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ISE 195Introduction to Industrial &

Systems Engineering

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Queuing Theory

(Topics in ISE 484)

Queuing Theory

(Topics in ISE 484)

Ref: Applied Management Science, 2nd Ed., Lawrence & PasternackSlides based on textbook slides

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Queuing is Familiar

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Queuing Occurs in Service Systems

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Introduction

• 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 or Minimize Costs

– Desired Service Level.

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Application Areas for Queuing

• Transportation Systems– Cars waiting in traffic– Planes waiting on runways

• Factories– Jobs waiting for machines– Machines waiting for repairs

• Service Systems– Customers at retails stores or restaurants– Paperwork in a business process– Patients at a hospital emergency room or waiting for

an operating room

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Essential Tradeoff in Queuing

• What is the value of improved service to a customer?– Higher levels of service either require more

resources (people, machines, technology) at higher cost

– Saving money on resources often leads to longer waiting times, and lower levels of service

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Introduction

• 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 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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Fundamental Queuing Insights

• Over long periods of time, customers cannot arrive at a rate that is higher than the overall rate of service, or queues get large!– Queuing theory helps us estimate the waiting time– How many resources are enough to guarantee

reasonable queues?

• During short busy periods, rate of arrivals can outpace rate of service, but service will suffer.– How will we manage busy periods?

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Ways to Manage Queues

• Segment the customers– Split off customers that can be handled

quickly and uniformly

• Inform customers of what to expect• Divert the customer’s attention while

waiting• Encourage customers to come during slow

times• Train servers to be friendly

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The 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 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 Process

Note: Arrivals according to Poisson Process have exponential inter-arrivals which turns out to provide quite nice theoretical results.

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HANK’s HARDWARE

• Arrival 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.

– Hank would like to know 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.

X = Number of Arrivals in t = 3 time units

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

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HANK’s HARDWARE

k

P(X = k)

Illustration of the Poisson distribution.

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• Factors that influence the modeling of queues

–Line configuration

–Jockeying

–Balking

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 Configuration

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Waiting Line Configurations

Single Server Multiple ServerSingle Line

Multiple ServerMultiple Lines Tandem Queue

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• Jockeying occurs when customers switch lines once they perceive 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 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 Rules

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Tandem 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 (such as

hospital)

Homogeneity

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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. – Simple model; useful for analysis purposes.

The Service Process

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HANK’s HARDWARE

• Service time – Hank’s estimates the average service time to

be 1/ = 4 minutes per customer.– Service time follows an exponential

distribution.– Define:

• X = Amount of time it takes to serve the next customer

– What is the probability that it will take less than 3 minutes to serve the next customer?

• What is Pr( X < 3 minutes )?

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• Using Excel for the Exponential Probabilities – 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

– In ISE 484 you will learn the details of these calculations; for now we are simply illustrating how the results would be used

HANK’s HARDWARE

3 minutes = .05 hours

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HANK’s HARDWARE

• What is your recommendation to HANK’s management based on this?– Management wants to have 95% of

customers served in less than 3 minutes– You recommend a detailed study of the

service process to implement some lean techniques to improve the level of customer service

– Notice, this is just the service time, and doesn’t include any of the queuing time

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• Characteristics• The “Lack of Memory” 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

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Performance Measures in Queuing Systems

• 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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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 Measures

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MARY’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 SHOES -

• Solution– 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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Excel Queueing Spreadsheet

System 1 System 2Input Parameters

Number of Servers 1 1

Average Processing Rate (jobs per unit time)

1.00 1.00

Average Processing Time (time per job)

1.00000000 1.00000000

Average Arrival Rate (jobs per time unit)

0.50 0.50

Measures of PerformanceUtilization 0.5000 0.5000

Average Time in Queue (time units)

1.00000000 1.00000000

Average Time in System (time units)

2.00000000 2.00000000

Average Number in Queue

0.50000000 0.50000000

Average Number in System

1.00000000 1.00000000

Queueing Model Calculator

Cells that are this color can be changed. Results for the Queueing model are displayed in the other cells.

The formulas used here are for the M/M/m system. The arrival process is assumed to be Poisson (exponential inter-arrival times).

The service process is assumed to have exponential processing times.

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MARY’s SHOES

• Recommendations to management– 16 minutes of time in queue seems excessive– Question 1: Is the model correct?

• Compare model predictions to reality

– How does the 16 minute waiting time compare to the goals for customer service for Mary’s shoes?

• Make recommendations to improve waiting time– More resources

– Better processes

– Better information systems

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Tandem 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

Tandem Queuing Systems

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

• 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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Applications of Queuing

• Determine number of servers

• Examine line configurations

• Evaluate efficiency of process

• Determine storage requirements

• Various cost benefit analyses

• Bound performance of complex systems

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Other Applications

• In ISE 483 (Integrated Systems for Manufacturing)– Use Queuing models to decide number of machines,

workers in a system– Extend queuing models to include impact of downtime

(breakdowns, maintenance)

• In ISE 471 (Simulation)– Use queuing models to get a quick “guess” at what

the result of a simulation should be close to– Helps in debugging and validating models

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Questions?Questions?

Ref: Applied Management Science, 2nd Ed., Lawrence & PasternackSlides based on textbook slides