Queue Decomposition and its Applications in State-Dependent … · 2013-12-06 · These applications can be analyzed using our results. Therefore our results shed light on these applications

Submitted to Manufacturing & Service Operations Managementmanuscript (Please, provide the mansucript number!)

Authors are encouraged to submit new papers to INFORMS journals by means ofa style file template, which includes the journal title. However, use of a templatedoes not certify that the paper has been accepted for publication in the named jour-nal. INFORMS journal templates are for the exclusive purpose of submitting to anINFORMS journal and should not be used to distribute the papers in print or onlineor to submit the papers to another publication.

Queue Decomposition and its Applications inState-Dependent Queueing Systems

Hossein Abouee-MehriziDepartment of Management Sciences, University of Waterloo, Waterloo, ON, N2L 3G1, CANADA,

[email protected]

Opher BaronJoseph L. Rotman School of Management, University of Toronto, Toronto, M5S 3E6, CANADA,

[email protected]

A main idea in the analysis of Markovian queueing systems is that at each point in time the system can

be decomposed into two systems: in one system there are n or less people and in the other one there are

n+1 or more people. In this paper we introduce the idea of Queue Decomposition (QD) into non-Markovian

systems. Specifically, we consider an Mn/Gn/1 state-dependent single server queue. We allow the arrival rate

of customers to depend on the number of people in the system. Service times are also state-dependent and

service rates can be modified at both arrivals and departures of customers as is the case in many applications.

We show that the steady-state solution of this system at arbitrary times can be derived by QD using the

supplementary variable method, and that the system’s state at arrival epochs can be decomposed using an

embedded Markov chain. For the queueing system with infinite buffer size, we first obtain an expression

for the steady-state distribution of the number of customers in the system at both arbitrary and arrival

times. Then, we derive the rate at which the number of customers in the system decreases by one (i.e., the

system moves from state n to n− 1) at both arbitrary times and arrival epochs. We show that using these

transition rates, our state-dependent queueing system is equivalent to a Markovian birth-and-death process.

This equivalency demonstrates our main insight that the Mn/Gn/1 system can be decomposed at any given

state as a Markovian queue. Thus, many of the existing results for systems modeled as M/M/1 queue can

be carried through to the much more practical M/G/1 model with state dependence. We discuss several

service and inventory problems that can be modeled as a state-dependent queueing system with either finite

or infinite buffer size and demonstrate our results for these applications.

Key words : Mn/Gn/1 queue, birth-and-death process, state-dependent service times, state-dependent

arrivals

1

mailto:[email protected]

mailto:[email protected]

Abouee-Mehrizi and Baron: State-Dependent Queues2 Article submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the mansucript number!)

1. Introduction

Markovian queues have been used to model and analyze the congested queuing systems in a vast

body of literature. The beauty of the single stage Markovian queues is that the system can be

modeled as a Birth-and-Death (B&D) process and decomposed to new Markovian queues at any

given state. These properties make the problem tractable even when the arrivals and service rates

are state-dependent or when the control of the queue dynamically changes. This fact made the

M/M/1 queue a preferable model for many theoretical studies in management science.

However, in many applications assuming Markovian service times is not realistic. Thus, much

attention has been given to analysis of queues with generally distributed service times. An impor-

tant limitation of such queueing systems however is that their analysis is not straightforward. This

limitation is especially true when the arrival rates and the service times depend on the state of

the system. Moreover, since such systems are not regenerative at arrival epochs, the solution of

the system is not simple when the control of the queue is adjusted at such epochs. This apparent

difficulty in the analysis of such queueing systems limited their study, even when they are a more

appropriate model for the application considered. We believe that this difficulty therefore limited

the insights generated by many papers.

In this paper we consider a queueing system with state-dependent arrival and service rates,

denoted as Mn/Gn/1. We show that such state-dependent queueing systems can be decomposed at

any given state of the system and modeled and solved as a B&D process. We next explain the state-

dependent Mn/Gn/1 queueing model we consider and discuss several interesting applications of this

model in the control of service systems, security screening, healthcare, and inventory management.

These applications can be analyzed using our results. Therefore our results shed light on these

applications and can be further used to study these and other systems.

1.1. Model and Several Applications

We study a single server queue with state-dependent arrival rates and service times. Customers

arrive to the system according to a Poisson process with a rate of λn when there are n customers

in the system. The service times are also state-dependent. Specifically, when there are n customers

in the system and a new service time starts, it is generally distributed with a mean of 1/µn, a

density function of bn(·), and Laplace Transform (LT) of bn(·). We assume that bn(·) is absolutely

continuous with finite moments. The density function bn(·) may be completely different than bn+1(·),

e.g., bn+1(·) can be uniformly distributed while bn(·) is exponentially distributed.

We also allow the service rate to change when a new customer arrives to the system as follows:

when there are n ≥ 1 customers in the system and a new arrival occurs, the rate of the service

increases by a factor of αn+1 > 0, so that the residual service time of the customer in service

Abouee-Mehrizi and Baron: State-Dependent QueuesArticle submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the mansucript number!) 3

decreases by this factor. (Note that if αn+1 < 1 the service rate decreases and the residual service

time of the customer in service actually increases.) Specifically, an arrival that sees n customers in

the system and a remaining service time of η for the customer in service, causes the service rate

to immediately change such that the remaining service time becomes ηαn+1

(assuming no future

arrivals before the end of the current service). That is, if the residual service time when a new

customer arrives, η, has a density f(η), the remaining service time after this arrival will change to

1αn+1

η and therefore it will have a density of

1

αn+1

f(1

αn+1

η). (1)

We assume that both µn and αn are finite and greater than zero, which implies that the mean

of all service times are finite. We further assume a non-idling policy, i.e., the server starts working

as soon as there is a customer in the system and is only idle when the system is empty.

The Mn/Gn/1 system that we consider is very flexible and therefore can be used to model many

applications. We next discuss several such applications and their respective Mn/Gn/1 model. We

will derive solutions to several of these applications in Section 4.

1.1.1. Service Systems

Parkinsons Law: “Work expands to fill the time available for its completion” Parkinson (1955)

Since this paper there have been several empirical support for this law, three examples are given

next. Marin et al. (2007) empirically analyze the behavior of security screening at an airport and

show that service times depend on the queue length. They recommend to include such service time

dependency in queueing models. The stylized M/G/1 has been established as a good approximation

for the real world security-check queues in Zhang et al. (2011). Thus, the Mn/Gn/1 we consider

allows the inclusion of such service time dependency in security screening systems. Hasija et al.

(2010) observe a similar behavior in a contact center. In another study, Zhao et al. (2012) conduct

an experimental study in a supermarket modeled as a single server queue and observe that the

service rate depends on the number of customers in the queue. They notice that the service rate

is non-monotone and it converges. In these cases it appears that arrival rate should also converge.

Indeed, considering the changes of arrival and service rates (λn and αn) in the Mn/Gn/1 allows

to simultaneously examine customers and the servers’ behaviors as a function of the queue length.

This leads to our first Mn/Gn/1 model:

Model 1: the Mn/Gn/1 system when state-dependence is for a finite number of states. In this

model we assume that there exists a k <∞ such that for any i ≥ k, arrival rates and service

times are independent of the number of customers in the system, λi = λk, bi(·) = bk(·), αi = 1, andλiµi+1

< 1 for every i≥ k. This model is appropriate for analyzing cases where server’s rate depends

on the number of people in the system as in the security system and supermarket cases discussed

above. We analyze this model in Section 4.1.


Arrival Rates and Service Times Control in Service Systems In recent years dynamic

control mechanisms have been studied in the literature as an important tool for achieving com-

petitive advantage, cost reduction, and added value for service systems. For example, George and

Harrison (2001) consider an M/M/1 queueing system with adjustable service rate with the objec-

tive of minimizing the long-run average holding (congestion) and service level costs. Our results

allow the arrival and service rates in an M/G/1 queueing system to be dynamically controlled.

Therefore, we can analyze the M/G/1 queueing system with adjustable arrival and service rates

under different control policies. This leads to our second and third models as follow:

Model 2: Mn/Gn/1/K queues. In this model we assume that there exists a K <∞ such that for

any i≥K the arrival rate λi to the system is zero. Alternatively, we can allow λK > 0, but assume

that the manager rejects all customers as long as there are K people in the system. We further let

αK = 1 (and 0<αj <∞ for j = 0, ...,K − 1). We analyze this model in Section 4.2 and obtain the

closed form expression for the steady-state probability of having n customers in the system, PF (n),

and the steady-state service rate µn when there are n people in the system. These closed form

results enable the effective performance evaluation of any service and arrival rate control policy in

the presence of general service times.

1.1.2. Performance Evaluation

Model 3: Control of arrival and service rates in M/G/1 queues Ata and Shneorson (2006)

consider an M/M/1 service facility where the system manager makes service and arrival rates

decisions at both arrivals and service terminations epochs. They model the system as an M/M/1

queueing system. They show that it is optimal to operate the system as one with a finite buffer.

They mention that “Arguably, the M/G/1 queue would be a better model, but it is not amenable

to dynamic analysis.” As the third application, we apply the results obtained for the M/n/Gn/1

queueing systems to control the arrival and service rates in an M/G/1 queueing system. This model

combines optimization with performance evaluation based on Model 1.

1.1.3. Inventory Systems: Multi-Class of Customers with Priority and Brute-Force

Optimization

Multi-Echelon Inventory Management Multi-echelon inventory management with ample

supply has been considered since Clark and Scarf (1960) under different settings; see Zipkin (2000),

Axsater (2006), and references therein. But, few papers in the literature consider multi-echelon

inventory systems where the supplier faces congestion; see Abouee-Mehrizi et al. (2013) and ref-

erences therein. The main reason is that to obtain the distribution of the number of orders and

inventory level at each echelon, the distribution of the number of orders and inventory level at

the supplier is required, see e.g., Svoronos and Zipkin (1991). Using the Queue Decomposition


(QD) approach given in this paper, we can extend the results of Svoronos and Zipkin (1991) to

systems with general production and transportation times under different allocation policies as in

Abouee-Mehrizi et al. (2013). Moreover, we can examine multi-echelon systems where some arriv-

ing customers may be rejected based on the congestion in the system (see e.g., Ceryan et al. (2012)

for systems with the customer rejection). In addition, the QD approach can be used to analyze

multi-echelon inventory systems with several priorities.

Multi-Class Make-to-Stock Systems with Priority and Backlogs Multilevel Rationing

(MR) inventory policy has been studied in multi-class make-to-stock queueing systems with priority

since Ha (1997a). Based on this policy, the inventory control changes dynamically over time. More

specifically, under the MR policy in a system with n classes of customers, there are n+ 1 rationing

levels, R1, ...,Rn+1 where customers of classes 1, ..., r are served as long as the inventory level is

between Rr and Rr+1. During this period when inventory is between Rr and Rr+1 the customers of

other classes are backlogged. To obtain the total cost of a M/M/1 make-to-stock system under the

MR policy, Dynamic Programming (DP) has been used (see e.g., de Vericourt et al., 2002). But,

when production times are not exponentially distributed, it is not practical to use DP. Fortunately,

the QD approach given in this paper can be used to capture the dynamic changes of the control

policy at any given inventory level (see Abouee-Mehrizi et al., 2012). Moreover, the QD approach

can be used to extend these results to a system where the arrival rate decreases or the production

rate increases as the congestion in the system increases.

