Queuing Analysis - jeryfarel · Decisions concerning the management of waiting lines are ......

Introduction to Management Science

(8th Edition, Bernard W. Taylor III)

Chapter 15

Chapter 15 - Queuing Analysis 1

Queuing Analysis

Elements of Waiting Line Analysis

The Single-Server Waiting Line System

Undefined and Constant Service Times

Chapter Topics

Undefined and Constant Service Times

Finite Queue Length

Finite Calling Problem

The Multiple-Server Waiting Line

Addition Types of Queuing Systems


Significant amount of time spent in waiting lines by people, products, etc.

Providing quick service is an important aspect of quality

Overview

Providing quick service is an important aspect of quality customer service.

The basis of waiting line analysis is the trade-off between the cost of improving service and the costs associated with making customers wait.

Queuing analysis is a probabilistic form of analysis.


The results are referred to as operating characteristics.

Results are used by managers of queuing operations to make decisions.

Waiting lines form because people or things arrive at a service faster than they can be served.

Most operations have sufficient server capacity to handle

Elements of Waiting Line Analysis

Most operations have sufficient server capacity to handle customers in the long run.

Customers however, do not arrive at a constant rate nor are they served in an equal amount of time.

Waiting lines are continually increasing and decreasing in length.and approach an average rate of customer arrivals and an average service time, in the long run.


average service time, in the long run.

Decisions concerning the management of waiting lines are based on these averages for customer arrivals and service times.

They are used in formulas to compute operating characteristics of the system which in turn form the basis of decision making.

Components of a waiting line system include arrivals (customers), servers, (cash register/operator), customers in line form a waiting line.

The Single-Server Waiting Line System (1 of 2)

line form a waiting line.

Factors to consider in analysis:

The queue discipline.

The nature of the calling population

The arrival rate

The service rate.


The service rate.

The Single-Server Waiting Line System (2 of 2)


Figure 15.1The Fast Shop Market Queuing System

Queue Discipline: The order in which waiting customers are served.

Calling Population: The source of customers (infinite or

Single-Server Waiting Line SystemComponent Definitions

Calling Population: The source of customers (infinite or finite).

Arrival Rate: The frequency at which customers arrive at a waiting line according to a probability distribution (frequently described by a Poisson distribution).

Service Rate: The average number of customers that can


Service Rate: The average number of customers that can be served during a time period (often described by the negative exponential distribution).

Assumptions of the basic single-server model:

An infinite calling population

Single-Server Waiting Line SystemSingle-Server Model

A first-come, first-served queue discipline

Poisson arrival rate

Exponential service times

Symbology:

= the arrival rate (average number of arrivals/time period)


= the arrival rate (average number of arrivals/time period)

= the service rate (average number served/time period)

Customers must be served faster than they arrive ( < ) or an infinitely large queue will build up.

Probability that no customers are in the queuing system:

1Po

Single-Server Waiting Line SystemBasic Single-Server Queuing Formulas (1 of 2)

Probability that n customers are in the system:

Average number of customers in system: and

1Po

1

nPo

nPn

L


Average number of customers in system: and waiting line:

2

Lq

L

Average time customer spends waiting and being served:

LW

1

Single-Server Waiting Line SystemBasic Single-Server Queuing Formulas (2 of 2)

Average time customer spends waiting in the queue:

Probability that server is busy (utilization factor):

LW

1

Wq

U


Probability that server is idle:

11 UI

= 24 customers per hour arrive at checkout counter

= 30 customers per hour can be checked out

24/30)-(11 Po

Single-Server Waiting Line SystemCharacteristics for Fast Shop Market (1 of 2)

systemtheincustomersnoofy probabilit.20

24/30)-(11

Po

systemtheinavgtheoncustomers424)-24/(30

L


line waitingtheinavgtheoncustomers3.224)]-30(24)2/[30(

2

Lq

Single-Server Waiting Line SystemCharacteristics for Fast Shop Market (2 of 2)

customerpersystemtheintimeavgmin)(10hour0.167

24]-1/[30

LW 1

line waitingtheintimeavgmin)(8hour0.133

24)]-24/[30(30

Wq

idlebe willservery probabilit.20busy,servery probabilit.80

24/30

U


idlebe willservery probabilit.20busy,servery probabilit.80

Single-Server Waiting Line SystemSteady-State Operating Characteristics

Because of steady-state nature of operating characteristics:

Utilization factor, U, must be less than one: U < 1,or / < 1 and < . / < 1 and < .

