D S Lecture #3 - Queing Theory

8/7/2019 D S Lecture #3 - Queing Theory

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USE OF PROBABILISTIC

MODELS TO SOLVE REALWORLD PROBLEMS:

QUEUING THEORY

DECISION SCIENCE

LECTURE # 3


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WHAT IS QUEUING THEORY ? Instances of waiting in a line:

Models developed to better understand& make decisions on the problem ofwaiting in lines.

A waiting line is called a queue

The body of knowledge dealing withwaiting lines is called QUEUINGTHEORY


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Waiting Line/Queuing Theory Modelsconsist of Mathematical Formulas and

relationships that can be used todetermine the OPERATINGCHARACTERISTICS (performancemeasures) for a waiting line.

There are many such operatingcharacteristics only some will bestudied

WHAT IS QUEUING THEORY ?


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OPERATING

CHARACTERISTICS(PERFORMANCE MEASURES)

1. The Probability that no units are in the system.

2. The Average Number of units in the waiting line3. The Average Number of units in the system (the

number of units in the waiting line plus the number ofunits being served)

4. The Average Time a unit spends in the waiting line5. The Average Time a unit spends in the system (the

waiting time plus the service time)

6. The Probability that an arriving unit has to wait forservice.


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Single-Channel Waiting Line

STRUCTURE OF A WAITING

LINE SYSTEM


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STRUCTURE OF A WAITING LINE

SYSTEM SINGLE CHANNEL

a server takes a customers order

Determines the total cost of the orderTakes the money from the customer

Fills the order

The server then takes the order of the next customerwaiting for service


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STRUCTURE OF A WAITING LINE

SYSTEM MULTIPLE CHANNEL


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Distribution of Arrivals

Defining the arrival process for a waiting line

involves determining the probabilitydistribution for the number of arrivals in agiven period of time

Arrivals usually occur randomly and

independently of other arrivals The Poisson Probability Distribution

provides a good description of the

arrival pattern.


LINE SYSTEM


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Distribution of Service Times

The time a customer spends at the service

facility once the service has started startswhen the customer begins to place the orderwith the server and ends when the customerhas received the order

Probability distribution is assumed to followan Exponential Probability distribution


LINE SYSTEM


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Queue Discipline

In general, for most customer-oriented

waiting lines, the units waiting forservice are arranged on a first-come,first served basis: referred to as anFCFS queue discipline.

Other types: Entering and leaving a elevator

Priority cases ( the aged, the disabled)


LINE SYSTEM


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Steady-State Operation

I.e. normal state.

When Best Patty Co. Ltd opens in themorning, no customers are in therestaurant. Gradually, activity builds up

to a normal or steady state.

Start-up period transient period

Normal period Steady-state


LINE SYSTEM


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QUEUING MODELS 5 TYPES

1. Single-Channel Waiting Line Modelwith Poisson Arrivals and ExponentialService Times (infinite callingpopulation - M/M/1) - Basic;

2. Multiple-Channel Waiting Line Model with PoissonArrivals and Exponential Service Times;

3. Single-Channel Waiting Line Model with PoissonArrivals and Arbitrary Service Times; (M/G/1)

4. Multiple-Channel Model with Poisson Arrivals,Arbitrary Service Times, and no waiting line;(M/G/k)

5. Waiting Line Models with Finite Calling Populations(M/M/1)


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Operating Characteristics are calculatedbased on the method chosen as a result of

the queuing/waiting line trend; If the Operating Characteristics are

unsatisfactory in terms of meeting thecompany standards of service, thenManagement should consider alternativedesigns or plans for improving the waitingline operation.

QUEUING MODELS USE OF

THE RESULTS


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Waiting line models often indicate whereimprovements in Operating

Characteristics are desirable. The decision of how to modify the waiting

line configuration to improve the operatingcharacteristics must be based on the

insights and creativity of the analyst. To make improvements in the waiting line

operation, analysts often focus on ways toimprove the service rate


THE RESULTS


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Generally, service rate improvementsare obtained by making either or both

the following changes:1. Increase the mean service rate by

making a creative design change or byusing new technology

2. Add service channels so that morecustomers can be servedsimultaneously.


THE RESULTS


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OPERATING

CHARACTERISTICS(PERFORMANCE MEASURES)

1. The Probability that no units are in the system.

2. The Average Number of units in the waiting line

3. The Average Number of units in the system (thenumber of units in the waiting line plus the number ofunits being served)

4. The Average Time a unit spends in the waiting line

5. The Average Time a unit spends in the system (thewaiting time plus the service time)


7. The probability of n units in the system


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Single-Channel Waiting Line Model

with Poisson Arrivals andExponential Service Times -infinitepopulation M/M/1) - Basic

i.e.1. No finite calling populations

2. Steady-state operating characteristics fora single-channel waiting line;

3. Arrival follows a Poisson probabilitydistribution

4. Service times follow an Exponential

probability distribution


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Single-Channel Waiting

Line Model with Infinitepopulation - M/M/1) - Basic Applicable only when the the mean service rate is greater than the mean arrival rate ,

I.e./ < 1. If this condition does not exist, the waiting

line will continue to grow without limitbecause the service facility does not have

sufficient capacity to handle the arrivingunits.