Multi-Class Make-to-Stock Systems with Lost Sales Multilevel Rationing (MR) inventory

policy has been also considered in multi-class make-to-stock queueing systems with lost sales. Ha

(1997) shows that for an M/M/1 make-to-stock system with multi-class of customers and lost

sales, the MR policy is optimal in which it minimizes the total cost. Then, he provides a static

method to obtain the total cost of the system with two classes of customers. Ha (2000) consider

an M/E/1 make-to-stock system with lost sales and show that the MR policy is optimal in this

setting. However, finding the optimal rationing levels when the number of customers type is large

(i.e., there are more than three) using DP is very hard due to curse of dimensionality. To the best

of our knowledge, the analysis of this problem for M/G/1 make-to-stock queues with lost sales has

not been addressed in the literature before. This is the third model we consider:

Model 4: MR policy in multi-class make-to-stock systems with lost sales. In Section 4.4 we

model the make-to-stock inventory system with several classes of customers where the customers

have different priority levels as a state-dependent Mn/Gn/1 queueing system. Here, λn is the sum

of demands of all customers types that are served when the shortfall is of n customers (there are

n orders in the system), and αj = 1 for j = 1, ...,K. This application demonstrates that the QD

results provided in this paper can be used to analyze systems with several classes of customers


where customers have different priorities. Moreover, the optimal controls of such systems can be

obtained using the QD results. Although this optimization is done using a “brute force” approach,

it is still effective since the solution of the Mn/Gn/1/K using QD is computationally efficient.

Moreover, the QD approach can be used to model systems where customers may decide not to

join the system when the number of people in the system is large, or the supplier may outsource

some customers to manage the congestion in the system. The former can be modeled by having λn

nonincreasing in n with a possible of λn = 0 for large n.

1.1.4. Summary of Models The models discussed above show that the QD approach can be

used to model and analyze different queueing systems with finite and infinite buffer sizes as well as

single and multi-class customers under the FCFS and priority policies. Moreover, they demonstrate

that the results can be used to find the optimal controls of a system such as rationing levels in the

multi-class inventory system for a given policy.

1.2. Contributions and Results

In this paper, we introduce the Queueing Decomposition (QD) approach for the analysis of the

Mn/Gn/1 system with infinite and finite buffer sizes. Since the arrival process depends on the

number of customers in the system, the Poisson Arrival Sees Time Average (PASTA) property

does not hold. Therefore, we consider the system in both continuous time and at arrival epochs.

To analyze the system in continuous time, we use the supplementary variable method introduced

by Cox (1955) to model the system as a continuous time Markov Chain (MC). Using this method,

we derive a closed form expression for the steady-state distribution of the number of customers in

the system (assuming, of course, that the system is stable). Then, we obtain the steady-state rate

at which the Mn/Gn/1 system moves from state n to state n− 1 (where n denotes the number of

people in the system). We show that these transition rates can be used to decompose the state-

dependent queueing systems as a B&D process. That is, we show that the Mn/Gn/1 system can

be decomposed to several new Mn/Gn/1 queues at every given state, and we characterize these

queues and their solutions.

We also analyze the system at arrival epochs. For this analysis, we define an embedded MC and

obtain the transition probabilities in these MCs. Then, we derive the steady-state distribution of

the number of customers in the systems observed by an arrival. We show that the probability of

having n customers in the system at an arbitrary time and the probability of observing n people in

the system by an arrival are closely related. Specifically, the ratio between these two distributions

is identical to the ratio of the arrival rate when there are no customers in the system to the arrival

rate when there are n customers in the system.


Similar to the continuous time analysis, we obtain the steady-state rate at which the Mn/Gn/1

system moves from state n to state n− 1 and show that these rates can be used to decompose the

state-dependent queueing systems at arrival epochs as a B&D process.

The results obtained using our QD approach allow to study very general single server queues with

a similar fashion to the study of the tractable M/M/1. Thus, our results are helpful to generalize

the insights from earlier studies to more practical cases. To demonstrate the application of QD

and the steady-state transition rates in the Mn/Gn/1 system, we consider in detail four models:

1) The Mn/Gn/1 system where the arrival and service processes depend on the number of people

in the system for a finite number of states. Using QD we provide exact closed form expression for

the distribution of the number of people in this system. 2) The state-dependent queueing system

with a finite buffer, Mn/Gn/1/K. We show that the distribution of the number of people in the

system is similar to the one in the Mn/Gn/1 system and derive the exact closed form expression

for number of people in this queue. 3) control of the arrival and service rates in the M/G/1 system

to maximize the total welfare of the system. And 4) a multi-class M/G/1 make-to-stock system

with lost sales where the inventory is managed under the MR policy. We model this problem as a

state-dependent queueing problem and present some numerical examples.

To summarize the main contribution of this paper is the introduction of the QD approach to

systematically study systems that can be adequately modeled as a single server queue. For this

we: (i) provide the exact analysis of the Mn/Gn/1 state-dependent queueing systems with general

service times distributions. We analyze this system at both arbitrary times and arrival epochs and

obtain closed form expressions for the distribution of the number of customers in the system. (ii)

derive the steady-state transition rates at which the state-dependent queueing systems move from

state n to state n− 1. We show that the Mn/Gn/1 system can be decomposed at any given state

similar to the Markovian systems. And (iii) demonstrate the use of the QD results to analyze

several single server models. We show that the results provided in this paper can be used to model

and analyze queueing systems with single and multiple classes of customers. The analyses can be

used to derive the objective functions and obtain the optimal controls of these systems.

1.3. Literature Review

Queueing systems with state-dependent arrival and service rates have been studied in the literature

since Harris (1967). He provides the probability distribution of the number of people in the system

for M/Mn/1 queueing systems where the rate of the service is µn = nµ. He also derives the proba-

bility distribution of the number of people in two-state M/M/1 systems where the service time of

a customer depends on whether there are any other customers in the system or not at the onset of

their service. Shanthikumar (1979) considers a two-state state-dependent M/G/1 queueing system


and obtains the Laplace Transform (LT) of the steady-state waiting time distribution in such a

system. Regterschot and Smit (1986) analyze an M/G/1 queueing system with Markov modulated

arrivals and service times. Gupta and Rao (1998) consider a queueing system with finite buffer

where the arrival rates and service times depend on the number of people in the system. They

assume that the service times can be adjusted only at the beginning of the service and obtain the

distribution of the number of people in the system in continuous time and at arrival epochs. Kerner

(2008) considers a state-dependent Mn/G/1 queueing system and provides a closed form expression

for the probability distribution of the of the number of people in the system as a function of the

probability that the server is idle. But in contrast to our results, he could not derive this prob-

ability and doesn’t allow state-dependent service times. The derivations in both Gupta and Rao

(1998) and Kerner (2008) are special restrictive cases of our results. Abouee-Mehrizi et al. (2012)

consider a multi-class M/G/1 make-to-stock system with backlogs where customers are prioritized

according to their backlog costs. They use the QD for the special case of the M/G/1 queue to

obtain a closed form expression for the total cost of the system under the MR inventory policy.

Abouee-Mehrizi et al. (2013) follow the same approach to analyze a two-echelon make-to-stock

inventory system with general production and transportation times.

Workload-dependent queueing systems in which the arrivals and service times depend on the

workload of the system rather than the number of people in the system have been also studied in

the literature (see Bekker and Boxma, 2007, and references therein).

The rest of the paper is organized as follows. In Section 2 we explain the problem. In Section

3 we model and analyze the Mn/Gn/1 system at both arbitrary times and arrival epochs. Then,

we demonstrate the results obtained for the three applications discussed above in Section 4. We

summarize the paper in Section 5. All proofs not in the body of the paper appear in the Appendix.

2. State-Dependent Queueing System

In this section we first explain some preliminaries of the state-dependent queueing system that we

consider in this paper. Then, we present the known results of a Markovian system to highlight the

parallelism between the results we obtain for the Mn/Gn/1 and those for the Mn/Mn/1.

2.1. Preliminaries

Assuming the system detailed in Section 1.1 is stable, let P (i) denote the steady-state probability

of having i customers in the Mn/Gn/1 system, and 1µi

denote the expected service time given there

are i customers in the system. Note that expressing µi in our setting is not trivial because of the

service rate changes allowed. We will characterize it in Section 3. In principle µi is not equal to

µi. Note that assuming the system is stable is equivalent to assuming that the utilization of the


system is less than 1 or that the probability that the system is idle, P (0), is greater than zero (e.g.,

Asmussen, 1991)

P (0) = 1−∞∑i=0

P (i)λiµi+1

> 0. (2)

Condition (2) is based on the steady-state probabilities P (i) that requires the stability condition

to be obtained. For the Mn/Gn/1 system, we present the necessary and sufficient condition for the

stability of this system. A sufficient condition for the stability of this system is to have λiµi+1

< 1 for

every i≥C where C is a finite positive number (see e.g., Wang, 1994). The reason is that λiµi+1

< 1

ensures that the transient probability of being in state C of the system is positive (since 1µi<∞

and the arrival process is Poisson the transit probability of being in any state i < C is positive

as well). This sufficient condition is relevant for systems where the arrival and service rates are

state-dependent only for a finite number of states as in Model 1 given in Section 4.1.

Note that the PASTA property does not hold in the Mn/Gn/1 system that we consider. The

reason is that the rate of the arrival process depends on the number of people in the system, so

that future arrivals are not independent of the past and present states of the system. Therefore,

the Lack of Anticipation Assumption (LAA) required for PASTA does not hold in such systems

(For more detail of LAA assumption and its essentiality to PASTA, see Bertsimas, 2007). However,

conditioning on the number of customers in the system, PASTA does hold. This property that is

called conditional PASTA (Doorn and Regterschot, 1988) will help us to analyze the systems at

arrival epochs.

2.2. Markovian Systems

To highlight the parallelism between the Mn/Gn/1 and Mn/Mn/1 queues, we present the well

known analysis of the standard Mn/Mn/1 queueing system in this section.

Suppose that customers arrive to the system according to a Poisson process with rate λj when

there are j customers. Service times are exponentially distributed with a rate µj whenever there

are j customers and let αj =µjµj−1

. This queue can be modeled as a standard B&D process and

analyzed using the basic relation

λiP (i) = µi+1P (i+ 1) i≥ 0, (3)

see e.g., Gross and Harris (2011). Leading to the solution:

Observation 1. Consider the Mn/Mn/1 system with arrival rate λj and service rate µj when

there are j people in the system. Suppose that

∞∑i=1

λ0

λi

i−1∏j=0

λj+1

µj+1

<∞. (4)

Abouee-Mehrizi and Baron: State-Dependent Queues10Article submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the mansucript number!)

