Lecture 11Lecture 11Lecture 11Lecture 11

Queueing ModelsQueueing ModelsQueueing ModelsQueueing Models

2

Queueing SystemQueueing System

Queueing System: A system in which items (or customers) arrive at a station, wait in

a line (or queue), obtain some kind of service, an then leave the system.

Managerial Problem Related to Queueing Systems: Analysis Problems

Need to know if a given system is performing satisfactorily.

Design Problems Want to design the features of a system to

accomplish an overall objective.

Queueing System: A system in which items (or customers) arrive at a station, wait in

a line (or queue), obtain some kind of service, an then leave the system.

Managerial Problem Related to Queueing Systems: Analysis Problems

Need to know if a given system is performing satisfactorily.

Design Problems Want to design the features of a system to

accomplish an overall objective.

3

Characteristics of Characteristics of Queueing SystemQueueing System

A customer population, which is the collection of all possible customers.

An arrival process, which is how the customers from this population arrive.

A queueing process, which consists of the way in which the customers wait for service and the queueing discipline, which is how they are then selected for

service.

A customer population, which is the collection of all possible customers.

An arrival process, which is how the customers from this population arrive.

A queueing process, which consists of the way in which the customers wait for service and the queueing discipline, which is how they are then selected for

service.

4

Characteristics of Characteristics of Queueing System Queueing System (cont’)(cont’)

A service process, which is the way and rate at which customers are served.

A departure process, which is one of the following: Items leave the system completely after being served, resulting in a

one-step queueing system. Items, once finished at a station, proceed to some other work

station to receive further service, resulting in a network of queues.

A service process, which is the way and rate at which customers are served.

A departure process, which is one of the following: Items leave the system completely after being served, resulting in a

one-step queueing system. Items, once finished at a station, proceed to some other work

station to receive further service, resulting in a network of queues.

5

Queueing SystemQueueing System

CustomerPopulatio

n

ArrivalProcess

SystemSystem

.

.

.

WaitingCustomers

QueueingProcess

Depart

Network of queuesNetwork of queues

Arrival Depart

One-stepqueueing system

One-stepqueueing system

ArrivalDepart

Components of a Queueing System:

Departure Processes:

Servers

ServiceProcess

6

Arrival ProcessArrival Process

Classes of Interarrival Times: Deterministic, in which each successive customer arrives after the

same fixed and known amount of time. A classic example is an assembly line, where the jobs arrive at a station at unvarying time intervals (called the cycle time).

Probabilistic, in which the time between successive arrivals is uncertain and varies. Probabilistic interarrival times are described by a probability distribution.

Classes of Interarrival Times: Deterministic, in which each successive customer arrives after the

same fixed and known amount of time. A classic example is an assembly line, where the jobs arrive at a station at unvarying time intervals (called the cycle time).

Probabilistic, in which the time between successive arrivals is uncertain and varies. Probabilistic interarrival times are described by a probability distribution.

7

Arrival Process Arrival Process (Example)(Example)

Exponential Distribution

where is the average number of arrivals per unit of time.

Probability of next customer arriving within T units of previous arrival:P (interarrival time T) = 1- e-T

For example:P (interarrival time 1/6 hour) = 1- e-20(1/6)

= 0.964

Exponential Distribution

where is the average number of arrivals per unit of time.

Probability of next customer arriving within T units of previous arrival:P (interarrival time T) = 1- e-T

For example:P (interarrival time 1/6 hour) = 1- e-20(1/6)

= 0.964

tetf )(

8

Queueing ProcessQueueing Process

Single-line Queueing System: A queueing system in which the customers waiting in a single line

for the next available server.

Multiple-line Queueing System: A queueing system in which arriving customers may select one of

several lines in which to wait for service.

Single-line Queueing System: A queueing system in which the customers waiting in a single line

for the next available server.

Multiple-line Queueing System: A queueing system in which arriving customers may select one of

several lines in which to wait for service.

