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IntroductionIntroduction

• Queuing is the study of waiting lines, or queues.• The objective of queuing analysis is to design

systems that enable organizations to perform optimally according to some criterion.

• Possible Criteria– Maximum Profits.

– Desired Service Level.

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IntroductionIntroduction

• Analyzing queuing systems requires a clear understanding of the appropriate service measurement.

• Possible service measurements– Average time a customer spends in line.– Average length of the waiting line.– The probability that an arriving customer must wait

for service.

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Elements of the Queuing ProcessElements of the Queuing Process

• A queuing system consists of three basic components:– Arrivals: Customers arrive according to some arrival

pattern.

– Waiting in a queue: Arriving customers may have to wait in one or more queues for service.

– Service: Customers receive service and leave the system.

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The Arrival ProcessThe Arrival Process

• There are two possible types of arrival processes

– Deterministic arrival process.

– Random arrival process.• The random process is more common in

businesses.

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• Under three conditions the arrivals can be modeled as a Poisson process– Orderliness : one customer, at most, will arrive during any

time interval.

– Stationarity : for a given time frame, the probability of arrivals within a certain time interval is the same for all time intervals of equal length.

– Independence : the arrival of one customer has no influence on the arrival of another.

The Arrival ProcessThe Arrival Process

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P(X = k) =

Where = mean arrival rate per time unit.

t = the length of the interval.

e = 2.7182818 (the base of the natural logarithm).k! = k (k -1) (k -2) (k -3) … (3) (2) (1).

tke- t

k!

The Poisson Arrival ProcessThe Poisson Arrival Process

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HANK’s HARDWARE – HANK’s HARDWARE – Arrival ProcessArrival Process

• Customers arrive at Hank’s Hardware according to a Poisson distribution.

• Between 8:00 and 9:00 A.M. an average of 6 customers arrive at the store.

• What is the probability that k customers will arrive between 8:00 and 8:30 in the morning (k = 0, 1, 2,…)?

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k

• Input to the Poisson distribution

= 6 customers per hour.t = 0.5 hour.t = (6)(0.5) = 3.

t e- t

k !

0

0.049787

0

1!

1

0.1493612

2!0.224042

3

3! 0.224042

1 2 3 4 5 6 7

P(X = k )=

8

123

HANK’s HARDWARE – HANK’s HARDWARE – An illustration of the Poisson distribution. An illustration of the Poisson distribution.

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HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Poisson probabilitiesUsing Excel for the Poisson probabilities

• Solution– We can use the POISSON function in Excel to

determine Poisson probabilities.– Point probability: P(X = k) = ?

• Use Poisson(k, t, FALSE)• Example: P(X = 0; t = 3) = POISSON(0, 1.5, FALSE)

– Cumulative probability: P(Xk) = ?• Example: P(X3; t = 3) = Poisson(3, 1.5, TRUE)

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HANK’s HARDWARE – HANK’s HARDWARE – Excel PoissonExcel Poisson

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• Factors that influence the modeling of queues

– Line configuration

– Jockeying

– Balking

The Waiting Line CharacteristicsThe Waiting Line Characteristics

– Priority

– Tandem Queues

– Homogeneity

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• A single service queue.• Multiple service queue with single waiting line.• Multiple service queue with multiple waiting

lines.• Tandem queue (multistage service system).

Line ConfigurationLine Configuration

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• Jockeying occurs when customers switch lines once they perceived that another line is moving faster.

• Balking occurs if customers avoid joining the line when they perceive the line to be too long.

Jockeying and BalkingJockeying and Balking

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• These rules select the next customer for service.• There are several commonly used rules:

– First come first served (FCFS).– Last come first served (LCFS).– Estimated service time.– Random selection of customers for service.

Priority RulesPriority Rules

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Tandem QueuesTandem Queues

• These are multi-server systems. • A customer needs to visit several service

stations (usually in a distinct order) to complete the service process.

• Examples– Patients in an emergency room.– Passengers prepare for the next flight.

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• A homogeneous customer population is one in which customers require essentially the same type of service.

• A non-homogeneous customer population is one in which customers can be categorized according to: – Different arrival patterns– Different service treatments.

HomogeneityHomogeneity

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• In most business situations, service time varies widely among customers.

• When service time varies, it is treated as a random variable.

• The exponential probability distribution is used sometimes to model customer service time.

The Service ProcessThe Service Process

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f(t) = e-t

= the average number of customers who can be served per time period.Therefore, 1/ = the mean service time.

The probability that the service time X is less than some “t.”

P(X t) = 1 - e-t

The Exponential Service Time DistributionThe Exponential Service Time Distribution

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Schematic illustration of the exponential Schematic illustration of the exponential distributiondistribution

The probability that service is completed within t time units

P(X t) = 1 - e-t

X = t

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HANK’s HARDWARE – HANK’s HARDWARE – Service timeService time

• Hank’s estimates the average service time to be 1/ = 4 minutes per customer.