Then, the steady-state distribution of the number of people in the system is

P (i) =λ0P (0)

λi

i−1∏j=0

λj+1

µj+1

, (5)

where from (5) and the fact that∑∞

i=0P (i) = 1, we obtain

P (0) =1

1 +∑∞

i=1λ0λi

i−1∏j=0

λj+1

µj+1

. (6)

Note that, due to memoryless, relation (3) can be interpreted as both (i) the rate of moving from

state i to state i+ 1 at any time t while the B&D process is in state i, and (ii) the average rate of

moving from state i to state i+1 while the B&D process is in state i. Note however that while the

first interpretation is motivating the solution of B&D process, the second interpretation is sufficient

to express the steady-state probabilities P (i). Indeed from Level Crossing Theory (LCT) (Perry

and Posner, 1990) for any stable system that its states change by jumps of −1, 0, and 1, we have

average arrival rate while in state i ∗ P (i) = average service rate while in state i+1 ∗ P (i+ 1) .

(7)

Equation (7) implies that the solution for any queueing systems where service and arrivals are done

a single job at a time can be decomposed as in any B&D processes. Specifically, defining λi as the

average arrival rate while in state i and µi as the average service rate while in state i+ 1, for the

Mn/Gn/1 system (7) transforms into (3).

In Section 3.1.2, we derive the steady-sate rate at which the Mn/Gn/1 system moves from state

n to state n−1 and demonstrate that the Mn/Gn/1 system can also be analyzed as a B&D process.

Then, we show that the stability condition and the distribution of the number of customers in the

Mn/Gn/1 system is similar to the one in the Mn/Mn/1 system given in Observation 1.

An important implication of the B&D representation for Markovian queueing systems is that the

system can be easily decomposed to new Markovian queues at any given state. The B&D structure

implies that we can decompose the queue such that during time intervals with κ or more people in

the system the queue is also an Mn/Mn/1. Specifically, conditioning on that there are κ or more

people in the original system, the decomposed queue has arrival rate λκ+i and service rate µκ+i

whenever there are i people in this decomposed queue. We call this Mn/Mn/1 queue with arrival

rate λκ+i and service rate µκ+i the Auxiliary queue. Let PA(i) denote the steady-state probability

of having i people in the auxiliary queue. Let F (i) := 1−∑i

j=0P (j), then,

Observation 2. QD in Markovian systems: The steady-state probability of having κ+ i

(i≥ 0) customers in this system and the steady-state probability of having i≥ 0 customers in the

auxiliary queue are related as:

P (κ+ i) = (1−F (κ− 1))PA(i), i= 0,1, ... (8)

Abouee-Mehrizi and Baron: State-Dependent QueuesArticle submitted to Manufacturing & Service Operations Management; manuscript no. (Please, provide the mansucript number!)11

Observation 2 can be used to analyze queueing systems where the control of the queue dynami-

cally changes. In the M/G/1 setting this has been demonstrated in Abouee-Mehrizi et al. (2012).

We obtain a similar result for the Mn/Gn/1 system in Section 3.1.3.

3. Analysis of Mn/Gn/1 Queues

In this section, we first analyze the state-dependent queueing systems with general service time

distribution at both arbitrary times and arrival epochs. Then, we demonstrate that the number

of customers in the Mn/Gn/1 system is identical in distribution to this number at a specific B&D

process and obtain the transition rates of this process. We also show that the system can be

decomposed to new queues at any given state.

3.1. Time Average Analysis

In this section, we analyze the Mn/Gn/1 system at an arbitrary time. We use the method of

supplementary variable introduced by Cox (1955), see also Chapter II.6 in Cohen (1982) and

Hokstad (1975). To model the system using this method, we consider a pair of variables nt and ηt

where nt and ηt denote the number of customers in the system and the remaining service time of

the customer in service, respectively. Note that ηt is called the supplementary variable; but as you

will see next, it has an important role in characterizing the distribution of the number of customers

in the system. Let pn(η, t) denote the probability-density of having n customers in the system when

the residual service time of the customer in service is η at time t so that:

p0(t) = P (nt = 0), (9)

pn(η, t)dη = P (nt = n)∩ (η < ηt ≤ η+ dη) n= 1,2,3, ... (10)

We have:

Theorem 1. The Chapman-Kolmogorov equations that describe the dynamic of the pair

{(nt, ηt), t∈ [0,∞)} in our Mn/Gn/1 system are given by:

p0(t+ dt) = p0(t)(1−λ0dt) + p1(0, t)dt+ o(dt). (11)

p1(η− dt, t+ dt) = p1(η, t)(1−λ1dt) + p2(0, t)b1(η)dt+ p0(t)λ0b1(η)dt+ o(dt). (12)

pj(η− dt, t+ dt) = pj(η, t)(1−λjdt) + pj+1(0, t)bj(η)dt+αjpj−1 (ηαj, t)λj−1dt+ o(dt).

(13)

To get the intuition behind the Chapman-Kolmogorov equations given in Theorem 1, consider (13).

Suppose that there are j > 1 customers in the system at time t+dt and the remaining service time

is η. Then, at time t one of the following has happened: 1) there were j customers in the system


and no new customer arrived during the next dt units of time with probability of 1− λjdt (first

term on the right side of (13)); 2) there were j+1 customers in the system, service was completed,

and the service time of the next customer was between η and η + dt with probability of bj(η)dt

(second term on the right side of (13)); 3) there were j − 1 customers in the system when the

remaining service time was ηαj and a new customer arrived during the next dt units of time with

probability of λj−1dt (third term on the right side of (13)). Similar logic leads to (11) and (12) and

the rigorous derivation is provided in the proof.

Theorem 1 generalized the continuous time MC for the Mn/G/1 from Kerner (2008) to the

Mn/Gn/1 system that we consider.

3.1.1. Distribution of Number of People in the System In this section, we investigate

the MC presented in Theorem 1 and express the steady-state distribution of the number of people

in the corresponding Mn/Gn/1 system. Let P (i) denote the steady-state probability of having i

customers in the Mn/Gn/1 system. In Theorem 2 we present a closed form expression for P (i) (i≥

0) for each i≥ 0 using the continuous MC obtained in Theorem 1. Assuming that a steady-state

for the system exists, we let hj (·) denote the LT of the steady-state residual service time given

that there are j ≥ 0 customers in the system. Thus, for example h0(·) = b1(·). The solution below

assumes that hj(·) are given and for convenience sets µ0 = µ1 and α1 = 1. Then,

Theorem 2. Suppose that

∞∑i=1

λ0

λi

i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

<∞. (14)

Then, the steady-state distribution of the number of people in an Mn/Gn/1 queue is

P (i) =λ0P (0)

λi

i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

. (15)

where from (15) and∑∞

i=0P (i) = 1, we have

P (0) =1

1 +∑∞

i=1λ0λi

i−1∏j=0

1−hj(λj+1αj+1

)bj+1(λj+1)

. (16)

Note that condition (14) guarantees that the steady-state probability of having no customers in

the system, P (0), is larger than zero. Therefore, it is the necessary and sufficient condition for the

stability of our Mn/Gn/1 system.

We next assume that the system is stable, and obtain hi (·) recursively.


Theorem 3. Suppose that the Mn/Gn/1 system is stable. Then, the LT of the steady-state

distribution of the residual service time given that there are i customers in the system can be

calculated recursively from:

hi(s) =λi

s−λi[bi(λi)

1− hi−1( sαi

)

1− hi−1( λiαi )− bi(s)] i≥ 1, (17)

where h0(·) = b1(·).

Note that to characterize hi(·) we assume that the system is stable. But, the stability condition

in (14) is a function of hi(·). To verify the stability condition, one should assume that hi(·) obtained

in Theorem 3 are well defined, calculate them recursively, and then check if (14) holds. If (14)

holds, then the system is stable and therefore hi(·) are well defined. Of course, (14) requires the

evaluation of an infinite series of LTs, which can only be recursively calculated using (17); thus, in

application, the value of (14) may be limited. Still, we show below that Theorems 2 and 3 can be

used in several relevant cases.

3.1.2. Modeling the Mn/Gn/1 Queues as a Birth-and-Death Process In this section,

we demonstrate that the state-dependent queueing systems with general service time distribution

can be analyzed as a B&D process.

The rate at which the number of customers in the system decreases in the Mn/Mn/1 system is

equal to the service rate, µi, because of the memoryless property. In the following observation, we

obtain the steady-state rate at which the number of customers in the system decreases by one in

the Mn/Gn/1 system. Let µi denote the steady-state rate at which the system moves from state i

to i− 1 when i is the number of customers in the system. Then,

Observation 3. The steady-state number of customers in the Mn/Gn/1 system has the same

distribution as the steady-state number of customers in a B&D Mn/Mn/1 process with arrival rate

λi and service rate

µi =λibi (λi)

1− hi−1(λiαi

) (18)

when there are i people in the system.

Using Observation 3 and (18) we can rewrite the stability condition (14) for the Mn/Gn/1 system

as∞∑i=1

λ0

λi

i−1∏j=0

λj+1

µj+1

<∞, (19)

the probability that there are i customers in the system as

P (i) =λ0P (0)

λi

i−1∏j=0

λj+1

µj+1

, (20)


and the probability that the Mn/Gn/1 system is idle as

P (0) =1

1 +∑∞

i=1λ0λi

i−1∏j=0

λj+1

µj+1

. (21)

Comparing (19-21) with (4-6) we observe that the stability condition and the distribution of the

number of people in the Mn/Gn/1 has the same structure as the one in the Mn/Mn/1. This

similarity indicates that the solution of the Mn/Gn/1 can be decomposed as in any B&D processes.

Although expressing µi is not trivial whenever service times are not Markovian, fortunately, we

could obtain it for the state-dependent Mn/Gn/1 system in Observation 3. Having µi and relation

(3), we can decompose the Mn/Gn/1 system as in the B&D process and analyze the system. We

will use these results to analyze several problems as a B&D process in Section 4.

3.1.3. Queue Decomposition at a Given State In Observation 2, we showed that the

Mn/Mn/1 queueing system can be decomposed at any given state. In this section we extend this

result to the Mn/Gn/1 system. More specifically, we show that the time intervals when there are

κ or more people in the Mn/Gn/1 can be decomposed as in the Mn/Mn/1.

Specifically, as in Abouee-Mehrizi et al. (2012a) we define an auxiliary queue such that the

steady-state probability of having κ+ i customers in the original system given that there are κ

or more people in this system is identical to the steady-sate distribution of having i jobs in this

auxiliary queue. To distinguish between the original queue and the auxiliary queue, we use the

term “job” in the auxiliary queue. We define the auxiliary queue as an Mn/Gn/1 queue with the

following (a) arrival and (b) service processes:

Step (a): jobs arrive to the auxiliary queue according to a Poisson process with rate λκ+i for

i≥ 0 when there are i people in the auxiliary queue.

Step (b): the distribution of the first service time in each busy period of the auxiliary queue

is the conditional residual service time in the original queue given that there are κ customers in

the system, i.e., the equilibrium remaining service times given that there are κ customers in the

system. The distribution of the rest of the service times in each busy period of the auxiliary queue

is identical to the original queue, i.e., bκ+i(·) for i≥ 0 when there are i people in the auxiliary queue.

Moreover, when there are i customers in the auxiliary queue and a new arrival occurs, the rate of

the service increases by a factor of ακ+i+1 > 0, so that the residual service time of the customer in

service decreases by this factor.

