaminated on January 1st,

in the simulation model a unit of the same type and age was randomly selected and removed

from the inventory in the first simulation cycle. If no unit with the same age and type was

available in inventory, a unit of the same type was randomly selected.

Emergency/trauma cases. Using the trace-driven simulation model we identified 769

instances where a compatible unit was substituted for an exact match. In 689 cases, however,

an exact match was available in the inventory. The majority of these cases corresponded to

substitution of O– and O+ blood types, suggesting that the demand points were associated with

emergency and trauma cases. We flagged these demand points in all our scenarios and forced

the model to allocate the same type observed in the data regardless of the available inventory.

If that specific type was not available in inventory, the demand was counted as unmet.

Allocation policy. We observed that the hospital does not strictly follow a FIFO policy.

In particular, among 10,349 transfused units there were 8709 (84%) instances where an older

unit of the same type was available in the inventory. Nevertheless, in the majority of cases the

allocated unit was among the older ones available in the inventory. That is, older units were

allocated with a higher probability specially if there were units in the inventory that had a

short remaining shelf-life. We also analyzed how ABO/Rh compatible units were substituted in

cases where the demand could not be satisfied using an exact match. There were 80 instances

corresponding to cases where an exact match was not available in the inventory. We analyzed

these cases and identified the substitution rule given in Table 3.1. The rule is consistent with

what we observed from data except in 26 cases, 25 of which corresponded to substituting

compatible units for AB+ and 1 of them for AB–. For these types, the hospital did not follow

any specific rule and allocated one of the compatible units. To capture the allocation policy

of the hospital in our simulation, after selecting the type of the unit to be allocated (by giving

preference to an exact match and using Table 3.1 if an exact match is not available) one unit

was randomly selected from the n oldest units except if the oldest unit was older than Z days.

In the latter case, the oldest unit was allocated. We then estimated the parameters to be n = 8


Daily demand in 2011

Dai

ly D

eman

d

1030

5070

Jan. Mar. May July Sep Nov

Figure 3.2: A time series plot of daily demand in 2011. The dotted lines mark the first, secondand third quartiles.

Demand blood type Compatible types in order of preference

First Second Third Fourth Fifth

A+ O+ A– O–

A– O– A+ O+

B+ O+ B– O–

B– O– B+ O+

AB+ A+ B+ AB- O– B–

AB- A– O+ O– B– AB+O+ O–

O– O+

Table 3.1: ABO/Rh substitution rule inferred from data.

and Z = 34 using the data.

Ordering policy. Inventory was reviewed at the end of each day and orders were made as

needed (typically no orders were received on Sundays). Non-routine orders were also made if

the inventory level of any blood type was “too low”. The hospital had specific target inventory

levels for each blood type to replenish the inventory to. Analysis of the data revealed that the

hospital did not strictly follow the provided target levels and held higher inventory levels. In the

data we could not observe the type of orders. Therefore, we used the average inventory-on-hand

at the beginning of the day as order-up-to levels for each blood type. In the simulation model,

each day (except on Sundays) the inventory of each blood type was updated to the order-up-to

levels.


Average Issue age Number Transfused

Product Data Simulation Relative error Data Simulation Relative error

Mean SD Mean (%) SD(%) Mean SD Mean(%) SD(%)

A+ 20.78 20.54 0.07 1.14 0.32 3371 3406.9 0.07 1.08 0.27A– 21.78 21.44 0.20 1.68 0.65 691 685.7 0.20 2.03 0.60B+ 27.23 27.31 0.24 0.77 0.45 789 784.5 0.24 0.58 0.16B– 17.17 17.73 0.31 3.28 1.71 139 136.9 0.31 1.99 1.20

AB+ 25.63 25.15 0.62 2.47 1.79 253 228.9 0.62 9.68 3.46AB– 24.75 22.10 1.21 10.71 4.88 32 33.6 1.21 5.12 2.14O+ 19.03 18.96 0.07 0.43 0.33 4032 4029.6 0.07 0.10 0.07O– 19.97 20.11 0.12 0.78 0.45 1042 1042 0.12 0.50 0.29

All types 20.66 20.52 0.06 0.67 0.27 10349 10349 - 0.00 -

Table 3.2: The output of simulation (baseline ) versus empirical data.

3.3.3 Validation

To validate the model we conducted simulations representing the status quo operations of

the hospital and compared the outputs to the data. We conducted 30 simulations of the

baseline scenario and compared the average outputs with data. The average issue age and

the number of transfusions for each blood type are presented and compared to data in Table

3.2. The aggregate distribution of the issue age (across all blood types) obtained from the

simulation is also compared with the empirical distribution obtained from data in Figure 3.3.

Results indicate that the simulation outputs are in close agreement with the historical data

in terms of the issue age. We also validated our simulation model in terms of the number

of outdated units. The average number of outdated units across simulation outputs was 13.9

units (standard deviation: 3.27), which is higher than the actual number of outdates in 2011

(6 units). However, this difference (which is less than %0.08 of total units considered in the

simulation) can be attributed to our simplified model of the hospital ordering policy that does

not include more flexible practices such as emergency orders. Overall, minor discrepancies

between the simulation and reality are expected as the ordering and allocation policies followed

by the hospital are not as systematic as modeled in the simulation. Nevertheless, the proposed

model seems to adequately capture the main features of the hospital RBC inventory and hence

is useful in evaluating the effect of alternative (and more systematic) ordering and allocation

polices.


1−7 8−14 15−21 22−28 29−35 36−42

DataSimulation

Issue Age

Days

% o

f RB

C u

nits

010

2030

4050

60

Figure 3.3: Issue age distribution obtained from simulation compared to the empirical distribution

obtained from data.

3.4 Policies

In all scenarios the ordering policy was to update the inventory level on a daily basis (except

Sundays) to specified order-up-to levels. Order-up-to levels in hospitals are usually set according

to inventory targets measured in terms of the average days of demand on hand. We followed

the same approach in setting the inventory targets and varied the order-up-to levels across

scenarios.

A modified version of the threshold-based allocation policy was considered and the threshold

value was varied across the scenarios. Under a threshold policy with parameter T , first the type

of blood to be allocated is selected by giving the highest preference to an exact match, and

selecting a compatible unit in the order specified in Table 3.1 (also used in Simonetti et al. 2014)

if an exact match is not available. Next, the oldest unit of the selected type that is younger

than the threshold T is allocated. If all units are older than T , the freshest unit available is

allocated.

We considered 4 different ordering policies by varying the order-up-to levels. Specifically, we

considered the current ordering policy (i.e., with order-up-to levels set to the average inventory

level for each type) in addition to the alternatives of keeping 7, 5, and 3 times the average

daily required inventory in the hospital. For each of the four ordering policies, we considered

6 allocation policies by setting the threshold value to T = 42, 28, 21, 14, 7, and 0. We then

simulated each of the 24 scenarios and compared their performances.


Type Compatible type in order of preferenceFirst Second Third Fourth Fifth Sixth Seventh

A+ O+ A– O–A– O– A+ O+B+ O+ B– O–B– O– B+ O+

AB+ A+ B+ O+ AB– A– B– O–AB– A– B– O– AB+ A+ B+ O+O+ O–O– O+

Table 3.3: ABO/Rh compatibility in order of preference used in simulation scenarios when anexact match is not available.

3.5 Results of the simulation experiments

3.5.1 Trade–off between the average issue age and the outdate rate

The output of scenarios were compared to the current performance in terms of the resulting

issue age, proportion of outdated units and the proportion of unsatisfied demand. No unsatisfied

demand was observed under any of the allocation policies when 5 or more days of inventory was

kept (data not shown). When the policy was to keep 3 days of inventory, a small proportion of

demand was unsatisfied (maximum 0.65% for LIFO). Figure 3.4 illustrates the tradeoff between

the average issue age and proportion of outdates across all scenarios. Comparing to the current

performance (average issue age: 20.66 days, outdate rate: 0.06%), by adopting a strict FIFO

policy (T = 42) and keeping 7 or 5 days of inventory, the average issue age is reduced by

19.9% (to 16.55) and 29.4% (to 14.59), respectively. This is achieved without an increase in

the outdate rate (0.06% for 7 days of inventory and 0.05% for 5) and without resulting in any

unmet demand. By adopting a FIFO policy and keeping 3 days of inventory, the average issue

age is reduced by 38.3% (to 12.74), again without an increase in the output rate (0.05%) but

leading to 0.64% of unsatisfied demand. Further reduction of average issue age for each of the

ordering policies is observed under alternative allocation policies (lower threshold values). This

is however achieved at the cost of an increased outdate rate. When inventory levels are high,

both the reduction of average issue age and the increase in outdate rate are more significant

(Figure 3.4). Under current inventory levels, following a threshold policy leads to unacceptably

high outdate rates (e.g. 18% for T = 14). With lower inventory levels the increase in output

rate is smaller, but at the same time the reduction of average issue age is less significant. When

the threshold value is 7, the performance is very close to that of LIFO for all ordering policies.


Outdate rate (%)

Aver

age

issu

e ag

e

1012

1416

1820

0 3 6 9 12 15 18 21 24

Hospital's ordering policyKeeping 7 days of inventoryKeeping 5 days of inventoryKeeping 3 days of inventory

Figure 3.4: Tradeoff between average issue age and proportion of outdates under different ordering and allo-

cation policies. Each line corresponds to the specified ordering policy. The tick marks on each line (from left to

right) correspond to allocation policies with T = 42 (FIFO), 28, 21, 14, 7, and 0 (LIFO). The symbol (×) marks

the performance of the hospital in 2011 obtained from the data.

3.5.2 Distribution of the issue age

We also evaluated the outcome of scenarios in terms of their resulting distribution of issue age.

The results for selected scenarios are presented in Figure 3.5. The empirical distribution of

issue age obtained from data is also presented as a benchmark. The empirical distribution has

a higher fraction of “older” units (e.g., above 28 or 35 days) compared to alternative scenarios

including the ones with FIFO allocation policy. As expected, allocated units tend to be older

when higher levels of inventory are kept. Non FIFO allocation polices significantly reduce the

fraction of units with issue age above the threshold, as they only issue units with age above

the threshold as a recourse action. An exception is when T = 7. This is because about 67% of

units have age higher than 7 at the time of receipt by the hospital (see Figure 3.1). For higher

threshold values, however, the majority of units have issue age below the threshold specially

when high levels of inventory are kept. This is further illustrated in Figure 3.6. Observe

that with current levels of inventory a significant majority of issued units have age below the

threshold (expect when T = 7). When the ordering policy is to keep 3 days of inventory, a

higher proportion of units have age above the threshold under each of the allocation policies.


1−7 8−14 15−21 22−28 29−35 36−42

DataFIFOT=21T=14T=7

Issue age distribution (a)

Days

% o

f RB

C u

nits

020

4060

8010

0 Current inventory levels

1−7 8−14 15−21 22−28 29−35 36−42

DataFIFOT=21T=14T=7

Issue age distribution (b)

Days

% o

f RB

C u

nits

020

4060

8010

0 Keeping 5 days of inventory

1−7 8−14 15−21 22−28 29−35 36−42

DataFIFOT=21T=14T=7

Issue age distribution (c)

Days

% o

f RB

C u

nits

020

4060

8010

0 Keeping 3 days of inventory

Figure 3.5: Issue age distribution for selected allocation policies when keeping (a) current levelsof inventory, (b) 5 days of inventory and (c) 3 days of inventory.


FIFO T=28 T=21 T=14 T=7 LIFO

Issue age above T Issue age below or equal to T

Days

% o

f RB

C u

nits

020

4060

8010

0 Current inventory levels (a)

FIFO T=28 T=21 T=14 T=7 LIFO

Issue age above T Issue age below or equal to T

Days

% o

f RB

C u

nits

020

4060

8010

0 Keeping 3 days of inventory (b)

Figure 3.6: Fraction of units issued with age below or above the threshold under selectedpolicies.

3.5.3 Order size variability

In the data, we observed a much higher variability in order sizes compared to demand. Specif-

ically, average daily order size was 28.5 with an standard deviation of 25.1, while the average

daily demand was 28.3 with an standard deviation of 12.2 (see Figure 3.7 (a)). This amplifi-

cation of demand variability from the hospital to the supplier (CBS) can be recognized as the

Bullwhip effect observed in other industries (see, Lee et al. 1997). Assuming that this observa-

tion is not specific to our hospital, one would expect the demand variability faced by CBS to be

much higher than the demand at the hospitals. One consequence of this amplified variability,

among others, is the added levels of inventory required to be kept by CBS to cope with the

demand variability, which in turn increase the initial age of units received by the hospitals.

We observed, in our simulation experiments, that following a systematic order-up-to level

ordering policy significantly reduces the order size variability at the hopsital. For example,

keeping 5 days of inventory and following a FIFO allocation policy leads to an average order

size of 27.9 with an standard deviation of 16.9 (see Figure 3.7 (b)). That is a 33% reduction

in the standard deviation of the daily order size. Using the same ordering policy with Non-

FIFO allocation policies slightly increased the order variability. For example, keeping 5 days

of inventory and using a threshold policy with T = 21 leads to an average order size of 28.8

with standard deviation 17.3 (see Figure 3.7 (c)). The variability increased as we decreased

the threshold value, with maximum standard deviation 18.4 under the LIFO policy. The added

variability can be attributed to the increase in outdates under the threshold policy, which lowers

the inventory levels at certain points, requiring higher order sizes.


Figure 3.7: Daily demand and orders observed (a) in empirical data; (b) when keeping 5 daysof inventory and using a FIFO policy; and (c) when keeping 5 days of inventory and using athreshold policy with T = 21.


A+ A− B+ B− AB+ AB− O+ O−

FIFOT=21T=14

Type distribution among outdates

Days

% o

f out

date

d R

BC

uni

ts

010

2030

4050

60

Keeping 5 days of inventory

Figure 3.8: Type distribution of outdated units for selected scenarios.

3.5.4 Type distribution among outdated units

We also investigated the type distribution among outdated units for each scenario. In particular,

we calculated the fraction of each ABO/Rh type among the outdated units for all scenarios. We

observed that when the outdate rate is small (e.g., with lower inventory and under FIFO policy)

outdates are among types with lower demand (i.e., AB–, AB+). However, under scenarios with

significant outdate rate, the outdates were mainly among the more prevalent types. The types

with the highest outdate rate were O+, A+, and B+ , also suggesting a higher fraction of

outdates for Rh+ blood. This is exemplified in Figure 3.8 where the type distribution of

outdates is presented for selected allocation policies and when keeping 5 days of inventory.

3.6 Discussion

The objective of this research was to study the effectiveness of practical inventory policies in

reducing the age of transfused RBCs. We built a simulation model based on the operations of

the RBC inventory of a Canadian acute care teaching hospital. We used our model to estimate

and compare the performance of various ordering and allocation policies.

Our analysis revealed that the hospital did not strictly follow a FIFO policy. In contrast,

we observed that the allocated units were among the “older” units available in the inventory.

However, if a unit was close to its expiration (around last week of shelf-life) then it would

most likely be selected for transfusion. We speculate that this is linked to the current practice

whereby upon receiving a demand, RBC units are retrieved from the front of the fridge, where


the older units are kept, without necessarily ensuring that the oldest available unit is being

allocated unless a unit is soon to be expired. Another possible explanation could be related to

RBC units kept in the Operating Room (OR). Usually the required RBC units for surgeries

are packed in coolers and sent to the OR in the morning. The unused units however are not

returned to the inventory and are kept in the OR for future use. As a result, if there is demand

for RBC units in the ward or ICU it will be satisfied using the oldest units in the inventory

which could be fresher than those in the OR. Regardless of the exact reason, we found that

while this deviation from FIFO policy had little effect on the average issue age, its impact on

the distribution of issue age was significant. In particular, it resulted in significantly higher

proportion of units older than 28 days compared to an strict FIFO policy.

We observed that by following a strict FIFO policy and lowering the order-up-to levels, the

hospital could significantly reduce the average issue age. The reduction of average issue age

was achieved without resulting in unsatisfied demand or increasing the outdate rate, suggesting

that the hospital inventory levels were unnecessarily high.

Further reduction of the average issue age was observed under different threshold policies.

The threshold policy was particularly effective in limiting the proportion of units older than the

threshold. Nevertheless, it was observed that the tradeoff between the average issue age and the

outdate rate highly depends on the ordering policy. In particular, with high levels of inventory

the increase in outdate rate was usually unacceptably high. This could be explained using our

results on the distribution of issue age. Under a threshold policy with parameter T , units with

age above T are only issued when there are no units available below that age. With high levels

of inventory (e.g., 7 times the average daily demand) and daily replenishment, the availability

of units with age below the threshold is high (an exception is when T is very small e.g. 7 which

we shall discuss shortly). The excess inventory with age above T is therefore mostly not used

and eventually expires. In contrast, with lower levels of inventory, a lower proportion of units

have age above the threshold but a higher proportion of allocated units are among such units.

As a result, the added outdate rate is smaller, yet the decrease in average issue age is also less

significant. When the threshold value was very small (e.g., equal to 7) the performance of the

threshold policy was close to LIFO. This is linked to the distribution of the age of RBCs at

the time of receipt by the hospital (Figure 3.1). About 67% of units are already older than

7 days when delivered to the hospital. Therefore, even when large amounts of inventory are

kept, the majority of issued units have age above 7 days. Since under the threshold policy the


freshest unit is issued in case there are no units younger than the threshold, the policy results

in approximately the same performance as LIFO which always allocates the freshest unit.