9

Classifications of Classifications of Queueing ModelsQueueing Models

Queueing Disciplines First-In-First-Out (FIFO) Last-In-First-Out (LIFO) Priority Selection

Notations (Kendall)A / B / c / K / L

where A: interarrival time distribution

B: service time distribution

c: no. of services

K: the system capacity

L: the size of the calling population

Queueing Disciplines First-In-First-Out (FIFO) Last-In-First-Out (LIFO) Priority Selection

Notations (Kendall)A / B / c / K / L

where A: interarrival time distribution

B: service time distribution

c: no. of services

K: the system capacity

L: the size of the calling population

10

Classifications of Classifications of Queueing Models Queueing Models (cont’)(cont’)

The arrival process: Symbol A to describe interarrival distribution:

D = deterministic interarrival time M = probabilistic interarrival time with exponential distribution G = probabilistic interarrival time with general distribution other

than exponential. The service process: Symbol B to describe the service time

distribution: D = deterministic service time M = probabilistic service time with exponential distribution G = probabilistic service time with general distribution other than

exponential.

The arrival process: Symbol A to describe interarrival distribution:

D = deterministic interarrival time M = probabilistic interarrival time with exponential distribution G = probabilistic interarrival time with general distribution other

than exponential. The service process: Symbol B to describe the service time

distribution: D = deterministic service time M = probabilistic service time with exponential distribution G = probabilistic service time with general distribution other than

exponential.

11

Classifications of Classifications of Queueing Models Queueing Models (cont’)(cont’)

The queueing process: this number c denotes how many parallel stations or channels are in the system (Servers are assumed to be identical in rate of service).

When the waiting space and/or customer population size is finite:

K = Maximum number of customers that can be in the system at any one

time

= number of parallel stations plus total number of customers that can

wait for service L = Total number of customers in the population

Example: M/M/3//10 = a system that has room for an infinite number of customer

(K has been left off) but only 10 possible customers exists.

The queueing process: this number c denotes how many parallel stations or channels are in the system (Servers are assumed to be identical in rate of service).

When the waiting space and/or customer population size is finite:

K = Maximum number of customers that can be in the system at any one

time

= number of parallel stations plus total number of customers that can

wait for service L = Total number of customers in the population

Example: M/M/3//10 = a system that has room for an infinite number of customer

(K has been left off) but only 10 possible customers exists.

12

Performance MeasuresPerformance Measures

Performance Measures for Evaluating a Queueing System

Performance Measures for Evaluating a Queueing System

Transient Phase Steady State Phase

Customer number

(in order of arrival)

13

Common performance Common performance MeasuresMeasures

Arrival rate – Service rate of one server – Server utilization– Mean number in the system – L Mean queue length – Lq Average time in the system – W Average waiting time – Wq

Steady-state probability distribution – Pn, n = 0,1,2,…

Arrival rate – Service rate of one server – Server utilization– Mean number in the system – L Mean queue length – Lq Average time in the system – W Average waiting time – Wq

Steady-state probability distribution – Pn, n = 0,1,2,…

14

Lq = W qLq = W q

L = WL = W

W = Wq + (1 / )W = Wq + (1 / )

Relationships among Relationships among Performance MeasuresPerformance Measures

Average timein the system

Average timein the system = Average

waiting time

Averagewaiting time + Average

service time

Averageservice time

Average no. of customersin the system

Average no. of customersin the system = Average no. of arrivals

per unit of time

Average no. of arrivalsper unit of time

Average timein the system

Average timein the system

Average no. of customersin the queue

Average no. of customersin the queue = Average no. of arrivals

per unit of time

Average no. of arrivalsper unit of time

Average timein the queue

Average timein the queue

15

Relationships among Relationships among Performance Measures Performance Measures (Example)(Example)

Suppose = 12, = 4, Lq = 3 Suppose = 12, = 4, Lq = 3

4

1

12

3

q

q

LW

2

1

4

1

4

11

qWW

62

112 WL

16

MM / / MM / 1 Queueing / 1 Queueing SystemSystem

Analyzing a Single-Line Single-Channel Queueing System with Exponential Arrival and Service Processes (M / M / 1)

Queueing System for Weigh Station = the average number of trucks

arriving per hour

= 60

= the average number of trucks

that can be weighed per hour

= 66

Analyzing a Single-Line Single-Channel Queueing System with Exponential Arrival and Service Processes (M / M / 1)

