Supplement CWaiting Line Models

Operations Managementby

R. Dan Reid & Nada R. Sanders4th Edition © Wiley 2010

Learning Objectives

Describe the elements of a waiting line problem.

Use waiting line models to estimate system performance.

Use waiting line models to make managerial decisions.

Elements of Waiting Lines “Queue” is another name for a waiting

line. A waiting line system consists of two

components: The customer population (people or objects

to be processed) The process or service system

Whenever demand exceeds available capacity, a waiting line or queue forms

There is a tradeoff between cost and service level.

Customer Population Characteristics

Finite versus Infinite populations: Is the number of potential new customers materially

affected by the number of customers already in queue? Balking

When an arriving customer chooses not to enter a queue because it’s already too long.

Reneging When a customer already in queue gives up and exits

without being serviced. Jockeying

When a customer switches between alternate queues in an effort to reduce waiting time.

Service System

The service system is defined by: The number of waiting lines The number of servers The arrangement of servers The arrival and service patterns The waiting line priority rules

Number of Lines Waiting lines systems can have

single or multiple queues. Single queues avoid jockeying

behavior and perceived fairness is usually high.

Multiple queues are often used when arriving customers have differing characteristics (e.g. paying with cash, less than 10 items, etc.) and can be readily segmented.

Servers Single servers or multiple, parallel

servers providing multiple channels Arrangement of servers (phases)

Multiple phase systems require customers to visit more than one server

Example of a multi-phase, multi-server system:

C C C CC Depart

Arrivals

1

2

3 6

5

4

Phase 1 Phase 2

Example Queuing Systems

Arrival & Service Patterns

Arrival rate: The average number of customers arriving

per time period Modeled using the Poisson distribution Arrival rate usually denoted by lambda () Example: =50 customers/hour; 1/=0.02

hours between customer arrivals (1.2 minutes between customers)

Arrival & Service Patterns con’t

Service rate: The average number of customers that can be

served during the period of time Service times are usually modeled using the

exponential distribution Service rate usually denoted by mu (µ) Example: µ=70 customers/hour; 1/µ=0.014 hours

per customer (0.857 minutes per customer). Even if the service rate is larger than the

arrival rate, waiting lines form! Reason is the variation in specific customer

arrival and service times.

Waiting Line Priority Rules First come, first served Best customers first (reward loyalty) Highest profit customers first Quickest service requirements first Largest service requirements first Earliest reservation first Emergencies first Etc.

Waiting Line Performance Measures

Lq = The average number of customers waiting in queue

L = The average number of customers in the system

Wq = The average waiting time in queue W = The average time in the system p = The system utilization rate (% of

time servers are busy)

Single-Server Waiting Line Assumptions

Customers are patient (no balking, reneging, or jockeying)

Arrivals follow a Poisson distribution with a mean arrival rate of . This means that the time between successive customer arrivals follows an exponential distribution with an average of 1/

The service rate is described by a Poisson distribution with a mean service rate of µ. This means that the service time for one customer follows an exponential distribution with an average of 1/µ

The waiting line priority rule is first-come, first-served

Infinite population

Formulas: Single-Server Case

form. eventually willline longinfinitly an

case, not the is thisIf stability. systemfor :Note

nutilizatiosystemaverage

rateservicemeanmu

ratearrivalmeanlambda

p

Formulas: Single-Server Case con’t

in timepoint given aat

systemtheincustomersofyprobabilit1

waitingspenttimeaverage

serviceincludingsystemintimeaverage1

lineincustomersofnumberaverage

systemin customers ofnumber average

nppP

pWW

W

pLL

L

nn

q

q

State Univ Computer Lab A help desk in the computer lab serves

students on a first-come, first served basis. On average, 15 students need help every hour. The help desk can serve an average of 20 students per hour.