Our study has some limitations. We evaluated the policies in meeting the same demand

as observed in historical data. Drastic changes in demand are expected to affect some of our

results. For example, episodes of high demand are expected to increase the proportion of unmet

demand, and episodes of low demand are likely to increase the issue age. Robustness of our

results to different demand profiles should therefore be investigated in future research. We

also assumed that the hospital orders are fully satisfied. This is not always true as sometimes

the orders are partially satisfied if the supplier is facing a shortage. Nevertheless, our study

provides insights and quantifies the impact of inventory levels on the outdate rate, issue age

and availability of RBCs.

The impact of inventory levels in hospitals on the age of transfused RBCs has been previously

highlighted in the literature (Dzik et al. 2013). To the best of our knowledge, however, this

is the first study that investigates the joint effect of ordering and allocation policies on the

age and availability of RBCs. Our study differs from that of Atkinson et al. (2012) both with

respect to the modeling assumptions and the specifics of the threshold-based allocation policy.

In Atkinson et al. (2012), the suggested policy is to allocate the oldest unit among all compatible

units that are younger than the threshold if an exact match is not available. This is in contrast

to our policy, which is to first choose the compatible type using Table 3.3 and then allocate the

oldest unit of that type which is younger than the threshold. While simulation results reviled no

significant difference between the outcomes of the two variations of the threshold policy (data

not shown), our policy seems to be more practical as it could be easily implemented by dividing

the shelf for each blood type into two categories (younger and older than the threshold) and

it does not require information on the age of all units in the inventory. Our results on the

performance of the threshold policy also differ from those reported by Atkinson et al. (2012).

In their study they estimated that by using a policy with a threshold of 14 days, the average

issue age can be reduced by 10 to 20 days at the cost of increasing the proportion of unmet

demand by 0.05% (outdate rate not reported). Our study predicts a less significant reduction of

average issue age and identifies the potential risk of increased outdate rate under the threshold

policy when inventory levels are high. The discrepancy between our results can be linked to

the modeling assumptions, in particular those on the supply side. In their model RBC units

arrive at the hospital one by one according to a random process. More importantly, all units


are 2 days old at the time of receipt by the hospital. In our model, inventory is replenished

according to an order-up-to policy and the initial age of units is randomly drawn from an

empirical distribution. Our assumptions are more representative of the operations of the blood

supply chain in North America.

Our study identifies a potential in reducing and controlling the age of transfused RBCs

through practical inventory policies at the level of hospitals. Future work should investigate

the effect of the size of the hospital, frequency of deliveries to the hospital, and alternative

ordering and allocation practices on the issue age and availability of RBCs. Furthermore, a

broader study of the blood supply chain is required to understand the effects of adjusting

inventory policies at a group of hospitals sharing the same supplier. For instance, we assumed

that modifying ordering policies of the hospital does not affect the initial age of units at the

time of receipt by the hospital. However, reduction of inventory levels at a group of hospitals

in the region could affect the initial age of units due to an increased level of inventory at the

supplier level. In this case, the supplier should also accordingly modify its inventory policies to

balance the risk of shortage and the age of supplied units to the hospitals. It should however

be noted that, as discussed in Subsection 3.5.3, following a systematic ordering policy at the

hospitals could reduce the demand variability for the supplier and allow it to better manage its

inventory.

Chapter 4

Rational Abandonment from

Priority Queues: Equilibrium

Strategy and Pricing Implications

4.1 Introduction

Priority queues are prevalent in today’s service industry. Many service firms assign customers

to different priority levels, or provide them with an option to purchase priority and receive

faster service. Such priority systems create a natural incentive for low-priority customers to

abandon the queue after joining, if they observe “too many” higher-priority customers arrive

and overtake them while they wait for service.

In a recent empirical study, Batt et al. (2013) investigate the factors affecting customer

abandonment from a hospital Emergency Department (ED), where upon arrival, patients are

assigned a priority level based on the severity of their conditions. They find that the observable

aspect of the queue affects the abandonment behavior of patients. In particular, they report

that observing additional arrivals, especially that of sicker (and higher-priority) patients, in-

creases the abandonment rate of lower-priority patients from the ED. Priority queues are also

common in the entertainment industry. One example is the implementation of priority queues

in amusement parks such as Six Flags America. Customers who purchase a gold Flash Pass are

given priority in taking the rides. Therefore, ordinary customers in the queue have an incentive

to abandon and look into other rides if they observe “too many” Flash Pass customers arriving

65

Chapter 4. Rational Abandonment from Priority Queues 66

at the ride. Similarly, visitors to The London Eye (a Ferris wheel in London, England) can

purchase a Fast Track ticket online and avoid the long queue at the location by entering a

separate line. Again, ordinary customers who observe the arrival of Fast Track customers may

decide to abandon the system if they feel that the wait is going to be “too long”.

These priority systems give rise to two fundamental questions: (1) How do rational cus-

tomers make abandonment decisions from such priority queues? (2) Given this abandonment

behavior, how should a service provider structure its pricing policy to maximize its revenue or

the system welfare?

As detailed in Section 4.2, these questions have remained unanswered so far. We answer

these questions for an observable M/M/1 queue with two priority classes. We assume that

customers are forward-looking upon arrival and make join/balk/abandon decisions to maximize

their expected utility, which is the difference of expected service reward minus linear delay

cost minus possible fees for order placement and service delivery. We characterize customers’

equilibrium join/balk/abandon strategy and investigate its implications on the social welfare,

as well as the service provider’s revenue and pricing decisions. Whereas high-priority customers

have no incentive to abandon the queue, we show that the equilibrium join/balk/abandon

strategy of low-priority customers has a threshold structure that depends both on the queue

composition and on the pricing structure. We then consider pricing as a means to control the

balking and abandonment behavior of low-priority customers. A distinguishing feature of our

model is that in the presence of abandonment the timing of payment matters. In particular, we

demonstrate that charging the customer upon entering the system or service completion leads

to different outcomes. We show that welfare maximization requires charging only a service

fee, and no entrance fee (i.e., for order placement or joining the queue). In contrast, revenue

maximization requires charging not only a service fee, but also an entrance fee. Moreover,

charging only an entrance fee may generate more or less revenue than charging only a service

fee.

This chapter reports the following main contributions to the analysis and optimization of

queueing systems with strategic customers:

1. Rational abandonment in priority queues. We establish the structure of the equilibrium

join/balk/abandon strategy of low-priority customers, and recursively characterize the aban-

donment thresholds and customer utility. We develop these results by solving inter-related

optimal stopping problems.


2. Pricing in the presence of rational abandonment. This work is the first to study pricing

for any queueing model that accounts for customers’ rational abandonment decisions. We

provide guidelines on how services should be priced in presence of customer abandonment. In

particular, we demonstrate, both analytically and numerically, the importance of the timing of

payment.

This work may also stimulate more research, both on further models of rational abandon-

ment in priority queues, and on pricing and operational controls of such systems. As detailed

below, the literature on priority queues has so far focused on exogenous abandonment models

or outright ignored customers’ abandonment behavior.

The rest of this chapter is organized as follows. After reviewing the literature in the next

section, we describe the model in Section 4.3. Section 4.4 characterizes the equilibrium behavior

of low-priority customers. Section 4.5 discusses preliminaries for our analysis of pricing under

rational abandonment. Section 4.6 discusses pricing for welfare maximization, and Section 4.7

discusses pricing for revenue maximization. Section 4.8 offers concluding remarks. All proofs

are presented in the Appendix.

4.2 Related Literature

Our study relates to several branches of the literature, briefly summarized below.

Modeling customer abandonment through exogenous deadlines. The classical ap-

proach in modeling customer abandonment in queueing systems is to assume that customers

arrive at the system with i.i.d. patience thresholds, and abandon once their waiting time ex-

ceeds their threshold. There are numerous examples in the performance evaluation and control

literature, especially with applications to design and control of contact centers. See for instance

Garnett et al. (2002), Basamboo and Randhawa (2010, 2013), Zeltyn and Mandelbaum (2005),

Down et al. (2011), Baron and Milner (2009) for examples of single-class queues and Iravani

and Balcıoglu (2008), Sarhangian and Balcıoglu (2013), Jouini and Roubos (2013), Jennings

and Reed (2012) for multi-class queues. Pricing is generally ignored in this stream of literature.

One exception is Lee and Ward (2014) where the authors study the joint pricing and capac-

ity decision for a GI/GI/1 + GI queue using a diffusion approximation. The abandonment

behavior is however modeled exogenously and is independent of the service price.

Models of rational abandonment. A smaller set of papers endogenize the abandonment


behavior of customers. However, they consider service disciplines that differ from the priority

policy considered in this chapter, and do not consider pricing (see Hassin and Haviv 2003,

Chapter 5).

In the unobservable setting, i.e., when the queue is hidden from the customers such as in

contact centers, this is usually done by assuming a nonlinear waiting cost or service reward.

See, e.g., Hassin and Haviv (1995), Haviv and Ritov (2001), Shimkin and Mandelbaum (2004).

Mandelbaum and Shimkin (2000) assume that the reward is constant and the waiting cost is

linear, but with a certain probability and without knowing so, some customers may enter a

“fault position” and never get served. Aksin et al. (2013) and Ata et al. (2015) consider a dy-

namic abandonment model. A customer’s utility is comprised of a waiting cost, service reward,

and exogenous random shocks. In each waiting period, after observing the realizations of the

random shocks, a customer decides whether to abandon or stay in the system given that she

knows the probability of getting served in that period. The random shocks play a key role in

this model in that customers have no incentive to abandon without these shocks. Aksin et al.

(2013) use a structural estimation approach to estimate callers’ parameters using call center

data and implement a simulation-based approach to compute the equilibrium abandonment

distribution. Ata et al. (2015) prove the existence and uniqueness of the equilibrium abandon-

ment distribution by studying the corresponding virtual waiting time distribution, using exact

analysis for a system with a single customer class, and approximate analysis in the heavy traffic

regime for a multi-class system.

There are also a few papers that study the rational abandonment of customers in the observ-

able setting. For example, Assaf and Haviv (1990) study equilibrium abandonment strategies

in a processor sharing queue and Hassin (1985) considers a Last-Come, First-Served (LCFS)

preemptive–resume discipline. Maglaras et al. (2015) present a model of customer abandon-

ment for an observable First-Come, First-Served (FCFS) queue assuming that customers do

not know the service rate but learn it through observing the service times of other customers

while they wait.

A few recent studies have attempted to empirically investigate the factors driving the aban-

donment behavior of customers. Beside Batt et al. (2013), in a parallel study Bolandifar et al.

(2014) also investigate the abandonment behavior of customers from an ED. They find that be-

side waiting time, the number of patients in the ED and service rate also affect the abandonment

behavior of customers. They also discuss modeling implications of their findings.


Expulsion/termination control. When considered from the perspective of the social

planner, our problem relates to the literature on expulsion and termination control in queueing

systems, where customers can be removed from the system by the central planner after joining.

Xu and Shanthikumar (1993) study the optimal admission control policy for a FCFS M/M/m

system with identical servers by considering the corresponding optimal expulsion control policy

for the dual system where all customers are admitted but served with the preemptive-resume

LCFS discipline. Xu (1994) applies the same duality approach to obtain the optimal admission

and scheduling control of anM/M/2 queue with nonidentical servers and Righter (2000) extends

the analysis to multiple classes of customers. Brouns and van der Wal (2006) consider admission

and termination control for a preemptive priority queue with two-classes. They show that the

optimal policy for both decisions has a threshold structure. Our welfare maximization analysis

(see Section 4.5) differs from that of Brouns and van der Wal (2006), in that (a) we focus on

controlling the low-priority segment, (b) our results relate the optimal admission and expulsion

thresholds to each other (theirs do not), and (c) we also investigate pricing as a means to achieve

the socially optimal policy.

Pricing for queues with rational customers. Another related stream is the literature

on pricing for queues with rational customers, cf. Hassin and Haviv (2003) for an overview.

To the best of our knowledge, this paper is the first to investigate pricing for such systems

in the presence of rational customer abandonment. Studies of optimal pricing and scheduling

decisions for priority queues consider customers’ rational join/balk and class choice decisions

but preclude abandonment. For examples in unobservable queues, we refer to Mandelson and

Whang (1990) for welfare maximization and Afeche (2013, 2004) and the references therein for

revenue maximization. Adiri and Yechiali (1974) and Hassin and Havi (1997) study an observ-

able queue with multiple priority classes where upon arrival a customer can decide whether to

purchase any of the priority levels or balk from the system. Alperstein (1998) characterizes the

optimal number of priority classes offered to customers. In contrast, we assume that customers

arrive at the system with preassigned priority levels but are strategic with respect to their

join/balk/abandon decisions. This applies to services in which the priority level is an inherent

characteristic or attribute of customers, e.g., the hospital ED. Indeed, private hospitals charge

patients for ED services and differ in terms of whether they charge patients after they see a

physician or before being admitted to the ED. 1 The assumption also makes sense in cases

1For example, the ED of Mater Misericordiae University Hospital in Dublin, Ireland charges patients for


where the customers chose their priority level prior to observing the system, e.g., London Eye

where customers must purchase the Fast Track ticket online.

Pricing for service cancellation. Finally, our work also relates to a few papers in the

marketing literature, which study service cancellation without queueing considerations. Xie and

Gerstner (2007) consider services with capacity constraints and explore the benefits of allowing

customers to cancel their tickets in return for a refund. They show that the provider can profit

by offering refunds to canceling customers and reselling the capacity to new customers. Guo

(2009) investigates the same problem in a competitive setting with multiple providers. In our

setting, customer cancellation corresponds to abandoning the queue and refund for cancellation

can be implemented by charging a combination of entrance and service fees.

4.3 The Model

We consider an observable M/M/1 queue with two customer types; high- and low-priority.

Customers are served according to a preemptive priority discipline in favor of the high-priority

type. Within each priority class the service discipline is First-Come, First-Served (FCFS). We

assume that the firm can distinguish between low- and high-priority customers. This implies

that customers do not get to choose their priority class. However, as detailed below, customers

can choose whether to join or balk upon arrival and whether to stay or abandon while in the

queue. The service rate is μ for both customer types. Low-priority (high-priority) customers

arrive to the system with rate λl (λh), incur a delay cost cl (ch) per unit of time in the system

(including service) and have service valuation Rl (Rh). We assume that cl < Rlμ and ch < Rhμ,

so that a customer of either type prefers receiving service to balking or abandoning if there is no

other customer ahead of her in the system. All parameters are common knowledge. Customers

can observe the number and priority level of customers in the system.

The arrival process to the system is exogenous. However, once they arrive to the sys-

tem, customers are forward-looking and maximize their expected utility with respect to their

join/balk/abandonment decisions. Upon arriving to and observing the state of the system, each

customer decides whether to join or balk. A joining customer may decide to later abandon the

queue if that maximizes her expected utility. In other words, each customer, starting from the

time of her arrival to the system until the end of service, has the option of abandoning the

entering the ED and being triaged by a nurse. See, http://www.mater.ie/patients/emergency-services/ed/ed-charges/ (Accessed August 13 2015).


queue. Accordingly, abandoning the queue upon arrival corresponds to balking. We assume

that customers join (stay in) the system if they are indifferent between joining and balking

(staying and abandoning).

4.4 The Equilibrium Strategy of Low-priority Customers

Due to the preemptive priority discipline, the waiting time of high-priority customers is inde-

pendent of low-priority customers. From the viewpoint of high-priority customers the system

operates as a single-class queue with only high-priority customers. Hence, the strategy of high-

priority customers is readily available from Naor (1969). A high-priority customer joins the

system if and only if the number of high-priority customers (including herself) does not exceed

nh ≡ �Rhμ/ch . Furthermore, high-priority customers who join the system have no incentive

to later abandon.

In contrast, the waiting time of a low-priority arrival depends both on the number of cus-

tomers she finds in system upon arrival, and on future arrivals of high-priority customers.

Therefore, a low-priority customer has an incentive to abandon if “too many” high-priority

customers arrive during her sojourn time. Further, the waiting time of a low-priority customer

is not affected by future low-priority arrivals, but it depends on the stay/abandon decisions of

low-priority customers in front of her. In equilibrium, we require each customer to take actions

that maximize her expected utility, given that all other customers also behave as rational utility

maximizers.

Before characterizing the equilibrium abandonment strategy, we need the following defini-

tion.

Definition We say a low-priority customer is in position (m,n) with m,n ≥ 0 and m ≤ n

where m is her service order among low-priority customers and n is her service order among all

customers. Upon finishing service the customer moves to position (0, 0). We also refer to m as

her low-priority position and to n as her system position.

4.4.1 Structure of the Equilibrium Strategy

Let the value function v(m,n) denote the maximum expected utility of the customer at position

(m,n) under the equilibrium strategy. As detailed in the Appendix, these value functions are

obtained as the solutions of inter-related optimal stopping problems: Specifically, the value


function of the first low-priority customer, v(1, n), is the solution of her stopping problem that

only accounts for the behavior of high-priority customers. The value function of the customer

in low-priority position m > 1 is the solution of her stopping problem that accounts both for the

behavior of high-priority customers and for the behavior of low-priority customers in positions

i < m. The following result proves the existence and characterizes the structure of a unique

equilibrium strategy for low-priority customers.

Proposition 4.4.1 There is a unique equilibrium join/balk/abandon strategy for low-priority

customers which has the following threshold structure.

1. A low-priority customer joins/stays in the system if and only if her position (m,n)

satisfies m ≤ L < ∞ and n ≤ n(m) ≤ nh+m, where the thresholds L and {n(1), . . . , n(L)} are

functions of the problem parameters.