Queueing System for Weigh Station = the average number of trucks

arriving per hour

= 60

= the average number of trucks

that can be weighed per hour

= 66 Weigh Station

17

Formulas (Formulas (MM / / MM / 1) / 1)

Performance Measure General Formula

Utilization

Average number in line

Average waiting time in queue

Average waiting time in system

Performance Measure General Formula

Utilization

Average number in line

Average waiting time in queue

Average waiting time in system

1

2

qL

)1(

q

q

LW

)1(

11

qWW

18

Formulas (Formulas (MM / / MM / 1) / 1) (cont’)(cont’)

Performance Measure General Formula

Average number in system

Probability that no customers are in system

Probability that an arriving customer has to wait

Probability of n customers in the system

Performance Measure General Formula

Average number in system

Probability that no customers are in system

Probability that an arriving customer has to wait

Probability of n customers in the system

01 PPw

)1(0 nnn PP

10P

1

WL

19

ComputationComputation

Utilization (): = / = 60 / 66 = 0.9091

Prob.(no customers are in system) (P0): P0 = 1 - = 1 – 0.9091 = 0.0909

Average no. in line (Lq): Lq = 2 / (1 - ) = 9.0909

Average waiting time in the queue (Wq): Wq = Lq / = 9.0909 / 60 = 0.152

Average waiting time in the system (W): W = Wq + 1 / = 0.1667

Average no. in the system (L): L = W = 60 0.1667 = 10

Prob.(arriving customer having to wait) (Pw): Pw = 1 - P0 = = 0.9091

Prob.(n customers in the system) (Pn): Pn = n P0

Utilization (): = / = 60 / 66 = 0.9091

Prob.(no customers are in system) (P0): P0 = 1 - = 1 – 0.9091 = 0.0909

Average no. in line (Lq): Lq = 2 / (1 - ) = 9.0909

Average waiting time in the queue (Wq): Wq = Lq / = 9.0909 / 60 = 0.152

Average waiting time in the system (W): W = Wq + 1 / = 0.1667

Average no. in the system (L): L = W = 60 0.1667 = 10

Prob.(arriving customer having to wait) (Pw): Pw = 1 - P0 = = 0.9091

Prob.(n customers in the system) (Pn): Pn = n P0

n 0 1 2 3 …

pn 0.0909 0.0826 0.0751 0.0683 …

20

M M / / M M / / cc Queueing Queueing SystemSystem

Analyzing a Single-Line Multiple-Channel Queueing System with Exponential Arrival and Service Processes (M / M / c)

Queueing System for

Weigh Station

with Two Scales c = 2 servers

= 70 trucks per hour

= 40 trucks per hour on each scale

Analyzing a Single-Line Multiple-Channel Queueing System with Exponential Arrival and Service Processes (M / M / c)

Queueing System for

Weigh Station

with Two Scales c = 2 servers

= 70 trucks per hour

= 40 trucks per hour on each scaleWeigh Station

21

Formulas (Formulas (MM / / MM / / cc))

Performance Measure General Formula

Server utilization

Probability that no customers are in system

Average number in line

Performance Measure General Formula

Server utilization

Probability that no customers are in system

Average number in line

11

!1

!)(

11

0

0

cc

nc

Pc

c

n

n

20

1

)1(!

)(

cc

PcL

c

q

c

22

Formulas (Formulas (MM / / MM / / cc) ) (cont’)(cont’)

Performance Measure General Formula

Average waiting time in queue

Average waiting time in system

Average number in system

Performance Measure General Formula

Average waiting time in queue

Average waiting time in system

Average number in system

1

qWW

q

q

LW

20

1

)1(!

)(

cc

PccL

c

W

23

ComputationComputation

Utilization (): = / c = 70 / (2 40) = 0.875

Prob.(no customers are in system) (P0):

Utilization (): = / c = 70 / (2 40) = 0.875

Prob.(no customers are in system) (P0):

75.275.11!1

75.1

!0

75.1

!)1(

)(

!1

)(

!0

)(

!

)( 101101

0

c

ccc

n

c cc

n

n

25.12853125.1875.01

1

!2

75.1

1

1

!