Based on this description, we know: µ = 20 students/hour (average service time is

3 minutes) = 15 students/hour (average time between

student arrivals is 4 minutes)

Average Utilization

%7575.020

15orp

Average Number of Studentsin the System, and in Line

studentsL 31520

15

studentspLLq 25.2375.0

Average Time in the System & in Line

minutes12

hours2.01520

11

or

W

minutes9

hours15.02.075.0

or

pWWq

Probability of nStudents in the Line

079.075.075.011

105.075.075.011

141.075.075.011

188.075.075.011

25.0175.011

444

333

222

11

00

ppP

ppP

ppP

ppP

ppP

Single Server: Spreadsheet Approach

123456789

10111213141516171819202122

A B C

Queuing Analysis: Single Server

InputsTime unit hourArrival Rate (lambda) 15 customers/hourService Rate (mu) 20 customers/hour

Intermediate CalculationsAverage time between arrivals 0.066667 hourAverage service time 0.05 hour

Performance MeasuresRho (average server utilization) 0.75P0 (probability the system is empty) 0.25L (average numberin the system) 3 customersLq (average number waiting in the queue) 2.25 customersW (average time in the system) 0.2 hourWq (average time in the queue) 0.15 hour

Probability of a specific number of customers in the systemNumber 2Probability 0.140625

Key Formulas

B9: =1/B5

B10: =1/B6

B13: =B5/B6

B14: =1-B13

B15: =B5/(B6-B5)

B16: =B13*B15

B17: =1/(B6-B5)

B18: =B13*B17

B22: =(1-B$13)*(B13^B21)

Use Data Table (tracking B22) to easily compute the probability of n customers in the system.

Single Server: Probability of n Students in the System

Probability of Number in System

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

0.30000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number in System

Pro

bab

ilit

y

Multiple Server Case

Assumptions Same as Single-Server, except here

we have multiple, parallel servers Single Line When server finishes with customer,

first person in line goes to the idle server

All servers are identical

Multiple Server Formulas

form. eventually willline longinfinitly an

case, not the is thisIf stability. systemfor :Note

nutilizatiosystemaverage

servers identical parallel, ofnumber

server for rateservicemeanmu

ratearrivalmeanlambda

s

sp

s

one

Multiple Server Formulas con’t

in timepoint given aat system in the

customers ofy probabilit

for !

/

for !

/

in timepoint given aat system in the customers

zero ofy probabilit 1

1

!

/

!

/

0

0

11

00

n

snPss

snPnP

psnP

sn

n

n

n

s

n

sn

Multiple Server Formulas con’t

systemin customers ofnumber average

serviceincludingsystemintimeaverage1

linein waitingspenttimeaverage

linein customers ofnumber average 1!

/2

0

WL

WW

LW

ps

pPL

q

qq

s

q

Example: Multiple Server Computer Lab Help Desk Now 45 students/hour need help. 3 servers, each with service rate of

18 students/hour Based on this, we know:

µ = 18 students/hour s = 3 servers = 45 students/hour

Flexible Spreadsheet Approach Formulas are somewhat complex to set up initially, but

you only need to do it once!

For other multiple-server problems, can just change the input values.

This approach also makes sensitivity analysis possible.

123456789

101112131415161718192021222324

A B C

Queuing Analysis: Multiple Servers

InputsTime unit hourArrival Rate (lambda) 45 customers/hourService Rate per Server (mu) 18 customers/hourNumber of Servers (s) 3 servers

Intermediate CalculationsAverage time between arrivals 0.022222 hourAverage service time per server 0.055556 hourCombined service rate (s*mu) 54 customers/hour

Performance MeasuresRho (average server utilization) 0.833333P0 (probability the system is empty) 0.044944L (average numberin the system) 6.011236 customersLq (average number waiting in the queue) 3.511236 customersW (average time in the system) 0.133583 hourWq (average time in the queue) 0.078027 hour

Probability of a specific number of customers in the systemNumber 5Probability 0.081279