2. The thresholds n(m) satisfy the following properties: (i) If n(m) = nh +m, then n(i) =

nh + i for i ∈ {1, 2, ...,m− 1}, and (ii) if n(m) < nh +m for some m < L, then n(i) = n(m)

for i ∈ {m+ 1, ..., L}.

The first part of Proposition 4.4.1 implies that there exists a set of positions S ≡ {(m,n);m ≤L, n ≤ n(m)} that characterizes the equilibrium behavior of low-priority customers as follows:

A customer balks from the system if the position that she would occupy upon joining the system

does not belong to the set S. Otherwise, the customer joins the system and abandons if her

position falls outside of the set S due to new high-priority arrivals.

The second part of Proposition 4.4.1 characterizes the properties of the thresholds n(m)

that determine the set S. Recall that nh is the maximum number of high-priority customers

in the system. Thus, if n(m) = nh +m for some low-priority position m it means that a low-

priority customer does not abandon as long as she is in that low-priority position. In this case,

n(i) = nh + i for i < m in property (i) implies that if the customer at low-priority position m

does not abandon the system, then neither do any of the low-priority customers ahead of her

(in low-priority positions i < m). However, if n(m) < nh+m for some low-priority position m,

then that customer’s position can fall out of the set if enough high-priority customers arrive

during her sojourn time (so that n = n(m)+1). In this case, n(i) = n(m) for i > m in property

(ii) implies that low-priority customers in low-priority positions higher than m abandon from

the same system position as the customer in low-priority position m. It follows that the next

customer to abandon (if any) is always the customer at the end of the line. In other words, the


customers abandon in a last-come, first-abandon order.

The following proposition characterizes the equilibrium strategy in the limiting case where

the high-priority balking threshold nh is sufficiently large or infinite.

Proposition 4.4.2 For sufficiently large or infinite nh, the equilibrium abandonment strategy

of a customer only depends on her system position and the equilibrium is characterized by a

single finite threshold n such that n = L = n(1) = n(2) = · · · = n(L).

The result is intuitive. If high-priority customers do not balk or the high-priority balking

threshold is large enough such that all low-priority customers abandon before that threshold is

reached, then the strategy of low-priority customers is independent of the type composition of

the customers currently in the system. Each customer abandons once the number of remaining

service completions passes a certain threshold.

4.4.2 Explicit Characterization of the Equilibrium Thresholds and Customer

Utility

Given the structure of the equilibrium strategy we can compute the threshold values and the

maximal expected utility of a customer at a given position, namely, the value function v(m,n).

To present the results we need the following performance metrics of the birth-death processes

associated with high-priority arrivals and service completions.

First, consider the birth-death process N1(t) on {0, 1, · · · ,J } with birth rates λi = λh and

death rates μi = μ for i ∈ {1, · · · ,J − 1}, and λ0 = λJ = μ0 = μJ = 0. Let w(i,J ) denote

the expected time it takes for the process to reach state 0 or J given the starting point i. Also

let q(i,J ) denote the probability of reaching state 0 before J given the starting point i. Both

metrics can be obtained by considering the embedded random walk associated with this process.

The embedded random walk is defined on the same state space and moves from i to i+ 1 with

probability λh/(λh + μ) and from i to i− 1 with probability μ/(λh + μ). Accordingly, w(i,J )

and q(i,J ) can be viewed as the “expected length of the game” and the “ruin probability” in

a gambler’s ruin problem (see, e.g., Rosenthal 2010 page 75). We have

w(i,J ) ≡i(1− ρJh )− J

(ρJ−ih − ρih

)(μ− λh)

(1− ρih

) , (4.1)

q(i,J ) ≡ 1− ρJ−ih

1− ρJh. (4.2)


Next, consider the birth-death process N2(t) on {0, 1, · · · ,K} with birth rates λi = λh for

i ∈ {1, · · · ,K − 1}, and death rates μi = μ for i ∈ {1, · · · ,K}, and λ0 = μ0 = λK = 0. Let

w(i,K) denote the expected first passage time of the process to state 0 given the starting point

i. Then, we have

w(i,K) ≡i(μ− λh) + λh

(ρKh − ρK−i

h

)(μ− λh)2

. (4.3)

We are now ready to present the result.

Proposition 4.4.3 Let

u(i, h, C) =

⎧⎪⎨⎪⎩Cq(i, h+ 1)− clw(i, h+ 1), h < nh + 1,

C − clw(i, h), h = nh + 1,

(4.4)

for i ≤ h with w, q and w defined in equations (4.1),(4.2) and (4.3), respectively. Then, letting

v(0, 0) = Rl, the equilibrium threshold and the value function for 1 ≤ m ≤ L are given by the

recursive relations

n(m) = (m− 1) + max{h ∈ {1, ..., nh + 1};u(h;h, v(m− 1,m− 1)) ≥ 0}, (4.5)

v(m,n) = u(n− (m− 1); n(m)− (m− 1), v(m− 1,m− 1)), m ≤ n ≤ n(m), (4.6)

where L ≡ max {m ≥ 1; v(m− 1,m− 1) ≥ cl/μ}.

Proposition 4.4.3 specifies a recursive algorithm for computing the value function for each

low-priority position m, by relating it to the value of moving ahead by one low-priority position

with no high-priority customers in the system, i.e., v(m − 1,m − 1). From Proposition 4.4.1

we know that in equilibrium the customer at low-priority position m abandons if her system

position exceeds some threshold n(m). We obtain this threshold for each low-priority position

m using the function u(i, h, C) given in (4.4). The function returns the expected utility of

staying in system for a low-priority customer who currently faces i service completions before

advancing by one low-priority position and receiving the value C, and who abandons as soon as

the number of service completions before advancing by one low-priority position exceeds h. For

each low-priority position m, the equilibrium threshold n(m) is the maximum system position

such that the expected utility of staying in system is nonnegative. Hence, for each m, the

equilibrium threshold n(m) and the value function v(m,n) satisfy (4.5) and (4.6). To calculate


the equilibrium thresholds we start from m = 1, for which moving one position in queue is

equivalent to completing service and hence C = v(0, 0) = Rl. This allows us to recursively

compute the value function and equilibrium thresholds for higher positions.

Next, we elaborate on the structure of the utility function u(i, h, C) in (4.4). Observe that

u(i, h, C) for both cases of (4.4) is the difference of an expected reward minus an expected

waiting cost. In the first case, i.e., for h < nh + 1, a tagged low-priority customer advances

by one low-priority position if i, the number of service completions required do so, reaches 0

before h+1, at which point the customer abandons the system. Note that in this case, i evolves

according to the birth-death process N1(t) with J = h+ 1. Therefore, using (4.2), for each i,

the probability of advancing by one low-priority position before abandoning is q(i, h+ 1), and

the expected reward is Cq(i, h + 1). Also, using (4.1) the expected waiting cost incurred until

the customer either advances in line or abandons is clw(i, h + 1). In the second case, i.e., for

h = nh + 1, the customer does not abandon and hence eventually advances by one low-priority

position and receives the reward C. Note that in this case the number of remaining service

completions until the customer advances by one low-priority position can be expressed using

process N2(t) with K = h. It follows using (4.3) that the expected waiting cost for the customer

in state i is clw(i, h).

Limiting case with nh sufficiently large or infinite. In the limiting case, i.e., when

the high-priority customers do not balk or the balking threshold is large enough, the analysis

can be simplified since by Proposition 2 the strategy of low-priority customers only depends

on their system position. That is, it is independent of the queue composition. Let g(n, n)

denote the expected utility of a low-priority customer at system position n ∈ {0, 1, ..., n+ 1} if

all low-priority customers follow the threshold strategy whereby they balk/abandon at system

position n+ 1 and join/stay in the system otherwise. We have

g(n, n) = Rlq(n, n+ 1)− clw(n, n+ 1), (4.7)

where q(n, n + 1) is the probability that a low-priority customer in system position n reaches

system position 0 before n+1, and w(n, n+1) is the expected time until this customer reaches

system position 0 or n+1, whichever happens first. As in Proposition 3, these quantities can be

viewed as the ruin probability and the expected length of the game in a gambler’s ruin problem

with initial wealth n, target wealth n+ 1, and winning probability λh/(μ+ λh). We have from


(4.2) and (4.1):

q(n, n+ 1) =1− ρn+1−n

h

1− ρn+1h

, (4.8)

w(n, n+ 1) =n(1− ρn+1

h )− (n+ 1)(ρn+1−nh − ρn+1

h )

(μ− λh)(1− ρn+1h )

. (4.9)

Note that g(0, n) = Rl and g(n+ 1, n) = 0 for any n.

The equilibrium threshold n is uniquely determined by

n = max{n ≥ 0; g(n, n) ≥ 0}, (4.10)

and the value function is given by

v(n) = g(n, n), n ∈ {0, 1, ..., n+ 1} .

We conclude this subsection with the following example.

Example 1. Consider two cases for a system with parameters Rl = 10, μ = 1, λh = 0.4, and

cl = 0.5: (i) nh = 10 and (ii) nh = 15. In case (i) the equilibrium strategy is given by L = 12,

n(1) = 11, and n(m) = 12 for 2 ≤ m ≤ 12. In case (ii) the equilibrium threshold is n = 12.

Figure 4.1 illustrates the equilibrium strategies. When nh = 10, i.e., case (i), the equilibrium

thresholds depend on the low-priority position m. In particular, the low-priority customer does

not abandon in low-priority positions m = 1 and m = 2 as she receives a positive expected

utility in all possible system positions. In case (ii), however, all low-priority customers abandon

once their system position exceeds the threshold n = 12.

4.5 Pricing under Rational Abandonment: Preliminaries

We now turn to the analysis of pricing under rational abandonment. For simplicity and to

highlight the interaction between abandonment and pricing implications, we focus on controlling

the low-priority segment; the pricing analysis in this chapter assumes a fixed arrival rate λh < μ

of high-priority customers who do not balk (nh = ∞). This setting applies, for example, to

cases with a predetermined high-priority price and where high-priority customers either do not

observe, or do not react to, the queue length. Furthermore, our structural results on the pricing

implications of rational abandonment extend to cases with high-priority balking (nh < ∞), and


X X X X X X X X XX X X X X X X X X X

0 1 2 3 4 5 6 7 8 9 10 11 12m0

12345678910111213

n

X X X X X X X X X X X X

0 1 2 3 4 5 6 7 8 9 10 11 12m0

12345678910111213

n

Figure 4.1: An illustration of the equilibrium abandonment strategy of low-priority customersin Example 1 with Case (i) on the left and Case (ii) on the right. A customer joins/stays inpositions marked with � and � and abandons if her position reaches a position marked with×. Customers do not abandon once they reach the positions marked with �.

to the solution of the joint low- and high-priority pricing problem.

We study pricing for welfare maximization in Section 4.6 and for revenue maximization

in Section 4.7. The remainder of this section presents two important preliminaries for these

analysis; the equilibrium behavior of low-priority customers in the presence of pricing (Section

4.5.1), and steady-state performance measures (Section 4.5.2).

4.5.1 Equilibrium Behavior of Low-Priority Customers in the Presence of

Pricing

We consider three different fees that are commonly used in service systems, namely, an entrance

fee Pe which is charged for joining the system (placing an order), a cancellation fee Pc which

is charged for abandoning (canceling the order), and a service fee Ps which is charged upon

service completion (order delivery). It is straightforward to verify that the equilibrium behavior

of low-priority customers in the presence of pricing has a similar threshold structure as without

pricing (see Proposition 2). Specifically, given an entrance fee Pe, cancellation fee Pc, and service

fee Ps, there is an unique symmetric equilibrium characterized by an abandonment threshold

na < ∞ and a balking threshold nb ≤ na. That is, a low-priority customer joins the system

if and only if her system position does not exceed nb, and abandons if and only if her system

position reaches na + 1.

We assume that Rl − Ps > −Pc (otherwise no customer in the system is willing to stay)

and Rl > Pe + Ps (otherwise no customer is willing to join). The equilibrium thresholds are


determined as follows.

Given a cancellation fee Pc and service fee Ps, let g(n, na;Pc, Ps) denote the expected utility

of a low-priority customer at system position n ∈ {0, 1, ..., na + 1} if all low-priority customers

follow the threshold strategy whereby they abandon at system position na + 1 and stay in the

system otherwise. We have

g(n, na;Pc, Ps) = (Rl − Ps)q(n, na + 1)− Pc(1− q(n, na + 1))− clw(n, na + 1), (4.11)

with q and w given in (4.8) and (4.9), respectively. In the absence of service and cancellation

fees, Pc = Ps = 0 so that g(n, na; 0, 0) = g(n, na), i.e., (4.11) specializes to (4.7).

Then the equilibrium abandonment threshold is uniquely determined as the maximum sys-

tem position at which a low-priority customer (weakly) prefers to stay in the system versus

canceling her order at a cost of Pc:

na(Pc, Ps) ≡ max{n ≥ 0; g(n, n;Pc, Ps) ≥ −Pc}, (4.12)

which is well-defined if Rl − Ps > −Pc. The value function satisfies

v(n;Pc, Ps) = g(n, na(Pc, Ps);Pc, Ps).

Since joining the system requires an immediate payment of Pe, a low-priority customer joins at

system position n iff v(n;Pc, Ps) ≥ Pe. Therefore, the equilibrium balking threshold is uniquely

determined as the maximum system position at which the customer joins:

nb(Pe, Pc, Ps) ≡ max{n ≥ 0; g(n, na(Pc, Ps);Pc, Ps) ≥ Pe}, (4.13)

which is well-defined if Rl > Ps+Pe. Notice that Pe = 0 implies equal balking and abandonment

thresholds (nb = na). A positive entrance fee implies nb ≤ na.

4.5.2 Steady-State System Performance Measures

We characterize three steady-state system performance measures that are required to evaluate

the system welfare and the provider’s revenue, namely, the queue-length distribution (πi), the

joining probability (qJ), and the service probability (qS). These three measures are functions


of the thresholds na and nb.

Queue-length Distribution. Let πi be the steady-state probability of having i customers

in the system (in this section we suppress the dependence of πi on the thresholds na and nb).

Define λ ≡ λl + λh, and ρ = λ/μ. Then πi satisfies the balance equations

λπi = μπi+1, i = 0, 1, ..., nb − 1,

λhπi = μπi+1, i = nb, ..., na − 1.

It follows that

πi =

⎧⎪⎨⎪⎩π0ρ

i, i = 0, 1, ..., nb,

π0ρi−nbh ρnb , i = nb + 1, ..., na.

(4.14)

For i ≥ na + 1 since all the customers in system are high-priority, we have

πi = (1− ρh)ρih, i ≥ nb + 1.

The normalization condition∑∞

i=0 πi = 1 implies that

π0 =1− ρna+1

h

1−ρnb+1

1−ρ + ρnbρh−ρ

na−nb+1

h1−ρh

. (4.15)

Joining Probability. Let qJ(nb, na) denote the “joining probability”, i.e., the probability

that a low-priority arrival joins the system (does not balk) upon observing the queue, given the

thresholds na and nb. We have

qJ(nb, na) =

nb−1∑i=0

πi =

nb−1∑i=0

ρiπ0 = π01− ρnb

1− ρ. (4.16)

Service Probability. Let qS(nb, na) denote the “service probability”, i.e., the probability

that a customer is eventually served; that is, she joins the system and does not abandon before

completing service. Then

qS(nb, na) =

nb−1∑i=0

πiq(i+ 1, na + 1),


where q(i + 1, na + 1) is the probability that the customer joining at position i + 1, reaches

system position 0 before na + 1. Substituting for πi from (4.14) and for q(i + 1, na + 1) from

(4.8) yields after some algebra

qS(nb, na) = π01

1− ρhna+1

⎛⎜⎝1− ρnb

1− ρ− ρh

na

1−(

ρρh

)nb

1−(

ρρh

)⎞⎟⎠ . (4.17)

4.6 Pricing for Welfare Maximization

In this section we study the problem of maximizing the system welfare. Since we consider a

fixed high-priority arrival stream, this problem is equivalent to that of maximizing the surplus

from low-priority customers.

4.6.1 Welfare Maximization Requires Equal Balking and Abandonment Thresh-

olds

We start by considering how a welfare-maximizing service provider would control the system

through balking and abandonment thresholds. Suppose the social planner accepts a new low-

priority arrival up to system position mb and keeps a low-priority customer in system up to

position ma ≥ mb. Let S(mb, ma) denote the social welfare as a function of these thresholds.

We have

S(mb, ma) = λl

mb−1∑i=0

πi(mb, ma)g(i+ 1, ma), (4.18)

where πi(mb, ma), given in (4.14), is the steady-state probability of having i customers in

system, and g(i+ 1, ma), given in (4.7), is the expected utility of a low-priority customer who

joins at position i+ 1 and is removed if she reaches position ma + 1.

Proposition 4.6.1 establishes that such a threshold policy, with equal thresholds, is socially

optimal.

Proposition 4.6.1 The socially optimal policy has a threshold structure: Accept a new low-

priority arrival up to system position m∗b and keep a low-priority customer in system up to

the same system position m∗b = m∗

a. Furthermore, the socially optimal thresholds satisfy m∗b =

m∗a ≤ n.

First, the welfare-maximizing balking and abandonment thresholds are equal (m∗a = m∗

b).


This is intuitive, because the expected utility of a low-priority customer in a given position,

and her externality on other low-priority customers, are independent of how she reached this

position (i.e., by joining this position upon arrival, or by joining into a lower position and

“falling behind” later due to high-priority arrivals).