The ratio of the arrival rate to the service rate must be less than one or, the service rate must be greater than the arrival rate.

The server must be able to serve customers faster than the arrival rate in the long run, or waiting line will grow to infinite size.


to infinite size.

Manager wishes to test several alternatives for reducing customer waiting time:

Addition of another employee to pack up purchases

Single-Server Waiting Line SystemEffect of Operating Characteristics (1 of 6)

Addition of another employee to pack up purchases

Addition of another checkout counter.

Alternative 1: Addition of an employee (raises service rate from = 30 to = 40 customers per hour).

Cost $150 per week, avoids loss of $75 per week for each minute of reduced customer waiting time.


System operating characteristics with new parameters:

Po = .40 probability of no customers in the system

L = 1.5 customers on the average in the queuing system

System operating characteristics with new parameters (continued):

Lq = 0.90 customer on the average in the waiting line


Lq = 0.90 customer on the average in the waiting line

W = 0.063 hour average time in the system per customer

Wq = 0.038 hour average time in the waiting line per customer

U = .60 probability that server is busy and customer must wait

I = .40 probability that server is available

Average customer waiting time reduced from 8 to 2.25


Average customer waiting time reduced from 8 to 2.25 minutes worth $431.25 per week.

Alternative 2: Addition of a new checkout counter ($6,000 plus $200 per week for additional cashier).

= 24/2 = 12 customers per hour per checkout counter


= 24/2 = 12 customers per hour per checkout counter

= 30 customers per hour at each counter

System operating characteristics with new parameters:

Po = .60 probability of no customers in the system

L = 0.67 customer in the queuing system

Lq = 0.27 customer in the waiting line


Lq = 0.27 customer in the waiting line

W = 0.055 hour per customer in the system

Wq = 0.022 hour per customer in the waiting line

U = .40 probability that a customer must wait

I = .60 probability that server is idle

Savings from reduced waiting time worth $500 per week -$200 = $300 net savings per week.

After $6,000 recovered, alternative 2 would provide $300 -


After $6,000 recovered, alternative 2 would provide $300 -281.25 = $18.75 more savings per week.



Table 15.1Operating Characteristics for Each Alternative System




Figure 15.2Cost Trade-Offs for Service Levels

Single-Server Waiting Line SystemSolution with Excel and Excel QM (1 of 2)


Exhibit 15.1

Single-Server Waiting Line SystemSolution with Excel and Excel QM (2 of 2)


Exhibit 15.2

Single-Server Waiting Line SystemSolution with QM for Windows


Exhibit 15.3

Constant, rather than exponentially distributed service times, occur with machinery and automated equipment.

Constant service times are a special case of the single-

Single-Server Waiting Line SystemUndefined and Constant Service Times

Constant service times are a special case of the single-server model with undefined service times.

Queuing formulas:

1Po

/222

LqWq

1


/

/

12

222Lq

LqL

1WqW

U

Data: Single fax machine; arrival rate of 20 users per hour, Poisson distributed; undefined service time with mean of 2 minutes, standard deviation of 4 minutes.

Single-Server Waiting Line SystemUndefined Service Times Example (1 of 2)

302012

23020

2151

220

12

222

useinnotmachinethaty probabilit.33302011

/

//

/

/

Lq

Po

minutes, standard deviation of 4 minutes.

Operating characteristics:


machinetheusingandlineinemployees4.0

3020333

linein waitingemployees3.33

30201212

)/(.

//

LqL

time waitingminutes10hour1665020333

LqWq ..

Operating characteristics (continued):

Single-Server Waiting Line SystemUndefined Service Times Example (2 of 2)

nutilizatiomachine67%3020

systemtheinminutes12

hour0.1998301166501

20

U

WqW .


In the constant service time model there is no variability in service times; = 0.

Substituting = 0 into equations:

Single-Server Waiting Line SystemConstant Service Times Formulas

Substituting = 0 into equations:

All remaining formulas are the same as the single-server formulas.

22

12

2

12

2202

12

222

/

/

/

/

/

/Lq


formulas.

Car wash servicing one car at a time; constant service time of 4.5 minutes; arrival rate of customers of 10 per hour (Poisson distributed).