To summarize, the auxiliary queue is a state-dependent queue with general service time distribu-

tion, Mn/Gn/1. We next prove that the auxiliary queue is equivalent the original Mn/Gn/1 queue

during the time intervals when there are κ or more people in the queue.


Let PA(i) denote the steady-state probability of having i jobs in the Auxiliary queue. The

steady-state distribution of the number of jobs in the auxiliary queue is similar to the one in an

Mn/Gn/1 queue given in Theorem 2:

PA(i) = PA(0)i−1∏j=0

1− hκ+j(λκ)

bκ(λκ). (22)

Recall the in the original system we let F (i) := 1−∑i

j=0P (j) (for the original Mn/Gn/1 system).

In the following observation, we show that the probability of having i jobs in the auxiliary queue,

PA(i), is identical to the probability of having κ+ i customers in the original queue, P (κ+ i), given

that there are more than κ− 1 customers in the Mn/Gn/1 system.

Observation 4. QD in Mn/Gn/1 systems: The steady-state probability of having κ+ i (i≥

0) customers in the original Mn/Gn/1 system and the steady-state probability of having i ≥ 0

customers in the auxiliary queue are related as:

P (κ+ i) = (1−F (κ− 1))PA(i), i= 0,1, ... (23)

The intuition behind Observation 4 is that because the Mn/Gn/1 queue is similar to a B&D

process, this queue can be decomposed at any state just as any Markovian queue. Indeed because

in the Mn/Gn/1 settings the memoryless property does not hold, this decomposition requires a

careful definition of the first service time in the auxiliary queue; however, then all other service and

arrival times are defined as in the original queue. Observation 4 states that the steady-state number

of jobs in the auxiliary queue is identical to the steady-state number of customers in the original

state-dependent queue during the time intervals when there are more than κ − 1 customers in

the original Mn/Gn/1 system. This observation can be used to decompose the Mn/Gn/1 queueing

system at any given state. We use this observation and appropriate definitions of auxiliary queues

in Section 4 to analyze several applications.

3.2. Analysis at Arrival Epochs

In this section, we analyze the Mn/Gn/1 at arrival epochs. We remind that since in this system

the arrival process are state dependent, the PASTA property does not hold. Therefore, the steady-

state distribution of the number of customers seen by an arrival are typically different than the

steady-state distribution of the number of customers in the system at an arbitrary time.

Let P a(n) denote the steady-state probability that an Arrival observes n customers in the system.

To obtain P a(n), we consider that in the Mn/Gn/1 the distribution of the number of customers

seen by an arrival is identical to the steady-state distribution of the number of customers seen by

a departure (this easily follows by a level crossing argument as in Buzacott and Shanthikumar,

1993). Therefore, we analyze the system at departure epochs by defining an embedded MC.


Let Mn denote the number of customers left behind by the nth departing customer in the system.

Mn can be found by the MC embedded at the Departures. Let P d(n) denote the steady-state

distribution of being in state n of this MC. Then,

P a(n) = P d(n). (24)

To derive P d(n), we need to determine the one-step transition probabilities of Mn,

pjk = P (Mn+1 = k|Mn = j) . (25)

For state j ≥ 1, let vjk denote the probability of having k ≥ 0 arrivals during the service time

that starts when there are j customers in the system. For j = 0, the relevant service time starts

when there is one customer in the system. So we let v0k = v1k. Then, the probability that the next

departure leaves k customers behind given that there are no customers in the system, p0k, is v0k.

Similarly, the probability that the next departure leaves k customers behind given that there are

j ≥ 1 customers in the system, pjk, is vjk−j+1. To summarize, we denote the one-step transition

probabilities of Mn, the embedded MC, as

pjk =

{v0k, j = 0;vjk−j+1, j ≥ 1.

(26)

Note that in the standardM/G/1, these probabilities can be easily obtained since the distribution

of the service times are identical for all customers. Let λ and b(·) denote the arrival rate and the

LT of the service time distribution in the standard M/G/1, respectively. Then, considering that vjk

is independent of j, the probability generating function of vk := vjk (j = 0,1, ...) is (see e.g., Takagi

1991)

V (z) =∞∑k=0

vkzk = b(λ(1− z)). (27)

We next extend this results to the Mn/Gn/1 and derive the one-step transition probabilities of

the embedded MC. Note that the probability of the number of arrivals between two departures

cannot be obtained similar to the standard M/G/1 as in (27) since the distribution of the service

times are not identical, and any arrival may change the arrival and service rates. Therefore, instead

of considering the distribution of the number of arrivals during a service time, we consider the

probability of a new arrival during the residual service time observed by any customer upon arrival.

Fortunately, this distribution possesses the conditional PASTA property:

Corollary 1. The conditional steady-state distribution of the residual service time observed by

arrivals that find j customers in the system is identical to the conditional steady-state distribution

of the residual service time at an arbitrary time in the Mn/Gn/1 system that also observes j

customers in the system.


We next determine the transition probabilities, pjk, for j > 0. (The derivation for p0k is similar

but more detailed and it is provided in the proof of Theorem 4.) Consider pj,j−1. The probability

that the next departing customer leaves one less customer behind given that the last departing

customer left, P (Mn+1 = j− 1|Mn = j). This is equal to the probability of no arrival during the

next service time, bj(λj) (e.g., Conway 1967, page 171). Therefore,

pj,j−1 = bj(λj), j ≥ 1. (28)

With similar logic, the probability that the next departing customer leaves k customers behind

given that the last departing customer left j ≥ 1 customers behind, P (Mn+1 = k|Mn = j), is equal

to the probability of k − j + 1 arrivals during the next service time. This probability is equal to

the probability that one customer arrives after the next service time starts,(

1− bj(λj))

, and a

customer arrives during the remaining service time of all arrivals that see i < k customers in the

system,(

1− hi( λi+1

αi+1))

, and no arrival during the remaining service time once there are k customers

in the system, hk(λk+1

αk+1). Therefore,

pjk = hk

(λk+1

αk+1

)(1− bj(λj)

) k−1∏i=j

(1− hi

(λi+1

αi+1

)), j ≥ 1, k≥ j. (29)

Note that for the M/M/1 queue we have hj(λj) = bi(λi) = b(λ) = λµ+λ

so that

pjk =µ

µ+λ

(λ

µ+λ

)k−j+1

, j ≥ 1, k≥ j. (30)

As expected due to memoryless, PASTA, and that the minimum of two independent exponential

variables is an exponential variable, in the M/M/1 setting pjk has a binomial distribution with

parameter µµ+λ

.

Using the transition probabilities in (29) (and these for p0j), the steady-state distribution of the

number of customers in the system observed by an arrival is as follows.

Theorem 4. Suppose that∞∑i=1

i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

<∞. (31)

Then, the steady-state distribution of the number of people in an Mn/Gn/1 queue observed by an

arrival is

P a(i) = P a(0)i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

, (32)

where from (32) and∑∞

i=0 Pa(i) = 1, we have

P a(0) =1

1 +∑∞

i=1

i−1∏j=0

1−hj(λj+1αj+1

)bj+1(λj+1)

. (33)


Comparing (15) and (32), we can relate the steady-state probability of having i customers in the

system at an arrival epoch to the steady-state probability of having i customers in the system at

an arbitrary time.

Observation 5. The steady-state probability that an arrival observes i customers in the system

is equal to the fraction of intervals when there are i customers in the system given an arrival

occurs. More specifically, in an Mn/Gn/1 system, the relation between the steady-state probability

of having i customers in the system at arrival (or departure) epochs and at arbitrary times is given

by:

P a(i) =λiP (i)∑∞k=0 λkP (k)

. (34)

Note that when arrival rates are identical, λi = λ for i = 0,1, ..., the PASTA property holds and

therefore P a(i) = P (i).

Observation 5 indicates that P a(i) can be obtained using P (i) given in Theorem 2. More inter-

estingly, Observation 5 together with Theorem 4 provide an alternative proof for Theorem 2 using

that:

Lemma 1. In an Mn/Gn/1 system, the relation between the steady-state probability of having i

customers in the system at arbitrary times and at arrival (or departure) times is given by:

P (i) = P (0)λ0P

a(i)

λiP a(0)i≥ 0. (35)

Similar to the analysis in Section 3.1.2, we next obtain the rate at which the number of customers

in the Mn/Gn/1 system decreases by one at arrival epochs. Let µai denote the rate that the number

of people in the system decreases from i to i− 1. Then,

Observation 6. The rate at which the Mn/Gn/1 system moves in steady-state from state i to

state i− 1 considering the system at arrival epochs is

µai =λi−1λi

µi =λi−1bi (λi)

1− hi−1(λiαi

) . (36)

Observation 6 relates the utilization of the system in steady-state with the utilization of the

system as it would be measured by arrivals. Specifically, Observation 6 indicates that the steady-

state utilization of the system observed by arrivals who find j customers in the system,λj−1

µaj, is

identical to the steady-state utilization of the system when there are j customers in the system,λjµj

.

4. Applications of QD

In this Section, we analyze the four models discussed in Section 1.1 using the results from Section 3:

(1) the Mn/Gn/1 system when state-dependence is for a finite number of states, (2) Mn/Gn/1/K

queues, (3) control of arrival and service rates in the M/G/1 queues, and (4) MR policy in multi-

class make-to-stock systems with lost sales.


4.1. Application 1: Analysis of the Mn/Gn/1 System when State-Dependence is fora Finite Number of States

In Section 3.1, we obtained the steady-state probability of having i > 0 customers in the system

as a function of P (0), and provided an expression for P (0) that includes an infinite sum. In

this section we obtain a closed form expression for the steady-state probability that there are

no customers in the system, P (0), for the special case of Application 1. We assume that there

exists a k <∞ such that for any i≥ k, λi and bi are independent of the number of people in the

system, i.e., k = mini{λi, bi(·), αi : λi = λi+1 = ..., bi(·) = bi+1(·) = ..., αi = αi+1 = ...}. This queue is

state-dependent for i < k and has the same arrival rate and service time distribution for i≥ k. We

assume that λk/µk < 1 to ensure that the system is stable. As we will soon show, in this case the

infinite summation in (16) becomes finite; thus the probability that the system is idle can also be

simplified (see (39) below).

To obtain P (0), we use QD. Consider the auxiliary queue defined in Section 3.1.3 for κ = k.

Let ρb and µb denote the server utilization and the rate of the first exceptional service times in

this auxiliary queue, respectively. Note that the service time distributions, bi(·), in this system

are identical for i > k. Therefore, the rate of the service time in the auxiliary queue is µk with

probability ρb and µb with probability (1− ρb), so that

ρb = ρbλkµk

+ (1− ρb)λkµb

leading to

ρb =λkµk

µbµk +λk(µk−µb). (37)

To obtain the utilization of the auxiliary queue, ρb, we derive the average rate of the exceptional

first service times, µb.

Lemma 2. The average exceptional first service times in the busy period of the auxiliary queue

is,

1

µb=

1

µk− 1

λk+

k−1∑j=1

k−1∏i=j

1

αi+1

(1

µj− 1

λj

)bi+1(λi+1)

1− hi( λi+1

αi+1)− 1

µ1

b1(λ1)

1− h0(λ1α1

)

k−1∏i=1

1

αi

bi+1(λi+1)

1− hi( λi+1

αi+1). (38)

Note that Sigman and Yechiali (2007) provide the expected conditional stationary remaining service

time in an M/G/1 queue. Lemma 2 extends their result to the Mn/Gn/1 queue.