Second, the welfare-maximizing threshold is smaller than or equal to the equilibrium thresh-

old under “self-optimization”, that is, if the provider does not control the queue. The intuition

for this result extends the one for the FIFO case analyzed by Naor (1969) to settings with aban-

donment: Under self-optimization low-priority customers ignore the externality they impose on

customers that join after they do, and therefore join, and remain in the system, up to a higher

position.

4.6.2 Welfare-Maximizing Pricing Requires a Service Fee and No Entrance

Fee

The pricing that induces the welfare-maximizing operation consists of a single fee that is charged

upon completing service. Let P ∗s be the socially optimal service fee and m∗ = m∗

a = m∗b . Then

setting

P ∗s = Rl − cl

w(m∗, m∗ + 1)

q(m∗, m∗ + 1)(4.19)

induces the equilibrium balking/abandonment threshold m∗, as is evident from (4.11)-(4.12).

The service fee P ∗s reflects the system externality of a customer in position m∗.

This pricing result is in line with the classic FIFO analysis of Naor (1969), except for one

important distinction: If customers have an incentive to abandon after joining the system, as

in our model with priorities and unlike the FIFO case, the timing of payments plays a key

role for queueing control. Specifically, in our setting the welfare-maximizing operation typically

cannot be induced if the provider charges an entrance fee (upon joining the queue) instead

of a service fee (upon service completion). With such a fee the provider can only control the

balking threshold but not the abandonment threshold, and the two thresholds will typically

differ, which is suboptimal by Proposition 4.6.1. In contrast, by charging only a service fee the

provider can ensure equal balking and abandonment thresholds at the desired level.


4.7 Pricing for Revenue Maximization

In this section we study the problem of maximizing the provider’s revenue. Since we consider a

fixed high-priority arrival stream, this problem is equivalent to that of maximizing the revenue

from low-priority customers.

As shown in Section 4.6, welfare-maximization requires charging only a service fee, and no

entrance fee. In contrast, as we show in this section, revenue-maximization typically requires

both an entrance and a service fee. Furthermore, charging an entrance fee only may generate

more or less revenue than charging a service fee only.

4.7.1 Problem Formulation

Consider a revenue-maximizing service provider charging three different fees; Pe for entrance,

Pc for cancellation, and Ps for service completion. The provider solves

max{Pe≥0,Pc≥0,Ps≥0}

λl(Pe + Pc)qJ(na, nb) + λl(Ps − Pc)qS(na, nb) (4.20)

s.t. na = max{n ≥ 0; g(n, n;Pc, Ps) ≥ −Pc}, (4.21)

nb = max{n ≥ 0; g(n, na;Pc, Ps) ≥ Pe}. (4.22)

The revenue rate in (4.20) has two components. The first derives from all customers who join

the system, regardless of whether they get served; they each pay Pe + Pc, the sum of entrance

plus cancellation fees. The second derives from all customers who get served; they each pay

Ps −Pc, the service fee net of cancellation fee. The joining and service probabilities, qJ(na, nb)

and qS(na, nb), respectively, are given in (4.16) and (4.17) (see Section 4.5.2). The constraints

(4.21) and (4.22) specify, respectively, the abandonment threshold na and the balking threshold

nb that are implied in equilibrium by the fee triple (Pe, Pc, Ps), where g(n, na;Pc, Ps) is the

expected utility of a low-priority customer at system position n if all low-priority customers

abandon at system position na + 1 and stay in system otherwise (refer to Section 4.5.1).

We simplify and reformulate the revenue-maximization problem (4.20)-(4.22) into an equiv-

alent optimization problem over the thresholds na and nb. Let Π(na, nb) denote the revenue

rate from low-priority customers as a function of these thresholds.


Proposition 4.7.1 The problem (4.20)-(4.22) is equivalent to

max{nb≤na}

Π(na, nb) ≡ λlPeqJ(na, nb) + λlPsqS(na, nb) (4.23)

s.t. Ps = Rl − clw(na, na + 1)

q(na, na + 1), (4.24)

Pe =(Rl − Ps

)q(nb, na + 1)− clw(nb, na + 1), (4.25)

where Pe ≡ Pe + Pc denotes the “net” entrance fee and Ps ≡ Ps − Pc the “net” service fee.

The fees Pe and Ps fees have the following intuitive revenue equivalence interpretation:

charging a cancellation fee Pc upon abandonment is equivalent to charging Pc upon entrance

and then refunding it if the customer completes service. Therefore, the provider can achieve

maximum revenues using only these two fees, which give direct control over the abandonment

and balking thresholds. For simplicity we henceforth set Pc = 0 without loss of generality.

By (4.24) the service fee Ps extracts all surplus of the “marginal staying” customer, that is,

the one in the system position na, the highest position without abandonment. By (4.25) the

entrance fee Pe extracts all surplus of the “marginal joining” customer, that is, the one joining

into position nb, the highest position without balking.

4.7.2 Revenue-Maximizing Pricing Requires Service Fee and Entrance Fee

It is clear from (4.23)-(4.25) that the provider can typically achieve a higher revenue rate by

charging both an entrance and a service fee. We formalize this result as follows.

Proposition 4.7.2 It is generally not revenue-maximizing to charge only for entrance (with

Ps = 0), or only for service (with Pe = 0).

Charging an entrance fee, in addition to a service fee, has two countervailing revenue effects.

(1) It reduces the joining probability, which lowers the revenue. (2) It increases the expected

payment per joining customer, which increases the revenue. The second effect dominates, so

that charging both fees is optimal – unless the high-priority load exceeds some threshold. The

following example illustrates (and proves) this result.

Example 2. Consider a system with parameters Rl = 60, μ = 1, λl = 0.4, and cl = 1.

Figure 4.2 shows the optimal entrance and service fees as a function of the high-priority arrival

rate λh. It also shows the optimal service fee if the firm could only charge for service (see


Entrance fee

Service fee only Service fee

0.2 0.4 0.6 0.8 1.0Λh

10

20

30

40

50

Figure 4.2: Optimal service and entrance fees as a function of λh. System parameters: R =60, μ = 1, λl = 0.4, cl = 1.

Subsection 4.7.3). Observe that for most values of λh it is optimal to charge both an entrance

and a service fee. As λh increases to 1, however, the optimal entrance fee approaches zero and

the optimal pricing strategy is to charging a single fee upon service completion. Further observe

that the optimal service fee is greater than the entrance fee for all value of λh.

4.7.3 Maximizing Revenues with a Single Fee: Charge for Entrance or Ser-

vice Completion?

Although it is typically optimal to charge both an entrance and a service fee, charging only

a single fee is practically appealing due to its simplicity. We investigate whether the firm

can generate more revenue by charging customers only upon entering the system or only after

service completion. The answer lies in the following intuitive trade-off captured by the model:

On one hand, if customers are charged upon service completion, they are more willing to enter

the system but only those who do not abandon contribute to the firm’s revenue. On the other

hand, if customers are charged upon entering the system they are more reluctant to join the

system yet the firm collects revenue from all entering customers regardless of whether they

receive service or not.

Service Fee. Suppose the service provider charges a single fee Ps upon service completion,

that is, Pe = 0. It follows from (4.21) and (4.22) that a zero entrance fee induces equal balking

and abandonment thresholds. Let Πs denote the revenue rate under a service fee only. The


service provider maximizes revenues by solving

max{nb=na}

Πs(nb, na) ≡ λlPsqS(nb, na) (4.26)

s.t. Ps = Rl − clw(na, na + 1)

q(na, na + 1).

The service fee Ps is set such that the customer joining at system position na is indifferent

between joining or balking. Let n∗ denote the optimal balking and abandonment threshold in

the problem above, that is, the threshold corresponding to the optimal service fee.

Nonrefundable Entrance Fee. Assume that the service provider charges a single fee

Pe upon entering the system, that is, Ps = 0. In this case the firm can directly control only

the balking threshold, whereas the abandonment threshold is determined through (4.21). This

typically leads to unequal balking and abandonment thresholds; nb ≤ na. Let Πe denote the

revenue rate under an entrance fee only. The service provider maximizes revenues by solving

max{nb≤na}

Πe(na, nb) ≡ λlPeqJ(na, nb) (4.27)

s.t. na = max{n ≥ 0;Rlq(n, n+ 1)− clw(n, n+ 1) ≥ 0},

Pe = Rlq(nb, na + 1)− clw(nb, na + 1).

Denote by n∗a and n∗

b the optimal abandonment and balking thresholds, respectively, under the

entrance-fee-only problem.

Optimality Conditions. From our numerical experiments we find that, depending on the

problem parameters, either of these pricing strategies could be optimal. This is illustrated in

Figure 3. It shows, for two different values of the low-priority reward Rl, the combination of

arrival rates λh and λl for which the service/entrance fee is optimal. As Figure 3 demonstrates

the optimal pricing scheme does not have a monotone structure with respect to either of the

high- or low-priority arrival rates. This can be explained by the non-monotone behavior of

the optimal prices, as well as the joining and service probabilities (see Example 3 below). The

following proposition provides sufficient conditions under which either an entrance or a service

fee is optimal. (It seems infeasible to fully characterize analytically the parameter regimes

under which each of these pricing strategies are optimal).


Service fee optimal

Entrance fee optimal

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

Λh

Λl

Service fee optimal

Entrance fee optimal

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

Λh

Λl

Figure 4.3: Regions where a service/entrance fee is optimal for Rl = 40 (left) and Rl = 20(right) and cl = μ = 1.

Proposition 4.7.3 Assume λh is strictly positive but sufficiently small, i.e., let λh → 0+.

(i) Assume Rlμ/cl > 2 and let λl → 0+, then charging an entrance fee is optimal.

(ii) Assume Rlμ/cl > (n∗ + 1)2 − 1 where n∗ =⌊12(√

5 + 4Rlμ/cl − 1)⌋. Let ρl → 1+, then

charging an entrance fee is optimal.

(iii) Assume Rlμ/cl ≥ 2 and let λl → +∞, then charging a service fee is optimal.

Proposition 4.7.3 reinforces the observation made in Sections 4.6.2 and 4.7.2 by showing

that in the presence of (even a little) abandonment, the timing of the payment matters. More

specifically, it identifies three parameter regimes where charging an entrance or service fee is

optimal. Regime (i) corresponds to a very lightly loaded system with both low- and high-priority

arrival rates in the neighborhood of zero. In this case, given that the ratio of the reward to

the expected cost of a service completion is not too low, charging an entrance fee is optimal.

This condition is met in both examples presented in Figure 3. Our numerical experiments show

that an entrance fee is also the optimal choice if the load conditions of regime (i) are somewhat

relaxed, specifically, when the low- and high-priority arrival rates are not infinitesimal but still

sufficiently low. Regime (ii) corresponds to a system with low-priority load approaching one,

and high-priority arrival rate in the neighborhood of zero. In this case, a non-trivial condition

on the system parameters is required for either of the service or entrance fee to be optimal.

For example, the condition is met for the first example (left) presented in Figure 3, but not

for the second one (right). Regime (iii) corresponds to a system with high-priority arrival rate


na*

n*nb*

0.2 0.4 0.6 0.8 1.0 h

10

20

30

40

50

60Optimal thresholds

Ps

Pe

0.0 0.2 0.4 0.6 0.8 1.0 n

10

20

30

40

50

60Optimal fees

0.2 0.4 0.6 0.8 1.0 n

5

10

15

20

% Revenue gain or loss under service fee Service probability(service fee)Joining probability(entrance fee)

0.2 0.4 0.6 0.8 1.0 n

0.2

0.4

0.6

0.8

1.0

Figure 4.4: Comparison of service fee vs. entrance fee. System parameters: R = 60, μ = 1, λl =0.4, cl = 1.

in the neighborhood of zero and low-priority arrival rate sufficiently high. In this case, we can

show that Rlμ/cl ≥ 2 is a sufficient condition for the service fee to generate higher revenue.

This can be observed in both examples in Figure 3. In general, we observe that for sufficiently

high low- and high-priority loads the service fee is the optimal choice. Intuitively, under these

load conditions the added revenue due to higher joining rate under service fee dominates the

revenue loss associated with abandoning customers.

Revenue gain/loss of service vs. entrance fee. We are also interested in the amount

of revenue loss by choosing the suboptimal timing of payment. Based on extensive numerical

experiments, we observe that the percentage of loss due to charging a service fee when an

entrance fee is optimal is generally small. However, especially when the high-priority load is

high, charging a service fee results in significant revenue gain. This is because when the high

priority arrival rate and hence the probability of abandonment is high, charging an entrance

fee would discourage a large proportion of customers from joining the system.

We illustrate these observations with the following representative example.


Example 3. Consider a system with parameters Rl = 60, μ = 1, λl = 0.4, cl = 1. Figure

4.4 compares the optimal solution and the relevant system performance measures under the

optimal service and the optimal entrance fee. Observe that the optimal prices and join/service

probabilities are non-monotone with respect to the high-priority arrival rate. Further, notice

that for relatively small high-priority utilization, the revenue obtained under the optimal service

fee is slightly smaller than that achieved under the optimal entrance fee. However, for suffi-

ciently high utilization of high-priority customers, the optimal service fee yields a significant

revenue gain versus the optimal entrance fee. This observation can be explained as follows. As

illustrated in Figure 4.4, regardless of the high-priority load, the joining probability under the

optimal entrance fee is close to the service probability under the optimal service fee. However,

when the high-priority utilization is high, the optimal service fee is significantly larger than the

optimal entrance fee, making the service fee the optimal choice. In other words, by charging

the customers upon service completion, the firm can collect a higher fee from approximately

the same proportion of customers.

4.8 Concluding Remarks

This chapter studies the abandonment strategy of rational customers and its pricing implications

in an observable priority queue with two customer classes. Our main contributions are to

characterize the equilibrium abandonment strategy of low-priority customers, and to provide

insights on the optimal pricing strategies of a service provider in the presence of customer

abandonment.

Recent empirical studies (e.g., Bolandifar et al. 2014) have highlighted the need for models

of customer abandonment in observable queues that depend on factors other than time spent

in the system. Our study is a step in this direction. Accordingly, extensions of our model as

well as other models of customer abandonment in observable queues should be considered in

future.

One important problem is to consider multiple classes of customers with different cost and

reward parameters. The same approach used in this work can be applied to characterize the

equilibrium strategy of customers in a multi-class system, as the solution to inter-related optimal

stopping problems. The analysis is however more complicated, as for each class, one needs to

keep track of the number of customers in all lower priority classes. Further, the last-come,


first-abandon structure does no longer necessarily hold. It would therefore be interesting to

characterize the abandonment behavior in different parameter regimes.

We assume that all customers are forward-looking utility maximizers. One can relax this

assumption by assuming that all or some of the customers are myopic. That is, they ignore

future arrivals of high-priority customers and join/stay in system if, and only if, their expected

utility based on their current position is nonnegative. It is not hard to see that the abandonment

strategy of such customers has a threshold structure with the threshold being equal to the

balking threshold of Naor (1969). It follows that such customers join and stay in higher positions

compared to forward-looking ones and hence do not affect their abandonment strategy. Another

way of relaxing the rationality assumption is to assume that customers are boundedly rational.

That is, they do not always make the choice that maximizes their expected utility due to lack

of information or inability in estimating their expected waiting times; see Huang et al. (2013)

who study how bounded rationality affects join/balk decisions in M/M/1 FCFS systems.

Our model can also be modified to allow for customers to purchase priority. In this case,

a customer needs to decide which class to join, given that she may abandon the queue later

if she joins the low-priority queue. Note also that the waiting time of a low-priority customer

is also affected by priority purchasing decisions of future customers. Adiri and Yechiali (1974)

show that, when customers do not abandon, the priority purchasing decision of customers is

characterized by a single threshold such that if the number of low-priority customers in system is

below the threshold, a new arrival joins the low-priority queue and purchases priority otherwise.

However, the problem is more complicated when customers can abandon the queue after joining.

To see this, assume that customers follow the same strategy as that in Adiri and Yechiali (1974).

That is, they purchase priority if they observe more than a certain number of customers in the

low-priority line. However, as new arrivals start to purchase priority and join the high-priority

line, customers in the low-priority line may start to abandon the queue, making the low-priority

line the optimal choice for future arrivals.

An important assumption in our model is that service times are exponentially distributed.

Considering non-exponential service requirements complicates the analysis significantly and

may give rise to different results. This is apparent from the fact that even in a FIFO system

the queue joining strategy does not necessarily have a threshold structure (Altman and Hassin

2001). Moreover, even under FIFO service, customers have an incentive to abandon if for

example the hazard rate of the service time distribution is decreasing (Mandelbaum and Shimkin


2000). It would therefore be interesting to study for a priority queue how the equilibrium

join/balk/abandon strategy depends on the properties of the service time distribution.

Another interesting research direction is the control of queues with rational abandonment.

This work appears to be the first to investigate pricing as a means to control the abandonment

behavior of customers. Our findings identify new and important pricing implications that

call for more research on the interplay between pricing and abandonment, for example for

unobservable queues or for customers with nonlinear delay costs. Furthermore, there is a need

to investigate operational controls, such as scheduling, for systems in the presence of rational

customer abandonment. We hope that this research stimulates more work on the design and

control of queuing systems in the presence of rational customer abandonment.