Single-Server Waiting Line SystemConstant Service Times Example

waitingcars1.14103133132

2102

2

).)(.(

)()(

Lq

(Poisson distributed).

Determine average length of waiting line and average waiting time.

= 10 cars per hour, = 60/4.5 = 13.3 cars per hour


time waitingminutes6.84orhour0.11410141

1031331322

.

).)(.()(

LqWq

Undefined and Constant Service TimesSolution with Excel


Exhibit 15.4

Undefined and Constant Service TimesSolution with QM for Windows

Exhibit 15.5


Exhibit 15.5

In a finite queue, the length of the queue is limited.

Operating characteristics, where M is the maximum number in the system:

Finite Queue Length

1 11

111

Mnfor 11

1

PMLLqM

MML

nPoPn

MPo

)()/(

)/)((/

/

)()/(/

in the system:


1

1

WWq

PMLW

)(

Metro Quick Lube single bay service; space for one vehicle in service and three waiting for service; mean time between arrivals of customers is 3 minutes; mean service time is 2

Finite Queue Length Example (1 of 2)

arrivals of customers is 3 minutes; mean service time is 2 minutes; both inter-arrival times and service times are exponentially distributed; maximum number of vehicles in the system equals 4.

Operating characteristics for = 20, = 30, M = 4:

emptyissystemthaty probabilit.38530201

3020111

1

)/(/

)/(/M

Po


fullissystemthaty probabilit.0764

302038

53020111

)(.)(

)/()/(

MnPoPM

M

Average queue lengths and waiting times:

11

111

)/(

)/)((/

/

MMML

Finite Queue Length Example (2 of 2)

waitingcars0.6230

0761202411

systemtheincars1.24530201

53020530201

3020

111

).(.)(

)/()/)((

//

)/(/

PMLLq

L

M


linein waitinghour0.03330106701

systemthein waitinghours0.067076120

2411

.

).(.

)(

WWq

PMLW

Finite Queue Model ExampleSolution with Excel


Exhibit 15.6

Finite Queue Model ExampleSolution with QM for Windows

Exhibit 15.7


In a finite calling population there is a limited number of potential customers that can call on the system.

Operating characteristics for system with Poisson arrival

Finite Calling Population

2,...N1,nandsize,populationNwhere

0

1

nN

n nNN

Po

)!(!

Operating characteristics for system with Poisson arrival and exponential service times:


1 1

1

WqWLN

LqWqPoLqL

PoNLqPon

nNNPn

)()(

)()!(

!

Wheelco Manufacturing Company; 20 machines; each machine operates an average of 200 hours before breaking down; average time to repair is 3.6 hours; breakdown rate is Poisson distributed, service time is exponentially

Finite Calling Population Example (1 of 2)

is Poisson distributed, service time is exponentially distributed.

Is repair staff sufficient?

= 1/200 hour = .005 per hour

= 1/3.6 hour = .2778 per hour


N = 20 machines

65220

0 2778005

2020

1 .

..

)!(!

n

n

n

Po

Finite Calling Population Example (2 of 2)

repairfor waitinghours74100552020

169

systemtheinmachines5206521169

waitingmachines1696521005

277800520

0

.))(..(

.

.).(.

...

..

Wq

L

Lq

n


System seems inadequate.

systemtheinhours3352778

1741

00552020

..

.

))(..(

W

Finite Calling Population ExampleSolution with Excel and Excel QM (1 of 2)


Exhibit 15.8

Finite Calling Population ExampleSolution with Excel and Excel QM (2 of 2)


Exhibit 15.9

Finite Calling Population ExampleSolution with QM for Windows


Exhibit 15.10

In multiple-server models, two or more independent servers in parallel serve a single waiting line.

Biggs Department Store service department; first-come,

Multiple-Server Waiting Line (1 of 2)

Biggs Department Store service department; first-come, first-served basis.


Multiple-Server Waiting Line (2 of 2)

Figure 15.3Customer Service

Queuing System


Multiple-Server Waiting LineQueuing Formulas (1 of 3)

Assumptions:

First-come first-served queue disciplinePoisson arrivals, exponential service times Poisson arrivals, exponential service times Infinite calling population.