Using (15), (22), and (23) the probability of having no customers in the Mn/Gn/1 system of

Application 1 is:


Theorem 5. The steady-state probability of having no customers in the Mn/Gn/1 system of

Application 1 is

P (0) =1− ρb

λ0λk

k−1∏i=0

1−hi(λi+1αi+1

)bi+1(λi+1)

+ (1− ρb)

(1 +

∑k−1j=1

λ0λj

j−1∏i=0

1−hi(λi+1αi+1

)bi+1(λi+1)

) . (39)

Using Theorem 3 and substituting (39) from Theorem 5 into Theorems 2 provides a closed form

expression for all steady-state probabilities for systems where the arrival rates and service times

are state-dependent for a finite number of states as in Application 1.

4.2. Application 2: Analysis of Mn/Gn/1/K

In this section, we consider a state-dependent queue with a finite buffer K, Mn/Gn/1/K. This

queue is identical to an Mn/Gn/1 queueing system with λi = 0 for i ≥K (and, of course, it has

no issue of stability). It can be also considered as an Mn/Gn/1 queueing system with λi > 0 for

i ≥ 0 where the customers that find K people in the system are rejected. We prefer the view

above with rejection of customers arriving to a full queue with arrival rate λK > 0 to better relate

the Mn/Gn/1/K to the multi-priority make-to-stock system with lost demand system studied in

the next subsection. We analyze the system at both arbitrary times and arrival epochs using the

transition rates and QD results obtained for the Mn/Gn/1 system.

4.2.1. Time Average Analysis Here, we first use the transition probabilities obtained in

Observation 3 to derive the probability of having 0 < i < K customers in the system. Then, we

apply the QD result given in Observation 4 to obtain the normalization factors.

Let PF (i) denote the steady-state probability of having i (0≤ i≤K) customers in this system

with Finite buffer. If the buffer size of the system is 1, K = 1, there is no state dependency and

α1 = 1 by definition, thus the steady-state distribution of the number of people in the system can

be obtained from PF (0) +PF (1) = 1 and λ0PF (0) = µ1PF (1) as (see e.g., Gross and Harris, 2011)

PF (0) =µ1

λ0 +µ1

, (40)

PF (1) =λ0

λ0 +µ1

. (41)

Now consider a system with a buffer size larger than 1, K > 1. We make the following important

observation.

Observation 7. Comparing the transitions in the Mn/Gn/1/K with the ones in the Mn/Gn/1,

we observe that the transitions up to state K−1 in both systems are identical. The only difference

between these two systems is that all transitions that take the Mn/Gn/1 to a state greater than

K − 1 are lost in the Mn/Gn/1/K. This means that the transition rates in the states less than

K − 1 are identical in both systems.


Observation 7 emphasizes the equivalence between the transition rates in the Mn/Gn/1/K and

the one in the B&D process, discussed in Observation 3, for states i= 0, ...,K−1. This observation

enables us to solve the Mn/Gn/1/K using the derivations for the Mn/Gn/1 system in a similar

fashion that Mn/Mn/1/K can be solved using the Mn/Mn/1 ( similar to Gross and Harris, 2011,

page 76):

PF (i) =λ0PF (0)

λi

i−1∏j=0

λj+1

µj+1

, i= 1, ...,K. (42)

We next use Observation 7 to easily obtain the steady-state probability of having i <K people in

the Mn/Gn/1/K. We note that Gupta and Rao (1998) derive a direct solution of a special case of

our finite buffer queue using the supplementary variable method that we used to solve the infinite

buffer case. This method of course leads to the same solution for their special case, but is less

elegant given the analysis in Section 3 and Observation 7.

Corollary 2. The steady-state distribution of the number people in an Mn/Gn/1/K queue,

PF (i), is

PF (i) =λ0PF (0)

λi

i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

, i= 1, ...,K − 1 (43)

where µ0 = µ1 and hi (·) is given in (17).

Proof of Corollary 2.

Substituting (18) in (3) we get

λiPF (i) =λi+1bi+1 (λi+1)

1− hi(λi+1

αi+1

) PF (i+ 1) .

Therefore,

PF (i+ 1) =λiλi+1

1− hi(λi+1

αi+1

)bi+1 (λi+1)

PF (i) ,

which after some algebra leads to (43). �

We next obtain the steady-state probability of having exactly 0 and K customers in the system

using an auxiliary M/G/1/1 queue with the following (a) arrival and (b) service processes:

Step (a): jobs arrive to the auxiliary queue according to a Poisson process with a rate of λK−1.

Step (b): the distribution of the service time in the auxiliary queue is the conditional residual

service time in the original Mn/Gn/1/K queue given that there are K−1 customers in the system,

i.e., the equilibrium remaining service times given that there are K − 1 customers in the system.

Let 1/µFb and PAF (i) (i= 0,1) denote the mean service times and distribution of number of people

in the auxiliary queue, respectively. Then, using (40) and (41) we have,

PAF (0) =

µFbλK−1 +µFb

, (44)

PAF (1) =

λK−1λK−1 +µFb

. (45)


Therefore, to obtain PAF (i), we need to derive the mean of the service times, 1/µFb . Noting

that Observation 7 provides the equivalence between the Mn/Gn/1 and Mn/Gn/1/K for states

i= 0, ...,K − 1, 1/µFb can be obtained by substituting k=K − 2 in (38) as:

Corollary 3. The average of the exceptional first service times in the auxiliary queue is,

1

µFb=

1

µK−1− 1

λK−1+K−2∑j=1

K−2∏i=j

1

αi+1

(1

µj− 1

λj

)bi+1(λi+1)

1− hi( λi+1

αi+1)

− 1

µ1

b1(λ1)

1− h0(λ1α1

)

K−2∏i=1

1

αi

bi+1(λi+1)

1− hi( λi+1

αi+1). (46)

Let FF (i) :=∑i

j=0PF (j). Considering the equivalence between the Mn/Gn/1 and Mn/Gn/1/K

due to Observation 7 and following Observation 4, the probability of having i = 0,1 jobs in the

auxiliary queue, PAF (i), is identical to the probability of having K+ i− 1 customers in the original

queue, PF (K + i− 1), given that there are more than K − 2 customers in the system.

Corollary 4. The steady-state probability of having i=K−1, K customers in an Mn/Gn/1/K

is

PF (K − 1) = (1−FF (K − 2))PAF (0), (47)

PF (K) = (1−FF (K − 2))PAF (1). (48)

Considering (43), (44) and (47), the probability that the system is empty, PF (0), is

PF (0) =PAF (0)

λ0λK−1

K−2∏j=0

1−hj(λj+1αj+1

)bj+1(λj+1)

+PAF (0)

(1 +

∑K−2i=1

λ0λi

i−1∏j=0

1−hj(λj+1αj+1

)bj+1(λj+1)

) . (49)

Substituting (49) in (43), we can obtain PF (i) for i = 1, ...,K − 1. Therefore, PF (K) can be

obtained using (48).

4.2.2. Analysis at Arrival Epochs In this section, we obtain the steady-state distribution

of the number of customers seen by an arrival in an Mn/Gn/1/K queue using the results obtained

in Section 3.2.

Let P aF (n) denote the steady-state probability that an Arrival observes n customers in the

system. Unlike the Mn/Gn/1 system, in the Mn/Gn/1/K system the distribution of the number

of customers seen by an arrival is not identical to the steady-state distribution of the number

of customers seen by a departure. The reason is that not all arriving customers are accepted to

the system. Let P dF (n) denote the steady-state probability that a departure observes n customers

behind. We first determine the relation between P aF (n) and P d

F (n).


Lemma 3. The relation between the distribution of the number of customers in the system

observed by an arrival and the one seen behind a departure is

P aF (i) = P d

F (i)(1− P aF (K)) i= 0,1, ...,K − 1. (50)

Note that if the arrival rate to the system when there are K people in the system is zero, λK = 0,

the probability that an arrival observes K customers in the system is also zero, P aF (K) = 0. In this

case P aF (i) = P d

F (i) for i= 0, ...,K − 1.

Now we focus on the MC chain embedded at the departures and analyze the system. As we dis-

cussed in Section 4.2, the steady-state distribution of the number of people in the system observed

at arrival epochs can be obtained using the transition rates given in Observation 6 as in the B&D

process.

Corollary 5. The steady-state distribution of the number people in an Mn/Gn/1/K queue

observed by an arrival is

P dF (i) = P d

F (0)i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

i= 0,1, ...,K − 1 (51)

where from (51) and∑K−1

i=0 Pa(i) = 1, we have

P dF (0) =

1

1 +∑K−1

i=1

i−1∏j=0

1−hj(λj+1αj+1

)bj+1(λj+1)

. (52)

Considering Lemma 3 and Corollary 5, we can obtain the steady-state probability that an arrival

observes n (n < K) customers in the system, P aF (n). To obtain P a

F (K), we can use the auxiliary

queue defined in Section 4.2.1. Then, considering (48), we get

P aF (K) =

(1− FF (K − 2)

)PAF (1), (53)

where FF (i) := 1−∑i

j=0 PaF (j).

4.3. Application 3: Control of Arrival and Service Rates in M/G/1 Queues

Ata and Shneorson (2008) explain that dynamic control of the service and arrival rates in an

M/G/1 system to maximize revenue or welfare is not amenable. Therefore, the stylized Markovian

M/M/1 queueing system has been considered as an approximation. In this section, we apply the

results obtained in the previous sections to dynamically control the arrival and service rates in

an M/G/1 queueing system. We use the optimal arrival and service rates obtained in Ata and

Shneorson (2008) for the M/M/1 system as the arrival and service rates in the M/G/1 system.


Let b(λn) and c(µn) denote the value rate and cost rate when the arrival and service rates are

λn and µn, respectively. Moreover, let hn denote the holding cost when there are n customers in

the system. Similar to Ata and Shneorson (2008), we define the long-run average social welfare

generated per unit of time as

V =∞∑n=0

P (n) (b(λn)− c(µn)−hn) . (54)

We consider two heuristics to apply the optimal arrival and service rates of the M/M/1 system

as an approximation for the M/G/1 system. Note that in the M/M/1 system, the service rate

when there are i customers in the system, µi, is identical to the rate at which the system moves

from state i to i−1, µi. However, in the M/G/1 system these rates are different. Therefore, as the

first heuristic we set the service rate in the M/G/1 system equal to the optimal service rate in the

M/M/1 system. For the second heuristic, we set the transition rate µi in the M/G/1 system equal

to the optimal service rate µi in the M/M/1 system.

To obtain the required probabilities for this heuristic, we allow the service rate of the system

to be adjusted when a new service starts and at the arrivals. When a new service starts with n

customers in the system, we set the service rate equal to the optimal µn obtain in the M/M/1

system. Moreover, when a new customer arrives and there are n customers in the system with

change the rate of the service by αn+1 = µn+1/µn. To obtain the required probabilities for the

second heuristic, we set µi in (20) equal to optimal µi of the M/M/1 system.