4.9 Appendix A: Proofs.

Proof of Proposition 4.4.1. The proof is by Induction. We start with the customer at the first

low-priority position, whose strategy does not depend on that of other low-priority customers,

and show that her abandonment strategy has the claimed threshold structure. Then assuming

that the first m ≥ 1 customers follow the threshold abandonment strategy, we show that the

customer at low-priority position m+1 follows the same strategy with thresholds satisfying the

claimed properties.

Consider a tagged customer at low-priority position 1, that is there are no low-priority

customers in front of her. Note that any waiting cost incurred so far is sunk and hence given

that the customer is still in system she maximizes her expected future utility. Since there can

be a maximum of nh high-priority customers in system, the position of the tagged customer

is (1, n) with some n ∈ {1, ..., nh + 1}. For positions with n ∈ {1, ..., nh} the next event is

either a service completion, with probability μ ≡ μ/(μ + λh), or a new-high priority arrival

with probability λ ≡ λh/(μ + λh). For position (1, nh + 1) since the maximum number of

high-priority customers is reached, arrival of a high-priority does not change the position of

the tagged customer. Note that due to the memoryless property of the arrival and service

processes, no new information becomes available between these events and hence the customer

would only abandon at these discrete epochs. At each decision epoch, the customer can either

abandon and receive an immediate reward of 0 or stay until the next event. At each epoch if

the customer decides to stay in system she incurs an average waiting cost of c ≡ cl/(λh + μ).


At position (1, 1) if the customer decides to wait and the next event is a service completion,

she moves to (0, 0) and receives the reward Rl. Let v(n,m) denote the value function of the

customer at position (n,m) interpreted as the maximum future expected utility for a customer

at that position. Then the value function for the customer at low-priority position 1 satisfies

v(1, 1) = [−c+ λv(1, 2) + μv(0, 0)]+, (4.28)

v(1, n) = [−c+ λv(1, n+ 1) + μv(1, n− 1)]+, n ∈ {2, ..., nh}, (4.29)

v(1, nh + 1) = [−c+ λv(1, nh + 1) + μv(1, nh)]+, (4.30)

where g+ ≡ max(0, g) and v(0, 0) = Rl. Applying Lemma 4.10.1 in Appendix B for T = nh and

C = Rl, we know that v(1, n) is nonincreasing in n and strictly decreasing in n when positive.

Therefore, the customer’s strategy has a threshold structure. That is, there exists a threshold

n(1) ∈ {1, . . . , nh+1} such that if n(1) = nh+1 the customer stays in all possible positions and

never abandons the system, and otherwise stays in system in positions smaller than or equal to

n(1) and abandon at n(1) + 1.

We next characterize the threshold value n(1). Recall that for each position, the customer

decides to stay if the future expected utility of staying at that position is nonnegative. By

assumption v(0, 0) = Rl > cl/μ and hence −c + λv(1, 2) + μv(0, 0) > 0, that is the customer

stays at position v(1, 1). We now claim that

n(1) =

⎧⎪⎨⎪⎩1, if v(1, 1) < cl/μ,

max{n ∈ {2, ..., nh + 1}; v(1, n− 1) ≥ cl/μ}, otherwise.

(4.31)

This follows from the fact that the customer at position (1, n) with n ∈ {2, ..., nh + 1} chooses

to stay if and only if v(1, n− 1) ≥ cl/μ. The if part is clear from (4.29) and (4.30). To see the

only if part, assume to the contrary that v(1, n − 1) < cl/μ but the customer decides to stay

at position (1, n). This is a direct contradiction with (4.30) for n = nh + 1. For n ∈ {2, ..., nh}and using (4.29) this implies that

v(1, n) = −c+ λv(1, n+ 1) + μv(1, n− 1) ≥ 0. (4.32)

However, since v(1, n − 1) < cl/μ we have μv(1, n − 1) < c, or −c + μv(1, n − 1) < 0. Thus,

for (4.32) to hold we must have λv(1, n + 1) − v(1, n) > 0 which is a contradiction since


v(1, n+ 1) ≤ v(1, n) by Lemma 4.10.1. Note that the problem for the customer at low-priority

position 1 can now be expressed as

v(1, 1) = [−c+ λv(1, 2) + μv(0, 0)]+,

v(1, n) = [−c+ λv(1, n+ 1) + μv(1, n− 1)]+, n ∈ {2, ..., n(1)− 1},

v(1, n(1)) =

⎧⎪⎨⎪⎩[−c+ λv(1, n(1)) + μv(1, n(1)− 1)]+, n(1) = nh + 1,

[−c+ μv(1, n(1)− 1)]+, n(1) < nh + 1.

We now turn to the induction step. Assume the first m ≥ 1 low-priority customers follow

an abandonment strategy with thresholds {n(1), . . . , n(m)} such that if n(i) = nh + i then

the customer at low-priority position i ∈ {1, . . . ,m} stays in all possible positions and never

abandons the system, and otherwise stays in system positions smaller than or equal to n(i) and

abandons at n(i)+1. Further, assume that the threshold values satisfy the following properties.

First, if the customer at low-priority i does not abandon, then non of the customers in front of

her abandon either, that is, if n(i) = nh + i then n(j) = nh + j for all j ≤ i. Second, if the

customer at low-priority position i abandons, then the customer at low-priority position i + 1

also abandons from the same system position, that is, if for some i < m, n(i) < nh + i then

n(i + 1) = n(i). Note that the assumptions imply that non of the customers in front of the

customer at low-priority position m abandons before her, i.e., the next customer to abandon (if

any) is the customer at low-priority position m and from system position n(m) + 1. It follows

that the value function for the customer at position m satisfies

v(m,m) = [−c+ λv(m,m+ 1) + μv(m− 1,m− 1)]+, (4.33)

v(m,n) = [−c+ λv(m,n+ 1) + μv(m,n− 1)]+, n ∈ {m+ 1, ..., n(m)− 1},(4.34)

v(m, n(m)) =

⎧⎪⎨⎪⎩[−c+ λv(m, n(m)) + μv(m+ 1, n(m)− 1)]+, n(m) = nh +m,

[−c+ μv(m+ 1, n(m)− 1)]+, n(m) < nh +m.

(4.35)

We show that the customer at low-priority positionm+1 also follows the threshold abandonment

strategy such that if n(m + 1) = nh + (m + 1) then we also have n(m) = nh + m, and if

n(m) < nh+m then n(m+1) = n(m). Given the strategy of the first m low-priority customers,


the customer at low-priority position m+ 1 solves

v(m+ 1,m+ 1) = [−c+ λv(m+ 1,m+ 2) + μv(m,m)]+, (4.36)

v(m+ 1, n) = [−c+ λv(m+ 1, n+ 1) + μv(m+ 1, n− 1)]+, n ∈ {m+ 2, ..., n(m)}, (4.37)

v(m+ 1, n(m) + 1) =

⎧⎪⎨⎪⎩[−c+ λv(m+ 1, n(m) + 1) + μv(m+ 1, n(m))]+, n(m) = nh +m,

[−c+ μv(m+ 1, n(m))]+, n(m) < nh +m.

(4.38)

Note that (4.38) summarizes two cases for position (m+1, n(m)+1) depending on the strategy

of the customer at low-priority position m. If n(m) = nh +m, then a new high-priority would

balk from the system and hence does not affect the position of the tagged customer. For

n(m) < nh +m we have

v(m+ 1, n(m) + 1) =[−c+ λv(m, n(m) + 1) + μv(m+ 1, n(m))

]+(4.39)

= [−c+ μv(m+ 1, n(m))]+ . (4.40)

To see (4.39) note that upon arrival of a high-priority customer, the customer at low-priority

position m abandons as she moves to position (m, n(m) + 1). This in turn moves the tagged

customer to position (m, n(m) + 1). However, v(m, n(m) + 1) = 0 and hence (4.40) follows.

From Lemma 4.10.1 (Case 1) with T = n(m) and C = v(m,m), we know that v(m+ 1, n)

is nonincreasing in n and strictly decreasing when positive. Thus, similar to the case for low-

priority positions 1, if v(m,m) < cl/μ then no customer joins at low-priority position m+1 and

the maximum number of low-priority customers in system is L = m. Otherwise, the customer

at low-priority position m+ 1 follows a threshold strategy with threshold n(m+ 1) such that

n(m+ 1) =

⎧⎪⎨⎪⎩m+ 1, if v(m+ 1,m+ 1) < cl/μ,

max{n ∈ {m+ 2, ..., n(m) + 1}; v(m+ 1, n− 1) ≥ cl/μ}, otherwise.

(4.41)

In Lemma 4.10.2, part (i) we show that if n(m) = nh + m, then v(m + 1, n) ≤ v(m,n) for

n ∈ {m, . . . , nh+m}. This, together with (4.41) implies that if n(m+1) = nh+(m+1) then we

also have n(m) = nh +m. Further, in Lemma 4.10.2 part (ii) we show that if n(m) < nh +m,

then v(m + 1, n) = v(m,n) for n ∈ {m, . . . , n(m) + 1}. It follows that if n(m) < nh +m then

n(m + 1) = n(m). Let L ≡ max {m ≥ 1; v(m− 1,m− 1) ≥ cl/μ} be the maximum number of

low-priority customers in the system. The proof is complete. �


Proof of Proposition 4.4.2. Consider the customer at low-priority position 1. Recall from

equations (4.28) to (4.30) that for m = 1 and finite nh,

v(1, 1) = [−c+ λv(1, 2) + μRl]+,

v(1, n) = [−c+ λv(1, n+ 1) + μv(1, n− 1)]+, n ∈ {2, ..., nh},

v(1, nh + 1) = [−clμ+ v(1, nh)]

+.

Similarly, one can write the optimality equations for the case with nh = ∞, i.e., when high-

priority customers do not balk, as

v(1, 1) = [−c+ λv(1, 2) + μRl]+,

v(1, n) = [−c+ λv(1, n+ 1) + μv(1, n− 1)]+, n ∈ {2, 3, ...}.

Exploiting Lemma 4.10.1 with C = Rl, we know that in both cases there exist an n ≡ n(1) <

nh + 1 such that v(1, n) < cl/μ and v(1, n+ 1) = 0. Thus, v(1, n) solves

v(1, 1) = [−c+ λv(1, 2) + μRl]+, (4.42)

v(1, n) = [−c+ λv(1, n+ 1) + μv(1, n− 1)]+, n ∈ {2, ..., n− 1},

v(1, n) = [−c+ μv(1, n− 1)]+.

From Lemma 4.10.2, part (ii) we know that if the customer at position m ≥ 1 abandons, then

v(m+1, n) = v(m,n) for all n ∈ {m+1, . . . n(m)+1}. It follows that customers at higher system

positions also abandon from the same system position, i.e., n(1) = n(2) = · · · = n(L) = n.

Noting that L ≡ max {m ≥ 1; v(m− 1,m− 1) ≥ cl/μ} and v(m − 1,m − 1) = v(1,m − 1) for

all m ≥ 1 we also have L = n. �

Proof of Proposition 4.4.3. We show that the value function and the equilibrium thresholds

must indeed satisfy (4.6) and (4.5) respectively for all 1 ≤ m ≤ L. From Proposition 4.4.1 (see


also 4.33-4.35) in equilibrium the value function satisfies the recursion

v(m,m) = −c+ λv(m,m+ 1) + μv(m− 1,m− 1), (4.43)

v(m,n) = −c+ λv(m,n+ 1) + μv(m,n− 1), n ∈ {m+ 1, ..., n(m)− 1}, (4.44)

v(m, n(m)) =

⎧⎪⎨⎪⎩− cl

μ + v(m, n(m)− 1), n(m) = nh +m,

−c+ μv(m, n(m)− 1), n(m) < nh +m,

(4.45)

for 1 ≤ m ≤ L and with v(0, 0) = Rl, v(m, n(m) + 1) = 0 and v(m,n) ≥ 0 for all n ∈{m, ..., n(m)}. Observe that for each m the value function depends on the value of other low-

priority positions only through that of position (m − 1,m − 1), i.e., the position at which the

customer finds herself right after the next low-priority customer is served. Therefore, for each

low-priority position m, given the “reward-to-go” value C ≡ v(m−1,m−1) one can characterize

the equilibrium threshold by finding the maximum number of remaining service completions

before advancing to the next low-priority position, such that the customer does not abandon,

i.e., n(m)− (m− 1).

Define u(i, h, C) satisfying

u(1, h, C) = −c+ λu(2, h, C) + μu(0, h, C),

u(i, h, C) = −c+ λu(i+ 1, h, C) + μu(i− 1, h, C), i ∈ {2, ..., h− 1},

u(h, h, C) =

⎧⎪⎨⎪⎩− cl

μ + u(h− 1, h, C), h = nh + 1,

−c+ μu(h− 1, h, C), h < nh + 1,

with u(0, h, C) = C ≥ cl/μ, and u(i, h, C) ≥ 0. Note that u(i, h, C) is the expected utility of

staying in system for a tagged low-priority customer who currently faces i ≤ nh + 1 remaining

service completions before advancing one position in the queue and receiving reward C, andabandons once the number of high-priority customers in system passes h. The value function

thus satisfies

v(m,n) = u(n− (m− 1); n(m)− (m− 1), v(m− 1,m− 1)),

for 1 ≤ m ≤ L and m ≤ n ≤ n(m) as claimed in (4.6). The explicit expression for u(·;h, C) canbe obtained, e.g., by directly solving the above difference equation for each case (i.e., h = nh+1

and h < nh + 1). An intuitive derivation is provided in the main body of the paper and hence


the detailed derivation is omitted here. Finally, we show that the thresholds satisfy (4.5), i.e.,

h ≡ n(m)− (m− 1) = max{h ∈ {1, . . . , nh + 1};u(h, h, v(m− 1,m− 1)) ≥ 0}. (4.46)

First, consider the case where u(nh + 1, nh + 1, v(m − 1,m − 1)) ≥ 0. In this case since

w(i, h) is increasing in i the customer receives a nonnegative utility from staying in all system

positions and hence has no incentive to abandon, i.e., h = nh + 1. Otherwise from Proposition

(4.4.1) we know that the customer abandons once her system position passes some threshold

m ≤ n(m) < m + nh. To see that this threshold must indeed satisfy (4.46) note that using

(4.4),

u(i, h, v(m− 1,m− 1)) = v(m− 1,m− 1)q(i, h+ 1)− clw(i, h+ 1).

Hence, u(i, h, v(m − 1,m − 1)) ≥ 0 iff w(i, h + 1)/q(i, h + 1) ≤ v(m − 1,m − 1)/cl. From part

(ii) of Lemma (4.10.4) we know that w(i, h+1)/q(i, h+1) is increasing in i, thus u(h, h, v(m−1,m− 1)) ≥ 0 implies that u(i, h, v(m− 1,m− 1)) ≥ 0 for all 1 ≤ i ≤ h, i.e., the customer has

no incentive to abandon in any of the lower positions. Further, from part (i) of Lemma (4.10.4)

we know that w(h, h+1)/q(h, h+1) is increasing in h, hence u(h+1, h+1, v(m−1,m−1)) < 0

implies that u(h, h, v(m − 1,m − 1)) < 0 for all h + 1 ≤ h ≤ nh + 1. Therefore, the customer

has no incentive to stay in any higher positions either and therefore the unique threshold must

indeed satisfy (4.46). �

Proof of Proposition 4.6.1. To prove the first part of the proposition, we reformulate the

social planner’s problem as an infinite horizon Markov decision process. We first consider the

α-discounted problem and show that the claims are satisfied for any discount rate α > 0.

It is well-known that under certain conditions the long-run average optimal policy can be

obtained from the discounted optimal policy by letting α → 0 (Weber and Stidham 1987). It is

straightforward to verify that these conditions are satisfied for our problem. Thus, the average

optimal policy must also satisfy the claimed properties.

Let z(x1, x2) denote the maximum expected total discounted reward when the initial state

is (x1, x2), where x1 ≥ 0 and x2 ≥ 0 respectively denote the number of high- and low-priority

customers in the system. Defining Λ ≡ λl+λh+μ, and applying uniformization the optimality

equations are given by:

z(x1, x2) =1

Λ + α[−cx2 + λlT1z(x1, x2) + λhT2z(x1, x2) + μT3z(x1, x2)] ,


where

T1z(x1, x2) = max [z(x1, x2 + 1), z(x1, x2)] , (4.47)

T2z(x1, x2) = max0≤j≤x2

z(x1 + 1, j), (4.48)

T3z(x1, x2) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

z(x1 − 1, x2), x1 > 0, x2 ≥ 0,

Rl + z(0, x2 − 1), x1 = 0, x2 > 0,

z(0, 0), x1 = x2 = 0.

(4.49)

In the above formulation, (4.47) and (4.48) respectively correspond to arrival instances of

low- and high-priority customers and (4.49) corresponds to a service completion. The decision

epochs are arrival instances of customers to the system. Upon arrival of a low-priority customer,

the controller can decide whether to admit or reject the customer. Upon arrival of a high-priority

customer, the controller can remove low-priority customers (if any) from the system. Note that

in (4.48) we allow “batch removals”, i.e., the controller can remove more than one low-priority

customer upon arrival of a high-priority customer. However, under the optimal policy this can

only affect the transient states of the system (i.e., if the initial number of customers is above

the socially optimal threshold m∗) and hence does not change the steady-state probabilities.