)( 2

c

c c

06667.015

1

25.1275.2

10

P

11

!1

!)(

11

0

0

cc

nc

Pc

c

n

n

24

Computation (cont’)Computation (cont’)

Average no. in line (Lq):

Average waiting time in the queue (Wq):

Average waiting time in the system (W):

Average no. in the system (L):

Average no. in line (Lq):

Average waiting time in the queue (Wq):

Average waiting time in the system (W):

Average no. in the system (L):

7168.5

06667.0643398.1

06667.0875.01

1

!22

75.12

3

08167.070/7168.5/ qq LW

10667.040/108167.0)/1( qWW

4669.710667.070 WL

20

1

)1(!

)(

cc

PcL

c

q

25

Economic Analysis of Economic Analysis of Queueing SystemQueueing System

American Weavers, Inc. The plant has a large number of machines that jam frequently. Machines are repaired on a first-come-first-serve basis by one of

seven available repair persons. An average of 10 to 12 machines are out of operation at any one

time due to jams. Hiring more repair people will reduce the number of jammed

machines. How many should be hired?

American Weavers, Inc. The plant has a large number of machines that jam frequently. Machines are repaired on a first-come-first-serve basis by one of

seven available repair persons. An average of 10 to 12 machines are out of operation at any one

time due to jams. Hiring more repair people will reduce the number of jammed

machines. How many should be hired?

26

Economic Analysis of Economic Analysis of Queueing System Queueing System (cont’)(cont’)

Modeling Current System: Current number of repair persons (c = 7). The occurrence of jammed machines can be approximated by a

Poisson arrival process with an average rate of 25 per hour

( = 25). Each jammed machine requires a random amount of time for

repair that can be approximated by an exponential distribution with an average service time of 15 minutes, which is an average rate per server of four machines per hour ( = 4).

Modeling Current System: Current number of repair persons (c = 7). The occurrence of jammed machines can be approximated by a

Poisson arrival process with an average rate of 25 per hour

( = 25). Each jammed machine requires a random amount of time for

repair that can be approximated by an exponential distribution with an average service time of 15 minutes, which is an average rate per server of four machines per hour ( = 4).

27

Analyzing the Current Analyzing the Current SystemSystem

M / M / c Queue StatisticsNumber of identical servers 7

Mean arrival rate 25.0000

Mean service rate per server 4.0000

Mean server utilization (%) 89.2857

Expected number of customers in queue 5.8473

Expected number of customers in system 12.0973

Probability that a customer must wait 0.7017

Expected time in the queue 0.2339

Expected time in the system 0.4839

M / M / c Queue StatisticsNumber of identical servers 7

Mean arrival rate 25.0000

Mean service rate per server 4.0000

Mean server utilization (%) 89.2857

Expected number of customers in queue 5.8473

Expected number of customers in system 12.0973

Probability that a customer must wait 0.7017

Expected time in the queue 0.2339

Expected time in the system 0.4839

28

Cost AnalysisCost Analysis

cs = cost per server per unit of time = $50/hour

cw = cost per unit of time for a customer waiting in the system

= $100/hour for lost production, etc.

L = average number of customers in the system

Total Cost = cost of crew + cost of waiting

= cs c + cw L

= 50 7 + 100 12.0973

= $1559.73 per hour

cs = cost per server per unit of time = $50/hour

cw = cost per unit of time for a customer waiting in the system

= $100/hour for lost production, etc.

L = average number of customers in the system

Total Cost = cost of crew + cost of waiting

= cs c + cw L

= 50 7 + 100 12.0973

= $1559.73 per hour

29

Cost Analysis (cont’)Cost Analysis (cont’)

Crew size Expected no. in the system Hourly cost ($)

7 12.0973 50 7 + 100 12.0973 = 1559.73

8 7.7436 50 8 + 100 7.7436 = 1174.36

9 6.7863 50 9 + 100 6.7863 = 1128.63

10 6.7594 50 10 + 100 6.4594 = 1145.94

11 6.3330 50 11 + 100 6.3330 = 1183.30

0

200

400

600

800

1000

1200

1400

1600

1800

7 8 9 10 11

Service Cost

Waiting Cost

Total Cost

0

200

400

600

800

1000

1200

1400

1600

1800

7 8 9 10 11

Service Cost

Waiting Cost

Total Cost