We next numerically compare the total welfare given in (54) for an M/D/1 system using the two

heuristics. In the numerical examples we assume that b(λn) =Bλn−Cλ2n and hn = vn similar to Ata

and Shneorson (2008). We vary the parameters as B = {5,8}, C = {0.5,1}, and v = {1/2,1/2.5}.

We consider a maximum service rate 5 and the cost rate of the service either linear function,

c(µn) = µ or a nonlinear function, c(µn) = 1/2µ2.

Tables 1 and 2 show the total welfare of the system for different cases. Columns VM , VR, and

VT denote the total welfare of the system for the Markovian system, the M/D/1 system with the

service rate identical to the optimal service rate in the M/M/1 system, and the M/D/1 system

with the transition rate identical to the optimal service rate in the M/M/1 system, respectively.

As the numerical results show, when the cost of the service is linear (Table 1), the total welfare

when we set the transition rate in the M/D/1 system identical to the optimal service rate in the

M/M/1 system, VT , is greater than the total welfare when we set the service rate in the M/D/1

system identical to the optimal service rate in the M/M/1 system, VR, in the majority of the cases.

However, when the cost of the service is nonlinear (Table 2), VT is lower than VR.


Table 1 Total welfare when the cost rate of the service is linear, c(µn) = µ(µ≤ 5)

B C 1/v VM VR VT5 0.5 2 5.24 5.58 6.575 0.5 2.5 5.48 5.80 6.755 1 2 0.57 0.63 3.075 1 2.5 0.65 0.70 3.138 0.5 2 18.49 19.56 18.908 0.5 2.5 18.97 19.94 19.338 1 2 9.51 9.80 10.978 1 2.5 9.72 9.97 11.14

Table 2 Total welfare when the cost rate of the service is linear, c(µn) = 1/2µ2(µ≤ 5)

B C 1/v VM VR VT5 0.5 2 4.35 4.74 4.415 0.5 2.5 4.56 4.94 4.635 1 2 2.70 2.94 2.755 1 2.5 2.86 3.10 2.918 0.5 2 13.19 13.92 13.388 0.5 2.5 13.53 14.19 13.728 1 2 8.53 9.02 8.678 1 2.5 8.78 9.23 9.01

4.4. Application 4: MR Policy in Multi-Class Make-to-Stock Systems with LostSales

The queueing systems we consider in this paper enable the analysis of systems with multiple

customer classes and where the control of the queue changes dynamically. We demonstrate this by

studying the MR priority policy in make-to-stock systems with multiple classes of customers and

lost sales. This application uses the QD results to also optimize the controls of the make-to-stock

systems. The “brute force” search for the optimal controls we use is effective since the solution

based on QD is computationally efficient.

Consider a supplier that produces a single product to satisfy demand arising from n different

classes of customers. Demand from class r of customers follows Poisson process with rate βr and

each customer requests a single item. If demand is not satisfied immediately, it is lost. The cost of

losing a customer of class r is cr. Without loss of generality, we assume that c1 > c2 > ... (if two

distinct classes have the same lost sales cost, they can be aggregated to a single class). Customers

are prioritized based on their lost sales cost, class 1 to n from the highest to lowest priority.

The production times at the supplier are i.i.d and generally distributed with LT b(·). The supplier

keeps some inventory to satisfy the demand arising from different classes of customers. The system

incurs a holding cost of h per unit of time per unit of inventory. The stock allocation at the

supplier is based on the MR policy as follows: there are n+ 1 non-decreasing rationing levels Rr,

r = 1, ..., n+ 1 with R1 = 0; and Rn+1 = S, the base-stock level at the supplier. If the inventory


level, I, is between Rr+1 and Rr + 1 i.e., Rr < I ≤ Rr+1, only orders of class 1 to class r are

satisfied from the available stock on a FCFS basis, and orders of the other classes are lost. When

the inventory level is below R2 only orders of class 1 are satisfied. Note that in the special case

when R0 =R1 = ...=Rn = 0, all customers are satisfied on a FCFS basis.

Let PI(i) denote the probability that the inventory level at the supplier is i. Then, with the

rationing levels (R2, ...,Rn+1) the expected total cost of the system is

CMR(R2, ...,Rn+1) = h

Rn+1∑i=0

iPI(i) +n∑r=1

crβr

Rr∑j=0

PI(j). (55)

Therefore, to obtain the expected total cost of the system, we only need to obtain the distribution

of the inventory level at the supplier, I. A standard method to express I in the literature is to

consider the shortfall process (e.g., Baron, 2008, and references therein),

N :=Rn+1− I, I ≥Rn+1. (56)

We next show that the shortfall process N in the multi-class make-to-stock system with lost

sales described above is identical to the number of orders in a specific Mn/Gn/1/K and apply the

results of Section 4.2.1 to obtain the required probabilities.

For r = 1, ..., n when there are Rn+1 −Rr+1 ≤ i < Rn+1 −Rr orders in the system, the arrival

rate to the system is λi =∑r

j=1 βj. For example, if there are two classes of customers, n= 2, and

R2 = 4, R3 = 7, then the arrival rate to the system is β1 +β2 as long as the number of orders in the

system is less than 3, and it is β1 if the number of orders in the system is between 3 and 7. The

distribution of the production is independent of the number of orders in the system, bi(·) = b(·).

Considering (56) and using the above definition, we can rewrite the cost function (55) as

CMR(R2, ...,Rn+1) = h

Rn+1∑i=0

iPF (Rn+1− i) +n∑r=1

crβr

Rr∑j=0

PF (Rn+1− j), (57)

where PF (·) are given in Corollaries 2 and 4 with K = S and αi = 1.

Thus, with the exact cost function CMR(R2, ...,Rn+1) given in (57), we can search over different

vectors of rationing levels (R2, ...,Rn+1) to find the optimal one and the corresponding cost. In

our numerical examples, we search over Rn+1 = S by varying Rn+1 = 1, . . . ,M . For each Rn+1 we

look for the optimal rationing levels 0 =R1 ≤ · · · ≤Rn ≤Rn+1. We let M = SFCFS∗, where SFCFS

∗

denote the optimal base-stock levels under the FCFS policy. We note that, in all our numerical

results, we found that the optimal Rn+1 ≤ SFCFS∗.

We next apply these results to an example with a supplier and two classes of customers. Suppose

that the production times at the supplier is deterministic with a mean of 1. We compare the


expected total cost of the system under the MR policy with the expected total cost of the system

under the FCFS policy for a set of examples in Table 3 where ∆FCFS is defined as

∆FCFS =CFCFS −CMR(R2, ...,Rn+1)

CMR(R2, ...,Rn+1)× 100. (58)

Table 3 ∆FCFS for a system with two classes of customers, c1 = 5, and c2 = 1

.β1 +β2 β1/β2 ∆FCFS

h= 0.05 h= 0.11/2 0.00 0.00

0.7 1 4.70 3.152 6.53 3.03

1/2 11.09 6.540.9 1 14.33 11.72

2 13.79 10.72

The numerical results provided in Table 3 show that as the total arrival rate to the system

increases, the performance of the MR policy in comparison with this of the FCFS policy improves.

The reason is that the probability that a customer is lost increases as the total arrival rate to

the system increases. Therefore, keeping some stock for the high-priority customers reduces the

expected lost cost of the system. Moreover, as the holding cost of the system increases, the relative

gap between the optimal cost of the FCFS and MR policies decreases since the cost of keeping

stock for high-priority customers increases.

5. Summary

In this paper we considered state-dependent queueing systems where the arrival rates and service

times depend on the number of customers in the system. We allowed the service rate to change at

arrivals and the distribution of the service times to change when a new service starts. We analyzed

such systems at both arbitrary times and arrival epochs using QD and obtained the steady-state

distribution of the number of customers in the system.

We showed that the Mn/Gn/1 systems can be decomposed as in the standard B&D process and

derived the rate at which the number of customers in these systems decreases. We also demonstrated

that the state-dependent queueing system with general service time distribution can be decomposed

to new queues at any given state. Finally, we used these results to provide the exact analysis

of several models including the Mn/Gn/1/K queueing system and the single product multi-class

M/G/1 make-to-stock systems with lost sales.

We demonstrated that the QD is a methodology that can help analyzing queueing systems

with generally distributed service times. We believe that the methodology and the results of this

paper can be used to derive exact solutions for many single and multi-class queueing models. As


another example, consider an Mn/Gn/1 queueing system with Last-Come First-Serve policy and

preemptive repeat service times. Then, the distribution of the number of customers in this system

can be obtained using (15) by replacing hi(·) with bi(·).

In addition, we believe the QD approach can be used to extend the analysis and insights generated

by it for many problems that use the M/M/1 model to the more realistic M/G/1 (possibly with

state-dependent) settings. The analysis of state-dependent queues with finite buffer size in Section

4.1, Abouee-Mehrizi et al. (2012) of make-to-stock M/G/1 queues and Abouee-Mehrizi et al. (2013)

of multi-echelon inventory systems are three such examples.

As future research we would like to investigate additional models where the idea of the QD can

be used if the B&D structure we derive can be useful in deriving optimal control policies for the

M/G/1 settings, e.g., in the make-to-stock example. Another interesting subject is to see if QD

can be used in the Gn/Mn/1 settings as well.

References

Abouee-Mehrizi, H., B. Balcioglu, O. Baron 2012. “Strategies for a Single Product Multi-Class

M/G/1 Queueing System”, Operations Research, Vol. 60, 803–812.

Abouee-Mehrizi, H., O. Baron, O. Berman 2013. “Customer Differentiation in Capacitated Multi-

Echelon Inventory Systems”, Working paper.

Altman, E., K. Avrachenkov, C. Barakat, R. Nunez-Queija 2002. “State-Dependent M/G/1 Type

Queueing Analysis for Congestion Control in Data Networks”, Computer Network Vol. 39, 789–808.

Asmussen, S. 1991. “Ladder Heights and the Markov-Modulated M/G/1 Queue”, Stochastic Pro-

cesses and their Applications Vol. 37, 313–326.

Ata, B., S. Shneorson. 2006. “Dynamic Control of an M/M/1 Service System with Adjustable

Arrival and Service Rates”, Management Science , Vol. 52 1778–1791.

Axsater, S. 2006. Inventory Control, second edition, Springer, New York.

Baron, O. 2008. “Regulated random walks and the LCFS backlog probability: Analysis and appli-

cations”, Operations Research, Vol. 56, 471486.

Bekker, R., O. J. Boxma 2007. “An M/G/1 queue with adaptable service speed”, Stochastic Models

Vol. 23, 373–396.

Bertsimas, D. 2007. Introduction to Queueing Systems, monograph in preparation.

Ceryan, O., I. Duenyas, Y. Koren. 2012. “Optimal control of an assembly system with demand for

the end-product and intermediate components”, IIE Transactions , Vol. 44 386–403.

Clark, A. J., H. E. Scarf. 1960. “Optimal policies for a multi-echelon inventory problem”, Manage-

ment Science , Vol. 6 475–490.

Cohen, J. W. 1982. The Single Server Queue, North-Holland, Publishing Company.


Cox, D. R., 1955. “A Use of Complex Probabilities in the Theory of Stochastic Processes”, Pro-

ceedings of the Cambridge Philosophical Society, Vol. 51, 313–319.