Parameter definitions:

= arrival rate (average number of arrivals per time period) = the service rate (average number served per time


period) per server (channel)c = number of serversc = mean effective service rate for the system (must exceed arrival rate)

systemincustomersnoy probabilit11

01

1

ccc

ccn

n

n

n

Po

!!


systemtheincustomersaverage

systemincustomersnofy probabilitc nfor1

cnfor 1

Poc

L

Pon

nPn

Pon

cnccPn

)/(

!


systemtheinspendscustomertimeaverage

systemtheincustomersaverage21

LW

Pocc

cL

)()!()/(

queuetheiniscustomertimeaverage1

queuetheincustomersofnumberaverage

LqWWq

LLq


servicefor waitmustcustomery probabilit1

queuetheiniscustomertimeaverage1

Po

ccc

cPw

LqWWq

!


1043433

410

31

2

410

21

1

410

11

0

410

01

1

)()(

!!!!

Po

Multiple-Server Waiting LineBiggs Department Store Example (1 of 2)

= 10, = 4, c = 3

410045

2104313

3410410

customersnoofy probabilit045

1043410

31

410

21

410

11

410

01

)(.])([)!(

)/)()((

.

)(!!!!

L


departmentservicetheintimecustomeraveragehour600

106

departmentserviceinaverageoncustomers6

.

W

servedbeto waitingaveragetheoncustomers53

4106

.

Lq

Multiple-Server Waiting LineBiggs Department Store Example (2 of 2)

0451043

433

410

31

customerperlineintime waitingaveragehour350

1053

servedbeto waitingaveragetheoncustomers53

)(.)(

)(!

.

.

.

Pw

Wq


servicefor waitmustcustomery probabilit.703

045104343

)(.)(!

Pw

Multiple-Server Waiting LineSolution with Excel


Exhibit 15.11

Multiple-Server Waiting LineSolution with Excel QM


Exhibit 15.12

Multiple-Server Waiting LineSolution with QM for Windows

Exhibit 15.13


Additional Types of Queuing Systems (1 of 2)

Figure 15.4Single Queues with Single and Multiple

Servers in Sequence


Other items contributing to queuing systems:

Systems in which customers balk from entering system, or leave the line (renege).

Additional Types of Queuing Systems (2 of 2)

system, or leave the line (renege).

Servers who provide service in other than first-come, first-served manner

Service times that are not exponentially distributed or are undefined or constant

Arrival rates that are not Poisson distributed


Jockeying (i.e., moving between queues)

Problem Statement: Citizens Northern Savings Bank loan officer customer interviews.

Customer arrival rate of four per hour, Poisson distributed;

Example Problem Solution (1 of 5)

Customer arrival rate of four per hour, Poisson distributed; officer interview service time of 12 minutes per customer.

Determine operating characteristics for this system.

Additional officer creating a multiple-server queuing system with two channels. Determine operating characteristics for this system.


Solution:

Step 1: Determine Operating Characteristics for the Single-Server System


Server System

= 4 customers per hour arrive, = 5 customers per hour are served

Po = (1 - / ) = ( 1 – 4 / 5) = .20 probability of no customers in the system

L = / ( - ) = 4 / (5 - 4) = 4 customers on average in


the queuing system

Lq = 2 / ( - ) = 42 / 5(5 - 4) = 3.2 customers on average in the waiting line

Step 1 (continued):

W = 1 / ( - ) = 1 / (5 - 4) = 1 hour on average in the system


system

Wq = / (u - ) = 4 / 5(5 - 4) = 0.80 hour (48 minutes) average time in the waiting line

Pw = / = 4 / 5 = .80 probability the new accounts officer is busy and a customer must wait


Step 2: Determine the Operating Characteristics for the Multiple-Server System.

= 4 customers per hour arrive; = 5 customers


systemincustomersnoy probabilit429

11

01

1

.

!!

ccc

ccn

n

n

n

Po

= 4 customers per hour arrive; = 5 customers per hour served; c = 2 servers


systemtheincustomersofnumberaverage9520

21

.

)()!()/(

Po

cc

cL

LLq

Step 2 (continued):


1

queuetheiniscustomertimeaveragehour0380

1

queuetheincustomersofnumberaverage1520

.

.

cc

LqWWq


servicefor waitmustcustomery probabilit229

1

.

!

Po

ccc

cPw


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Queuing Analysis - jeryfarel · Decisions concerning the management of waiting lines are ......

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