Waiting Lines and Queuing Models

Queuing Theory

The study of the behavior of waiting lines Importance to business

There is a tradeoff between faster lines and increasedcosts faster lines suggests an increase in service, thus an increa

se in costs longer waiting times negatively affects customer satisfact

ion

What is the ‘ideal’ level of services that a firm should provide?

Management Uses from Queu ing Theory

Is it worthwhile to invest effort in reducing the serv ice time?

How many servers should be employed? Should priorities for certain types of customers be i

ntroduced? Is the waiting area for customers adequate?

Answers to these questions can be obtained with Analytic methods or queuing theory (formula ba

sed); and Simulation (computer based).

Queuing System Characteristics

Arrivals Waiting in Line Service Facility

Arrival Characteristics

Size of the calling population Finite: ex. 300 computers on campus maintained by 5 compute

r technicians (customers arriving for service are limited) Infinite: ex. c ars arriving at a highway tollbooth, shoppers arrivi

ng at a supermarket (the source is forever “abundant”)

Pattern of Arrivals Nonrandom: arrivals take place according to some known sche

dule (ex. assembly line) Random: arrivals are independent and cannot be predicted exa

ctly

Random Pattern of Arrival

Poisson Distribution a probability distribution that can be used to determine

the probability of X transactions arriving in a given timeinterval

P(X ) = for X = 0, 1, 2, 3, 4

where, P(X ) = probability of X arrivals X = number of arrivals per unit of time = average arrival rate e = 2.7183

e- X

X!

Examples of Poisson Distribution for Arrival Times

0 1 2 3 4 5 6 7 8 9

025

020.

015.

010.

005.

= 2 Distribution

Pro

bab

ility

Pro

bab

ility

0 1 2 3 4 5 6 7 8 9 10 11

025

020.

015.

010.

005.

4 4 444444444444

XX

Arrival Characteristics

Size of the calling population Finite: 300 5ex. computers on campus maintained by computer tec

44444444 Infinite: ex. 44444444 44 4 4444444 4444444444 44444444 44444444 44 4 ,

44444444444 Pattern of Arrivals

Random: 44444444 444 44444444444 444 444444 44 444444444 4444444 Nonrandom: ar r i val s t ake pl ace accor di ng t o some known schedul e

Behavior of the Arrivals Balking: customers who refuse to enter the system because the line is

too long Reneging: customers who enter the queue but leave without

completing their transactions Jockeying: swi t chi ng bet ween l i nes

Waiting Line Characteristics

Queue Length 444 444444 44 444 44444 44 444 4444 44 44444444 4444:

rictions ex. w aiting room

Unlimited: the length of the queue is not restricted

Queue Discipline Rule by which customers in the line are to receive service

Static: FCFS, first come first serve, FIFO, first in first out Dynamic: Priority e.g., rush jobs at a shop are processed

ahead of regular jobs

Service Facility Characteristics

Basic Queuing System Configurations Single Channel

one service provider per phase Multiple Channel

more than one service provider in a phase

Basi c Single 4444e 44444444444444

Service FacilityArrivals

Queue

Departures after Service

Single-Channel, Single-Phase System

Type 1 Service Facility

Arrivals

Queue Departures afterService

Single-Channel, Multiphase System

Type 2 Service Facility

Arrivals

Queue Departures

after

Service

Multichannel, Single-Phase System

Service Facility

1

Service Facility

2

Service Facility

3

Arrivals

Queue

Departures afterService

Multichannel, Multiphase System

Type 1 Service Facility

1

Type 1 Service Facility

2

Type2 Service Facility

1

Type 2 Service Facility

2

Multiple Queue Configurations

Arrivals

Departures

after

Service

Multiple Queue

Service Facility

1

Service Facility

2

Service Facility

3

Arrivals

Take a Number

Service Facility

4

7 3

Departures

after

Service

Service Facility

1

Service Facility

2

Service Facility

3

Service Facility

4

11

9

105

8

4

12

6

Service Facility Characteristics

Basic Queuing System Configurations Single Channel

one service provider per phase Multiple Channel

more than one service provider in a phase

Service Time Distribution Constant: it takes the same amount of time to service each

customer or unit Random: service times vary across customers or units

Examples of Exponential Distribu tion for Service Times

4444444 4444 44444440 30 60 90 120 150 180

s)

Probability (Service Takes Longer Than X Minutes) = e-uX for X > 0

Pro

bab

ility

(for

inte

rvals

of1

min

ute

)

u = Average Number Served per Minute

Average Service Time of 20 Minutes

Average Service Time of 1 hour

Assumptions of the Single-Channel, Single-Phase Model

Arrivals are served on a FIFO basis Every arrival waits to be served regardless of the length of the

line: that is there is no balking or reneging Arrivals are independent of preceding arrivals, but the

average number of arrivals (the arrival rate, λ) does not change over time

Arrivals are described by a Poisson probability distribution and come from an infinite or very large population

Service times also vary from one customer to the next and are independent of one another, but their average rate (μ) is known

Service times occur according to the negative exponential probability distribution

The average service rate is greater than the average arrival rate

Idea of Uncertainty

Note here that integral to queuing situation s is the idea of uncertainty in

Interarrival times (arrival of customers) Service times (service time per customer)

This means that probability and statistics are nee ded to analyze queuing situations.

System Performance Measures

Important to measuring the performance of the system are the parameters:

λ = the average number of arrivals per time period

μ = the average number of people or items served

per time period

System Performance Measures

Number of units in the system (customers) Average number in system (L or Ls) Average queue length (Lq)

Waiting Times Average time in the system (W or Ws) Average time in queue (Wq)

Utilization Rates Utilization factor () Probability of idle time (P0)

Queuing Equations

Average number in system (L or Ls)

Average queue length (Lq)

Average time in the system (W or Ws)

Average time in queue (Wq)

Utilization factor ()

Probability of idle time (P0)

L =

Lq =

W =

Wq=

=

P0 = 1 -

λμ - λ

λ2

μ (μ – λ)

1μ - λ

λμ (μ – λ)

λμ

λμ

Queuing Equations

Probability that the number of customers in the system is greater than k, Pn>k

where n = number of units in the system

Pn>k= ( )λ k + 1

μ

When to use what model?

Use Single-channel model, when you have Only one service provider Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist) FIFO

When to use what model? (2)

Use Multi-channel model, when you have More than one service providers Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying Random (inconstant) service times (Expo Dist)

but both channel must perform at the same rate

When to use what model? (3)

Use Constant-service time model, when you have Constant service times (a fixed cycle) Infinite source (calling population) Random pattern of arrivals (Pois Dist) No balking, reneging, jockeying

The question will be asking about either to choose the new or the old machines.

When to use what model? (4)

Use finite population model, when you have Finite source (calling population) Random (inconstant) service times (Expo Dist) Only one service providers Random pattern of arrivals (Pois Dist) FCFS

SUM

Finite Finite pop modelYes

No Constant ServTime

1 Channel >1 Channel

YesConstant Model

No

Single-Chn Model Multi-Chn Model