• Service time follows an exponential distribution.• What is the probability that it will take less than 3

minutes to serve the next customer?

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• We can use the EXPDIST function in Excel to determine exponential probabilities.

• Probability density: f(t) = ?– Use EXPONDIST(t, , FALSE)

• Cumulative probability: P(Xk) = ?– Use EXPONDIST(t, , TRUE)

Using Excel for the Exponential ProbabilitiesUsing Excel for the Exponential Probabilities

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• The mean number of customers served per minute is ¼ = ¼(60) = 15 customers per hour.

• P(X < .05 hours) = 1 – e-(15)(.05) = ?• From Excel we have:

– EXPONDIST(.05,15,TRUE) = .5276

HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Exponential ProbabilitiesUsing Excel for the Exponential Probabilities

3 minutes = .05 hours

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HANK’s HARDWARE – HANK’s HARDWARE – Using Excel for the Exponential ProbabilitiesUsing Excel for the Exponential Probabilities

=EXPONDIST(B4,B3,TRUE)

Exponential Distribution for Mu = 15

0.0002.0004.0006.0008.00010.00012.00014.00016.000

0.000 0.075 0.150 0.225 0.300 0.375

t

f(t)

=EXPONDIST(A10,$B$3,FALSE)Drag to B11:B26

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• The memoryless property.– No additional information about the time left for the completion of a

service, is gained by recording the time elapsed since the service started.

– For Hank’s, the probability of completing a service within the next 3 minutes is (0.52763) independent of how long the customer has been served already.

• The Exponential and the Poisson distributions are related to one another.

– If customer arrivals follow a Poisson distribution with mean rate , their interarrival times are exponentially distributed with mean time 1/

The Exponential Distribution -The Exponential Distribution - CharacteristicsCharacteristics

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9.3 Performance Measures of 9.3 Performance Measures of Queuing System Queuing System

• Performance can be measured by focusing on:– Customers in queue.

– Customers in the system.

• Performance is measured for a system in steady state.

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Roughly, this is a transient period…

n

Time

9.3 Performance Measures of 9.3 Performance Measures of Queuing System Queuing System

• The transient period occurs at the initial time of operation.

• Initial transient behavior is not indicative of long run performance.

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This is a steady state period………..

n

Time

9.3 Performance Measures of 9.3 Performance Measures of Queuing System Queuing System

• The steady state period follows the transient period.

• Meaningful long run performance measures can be calculated for the system when in steady state.

Roughly, this is a transient period…

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kEach with

service rate of

kEach with

service rate of

…

For k serverswith service rates

…

For k serverswith service rates

For one server

For one server

In order to achieve steady state, theeffective arrival rate must be less than the sum of the effective service rates .

In order to achieve steady state, theeffective arrival rate must be less than the sum of the effective service rates .

9.3 Performance Measures of 9.3 Performance Measures of Queuing System Queuing System

k servers

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P0 = Probability that there are no customers in the system.P0 = Probability that there are no customers in the system.

Pn = Probability that there are “n” customers in the system.Pn = Probability that there are “n” customers in the system.

L = Average number of customers in the system.L = Average number of customers in the system.

Lq = Average number of customers in the queue.Lq = Average number of customers in the queue.

W = Average time a customer spends in the system.W = Average time a customer spends in the system.

Wq = Average time a customer spends in the queue.Wq = Average time a customer spends in the queue.

Pw = Probability that an arriving customer must wait for service.

Pw = Probability that an arriving customer must wait for service.

= Utilization rate for each server (the percentage of time that each server is busy).

= Utilization rate for each server (the percentage of time that each server is busy).

Steady State Performance MeasuresSteady State Performance Measures

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• Little’s Formulas represent important relationships between L, Lq, W, and Wq.

• These formulas apply to systems that meet the following conditions:– Single queue systems,– Customers arrive at a finite arrival rate and– The system operates under a steady state condition.

L =W Lq = Wq L = Lq +

Little’s FormulasLittle’s Formulas

For the case of an infinite population

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• Queuing system can be classified by:– Arrival process.– Service process.– Number of servers.– System size (infinite/finite waiting line).– Population size.

• Notation– M (Markovian) = Poisson arrivals or exponential service time.– D (Deterministic) = Constant arrival rate or service time.– G (General) = General probability for arrivals or service time.

Example:

M / M / 6 / 10 / 20

Example:

M / M / 6 / 10 / 20

Classification of QueuesClassification of Queues

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MMMM1 Queuing System - 1 Queuing System - AssumptionsAssumptions

– Poisson arrival process.

– Exponential service time distribution.