To prove the result we show that z(x1, x2) satisfies the following:

z(x1, x2 + 2)− z(x1, x2 + 1) ≤ z(x1, x2 + 1)− z(x1, x2), (4.50)

z(x1 + 1, x2 + 1)− z(x1 + 1, x2) = z(x1, x2 + 2)− z(x1, x2 + 1). (4.51)

Inequality (4.50) states that z(x1, x2) is concave in x2. It follows from (4.47) that for each

x1 ≥ 0, it is optimal to accept new arrivals in states (x1, j) with j < x2(x1), where x2(x1) ≡min{x2 ≥ 0; z(x1, x2+1)−z(x1, x2) < 0}. Further, (4.51) implies that the value of an additional

low-priority customer only depends on the total number of customers in system. It follows that

there exist a threshold on the total number of customers in system m∗b , such that x2(x1) =

(m∗b − x1)

+ and it is socially optimal to accept new low-priority arrivals as long as the total

number of customers in system is less than m∗b , and reject otherwise. Next, (4.51) implies that

if it is optimal to reject a new low-priority arrival, it must also be optimal to remove a low-

priority customer upon arrival of a high-priority customer, and vice versa, i.e., m∗a = m∗

b ≡ m∗.

To see this, assume that it is optimal to reject a new low-priority arrival in state (x1, x2)


with x1 ≥ 0 and x2 ≥ 1, i.e., z(x1, x2 + 1) − z(x1, x2) < 0. It follows from (4.51) that

z(x1 + 1, x2) − z(x1 + 1, x2 − 1) < 0 and therefore it is also optimal to remove a low-priority

customer upon arrival of a high-priority one. A similar argument can be used to show that

the opposite holds as well. Finally, the finiteness of m∗ follows from the second part of the

proposition where we show m∗ ≤ n.

To prove that z indeed satisfies properties (4.50) and (4.51), we use the value iteration

algorithm. Let z0(x1, x2) = 0 for all x1, x2 ≥ 0, and

zn+1(x1, x2) =1

Λ + α[−cx2 + λlT1zn(x1, x2) + λhT2zn(x1, x2) + μT3zn(x1, x2)] ,

for n ≥ 0. Then, zn → z as n → ∞. The properties are clearly satisfied for n = 0. It is easy to

verify that the properties are preserved under summation and multiplication by a constant. It

can also be shown (See Lemma 4.10.3 in Appendix B) that T1, T2 and T3 preserve the properties.Thus, by induction the properties are satisfied for all n ≥ 0 and hence the result follows.

To prove the second part, i.e., that the socially optimal thresholds are not greater than the

individually optimal threshold, we show that individual optimization leads to joining the queue

at all positions where the social planner accepts new low-priority arrivals.

From the first part, we know that the socially optimal balking and abandonment thresholds

are equal. Hence, we can express the social welfare function (4.18) as a function of a single

threshold value m ≡ ma = mb:

S(m) = λl

m−1∑i=0

πi(m)g(i+ 1, m), (4.52)

where πi(m) is the steady-state probability of having i customers in system when using threshold

value m and g(i + 1, m) is the expected utility of a customer who joins at position i + 1 and

is removed when she reaches position m + 1. Recall from (4.10) that the self-optimization

threshold satisfies n = max{n ≥ 0; g(n, n) ≥ 0}. Thus, to show the result it suffices to show

that S(m)− S(m− 1) ≥ 0 implies g(m, m) ≥ 0.


Consider S(m) in (4.52) . Using the definition of g in (4.7) we have

S(m)− S(m− 1) = λl

m−1∑i=0

πi(m) (Rlq(i+ 1, m+ 1)− clw(i+ 1, m+ 1))

−λl

m−2∑i=0

πi(m− 1) (Rlq(i+ 1, m)− clw(i+ 1, m))

= Rl

(λl

m−1∑i=0

πi(m)q(i+ 1, m+ 1)− λl

m−2∑i=0

πi(m− 1)Rlq(i+ 1, m))

−cl

(λl

m−1∑i=0

πi(m)w(i+ 1, m+ 1)− λl

m−2∑i=0

πi(m− 1)Rlw(i+ 1, m))

= Rl(Am −Am−1)− cl(Im − Im−1), (4.53)

where,

Am ≡ λl

m−1∑i=0

πi(m)q(i+ 1, m+ 1), Im ≡ λl

m−1∑i=0

πi(m)w(i+ 1, m+ 1). (4.54)

It is not hard to verify that Am −Am−1 > 0. Therefore, S(m)− S(m− 1) ≥ 0 implies that

Rl

cl≥ Im − Im−1

Am −Am−1.

Also, note that g(m, m) = Rlq(m, m+ 1)− clw(m, m+ 1) ≥ 0 iff

Rl

cl≥ w(m, m+ 1)

q(m, m+ 1).

Hence, if we show that

Im − Im−1 ≥ (Am −Am−1)w(m, m+ 1)

q(m, m+ 1), (4.55)

the claim follows. First, observe that the following identities hold:

q(i+ 1, m+ 1) = q(i+ 1, m) + (1− q(i+ 1, m))q(m, m+ 1), (4.56)

w(i+ 1, m+ 1) = w(i+ 1, m) + (1− q(i+ 1, m))w(m, m+ 1). (4.57)

Note that the identities relate the probability of service and expected waiting time of a customer

starting from position i+ 1 and abandoning at position m+ 1, to those of a customer starting

from the same position but abandoning at position m. Both identities are easily obtained by


conditioning on whether the customer starting from position i+ 1 and abandoning at position

m+ 1, first reaches position m or not. Using (4.54) and the above identities we have

Im − Im−1 = λl

(πm−1(m)w(m, m+ 1) +

m−2∑i=0

πi(m)w(i+ 1, m+ 1)−m−2∑i=0

πi(m− 1)w(i+ 1, m))

= λl

(πm−1(m)w(m, m+ 1) +

m−2∑i=0

πi(m) (w(i+ 1, m) + (1− q(i+ 1, m))w(m, m+ 1))

−m−2∑i=0

πi(m− 1)w(i+ 1, m))

= λl

(πm−1(m)w(m, m+ 1) +

m−2∑i=0

πi(m) ((1− q(i+ 1, m))w(m, m+ 1))

+m−2∑i=0

[πi(m)− πi(m− 1)]w(i+ 1, m)), (4.58)

and similarly,

Am −Am−1 = λl

(πm−1(m)q(m, m+ 1) +

m−2∑i=0

πi(m)q(i+ 1, m+ 1)−m−2∑i=0

πi(m− 1)q(i+ 1, m))

= λl

(πm−1(m)q(m, m+ 1) +

m−2∑i=0

πi(m) (q(i+ 1, m) + (1− q(i+ 1, m))q(m, m+ 1))

−m−2∑i=0

πi(m− 1)q(i+ 1, m))

= λl

(πm−1(m)q(m, m+ 1) +

m−2∑i=0

πi(m) ((1− q(i+ 1, m))q(m, m+ 1))

+m−2∑i=0

[πi(m)− πi(m− 1)] q(i+ 1, m)). (4.59)

It follows that

(Am −Am−1)w(m, m+ 1)

q(m, m+ 1)= λl

(πm−1(m)w(m, m+ 1) +

m−2∑i=0

πi(m) ((1− q(i+ 1, m))w(m, m+ 1))

+

m−2∑i=0

[πi(m)− πi(m− 1)] q(i+ 1, m)w(m, m+ 1)

q(m, m+ 1)

). (4.60)

Comparing (4.60) and (4.58) and noting that πi(m)− πi(m− 1) < 0, it follows that for (4.55)


to hold, it is sufficient to have

w(i+ 1, m)

q(i+ 1, m)≤ w(m, m+ 1)

q(m, m+ 1),

for all 0 ≤ i ≤ m− 2. From Lemma (4.10.4) we know that w(i+1, m)/q(i+1, m) is increasing

in i and w(m, m+1)/q(m, m+1) is increasing in m. It follows that the above inequality holds

and hence the proof is complete. �

Proof of Proposition 4.7.1. Consider the constraint in (4.21). Using the definition of g in

(4.7) and noting that w(na, na + 1)/q(na, na + 1) is increasing in na (see Lemma (4.10.4), part

(i)), to induce the abandonment threshold na we must have

Rl − clw(na + 1, na + 2)

q(na + 1, na + 2)< Ps − Pc ≤ Rl − cl

w(na, na + 1)

q(na, na + 1). (4.61)

Similarly, from the second constraint (4.22), given na and to induce the balking threshold nb

we must have

(Rl − (Ps − Pc))q(nb + 1, na + 1)− clw(nb + 1, na + 1) < Pe + Pc

≤ (Rl − (Ps − Pc))q(nb, na + 1)− clw(nb, na + 1). (4.62)

Now let Pe ≡ Pe + Pc and Ps ≡ Ps − Pc. Observe that for fixed thresholds the revenue is

increasing in Pe and (4.61) is independent of this value. Thus, (4.62) must be satisfied with an

equality for any Ps satisfying (4.61). That is

Pe = (R− Ps)q(nb, na + 1)− clw(nb, na + 1). (4.63)

Substituting from (4.63) in the revenue function we get

Π(nb, na) = λl

[(Rl − Ps)q(nb, na + 1)− clw(nb, na + 1)

]qJ(nb, na) + λlPsqS(nb, na)

= λl(Rl − Ps)q(nb, na + 1)qJ(nb, na) + λlPsqS(nb, na)− λlclw(nb, na + 1)qJ(nb, na)

= λlRlq(nb, na + 1)qJ(nb, na) + λlPs [qS(nb, na)− q(nb, na + 1)qJ(nb, na)]

−λlclw(nb, na + 1)qJ(nb, na).


Now observe that

qS(nb, na) =

nb−1∑i=0

πiq(i+ 1, na + 1) ≥nb−1∑i=0

πiq(nb, na + 1) = q(nb, na + 1)qJ(nb, na),

where the inequality follows from the fact that q(i, na+1) is decreasing in i. It follows that the

revenue increases in Ps and thus Ps = R− clw(na, na + 1)/q(na, na + 1). �Proof of Proposition (4.7.3). We compare the optimal revenues under entrance and service

fee when λh is in the neighborhood of zero. To do so we compare the right derivative of the op-

timal revenue functions with respect to λh in the limit as λh → 0+. Then since the functions are

decreasing in λh and have the same value at λh = 0, the one with the larger derivative generates

more revenue for sufficiently small λh. Recall that n∗ is the optimal balking and abandonment

threshold under the service fee, and n∗b , n

∗a denote respectively the balking and abandonment

thresholds under the entrance fee. For clarity, here we state the optimal thresholds and the rev-

enue functions as functions of λh, i.e., we denote the thresholds by n∗(λh), n∗b(λh), n

∗a(λh) and the

optimal revenue functions by Π∗s(λh) = Πs(n

∗(λh), λh) and Π∗e(λh) = Πe(n

∗a(λh), n

∗b(λh), λh). It

can be shown that the optimal thresholds are piecewise constant, right-continuous functions of

λh. Hence, for sufficiently small λh, the optimal thresholds are constant and the same as the

optimal thresholds at λh = 0. It follows using (4.26) and (4.27) that

limλh→0+

dΠ∗s(λh)

dλh= lim

λh→0+

∂Πs(n∗(0), λh)

∂λh

= limλh→0+

∂

∂λh

[λl

(Rl − cl

w(n∗(0), n∗(0) + 1)

q(n∗(0), n∗(0) + 1)

)qS(n

∗(0), n∗(0))], (4.64)

and

limλh→0+

dΠ∗e(λh)

dλh= lim

λh→0+

∂Πe(n∗a(0), n

∗b(0), λh)

∂λh(4.65)

= limλh→0+

∂

∂λh[λl (Rlq(n

∗b(0), n

∗a(0) + 1)− clw(n

∗b(0), n

∗a(0) + 1)) qJ(n

∗b(0), n

∗a(0))] .

Also, using (4.26) and (4.27) we have

n∗ ≡ n∗(0) = n∗b(0) = argmaxn≥1λl(Rl − cln/μ)

1− ρnl1− ρn+1

l

, (4.66)

n∗ ≡ n∗a(0) = max{n ≥ 0;Rl − cln/μ ≥ 0}. (4.67)

Note that (4.66) and (4.67) are respectively the revenue maximizing and self-optimization

thresholds in the single class FIFO problem of Naor (1969) and hence n∗ ≤ n∗.


We proceed by computing the limits (4.64) and (4.65). As λh → 0+ we have

∂w(n, n+ 1)

∂λh→ −1/μ,

∂q(n, n+ 1)

∂λh→ −1,

∂qS(n, n)

∂λh→ −ρn−1

l (ρn+2l − (n+ 2)ρl + (n+ 1))

(1− ρn+1l )2

,

∂qJ(n, n)

∂λh→ −ρn−1

l (ρn+1l − (n+ 1)ρl + n)

(1− ρn+1l )2

.

Moreover, for nb ≤ na,

w(nb, na+1) → nb/μ, q(nb, na+1) → 1, qS(nb, na) →1− ρnb

l

1− ρnb+1l

, qJ(nb, na) →1− ρnb

l

1− ρnb+1l

,

(4.68)

and for nb < na,

∂w(nb, na + 1)

∂λh→ nb/μ,

∂q(nb, na + 1)

∂λh→ 0,

∂qJ(nb, na)

∂λh→ −ρnb−1

l (ρnb+2l − (nb + 1)ρl + nb)

(1− ρnb+1l )2

.

(4.69)

Substituting in (4.64) and simplifying we have

limλh→0+

dΠ∗s(λh)

dλh= lim

λh→0+λl

[(−cl

w′(n∗, n∗ + 1)q(n∗, n∗ + 1)− w(n∗, n∗ + 1)q′(n∗, n∗ + 1)

q(n∗, n∗ + 1)2

)qS(n

∗, n∗)

+

(Rl − cl

w(n∗, n∗ + 1)

q(n∗, n∗ + 1)

)q′S(n

∗, n∗)]

= λl

[(−cl

n∗ − 1

μ

)1− ρn

∗l

1− ρn∗+1

l

− (Rl − cln∗/μ)

ρn∗−1

l (ρn∗+2

l − (n∗ + 2)ρl + (n∗ + 1))

(1− ρn∗+1

l )2

],

(4.70)

For (4.65) we need to consider two cases:

limλh→0+

dΠ∗e(λh)

dλh= lim

λh→0+λl

[(Rlq

′(n∗, n∗ + 1)− clw′(n∗, n∗ + 1)

)qJ(n

∗, n∗)

+ (Rlq(n∗, n∗ + 1)− clw(n

∗, n∗ + 1)) q′J(n∗, n∗)

]

=

⎧⎪⎪⎨⎪⎪⎩λl

[(−cln

∗/μ) 1−ρn∗

l

1−ρn∗+1

l

− (Rl − cln∗/μ) ρn

∗−1l (ρn

∗+2l −(n∗+1)ρl+n∗)(1−ρn

∗+1l )2

], if n∗ < n∗,

λl

[(−Rl + cln

∗/μ) 1−ρn∗

l

1−ρn∗+1

l

− (Rl − cln∗/μ) ρn

∗−1l (ρn

∗+1l −(n∗+1)ρl+n∗)(1−ρn

∗+1l )2

], if n∗ = n∗.

(4.71)


Using (4.70) and (4.71) after some algebra we can write

limλh→0+

(dΠ∗

e(λh)

dλh− dΠ∗

s(λh)

dλh

)=

⎧⎪⎪⎪⎨⎪⎪⎪⎩λl

[(Rl − cln

∗/μ) ρn∗−1

l

(1−ρl

1−ρn∗+1

l

)2

− (cl/μ)1−ρn

∗l

1−ρn∗+1

l

], if n∗ < n∗,

λl (Rl − cln∗/μ)

(1−ρn

∗l +ρn

∗−1l (1−ρl)

1−ρn∗+1

l

), if n∗ = n∗.

(4.72)

Case 1: For n∗ < n∗ (which is typically the case) the entrance fee is optimal iff

(Rl − cln∗/μ) ρn

∗−1l

(1− ρl

1− ρn∗+1

l

)2

− (cl/μ)1− ρn

∗l

1− ρn∗+1

l

> 0. (4.73)

Case 2: For n∗ = n∗, it is easy to verify that for any ρl �= 1, the term(1− ρn

∗l + ρn

∗−1l (1−

ρl))/(1−ρn

∗+1l ) in (4.72) is positive. Thus, for ρl �= 1 the entrance fee is optimal iff Rl−cln

∗/μ >

0.

In the following we verify that the conditions provided in statements (i)− (iii) are sufficient

for either the service or entrance fee to be optimal.

(i) Let λl → 0+, then using (4.66) we find that n∗ = 1. Hence, (4.73) reduces to (Rl −cl/μ) − cl/μ > 0 or Rlμ/cl > 2. Also note that assuming Rlμ/cl > 2, we have n∗ ≥ 2. Hence,

we are in Case 1 and the entrance fee is optimal.

(ii) Let ρl → 1+, then we have (1 − ρl)/(1 − ρn∗+1

l ) → 1/(1 + n∗) and (1 − ρn∗

l )/(1 −ρn

∗+1l ) → n∗/(1 + n∗). Thus, (4.73) reduces to (Rl − cln

∗)/(1 + n∗)2 > cl(n∗/(1 + n∗)) which

simplifies to Rl/cl > (n∗ + 1)2 − 1. Note that since ρl �= 1, in both cases the sufficient

conditions for the entrance fee to be optimal are satisfied. Further, for ρl → 1+, n∗ is the

maximizer of f(n) ≡ λl(Rl − cln/μ)(n/(1 + n)). Therefore, it can be obtained by finding

the positive root of f(n) − f(n − 1) = 0 and applying the floor function to it, which yields

n∗ =⌊12(√

5 + 4Rlμ/cl − 1)⌋.

