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Systems Analysis MethodsDr. Jerrell T. Stracener, SAE Fellow

SMUEMIS 5300/7300

NTUSY-521-N

NTUSY-521-N

SMUEMIS 5300/7300

Queuing Modeling and Analysisupdated 11.16.01

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Queuing Models

• Several models exist, depending on the structure of the system, the nature of arrivals, the service policies, and the behavior of the customers in the queue.

• These queuing situations are commonly designated X/Y/Z where X indicates the arrival process, Y indicates the service process, and Z the number of servers.

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Queuing Models

• Some queuing situations are:1. Single server (1), Poisson arrivals (M), exponential service (M), called M/M/1.2. Single server, Poisson arrival, exponentialservice, with finite (limited) queue length:M/M/1 finite queue.3. M/M/1 finite source (a finite calling population).4. M/arbitrary/1 (arbitrary service time distribution,but mean and standard deviation are known).5. M/M/K (multiple servers: K).6. M/M/K finite queue.7. M/M/K finite source.8. M/constant/K (constant service times).

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Solution Approaches

There are two basic approaches to the solutionof queuing problems: analytical and simulation.

• The analytic approach - The measures of performance are determined through the use offormulas. Unfortunately, many queuing situationsare so complex that the analytic approach iscompletely impractical or even impossible.

• Simulation - For those situations in which theanalytic approach is unsuitable, the procedure ofsimulation can be used.

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Information Flow in Waiting Line Models

• It is helpful to use some measures of performancewhen evaluating service alternatives, particularlywhen a cost approach is planned.

• A solution to a queuing problem means computingcertain measures of performance

• These measures are computed from three inputvariables:

, the mean arrival rate, the mean service rate, the number of servers

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Information Flows

W = Average waiting time, per customer in the systemWq = Average waiting time, per customer in the queueL = Average number of customers in the systemLq = Average number of customers in the queueP(0) = Probability of the system being idlePw = Probability of the system being busyP(t > T) = Probability of waiting longer than time TP(n) = Probability of having exactly n customersin the systemP(n > N), P(n < N) - Probability of finding more than, or less than, N customers in the system

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Deterministic Queuing Systems

The simplest and the rarest of all waiting line situations involves constant arrival rates and constant service times. Three cases can bedistinguished:

1. Arrival rate equals service rate. Assume thatpeople arrive every 10 minutes, to a single server,where the service takes exactly 10 minutes. Thenthe server will be utilized continuously (100%utilization), and there will be no waiting line.

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Deterministic Queuing Systems

2. Arrival rate larger than service rate. Assume that there are six arrivals per hour (one every 10minutes) and the service rate is only five per hour(12 minutes each). Therefore, one arrival cannotbe served each hour, and a waiting line will buildup (at a rate of one per hour). Such a waiting linewill grow and grow as time passes and is termedexplosive.

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Deterministic Queuing Systems

3. Arrival rate smaller than service rate. Assumethat there are again six arrivals per hour but theservice capacity is eight per hour. In this case thefacility will be utilized only 6/8 = 75% of the time.There will never be a waiting line (if the arrivalscome at equal intervals).

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The Basic Poisson-Exponential Model (M/M/1)

The classical and probably best known of all waiting line models is the Poisson-exponentialsingle server model. It exhibits the followingcharacteristics.

Arrival rate - The arrival rate is assumed to be random and is described by Poisson distribution. The average arrival rate is designated by the Greekletter .

Service time - The service time is assumed to followthe negative exponential distribution. The averageservice rate is designated by the Greek letter , andthe average service time by 1/.

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The Major Ground Rules for the Operation of aSingle Server System

• Infinite source of population

• First-come, first-served treatment

• The ratio / is smaller than 1. This ratio isdesignated by the Greek letter . The ratio is ameasure of the utilization of the system. If theutilization factor is equal to or larger than 1, thewaiting line will increase without bound (will beexplosive), a situation which is unacceptable tomanagement.

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The Major Ground Rules for the Operation of aSingle Server System

• Steady state (equilibrium) exists. A system is in a ‘transient state’ when its measures ofperformance are still dependent on the initialconditions. However, our interest is in the ‘longrun’ behavior of the system, commonly known assteady state. A steady state condition occurs whenthe system becomes independent of time.

• Unlimited queuing space exists.

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Managerial Use of the Measures of Performance

Some of these measures can be used in a costanalysis, while others are used to aid in determining service level policies. For example:

a. A fast-food restaurant wants to design its service facility such that a customer will not wait,on the average, more than two minutes (i.e.,Wq 2 minutes) before being served.

b. A telephone company desires that the probability of any customer being without telephone service more than two days be 3% (i.e., P(t > 2 days) = 0.03

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Managerial Use of the Measures of Performance

c. A bank’s policy is that the number of customersat its drive-in facility will exceed 10 only 5% ofthe time (i.e., P(n > 10) = 0.05.

d. A city information service should be busy atleast 60% of the day (i.e., Pw > 0.6).

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Example - The Toolroom problem

The J.C. Nickel Company toolroom is staffed by oneclerk who can serve 12 production employees, onthe average, each hour. The production employeesarrive at the toolroom every six minutes, on the average. Find the measures of performance.

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Example - The Toolroom problem solution

It is necessary first to change the time dimensionsof and to a common denominator. is not givenin minutes, in hours. We will use hours as the common denominator.

1. Average waiting time in the system (toolroom)

hours, per employee

2. The average waiting time in the line.

hours,per employee

5.01012

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W

417.0101212

1010

qW

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Example - The Toolroom problem solution

3. The average number of employees in the toolroom area

employees

4. The average number of employees in the line.

employees

51012

10

L

17.4101212

1002

qL

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Example - The Toolroom problem solution

5. The probability that the toolroom clerk will beidle.

6. The probability of finding the system busy.

167.012

10110

P

833.012

10

wP

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Example - The Toolroom problem solution

7. The chance of waiting longer than 1/2 hour inthe system. That is T = 1/2.

8. The probability of finding four employees inthe system, n = 4.

368.012/11210

eeTtP

0.080412

2

12

10

μ

λ1

μ

λ4P

4n

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Example - The Toolroom problem solution

9. The probability of finding more than threeemployees in the system.

488.012

103

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N

nP