Doorn, E. A., G. J. K. Regterschot. 1988. “Conditional PASTA”, OR Letters, Vol. 7, 229–232.

George, J. M., J. M. Harrison. 2001. “Dynamic Control of a Queue with Adjustable Service Rate”,

Operations Research, Vol. 49, 720–731.

Gross, D., C. M.,Harris. 2011. Fundamentals of Queueing Theory, John Wiley & Sons, New York.

Gupta, U. C., T.S.S. S. Rao 1998. “On the Analysis of Single Server Finite Queue with State

Dependent Arrival and Service Processes: Mn/Gn/1/K”, OR Spektrum Vol. 20, 83–89.

Ha, A. 1997. “Inventory Rationing Policy in a Make-to-Stock Production System with Several

Demand Classes and Lost Sales”, Management Science Vol. 43, 1093–1103.

Ha, A. 2000. “Stock Rationing in an M/Ek/1 Make-to-Stock Queue”, Management Science, Vol. 46,

77–87.

Harris, C.M. 1967. “Queues with State-Dependent Stochastic Service Rates,” Operations Research,

Vol. 15, 117–130.

Hasija, S., E. Pinker, R. A. Shumsky. 2010. “Work Expands to Fill the Time Available: Capacity

Estimation and Staffing Under Parkinson’s Law”, Manufacturing & Service Operations Manage-

ment, Vol. 12, 1–18.

Hokstad, P. 1975. “A Supplementary Variable Technique Applied to the M/G/1 Queue,” Scand J

Statist, Vol. 2, 95–98.

Kerner, Y. 2008. “The Conditional Distribution of the Residual Service Time in the Mn/G/1

Queue,” Stochastic Models, Vol. 24, 364–375.

Kucukyazici, B., L. Green, V. Verter, R. Riopelle. 2012. “DESIGN AND OPERATION OF

STROKE UNITS,” working paper.

Martin, C. V., C. G. Drury, R. Batta, L. Lin 2007. “Human Factors Contributes to Queueing The-

ory: Parkinson’s Law and Security Screening,” Proceeding of the Human Factors and Ergonomics

Society, 51st Annual Meeting, 602–606.

Parkinson, D. F. 1955. “Parkinson’s law,” Economist Vol. 19 635–637.

Perry, D., M. J. M. Posner 1990. “Control of Input and Demand Rates in Inventory Systems of

Perishable Commodities”, Operations Research Vol. 37, 85–97.

Sigman, K., U. Yechiali. 2007. “Stationary remaining service time conditional on queue length”,

OR Letters, Vol. 35, 581–583.

Svoronos, A., P. Zipkin. 1991. “Evaluation of one-for-one replenishment policies for multiechelon

inventory systems”, Management Science, Vol. 37, 68–83.

Shanthikumar, J. G. 1979. “On a single-server queue with state-dependent service,” Naval Research

Logistics, Vol. 26, 305–309.


Regterschot, G. J. K., J. H. A. de Smit 1986. “The Queue M/G/1 with Markov Modulated Arrivals

and Services,” Mathematics of Operations Research, Vol. 11, 465–483.

Takagi, H. 1993. Queueing Analysis, Volume 2, Elsevier: North Holland, The Netherlands.

Tijms, H.C., Van Hoorn, M.H., Federgruen, A. 1981. “Approximations for the steady-state prob-

abilities in the M/G/c queue,” Advanced Applied Probability, Vol. 13, 186–206.

Upfold, J. 2002. “Emergency department overcrowding: ambulance diversion and the legal duty to

care”, Canadian Medical Association Journal, Vol. 166, 445–446.

de Vericourt, F., F. Karaesmen, Y. Dallery. 2002. “Optimal Stock Allocation for a Capacitated

Supply System”, Management Science, Vol. 48, 1486–1501.

Wang, P. P. 1996. “Queueing Models with Delayed State-Dependent Service Times”, European

Journal of Operational Research Vol. 88, 614–621.

Zhang, Z. G., H. P. Luh, C.H. Wang. 2011. “Modeling Security-Check Queues”, Management

Science, Vol. 57, 1979–1995.

Zhao, X., He, Q., Zhang, H. 2012. “On a Queueing System with State-dependent Service Rate,”

working paper.

Zipkin, P. 2000. Foundations of Inventory Management, McGraw Hill, Boston.


6. Proofs

Proof of Theorem 1.

Based on the definition of the supplementary variable that we use, we follow Hokstad (1975) to

derive a set of relations for a small time interval (t, t+dt) considering the pair {(nt, ηt), t∈ [0,∞)}.First consider p0(t+ dt). Note that the continuous time MC for the state of 0 is identical to the

one in the M/G/1 systems (the system is idle in both cases) and it is given in Hokstad (1975) as

(11).

We next consider the pair (1, η − dt). If at time t+ dt the system is in state (1, η − dt) where

η− dt≥ 0, at time t one of the following has happened: 1) the system was in state (1, η) and no

new customer arrived during the next dt units of time (with probability of 1−λ1dt+o(dt)); 2) the

system was in state (2,0) and the service time of the next customer to enter service was η (with

probability of b1(η)dt); 3) the system was in state (0) and a new customer arrived during the next

dt units of time (with probability of λ0dt+o(dt)); or 4) other possible events with probability o(dt)

or lower. Therefore,

p1(η− dt, t+ dt) = p1(η, t) (1−λ1dt+ o(dt)) + p2(0, t)b1(η)dt+ p0(t) (λ0dt+ o(dt)) b1(η) + o(dt).

Combining all terms with order of dt2 in o(dt), we get (12).

We finally consider the pair (j, η) for j > 1. If at time t+ dt the system state is (j, η − dt), at

time t one of the following has happened: 1) the system was in state (j, η) and no new customer

arrived during the next dt units of time (with probability of 1−λjdt+o(dt)); 2) the system was in

state (j+1,0) and the service time of the next customer was η (with probability of bj(η)dt); 3) the

system was in state (j−1, ηαj) and a new customer arrived during the next dt units of time (with

probability of λj−1dt+ o(dt));or 4) other possible events with probability o(dt) or lower. Thus,

pj(η− dt, t+ dt) = pj(η, t)(1−λjdt+ o(dt)) + pj+1(0, t)bj(η)dt+αjpt (j− 1, ηαj) (λj−1dt+ o(dt)) + o(dt).

Combining all terms with order of dt2 in o(dt), we get (13).

Note that the pair (nt, ηt) is a vector valued Markov process that represents the state of the

system at any given time t∈ [0,∞). Therefore, relations (12 -13) present a continuous time Markov

Chain (MC) for the Mn/Gn/1 system.

Before we prove Theorem 2, we prove the following lemma. Let pj(η) denote the steady-state

density of the residual service time of the customer on the server when there are j customers in

the system, assuming that such steady-state density exists. Then,

Lemma 4.∫ ∞u=η

e−λ1up′1(u)du=

∫ ∞u=η

e−λ1u (λ1p1(u)−λ0P (0)b1(u)− p2(0)b1(u))du (59)∫ ∞u=η

e−λjup′j(u)du=

∫ ∞u=η

e−λju (λjpj(u)−αjpj−1 (αju)λj−1− pj+1(0)bj(u))du. (60)


Proof of Lemma 4.

As we assume that bj(·) are all absolutely continuous, a sufficient condition for pj(η) to exist is

that the Mn/Gn/1 system is stable. Dividing both sides of (12) and (13) by dt and rearranging

them we get,

p1(η, t)− p1(η− dt, t)dt

= p1(η, t)λ1− p2(0, t)b1(η)−λ0p0(t)b1(η) +o(dt)

dt,

pj(η, t)− pj(η− dt, t)dt

= pj(η, t)λj − pj+1(0, t)bj(η)−αjpj−1 (ηαj, t)λj−1 +o(dt)

dt.

Taking the limit as dt→ 0, we get:

p′1(η, t) = λ1p1(η, t)−λ0p0(t)b1(η)− p2(0, t)b1(η),

p′j(η, t) = λjpj(η, t)−αjpj−1 (αjη, t)λj−1− pj+1(0, t)bj(η).

Assuming that the steady-state exists and taking the limit t→∞, we get:

p′1(η) = λ1p1(η)−λ0P (0)b1(η)− p2(0)b1(η), (61)

p′j(η) = λjpj(η)−αjpj−1 (αjη)λj−1− pj+1(0)bj(η). (62)

Multiplying both sides of (61) and (62) by e−λju and taking integral, we get (59) and (60).

Proof of Theorem 2.

We note that pj(∞) = 0, pj(0) = 0≥ 0 for j > 0, and P (0)> 0 because we assume that the system

is stable. Also,∫∞η=0

pj(η)dη= P (j) and similarly from (1)∫∞η=0

αjpj−1(αjη)dη= P (j−1). Then, by

setting η= λj = 0 in Lemma 4, from (59) and (60) we get

−p1(0) = λ1P (1)−λ0P (0)− p2(0)

−pj(0) = λjP (j)−λj−1P (j− 1)− pj+1(0).

Therefore, for j ≥ 1

λj−1P (j− 1)− pj(0) = λjP (j)− pj+1(0).

Note that λj−1P (j − 1)− pj(0) is independent of j and must go to zero in the limit j→∞ if the

system is stable (see e.g., Kerner, 2008). Therefore,

pj+1(0) = λjP (j), j ≥ 0. (63)

Note that (63) can be explained using level crossing as well. Considering level j (number of cus-

tomers in the system), λjP (j) is the rate of up-crossing this level and p(j + 1,0) is the rate of

down-crossing (i.e., the rate at which a departure leaves j customers behind).


Recalling that hj(·) denote the steady-state density of the residual service time observed by an

arrival that sees j customers in the system, we have

hj(η) =pj(η)∫∞

r=0pj(r)dr

=pj(η)

P (j). (64)

Therefore,

hj(s) =

∫∞r=0

e−srpj(r)dr

P (j). (65)

Substituting (63) in (61) and (62), and multiplying both sides by e−λjη, we get after some algebra,

e−λ1η(p′1(η)−λ1p1(η)) =−b1(η)e−λ1η(λ0P (0) +λ1P (1)) (66)

e−λjη(p′j(η)−λjpj(η)) =−e−λjη (αjpj−1 (αjη)λj−1 +λjP (j)bj(η)) . (67)

Considering (65), Lemma 4, and recalling bj(∞) = 0 for j ≥ 1 we get,

p1(η) = eλ1η∫ ∞u=η

(b1(u)e−λ1u(λ0P (0) +λ1P (1))

)du

pj(η) = eλjη∫ ∞u=η

(e−λju (αjpj−1 (αju)λj−1 +λjP (j)bj(u))

)du.

Substituting (63) in the above equations, we get for η= 0

λ0P (0) = (λ0P (0) +λ1P (1))

∫ ∞u=0

(b1(u)e−λ1u

)du

λj−1P (j− 1) =

∫ ∞u=0

(e−λju (αjpj−1 (uαj)λj−1 +λjP (j)bj(u))

)du.