(iii) Let λl → ∞, then it is easy to verify that the problem in (4.66) reduces to argmaxn≥1μ(Rl−cln/μ) and thus n∗ = 1. Assuming that Rlμ/cl ≥ 2, we have n∗ ≥ 2 and hence we are in Case 1.

Setting n∗ = 1, the lhs of (4.73) simplifies to (Rl − cl/μ)(

11+ρl

)2−(cl/μ)

11+ρl

, which approaches

zero from below as λl → ∞, proving the service fee is optimal. �


4.10 Appendix B: Additional Lemmas

Lemma 4.10.1 Fix m ≥ 1, C ≥ 0, and m ≤ T < ∞, and consider two cases:

• Case 1: The value function v(m,n) satisfies

v(m,m) = [−c+ λv(m,m+ 1) + μC]+, (4.74)

v(m,n) = [−c+ λv(m,n+ 1) + μv(m,n− 1)]+, n ∈ {m+ 1, ..., T}, (4.75)

v(m,T + 1) =

⎧⎪⎨⎪⎩[−c+ λv(m,T + 1) + μv(m,T )]+, T = nh +m− 1,

[−c+ μv(m,T )]+, T < nh +m− 1.

(4.76)

• Case 2: The value function v(m,n) satisfies

v(m,m) = [−c+ λv(m,m+ 1) + μC]+,

v(m,n) = [−c+ λv(m,n+ 1) + μv(m,n− 1)]+, n ∈ {m+ 1, ...}.

For all n ∈ {m, . . . , T} in case 1, and n ∈ {m,m+ 1, . . .} in case 2, we have

(i) v(m,n+ 1) ≤ v(m,n) ≤ C,

(ii) v(m,n+ 1)− v(m,n) ≤ −c if v(m,n+ 1) > 0.

Proof of Lemma 4.10.1. (i) Consider the problem in Case 1. We prove the first part using

value iteration. We approximate the value function using the following recursion for k ≥ 0 and

initialize the recursion with v0(m,n) = 0 for all n ≥ m:

vk+1(m,m) = [−c+ λvk(m,m+ 1) + μC]+, (4.77)

vk+1(m,n) = [−c+ λvk(m,n+ 1) + μvk(m,n− 1)]+, n ∈ {m+ 1, . . . , T}.(4.78)

vk+1(m,T + 1) =

⎧⎪⎨⎪⎩[−c+ λvk(m,T + 1) + μvk(m,T )]+, T = nh +m− 1,

[−c+ μvk(m,T )]+, T < nh +m− 1.

(4.79)

Then, by the convergence of the value iteration algorithm we have vk(m,n) → v(m,n) as

k → ∞. Thus, it suffices to show that the claim holds for all k ≥ 0. We prove this using

induction. The claim is clearly satisfied for k = 0. Assuming that vk(m,n+ 1) ≤ vk(m,n) for

all n ∈ {m, . . . , T}, we show that vk+1(m,n + 1) ≤ vk+1(m,n). To this end, we compare the


corresponding equations for different values of n. For n = m we have

vk+1(m,m) = [−c+ λvk(m,m+ 1) + μC]+, (4.80)

vk+1(m,m+ 1) = [−c+ λvk(m,m+ 2) + μvk(m,m)]+. (4.81)

For all n ∈ {m+ 1, . . . , T − 1} we have

vk+1(m,n) = [−c+ λvk(m,n+ 1) + μvk(m,n− 1)]+, (4.82)

vk+1(m,n+ 1) = [−c+ λvk(m,n+ 2) + μvk(m,n)]+, (4.83)

and for n = T ,

vk+1(m,T ) = [−c+ λvk(m,T + 1) + μvk(m,T − 1)]+,

vk+1(m,T + 1) =

⎧⎪⎨⎪⎩[−c+ λvk(m,T + 1) + μvk(m,T )]+, T = nh +m− 1,

[−c+ μvk(m,T )]+. T < nh +m− 1.

In all three cases comparing the equations and using the induction hypothesis the result directly

follows.

(ii) For n = T , assuming v(m,T + 1) > 0 and using (4.76) we have v(m,T + 1) = −c +

μv(m,T ). Hence, we can write v(m,T +1)−v(m,T ) ≤ v(m,T +1)− μv(m,T ) = −c. If m = T ,

then we are done. Otherwise for n ∈ {m, . . . , T − 1} and assuming v(m,n + 1) > 0 we have

using (4.75) that v(m,n+ 1) = −c+ λv(m,n+ 2) + μv(m,n). It follows using part (i) that

v(m,n+ 1)− μv(m,n) = −c+ λv(m,n+ 2) ≤ −c+ λv(m,n),

which yields v(m,n+ 1)− v(m,n) ≤ −c as claimed.

The proof for Case 2, i.e., when n is unbounded, is similar except there is no need to consider

the case with n = T . �

Lemma 4.10.2 Consider value functions v(m,n) and v(m+ 1, n) satisfying equations (4.33)-

(4.35) and (4.36)-(4.38), respectively. We have

(i) if n(m) = nh +m, then v(m+ 1, n) ≤ v(m,n) for n ∈ {m+ 1, . . . , nh +m}.(ii) if n(m) < nh +m, then v(m+ 1, n) = v(m,n) for n ∈ {m+ 1, . . . , n(m) + 1}.


Proof of Lemma 4.10.2. (i) The proof is by induction and convergence of the value iteration

algorithm. Note that since n(m) = nh + m, the value of staying at all system positions is

nonnegative for the customer at low-priority position m. Therefore, using (4.33)-(4.35), v(m,n)

satisfies

v(m,m) = −c+ λv(m,m+ 1) + μv(m− 1,m− 1),

v(m,n) = −c+ λv(m,n+ 1) + μv(m,n− 1), n ∈ {m+ 1, ..., nh +m− 1},

v(m, nh +m) = −c+ λv(m, nh +m) + μv(m, nh +m− 1).

Next we approximate the value function for the customer at low-priority position m+ 1 given

in (4.36)-(4.38) using the recursion

vk+1(m+ 1,m+ 1) = [−c+ λvk(m+ 1,m+ 2) + μv(m,m)]+,

vk+1(m+ 1, n) = [−c+ λvk(m+ 1, n+ 1) + μvk(m+ 1, n− 1)]+, n ∈ {m+ 2, ..., nh +m},

vk+1(m+ 1, nh +m+ 1) = [−c+ λv(m+ 1, nh +m+ 1) + μv(m+ 1, nh +m)]+,

for k ≥ 0 and with v0(m + 1, n) = 0 for all n ∈ {m + 1, ..., nh +m + 1}. The claim is clearly

satisfies for k = 0. Assuming

vk(m+ 1, n) ≤ v(m,n), (4.84)

we show that

vk+1(m+ 1, n) ≤ v(m,n), (4.85)

for all n ∈ {m + 1, ..., nh +m} and hence the result follows by induction. For n = m + 1 the

coresponding equations are

v(m,m+ 1) = −c+ λv(m,m+ 2) + μv(m,m),

vk+1(m+ 1,m+ 1) = [−c+ λvk(m+ 1,m+ 2) + μv(m,m)]+,


and for n ∈ {m+ 2, ..., nh +m− 1}, we have

v(m,n) = −c+ λv(m,n+ 1) + μv(m,n− 1),

vk+1(m+ 1, n) = [−c+ λvk(m+ 1, n+ 1) + μvk(m+ 1, n− 1)]+.

Finally, for n = nh +m,

v(m, nh +m) = −c+ λv(m, nh +m) + μv(m, nh +m− 1),

vk+1(m+ 1, nh +m) = [−c+ λvk(m+ 1, nh +m+ 1) + μvk(m+ 1, nh +m− 1)]+.

In each case, comparing the two equations and using (4.84) the claim directly follows.

(ii) For n(m) < nh + m, it is easy to verify that v(m + 1, n) satisfies the same equations

as v(m,n). Hence, by uniqueness of the value function we have v(m,n) = v(m + 1, n) for all

n ∈ {m+ 1, . . . , n(m) + 1}. �

Lemma 4.10.3 Let V denote the set of functions satisfying (4.50) and (4.51). If z ∈ V, then(i) T1z ∈ V, (ii) T2z ∈ V, and (iii) T3z ∈ V.

Proof of Lemma 4.10.3.

Proof of statement (i):

(a) For (4.50), we need to consider four cases.

(a.1.) If z(x1, x2 + 2) ≤ z(x1, x2 + 3), then by concavity we also have z(x1, x2 + 1) ≤z(x1, x2 + 2) and z(x1, x2) ≤ z(x1, x2 + 1). Thus, we can write

T1z(x1, x2 + 2)− T1z(x1, x2 + 1) = max [z(x1, x2 + 3), z(x1, x2 + 2)]

−max [z(x1, x2 + 2), z(x1, x2 + 1)]

= z(x1, x2 + 3)− z(x1, x2 + 2)

≤ z(x1, x2 + 2)− z(x1, x2 + 1)

= max [z(x1, x2 + 2), z(x1, x2 + 1)]

−max [z(x1, x2 + 1), z(x1, x2)]

= T1z(x1, x2 + 1)− T1z(x1, x2).

(a.2.) If z(x1, x2 + 2) > z(x1, x2 + 3), but z(x1, x2 + 1) ≤ z(x1, x2 + 2) and z(x1, x2) ≤


z(x1, x2 + 1), then

T1z(x1, x2 + 2)− T1z(x1, x2 + 1) = z(x1, x2 + 2)− z(x1, x2 + 2)

≤ z(x1, x2 + 2)− z(x1, x2 + 1)

= T1z(x1, x2 + 1)− T1z(x1, x2).

(a.3.) If z(x1, x2 + 2) > z(x1, x2 + 3) and z(x1, x2 + 1) > z(x1, x2 + 2) but z(x1, x2) ≤z(x1, x2 + 1), then

T1z(x1, x2 + 2)− T1z(x1, x2 + 1) = z(x1, x2 + 2)− z(x1, x2 + 1)

≤ z(x1, x2 + 1)− z(x1, x2 + 1)

= T1z(x1, x2 + 1)− T1z(x1, x2).

(a.4.) If z(x1, x2+2) > z(x1, x2+3), z(x1, x2+1) > z(x1, x2+2), and z(x1, x2) > z(x1, x2+1),

then

T1z(x1, x2 + 2)− T1z(x1, x2 + 1) = z(x1, x2 + 2)− z(x1, x2 + 1)

≤ z(x1, x2 + 1)− z(x1, x2)

= T1z(x1, x2 + 1)− T1z(x1, x2).

(b) For (4.51) we need to consider three cases.

(b.1.) If z(x1 + 1, x2 + 2) ≥ z(x1 + 1, x2 + 1), then by (4.51) z(x1, x2 + 3) ≥ z(x1, x2 + 2).

Also by concavity z(x1 + 1, x2 + 1) ≥ z(x1 + 1, x2). Thus, we can write

T1z(x1 + 1, x2 + 1)− T1z(x1 + 1, x2) = max [z(x1 + 1, x2 + 2), z(x1 + 1, x2 + 1)]

−max [z(x1 + 1, x2 + 1), z(x1 + 1, x2)]

= z(x1 + 1, x2 + 2)− z(x1 + 1, x2 + 1)

= z(x1, x2 + 3)− z(x1, x2 + 2)

= max [z(x1, x2 + 3), z(x1, x2 + 2)]

−max [z(x1, x2 + 2), z(x1, x2 + 1)]

= T1z(x1, x2 + 2)− T1z(x1, x2 + 1).


(b.2.) If z(x1 + 1, x2 + 2) < z(x1 + 1, x2 + 1), but z(x1 + 1, x2 + 1) ≥ z(x1 + 1, x2), then

T1z(x1 + 1, x2 + 1)− T1z(x1 + 1, x2) = z(x1 + 1, x2 + 1)− z(x1 + 1, x2 + 1)

= z(x1, x2 + 2)− z(x1, x2 + 2)

= T1z(x1, x2 + 2)− T1z(x1, x2 + 1).

(b.3.) If z(x1 + 1, x2 + 2) < z(x1 + 1, x2 + 1) and z(x1 + 1, x2 + 1) < z(x1 + 1, x2), then

T1z(x1 + 1, x2 + 1)− T1z(x1 + 1, x2) = z(x1 + 1, x2 + 1)− z(x1 + 1, x2)

= z(x1, x2 + 2)− z(x1, x2 + 1)

= T1z(x1, x2 + 2)− T1z(x1, x2 + 1).

Proof of statement (ii):

(a) For (4.50), we need to consider three cases.

(a.1.) If z(x1 + 1, x2 + 2) ≥ z(x1 + 1, x2 + 1), then

T2z(x1, x2 + 2)− T2z(x1, x2 + 1) = max0≤j≤x2+2

z(x1 + 1, j)− max0≤j≤x2+1

z(x1 + 1, j)

= z(x1 + 1, x2 + 2)− z(x1 + 1, x2 + 1)

≤ z(x1 + 1, x2 + 1)− z(x1 + 1, x2)

= max0≤j≤x2+1

z(x1 + 1, j)− max0≤j≤x2

z(x1 + 1, j)

= T2z(x1, x2 + 1)− T2z(x1, x2).

(a.2.) If z(x1 + 1, x2 + 2) < z(x1 + 1, x2 + 1), then let j∗ ≡ argmax0≤j≤x2+2z(x1 + 1, j). If

j∗ = x2 + 1, then

T2z(x1, x2 + 2)− T2z(x1, x2 + 1) = z(x1 + 1, x2 + 1)− z(x1 + 1, x2 + 1)

≤ z(x1 + 1, x2 + 1)− z(x1 + 1, x2)

= T2z(x1, x2 + 1)− T2z(x1, x2).


(a.3.) If z(x1 + 1, x2 + 2) < z(x1 + 1, x2 + 1) and j∗ ≤ x2, then

T2z(x1, x2 + 2)− T2z(x1, x2 + 1) = z(x1 + 1, j∗)− z(x1 + 1, j∗)

= T2z(x1, x2 + 1)− T2z(x1, x2).

(b) Next we verify (4.51).

(b.1.) If z(x1 + 2, x2 + 1) ≥ z(x1 + 2, x2) then by (4.51) we have z(x1 + 1, x2 + 2) ≥z(x1 + 1, x2 + 1). Hence we can write

T2z(x1 + 1, x2 + 1)− T2z(x1 + 1, x2) = max0≤j≤x2+1

z(x1 + 2, j)− max0≤j≤x2

z(x1 + 2, j)

= z(x1 + 2, x2 + 1)− z(x1 + 2, x2)

= z(x1 + 1, x2 + 2)− z(x1 + 1, x2 + 1)

= max0≤j≤x2+2

z(x1 + 1, j)− max0≤j≤x2+1

z(x1 + 1, j)

= T2z(x1, x2 + 2)− T2z(x1, x2 + 1).

(b.2.) If z(x1 + 2, x2 + 1) < z(x1 + 2, x2), then let j∗ ≡ argmax0≤j≤x2+1z(x1 + 2, j). If

j∗ = x2, then

T2z(x1 + 1, x2 + 1)− T2z(x1 + 1, x2) = z(x1 + 2, x2)− z(x1 + 2, x2)

= z(x1 + 1, x2 + 1)− z(x1 + 1, x2 + 1)

= T2z(x1, x2 + 2)− T2z(x1, x2 + 1).

(b.3.) If z(x1 + 2, x2 + 1) < z(x1 + 2, x2), and j∗ < x2, then

T2z(x1 + 1, x2 + 1)− T2z(x1 + 1, x2) = z(x1 + 2, j∗)− z(x1 + 2, j∗)

= z(x1 + 1, j∗ + 1)− z(x1 + 1, j∗ + 1)

= T2z(x1, x2 + 2)− T2z(x1, x2 + 1).

Proof of statement (iii):

(a) We first show (4.50).


If x1 > 0, x2 ≥ 0, we have using concavity,

T3z(x1, x2 + 2)− T3z(x1, x2 + 1) = z(x1 − 1, x2 + 2)− z(x1 − 1, x2 + 1)

≤ z(x1 − 1, x2 + 1)− z(x1 − 1, x2)

= T3z(x1, x2 + 1)− T3z(x1, x2).

If x1 = 0, x2 > 0, then

T3z(0, x2 + 2)− T3z(0, x2 + 1) = [Rl + z(0, x2 + 1)]− [Rl + z(0, x2)]

≤ z(0, x2)− z(0, x2 − 1)

= T3z(0, x2 + 1)− T3z(0, x2).

Finally, if x1 = x2 = 0,

T3z(0, 2)− T3z(0, 1) = [Rl + z(0, 1)]− [Rl + z(0, 0)]

≤ Rl + z(0, 0)− z(0, 0)

= T3z(0, 1)− T3z(0, 0),

where the inequality follows from the fact that the value of an additional low-priority customer

is bounded by Rl.

(b) Next we consider (4.51). If x1 > 0, x2 ≥ 0 we have

T3z(x1 + 1, x2 + 1)− T3z(x1 + 1, x2) = z(x1, x2 + 1)− z(x1, x2)

= z(x1 − 1, x2 + 2)− z(x1 − 1, x2 + 1)

= T3z(x1, x2 + 2)− T3z(x1, x2 + 1).

If x1 = 0, x2 > 0,

T3z(1, x2 + 1)− T3z(1, x2) = z(0, x2 + 1)− z(0, x2)

= [Rl + z(0, x2 + 1)]− [Rl + z(0, x2)]

= T3z(0, x2 + 2)− T3z(0, x2 + 1).