In the Operating Characteristics(formulas/performance measures, for this

model, >

.


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=> the mean number of arrivals pertime per time period (the mean arrivalrate)

=> the mean number of services pertime period ( the mean service rate)

Single-Channel Waiting

Line Model with Infinitepopulation


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1. The Probability that no units are in the system

Po= 1 2. The Average Number of units in the waiting line

Lq = 2 .( -

3. The Average Number of units in the system (the number ofunits in the waiting line plus the number of units being served)

L = Lq +

Single-Channel Waiting LineModel with Infinitepopulation


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4. The Average Time a unit spends in the waitingline

Wq = Lq

5.The Average Time a unit spends in thesystem (the waiting time plus the service time

W = Wq + 1


Model with Infinitepopulation


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Pw=


Pn = (/)nPo


Model with Infinitepopulation


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In the case of Best Patty Co. Ltd, the mean

arrival rate = 0.75 customers perminute and a mean service rate =1customer per minute. Thus >;

Thus, the equations above can be used

to provide Operating Characteristics forthe Best Patty Co. Ltd single-channelwaiting line - M/M/1) - Basic .

Single-Channel Waiting LineModel with Infinite

population Worked Example


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1. The Probability that no units are in the system

Po= 1 =1 0.75 = 0.25

1

2. The Average Number of units in the waiting line

Lq = 2 . = 0.752 . = 2.25 customers( ) 1(1 0.75)

3. The Average Number of units in the system (thenumber of units in the waiting line plus the number ofunits being served)

L = Lq + = 2.25 + 0.75 = 3 customers

1

Single-Channel Waiting LineModel with Infinite population

Worked Example


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4. The Average Time a unit spends in the waitingline

Wq = Lq = 2.25 = 3 minutes 0.75

5.The Average Time a unit spends in the system(the waiting time plus the service time

W = Wq + 1 = 3 + 1 = 4 minutes 1


Worked Example


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Pw= = 0.75=0.75

1


Pn =(/

)

nPo

This equation can be used to determine theprobability of any number of customers in thesystem.


Worked Example


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The probability of n units in the system

Pn = (/)n

Po

P1= (0.75/1)1 x 0.25= 0.75 x 0.25 = 0.1875

P2 = (0.75/1)2

x 0.25

= 0.5625 x 0.25 = 0.1406


Worked Example


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The probability of n units in the system

Pn = (/)n

Po

Number of Customers Probability

0 0.2500

1 0.1875

2 0.1406

3 0.1055

4 0.0791

5 0.0593

6 or more 0. 2373


Worked Example


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1. Customers have to wait an average of3 minutesbefore beginning to place an order, which

appears somewhat long for a business based onfast service

2. The facts that the average number of customerswaiting in line is 2.25 and that75% of the

arriving customers have to wait for service areindicators that something should be done toimprove the waiting line operation.


Worked Example -Interpretation


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3. The table shows 0.2373 probability that sixor more customers are in the Best Patty

system at one time. This condition indicates afairly high probability that Best Patty willexperience some long waiting times if itcontinues to use the single-channel operation.

4. If the operation characteristics are

unsatisfactory in terms of meeting companystandards for service, Best Pattys Managementshould consider alternative designs or plans forimproving the waiting line operation


Worked Example -Interpretation


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In the Single-Channel Infinite CallingPopulation Model, denotes the meanarrival for the system.

For the Single-Channel with a finite callingpopulation, the mean arrival rate for thesystem varies, depending on the numberof units in the system. Instead ofadjusting for the changing system arrivalrate, in this model indicates the mean

arrival rate for each unit.

Single-Channel Infinite Calling

Population vs Single-ChannelFinite Calling Populations (M/M/1)


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A TAXONOMY OF

QUEUING MODELSThere are many possible queuing models. For example, if the interarrival time in the basic model had been given a

different distribution (not the exponential) we would have had a different model, in the sense that the previous formulas for L,

Lq , and so on, would no longer hold. To facilitate communication among those working on queuing models, D. G.

Kendall proposed a taxonomy based on the following notation:

A/B/s

where A = arrival distribution

B = service distribution

s = number of servers

Different letters are used to designate certain distributions. Placed in the A or the B position, they indicate the arrival

or the service distribution, respectively. The following conventions are in general use:

M = exponential distribution

D = deterministic numberG = any (a general) distribution of service times

GI = any (a general) distribution of arrival times

We can see, for example, that the Xerox model is an M/M/1 model; that is, a single-server queue with exponential

inter-arrival and service times.

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D S Lecture #3 - Queing Theory