3456789

101112131415161718192021222324252627108109

E F G HWorking Calculations, mainly for P0 Calculation

lambda/mu 2.5s! 6

n (/)^n n! Sum0 1 1 11 2.5 1 3.52 6.25 2 6.6253 15.625 6 9.2291666674 39.0625 24 10.856770835 97.65625 120 11.670572926 244.14063 720 12.009657127 610.35156 5040 12.130758628 1525.8789 40320 12.168602849 3814.6973 362880 12.17911512

10 9536.7432 3628800 12.1817431911 23841.858 39916800 12.1823404812 59604.645 479001600 12.1824649213 149011.61 6.227E+09 12.1824888514 372529.03 8.718E+10 12.1824931215 931322.57 1.308E+12 12.1824938316 2328306.4 2.092E+13 12.1824939417 5820766.1 3.557E+14 12.1824939618 14551915 6.402E+15 12.1824939699 2.489E+39 9.33E+155 12.18249396100 6.223E+39 9.33E+157 12.18249396

Key Formulas for Spreadsheet

F10: =F$5^E10 (copied down) G10: =E10*G9 (copied down) H10: =H9+(F10/G10) (copied down) F5: =B5/B6 F6: =INDEX(G9:G109,B7+1) B10: =1/B5 B11: =1/B6 B12: =B7*B6 B15: =B5/B12 B16: = (INDEX(H9:H109,B7)+ (((F5^B7)/F6)*((1)/(1-B15))))^(-1)

B17: =B5*B19 B18: =(B16*(F5^B7)*B15)/(INDEX(G9:G109,B7+1)*(1-B15)^2) B19: =B20+(1/B6) B20: =B18/B5 B24: =IF(B23<=B7, ((F5^B23)*B16)/INDEX(G9:G109,B23+1),

((F5^B23)*B16)/ (INDEX(G9:G109,B7+1)*(B7^(B23-B7))))

Probability of n students in the system

Probability of Number in System

0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.16000 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number in System

Pro

bab

ilit

y

Changing System Performance Customer Arrival Rates

Try to smooth demand through non-peak discounts or price promotions

Number and type of service facilities Increase or decrease number of servers, or dedicate

specific servers for certain tasks (e.g., express line for under 10 items)

Change Number of Phases Can use multi-phase system instead of single phase.

This spreads the workload among more servers and may result in better flow (e.g., fast food restaurants having an order phase, pay phase, and pick-up phase during busy hours)

Changing System Performance Server efficiency

Add resources to each phase (e.g., bagger helping a checker at the grocery store)

Use technology (e.g. price scanners) to improve efficiency

Change priority rules Example: implement a reservation protocol

Change the number of lines Reduce multiple lines to single queue to

avoid jockeying Dedicate specific servers to specific

transactions

Waiting Lines Models within OM: How it all fits together Although it is unlikely that you calculate

performance measures for the lines you wait in on a day-to-day basis, you should now be aware of the potential for mathematical analysis of these systems. More importantly, management has a tool by which it can evaluate system performance and make decisions as to how to improve the performance while weighing performance against the costs to achieve that performance.

Waiting line models are important to a company because they directly affect customer service perception and the costs of providing a service.

Supplement C Highlights

The elements of a waiting line system include the customer population source, the patience of the customer, the service system, arrival and service distributions, waiting line priority rules, and system performance measures. Understanding these elements is critical when analyzing waiting line systems.

Waiting line models allow us to estimate system performance by predicting average system utilization, average number of customers in the service system, average number of customers waiting in line, average time a customer waits in line, and the probability of n customers in the service system.

The benefit of calculating operational characteristics is to provide management with information as to whether system changes are needed. Management can change the operational performance of the waiting line system by altering any or all of the following: the customer arrival rates, the number of service facilities, the number of phases, server efficiency, the priority rule, and the number of lines in the system. Based on proposed changes, management can then evaluate the expected performance of the system.

Homework Hints Problems C.3 and C.4: these are

based on the single-server model, the “additional server” is one who works within the single server system. C.3 asks for the utilization rate and average number of customers waiting in the system and in line. C.4 asks for the average time in the system and in line and the probability of more than 3 and 4 customers in the system