– A single server.

– Potentially infinite queue.

– An infinite population.

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The probability thata customer waits in the system more than “t” is P(X>t) = e-( - )t

The probability thata customer waits in the system more than “t” is P(X>t) = e-( - )t

P0 = 1 – ()

Pn = [1 – ()]()n

L = ( – )Lq = 2 [( – )]

W = 1 ( – )Wq = [( – )]

Pw =

=

M / M /1 Queue - Performance MeasuresM / M /1 Queue - Performance Measures

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MARY’s SHOESMARY’s SHOES

• Customers arrive at Mary’s Shoes every 12 minutes on the average, according to a Poisson process.

• Service time is exponentially distributed with an average of 8 minutes per customer.

• Management is interested in determining the performance measures for this service system.

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MARY’s SHOESMARY’s SHOES - - SolutionSolution– Input = 1/12 customers per minute = 60/12 = 5 per hour. = 1/ 8 customers per minute = 60/ 8 = 7.5 per hour.

– Performance Calculations

P0 = 1 - () = 1 - (57.5) = 0.3333

Pn = [1 - ()]()n = (0.3333)(0.6667)n L = ( - ) = 2Lq = 2[( - )] = 1.3333

W = 1( - ) = 0.4 hours = 24 minutesWq = ( - )] = 0.26667 hours = 16 minutes

P0 = 1 - () = 1 - (57.5) = 0.3333

Pn = [1 - ()]()n = (0.3333)(0.6667)n L = ( - ) = 2Lq = 2[( - )] = 1.3333

W = 1( - ) = 0.4 hours = 24 minutesWq = ( - )] = 0.26667 hours = 16 minutes

Pw = =0.6667= =0.6667

P(X<10min) = 1 – e-2.5(10/60)

= .565

– = 7.5 – 5 = 2.5 per hr.

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=A11-B4/B5

=1-B4/B5

=A11/B4

=C11-1/B5

=B4/B5

=H11*($B$4/$B$5)Drag to Cell AL11

=1-E11

MARY’s SHOESMARY’s SHOES - -

Spreadsheet solutionSpreadsheet solution

=B4/(B5-B4)

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Economic Analysis of Queuing Economic Analysis of Queuing SystemsSystems

• The performance measures previously developed are used next to determine a minimal cost queuing system.

• The procedure requires estimated costs such as: – Hourly cost per server .– Customer goodwill cost while waiting in line. – Customer goodwill cost while being served.

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Tandem Queuing SystemsTandem Queuing Systems

• In a Tandem Queuing System a customer must visit several different servers before service is completed.

Beverage Meats

• Examples– All-You-Can-Eat restaurant

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Beverage Meats

• In a Tandem Queuing System a customer must visit several different servers before service is completed.

Tandem Queuing Systems

• Examples– All-You-Can-Eat restaurant

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Beverage Meats

Tandem Queuing Systems

– A drive-in restaurant, where first you place your order, then pay and receive it in the next window.

– A multiple stage assembly line.

• Examples– All-You-Can-Eat restaurant

• In a Tandem Queuing System a customer must visit several different servers before service is completed.

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Tandem Queuing SystemsTandem Queuing Systems

• For cases in which customers arrive according to a Poisson process and service time in each station is exponential, ….

Total Average Time in the System = Sum of all average times at the individual stations

Total Average Time in the System = Sum of all average times at the individual stations

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BIG BOYS SOUND, INC.BIG BOYS SOUND, INC.

• Big Boys sells audio merchandise.• The sale process is as follows:

– A customer places an order with a sales person.

– The customer goes to the cashier station to pay for the order.

– After paying, the customer is sent to the pickup desk to obtain the good.

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• Data for a regular Saturday– Personnel.

• 8 sales persons are on the job.• 3 cashiers.• 2 workers in the merchandise pickup area.

– Average service times.• Average time a sales person waits on a customer is 10 minutes.• Average time required for the payment process is 3 minutes.• Average time in the pickup area is 2 minutes.

– Distributions.• Exponential service time at all the service stations.• Poisson arrival with a rate of 40 customers an hour.

BIG BOYS SOUND, INC.BIG BOYS SOUND, INC.

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What is the average amount of time, a customer who makes a purchase spends in the store?

Only 75% of the arriving customers make a purchase!

BIG BOYS SOUND, INC.BIG BOYS SOUND, INC.

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BIG BOYS SOUND, INC. – SolutionBIG BOYS SOUND, INC. – Solution

• This is a Three Station Tandem Queuing System

Sales ClerksM / M / 8

CashiersM / M / 3

Pickup deskM / M / 2

= 40

= 30

= 30

W1 = 14 minutesW2 = 3.47 minutes

W3 = 2.67 minutes

Total = 20.14 minutes.

(.75)(40)=30