If x1 = x2 = 0,

T3z(1, 1)− T3z(1, 0) = z(0, 1)− z(0, 0)

= [Rl + z(0, 1)]− [Rl + z(0, 0)]

= T3z(0, 2)− T3z(0, 1).

The proof is complete. �

Lemma 4.10.4 Consider w and q given in (4.1) and (4.2), respectively. We have

(i) w(n, n+ 1)/q(n, n+ 1) is increasing in n for n ≥ 1.

(ii) w(i, n+ 1)/q(i, n+ 1) is increasing in i for 0 ≤ i ≤ n.

Proof of Lemma 4.10.4. (i) Using (4.1) and (4.2) we have

w(n, n+ 1)

q(n, n+ 1)=

n(1− ρn+1h )− (n+ 1)(ρh − ρn+1

h )

μ(1− ρh)2.

=n(1− ρh)− ρh(1− ρnh)

μ(1− ρh)2. (4.86)

Noting that (4.86) is the same function on the left-hand-side of Eq. (22) in Naor (1969) divided

by μ, we know that the function is increasing in n.

(ii) Using (4.1) and (4.2) after simplifying we have

w(i, n+ 1)

q(i, n+ 1)=

i(1− ρn+1h )− (n+ 1)(ρn+1−i

h − ρn+1h )

μ(1− ρh)(1− ρn+1−ih )

.

Adding and subtracting n+ 1 in the numerator and simplifying we get

w(i, n+ 1)

q(i, n+ 1)=

i(1− ρn+1h )− (n+ 1)(ρn+1−i

h − ρn+1h ) + (n+ 1)− (n+ 1)

μ(1− ρh)(1− ρn+1−ih )

.

=i(1− ρn+1

h ) + (n+ 1)(1− ρn+1−ih )− (n+ 1)(1− ρn+1

h )

μ(1− ρh)(1− ρn+1−ih )

=(n+ 1)(1− ρn+1−i

h )− (n+ 1− i)(1− ρn+1h )

μ(1− ρh)(1− ρn+1−ih )

=n+ 1

μ(1− ρh)− (1− ρn+1

h )(n+ 1− i)

μ(1− ρh)(1− ρn+1−ih )

.

Hence, noting that ρh < 1, it suffices to show that −(n + 1 − i)/(1 − ρn+1−ih ) is increasing in

i. To this end, let ν ≡ n + 1 − i and note that ν decreases as i increases. Thus, it remains to


show that ν/(1− ρνh) is increasing in ν. The first difference is

ν + 1

1− ρν+1h

− ν

1− ρνh=

(1− ρνh)− (1− ρh)νρνh

(1− ρν+1h )(1− ρνh)

.

The denominator is clearly positive. To show the nominator is also positive, consider r(x) ≡ xν .

By the mean value theorem,

1− ρνh1− ρh

=r(1)− r(ν)

1− ν= r′(c0),

for some ρh < c0 < 1. But r′(c0) = νcν−10 > νcν0 > νρνh, implying that

1− ρνh1− ρh

> νρνh,

and hence the nominator is indeed positive and the proof is complete. �

Bibliography

Abate J, P Valko (2004) Multi-precision Laplace transform inversion. International Journal for

Numerical Methods in Engineering 60(5):979–993.

Adiri I, Yechiali U (1974), Optimal priority purchasing and pricing decisions in nonmonopoly

and monopoly queues, Operations Research 22:1051–1066.

Afeche P (2004) Incentive-compatible revenue management in queueing systems: optimal strate-

gic idleness and other delay tactics. Working paper, University of Toronto, Toronto, ON.

Afeche P (2013) Incentive-compatible revenue management in queueing systems: Optimal

strategic delay. Manufacturing & Service Operations Management 15(3):423–443.

Aksin Z, Ata B, Emadi S, Su C (2013) Structural Estimation of Callers’ Delay Sensitivity in

Call Centers, Management Science 59(12):2727–2746.

Alperstein H (1988) Optimal pricing policy for the service facility offering a set of priority

prices, Management Science 34:666–671.

Altman E, Hassin R (2001) Non-threshold equilibrium for customers joining an M/G/1 queue.

In Proceedings of 10th International Symposium on Dynamic Game and Applications.

Assaf D, Haviv M (1990) Reneging from Processor Sharing Systems and Random Queues,

Mathematics of Operations Research 15(1):129-138.

Ata B, Glynn PW, Peng X (2015) An Equilibrium Analysis of a Multiclass Queue with En-

dogenous Abandonments. Working paper. University of Chicago, Chicago, IL.

Atkinson MP, Fontaine MJ, Goodnough LT, Wein LM (2012) A novel allocation strategy for

blood transfusions: investigating the tradeoff between the age and availability of transfused

blood. Transfusion 52:108–117.

Baron O, Milner J (2009) Staffing to maximize profit for call centers with alternate service level

agreements. Operations Research 57(3):685–700.

115

BIBLIOGRAPHY 116

Baron O (2011) Managing Perishable Inventory. Wiley Encyclopedia of Operations Research

and Management Science.

Bassamboo A, Randhawa RS (2010) On the Accuracy of Fluid Models for Capacity Planning

in Queueing Systems with Impatient Customers. Operations Research 58(5):1398–1413.

Bassamboo A, Randhawa RS (2013) Using estimated patience Levels to Optimally Schedule

Customers. Working Paper. Northwestern University.

Batt RJ, Terwiesch C (2015) Waiting Patiently: An Empirical Study of Queue Abandonment

in an Emergency Department. Management Science 61(1):39–59.

Billingsley P (1995) Probability and Measure, (Third ed. WileyInterscience, New York).

Blake JT, Hardy M, Delage G, Myhal G (2012) Deja-vu all over again: using simulation to

evaluate the impact of shorter shelf life for red blood cells at Hema-Quebec. Transfusion

53:1544-1558.

Bolandifar E, Dehoratius N, Olsen T, Wiler JL (2014) Modeling the Behavior of Patients Who

Leave the Emergency Department Without Being Seen by a Physician. Working Paper.

University of Chicago, Chicago, IL.

Brandt A and Brandt M (2004). On the two-class M/M/1 system under preemptive resume

and impatience of the prioritized customers. Queueing Systems 47(1/2):147–168.

Brouns GAJF, van der Wal J (2006) Optimal threshold policies in a two-class preemptive

priority queue with admission and termination control. Queueing Syst. Theory Appl 54:21–

33.

Chen X, Pang Z, Pan L (2014) Coordinating inventory control and pricing strategies for per-

ishable products, Operations Research 62(2):284–300.

Cohen JW (1982) The single server queue (2nd edn. North-Holland, Amesterdam).

Deniz BI, Karaesmen F, Scheller-Wolf A (2010) Managing Perishables with Substitution: Inven-

tory Issuance and Replenishment Heuristics. Manufacturing & Service Operations Man-

agement 12(2):319–329.

Down DG, Koole G, Lewis ME (2011) Dynamic control of a single-server system with aban-

donments. Queueing Systems 67(1),63–90.

Dzik WH, Beckman N, Murphy EM, et al. (2013) factors affecting red blood cell storage at the

time of transfusion. Transfusion, 53:3110–3119.

BIBLIOGRAPHY 117

Eikelboom JW, Cook RJ, Liu Y, Heddle NM (2010) Duration of red cell storage before trans-

fusion and in-hospital mortality. Am Heart J. 159(5):737–743.

Feller W (1971) An Introduction to Probability Theory and its Applications (Second ed., Wiley).

Fergusson DA, Hebert P, Hogan DL, et al. (2012) Effect of Fresh Red Blood Cell Transfusions on

Clinical Outcomes in Premature, Very Low-Birth-Weight Infants: The ARIPI Randomized

Trial. JAMA 308(14):1443–1451.

Flegel WA, Natanson C, Klein HG (2014) Does prolonged storage of red blood cells cause harm?

Br J Haematol 165(1):3–16.

Fontaine MJ, Chung YT, Erhun F, Goodnough LT (2011) Age of blood as a limitation for

transfusion: potential impact on blood inventory and availability. Transfusion 51:662–663.

Garnett O, Mandelbaum A, Reiman M (2002) Designing a Call Center with Impatient Cus-

tomers. Manufacturing & Service Operations Management 4(3):208-227.

Goh CH, Greenberg BS, Matsuo H (1993) Perishable inventory systems with batch demand

and arrivals. Operations Research Letters 13(1):1–8.

Goh CH, Greenberg BS, Matsuo H (1993) Two-Stage Perishable Inventory Models. Management

Science 39(5):633–649.

Grasas A, Pereira A, Bosch MA, Ortiz P, Puig L (2014) Feasibility of reducing the maximum

shelf life of red blood cells stored in additive solution: a dynamic simulation study involving

a large regional blood system. Vox Sanguinis 108(3):233–242.

Graves S (1978) Simple Analytical Models for Perishable Inventory Systems, Technical Report

No. 141, Operations Res. Center, MIT, Cambridge, MA.

Gross D, Shortle JF, Thompson JM, and Harris CM (2008) Fundamentals of Queueing Theory

(Fourth ed., John Wiley & Sons).

Gross JL (2008) Combinatorial Methods with Computer Applications (Chapman & Hall/CRC).

Guo L (2009) Service cancellation and competitive refund policy. Marketing Science 28(5):901–

917.

Haijema R, van der Wal J, van Dijk NM (2007) Blood platelet production: opimization by

dynamic programming and simulation. Computers & Operations Research 34(3):760–779.

Hassin R (1985) On the Optimality of First Come Last Served Queues, Econometrica 53(1):201-

202.

BIBLIOGRAPHY 118

Hassin R, Haviv M (1995) Equilibrium strategies for queues with impatient customers. Oper.

Res. Lett. 17(1):41–45.

Hassin R, Haviv M (1997) Equilibrium threshold strategies: The case of queues with priorities,

Operations Research 45:966-973.

Hassin R, Haviv M (2003) To Queue or Not to Queue: Equilibrium Behavior in Queueing

Systems (Kluwer Academic Publishers, Norwell, MA).

Haviv M, Ritov Y (2001) Homogeneous Customers Renege from invisible queues at Random

Times under deteriorating waiting conditions. Queueing Systems 38:495-508.

Heddle NM (2011) Informing fresh versus standard issue red cell management - INFORM

International Standard Randomized Controlled Trial Number Register (ISRCTN).

Huang T, Allon G, Bassamboo A (2013) Bounded rationality in service systems. Manufacturing

Service Oper. Management 15(2):263–279.

Iravani F, Balcıoglu B (2008). On priority queues with impatient customers. Queueing Systems

58(4): 239–260.

Jennings OB, Reed JE (2012) An Overloaded Multiclass FIFO Queue with Abandonments.

Operations Research 60(5):1282–1295.

Jouini O, Roubos A (2013) On multiple priority multi-server queues with impatience. Journal

of the Operational Research Society 65:616-632.

Karaesmen IZ, Scheller-Wolf A, Deniz B (2011) Managing perishable and aging inventories: re-

view and future research directions. In: Kempf K, Keskinocak P, Uzsoy R (eds) Planning

production and inventories in the extended enterprise: a state of the art handbook, inter-

national series in operations research and management science, vol 151. Kluwer Academic

Publishers.

Kaspi H, Perry D (1983) Inventory systems of perishable commodities. Adv. Appl. Prob. 15:674–

685.

Kaspi H, Perry D (1984) Inventory systems of perishable commodities with renewal input and

poisson output. Adv Appl Prob. 16:402–421.

Keilson J, Seidmann A (1990) Product selection policies for perishable inventory systems. Uni-

versity of Rochester, Rochester, NY.

Koch CG, LI L, Sessler DI, Figuera P, Hoeltge GA, Mihaljevich T, Blackstone EH (2008)

BIBLIOGRAPHY 119

Duration of red-cell storage and complications after cardiac surgery. New Engl J Med

358:1229–1239.

Kopach R, Balcioglu B, Carter M (2008) Tutorial on constructing a red blood cell inventory

management system with two demand rates. European Journal of Operations Research

185:1051–1059.

Lacroix J, Hebert PC, Fergusson DA, Tinmouth A, et al. (2015) Age of Transfused Blood in

Critically Ill Adults. N Engl J Med 372:1410–1418.

Lee H, Padmanabhan V, Whang S (1997) Information distortion in a supply chain: The bull-

whip effect. Management Science 43(4):546–558.

Lee C, Ward AR (2014) Optimal pricing and capacity sizing when customers abandon. Working

paper. Colorado State University, Fort Collins, CO.

Lelubre C, Piagrenelli M, Vincent JL (2009) Association between duration of storage of trans-

fused red blood cells and morbidity in adult patients: myth or reality? Transfusion

49:1384–1394.

Li Y, Cheang B, Lim A (2012) Grocery Perishables Management. Production and Operations

Management 21:504–517.

Maglaras C, Yao J and Zeevi A (2015) Observational learning in queues with abandonments.

Working paper. Columbia University, New York, NY.

Mandelbaum A, Shimkin N (2000) A model for rational abandonment from invisible queues,

Queueing Systems: Theory and Applications 36:141–173.

Mendelson H, Whang S (1990) Optimal incentive-compatible priority pricing for the M/M/1

queue. Operations Research 38(5):870–883.

Nahmias S (1982) Perishable Inventory Theory: A Review. Operations Research 30(4):680–708.

Nahmias S (2011) Mathematical Models for Perishable Inventory Control. Wiley Encyclopedia

of Operations Research and Management Science.

Nahmias S (2011b) Perishable Inventory Systems (Springer, New York, USA).

National Blood Collection and Utilization Survey (NBCUS) (2011)

The HHS National Blood Collection and Utilization Survey.

http://www.aabb.org/programs/biovigilance/nbcus/Pages/default.aspx.

Naor P (1969) On the regulation of queue size by levying tolls. Econometrica 37(1):15–24.

BIBLIOGRAPHY 120

Offner PJ, Moore EE, Biffl WL, Johnson JL, Silliman CC (2002) Increased rate of infection

associated with transfusion of old blood after severe injury.Arch Surg 137:711–716.

Parlar M, Perry D, Stadje W (2011) FIFO versus LIFO issuing policies for stochastic perishable

inventory systems. Methodology and Computing in Applied Probability 13:405–417.

Pereira A (2013) Will clinical studies elucidate the connection between the length of storage of

transfused red blood cells and clinical outcomes? An analysis based on the simulation of

randomized controlled trials. Transfusion 53:34–40.

Perry D, Posner MJM (1990) Control of input and demand rates in inventory systems of per-

ishable commodities. Naval Research Logistics 37:85–97.

Perry D (1999) Analysis of a sampling control scheme for a perishable inventory system. Oper-

ations Research 47(6):966–973.

Pettila V, Westbrook AJ, Nichol AD, et al. (2011) Age of red blood cells and mortality in the

critically ill. Crit Care 15(2):R116.

Pierskalla WP, Roach CD (1972) Optimal issuing policies for perishable inventory. Management

Science 18(11):603–614.

Puterman ML (1994) Markov Decision Processes: Discrete Stochastic Dynamic Programming

(Wiley-Interscience).

R Development Core Team. 2013. R: a language and environment for statistical computing. R

Foundation for Statistical Computing, Vienna, Austria.

Righter R (2000) Expulsion and scheduling control for multiclass queues with heterogeneous

servers. Queueing Systems: Theory Appl. 34:289–335.

Rosenthal J.S. (2000) A first look at rigorous probability theory. World Scientific Publishing,

Singapore.

Sabouri A, Huh WT, Shechter SM (2015) Issuing Policies for Hospital Blood Inventory. Working

paper, University of British Columbia, Vancuver, Canada.

Sarhangian V, Balcıoglu B (2012) Waiting time analysis of multi-class queues with impatient

customers. Probability in the Engineering and the Informational Sciences 27(3): 333–352

Sayers M, Centilli J (2012) What if shelf life becomes a consideration in ordering red blood

cells. Transfusion 52:201–206.

BIBLIOGRAPHY 121

Shimkin N, Mandelbaum A (2004) Rational abandonment from tele-queues: Nonlinear waiting

costs with heterogeneous preferences. Queueing Systems: Theory Appl. 47(1–2):117–146.

Simonetti A, Forshee RA, Anderson SA, Walderhaug M (2014) A stock-and-flow simulation

model of the US blood supply. Transfusion 54:828–838.

Steiner ME, Ness PM, Assmann SF, et al. (2015) Effects of Red-Cell Storage Duration on

Patients Undergoing Cardiac Surgery. New Engl J Med 372:1419–1429.

Weber RR and Stidham S (1987) Optimal control of service rates in networks of queues. Ad-

vances in Applied Probability 19(1):202–218.

Xie J and Gerstner E (2007) Service Escape: Profiting from Customer Cancellations. Marketing

Science 26(1):18–30.

Xu SH (1994) A duality approach to admission and scheduling controls of queues. Queueing

Systems: Theory Appl. 18:273–300.

Xu SH and Shanthikumar JG (1993) Optimal expulsion control–a dual approach to admission

control of an ordered-entry system. Operations Research 41:1137–1152.

Xue Z, Ettl M, Yao DD (2012) Managing freshness inventory: Optimal policy, bounds and

heuristics. Working paper, IBM T. J. Watson Research Center, Yorktown Heights, NY.

Zeltyn S, and Mandelbaum A, Call centers with impatient customers: many-server asymptotics

of the M/M/n+G queue. Queueing Systems 51(3/4):361–402.

Documents

by Vahid Sarhangian - University of Toronto T-Space · 2016-02-19 · Acknowledgements Iwould liketoexpress my sincere gratitudetomy advisors, Professors OdedBerman, Opher Baron,