Solving the integral from right hand side of the above equations we get,

λ0P (0) = (λ0P (0) +λ1P (1))b1 (λ1)

λj−1P (j− 1) = λj−1P (j− 1)hj−1

(λjαj

)+λjP (j)bj (λj) ,

because from (65) we have∫ ∞u=0

e−λju (αjpj−1 (uαj))du= P (j− 1)hj−1(λjαj

). (68)

Finally, with h0(·) = b1(·) and α1 = 1 after some algebra and using induction we get for each j ≥ 1

P (i) =λ0P (0)

λi

i−1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

.

Standard arguments establish the rest of the theorem.


Proof of Theorem 3.

Setting η= 0 in Lemma 4, from (59) and (60) we get (similar to (68))

P (1)sh1(s)− p1(0) = λ1P (1)h1(s)− b1(s) (p2(0) +λ0P (0)) (69)

P (j)shj(s)− pj(0) = λjP (j)hj(s)−λj−1P (j− 1)hj−1

(s

αj

)− bj(s)pj+1(0). (70)

Substituting (63) in (69) and (70), we get

P (1)sh1(s)−λ0P (0) = λ1P (1)h1(s)− b1(s) (λ1P (1) +λ0P (0)) (71)

P (j)shj(s)−λj−1P (j− 1) = λjP (j)hj(s)−λj−1P (j− 1)hj−1

(s

αj

)− bj(s)λjP (j). (72)

Now considering (71) we get:

h1(s) =

(λ0P (0)− b1(s) (λ1P (1) +λ0P (0))

)(P (1)s−λ1P (1))

.

Substituting λ1P (1) from (15) and recalling that h0(·) = b1(·), we get

h1(s) =

(λ1

s−λ1

) b1 (λ1)

1− b1(λ1α1

) (1− b1(s))− b1(s)

.

Considering that h0(·) = b1(·) and α1 = 1, we get

h1(s) =λ1

s−λ1

[b1(λ1)

1− h0(sα1

)

1− h0(λ1α1

)− b1(s)

].

Similarly, we can obtain hj(s) for j > 1 using (72) as,

hj(s) =λj

s−λj

bj(λj)1− hj−1( sαj

)

1− hj−1(λjαj

)− bj(s)

, j ≥ 1.

Proof of Observation 3. From Theorem 2 we have

P (i) =λ0P (0)

λiΠi−1j=0

1− hj(λj+1

αj+1

)bj+1 (λj+1)

= λi−11− hi−1

(λiαi

)λibi (λi)

P (i− 1)

Multiplying both sides by λi−1 and comparing the result with (3), we get µi as in (18).

Proof of Observation 4. Consider the auxiliary queue. We define a continuous time MC

for the auxiliary queue similar to the one expressed in (12) and (13) with states (j, η) where j is

the number of jobs in the auxiliary queue, while η denotes the remaining service time. We define


gj(η, t) as the probability that there are j jobs in the auxiliary queue, and remaining service time

is η at time t. Therefore,

g1(η− dt, t+ dt) = g1(η, t)(1−λκ+1dt) + g2(0, t)bκ+1(η)dt+ gt(0,0)λκbA0 (η)dt, j = 1, (73)

gj(η− dt, t+ dt) = gj(η, t)(1−λκ+jdt) + gj−1(η, t)λκ+j−1dt+ gj+1(0, t)bκ+j(η)dt, j ≥ 1, (74)

where bA0 (·) is the steady-state service time of a job that finds 0 jobs in this queue.

Now consider the original system defined in (12) and (13). Given that there are more than κ−1

customers in the system, (12) and (13) reduced to the same equations as given in (73) and (74)

where bA0 (·) is the equilibrium service times of customers that find κ customers in the original

system (Step 2 of the definition of the auxiliary queue). Thus, the steady-state distribution of the

number of jobs in the auxiliary queue, PA(i), is identical to the steady-state state distribution of

the number of customers in the original queue given that there are more than κ customers in the

system, P (κ+ i).

Proof of Corollary 1. The proof is similar to the proof of Theorem 2.2.2 in Kerner (2008).

Doorn and Regterschot (1988) define the adapted LAA as the LAA conditioning on the state of the

system, and show that under the adapted LAA PASTA holds. We next establish that the adapted

LAA holds in our settings. Let X(t) denote the state of the system, and Ns be the Poisson process

that generates the future arrivals when the state of the system is X(t) = s. Then, for every s we

have {Ns(t+u)−Ns(t), u≥ 0} and X(t) are independent and the adaptive LAA holds. Therefore,

Theorem 1 in Doorn and Regterschot (1988) holds.

Proof of Theorem 4.

We first obtain the transition probabilities p0k. Consider p00, the probability that the next

departing customer leaves no customer behind given that the last departing customer left no

customer behind, P (Mn+1 = 0|Mn = 0). This probability is equal to the probability of having no

arrivals during the next service time. This probability is b1(λ1) (e.g., Conway 1967, page 171).

Therefore,

p00 = b1(λ1). (75)

Next consider P01. The probability that the next departing customer leaves one customer behind

given that the last departing customer left no customer behind, P (Mn+1 = 1|Mn = 0). This proba-

bility is equal to the probability of one arrival during the next service time. The only sample path

that would lead to this event is that there is exactly one arrival during the sojourn time of the

departing customer. Because this sojourn time is identical to the service time of this customer,

the probability p01 is equal to the probability that (i) a customer arrives after the service time

starts,(

1− b1(λ1))

and (ii) no customers arrives during the remaining service time observed by


this arrival. Noting that this arrival sees one customer in the system upon the arrival, the equi-

librium remaining service time observed by this arrival is h1(·). Considering that the rate of the

residual service time is modified by α2 after this customer joins the queue, the probability that no

customers arrives during the remaining service time observed by this arrival is h1(λ2α2

). Therefore,

p01 =(

1− b1(λ1))h1

(λ2

α2

). (76)

With a similar logic, the probability that the next departing customer leaves k customers behind

given that the last departing customer left no customer behind, P (Mn+1 = k|Mn = 0), is equal to

the probability of k arrivals during the next service time. This probability is equal to the probability

that a customer arrives after the first service time starts,(

1− b1(λ1))

, followed by a customer

arrival during the remaining service times of all arrivals that see i < k customers in the system,

each with probability(

1− hi( λi+1

αi+1))

, and no arrival during the remaining service time once there

are k customers in the system, hk(λk+1

αk+1). Recalling our definition h0(·) = b1(·), we have,

p0k = hk

(λk+1

αk+1

) k−1∏i=0

(1− hi(

λi+1

αi+1

)

). (77)

Using the transition probabilities in (29) derived in the body of the paper and (77) we next derive

the steady-state probabilities of the embedded MC. Considering (24), we need to derive the steady-

state distribution of the number of customers in system using the embedded MC. Note that the

steady-state probability that a departure leaves k customers behind satisfies Pd(k) =∑∞

j=0Pd(j)pjk.

First consider Pd(0):

Pd(0) = Pd(0)p00 +Pd(1)p10 = Pd(0)b1(λ1) +Pd(1)b1(λ1)

⇒ Pd(1) = Pd(0)1− b1(λ1)

b1(λ1). (78)

Next consider Pd(1):

Pd(1) = Pd(0)p01 +Pd(1)p11 +Pd(2)p21

= Pd(0)(

1− b1(λ1))h1

(λ2

α2

)+Pd(1)

(1− b1(λ1)

)h1

(λ2

α2

)+Pd(2)b2(λ2). (79)

(80)

Substituting Pd(1) from (78) to (80), we get

Pd(2) = Pd(0)1∏j=0

1− hj(λj+1

αj+1

)bj+1(λj+1)

.

The rest of the probabilities can be obtained similarly.


Proof of Lemma 1. (34) is equivalent to

P (i) =P a(i)

∑∞k=0 λkP (k)

λi. (81)

Using (81) we get P (0) as,

P (0) =P a(0)

∑∞k=1

λkλ0P (k)

1− P a(0). (82)

Using (81) and (82), we obtain P (i) as a function of P (0) given in ()35).

Proof of Observation 6. The proof is based on Theorem 4 and similar to the proof of

Observation 3.

Proof of Lemma 2.

Noting that 1/µb = −dhκ(s)

ds|s=0, by taking the derivative of (17) with respect to s and setting

s= 0, we get the result.

1

µb=−dhk(s)

ds|s=0 =

λk

(s−λk)2

[bk(λk)

1− hk−1( sαk

)

1− hk−1( λkαk )− bk(s)

]|s=0

− λk(s−λk)

[− bk(λk)

1− hk−1( λkαk )

dhk−1(sαk

)

ds− dbk(s)

ds

]|s=0

= − 1

λk+

1

µk+

[bk(λk)

1− hk−1( λkαk )

dhk−1(sαk

)

ds|s=0

].

We prove the result by induction. For k= 1 we have α1 = 1 so that

1

µb=− 1

λ1

+1

µ1

+

[b1(λ1)

1− b1( λ1α1)

db1(sα1

)

ds|s=0

]=− 1

λ1

+1

µ1

−

[b1(λ1)

1− b1( λ1α1)

1

µ1

],

which with h0(·) = b1(·) is equivalent to (38) for k= 1. Now suppose (38) holds for κ=m− 1, i.e.,

−dhm−1(s)ds

|s=0 =1

µm−1− 1

λm−1+m−2∑j=1

m−2∏i=j

1

αi+1

(1

µj− 1

λj

)m−2∏i=j

bi+1(λi+1)

1− hi( λi+1

αi+1)

− 1

µ1

m−2∏i=1

1

αi

m−2∏i=0

bi+1(λi+1)

1− hi( λi+1

αi+1)

(83)

We next show that it holds for k=m.

1

µb= − 1

λm+

1

µm+

[bm(λm)

1− hm−1( λmαm )

dhm−1(sαm

)

ds|s=0

]

= − 1

λm+

1

µm+

[bm(λm)

1− hm−1( λmαm )

1

αm

dhm−1(s)

ds|s=0

](84)

Substituting (83) in (84), we get (38) for κ=m which completes the proof.


Proof of Theorem 5. Based on Observation 4, we have

P (k) = (1−F (k− 1))PA(0).

Substituting P (k) and F (k− 1) from (15), we get

λ0P (0)

λk

k−1∏i=0

1− hi(λi+1

αi+1

)bi+1(λi+1)

=

1−k−1∑j=0

λ0P (0)

λj

j−1∏i=0

1− hi(λi+1

αi+1

)bi+1(λi+1)

(1− ρb)

resulting in (39) after some algebra.

Proof of Lemma 3. Let Na denote the number of customers in the system seen by an arrival.

Then,

P aF (i) = P (Na = i|accepted)P (accepted) +P (Na = i|not accepted)P (not accepted) . (85)

By level crossing argument for states i < K, the frequency of transitions from state i to state

i+ 1 is equal to the frequency of transitions from state i+ 1 to state i. Therefore, the probability

that an arriving customer observes i people in the system given that she is accepted to the queue,

P (Na = i|accepted), is identical to the probability that a departing customer leaves i people behind,

P dF (i). Moreover, for states i <K, P (Na = i|not accepted) is zero. Therefore,

P aF (i) = P d

F (i)(1− P a

F (K))

+ (0) P aF (K)

= P dF (i)

(1− P a

F (K)), i= 0, ...,K − 1.

Documents

Queue Decomposition and its Applications in State-Dependent … · 2013-12-06 · These applications can be analyzed using our results. Therefore our results shed light on these applications