D - 1

DD Waiting-Line ModelsWaiting-Line Models

PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e

PowerPoint slides by Jeff Heyl

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Outline Queuing Theory Characteristics of a Waiting-Line System

Arrival Characteristics Waiting-Line Characteristics Service Characteristics Measuring a Queue’s Performance

Queuing Costs Queuing Models

Model A(M/M/1): Single-Channel Queuing Model with Poisson Arrivals and Exponential Service Times

Little’s Law

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Learning Objectives

1. Describe the characteristics of arrivals, waiting lines, and service systems

2. Apply the single-channel queuing model equations

3. Conduct a cost analysis for a waiting line

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Queuing Theory

The study of waiting lines Waiting lines are common situations Useful in both

manufacturing and service areas

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Common Queuing Situations

Situation Arrivals in Queue Service Process

Supermarket Grocery shoppers Checkout clerks at cash register

Highway toll booth Automobiles Collection of tolls at booth

Doctor’s office Patients Treatment by doctors and nurses

Computer system Programs to be run Computer processes jobs

Telephone company Callers Switching equipment to forward calls

Bank Customer Transactions handled by teller

Machine maintenance

Broken machines Repair people fix machines

Harbor Ships and barges Dock workers load and unload

Table D.1

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Characteristics of Waiting-Line Systems

1. Arrivals or inputs to the system Population size, behavior, statistical

distribution

2. Queue discipline, or the waiting line itself Limited or unlimited in length, discipline of

people or items in it

3. The service facility Design, statistical distribution of service

times

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Parts of a Waiting Line

Figure D.1

Dave’s Car Wash

Enter Exit

Population ofdirty cars

Arrivalsfrom thegeneral

population …

Queue(waiting line)

Servicefacility

Exit the system

Arrivals to the system Exit the systemIn the system

Arrival Characteristics Size of the population Behavior of arrivals Statistical distribution

of arrivals

Waiting Line Characteristics

Limited vs. unlimited

Queue discipline

Service Characteristics Service design Statistical distribution

of service

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Arrival Characteristics

1. Size of the population Unlimited (infinite) or limited (finite)

2. Pattern of arrivals Scheduled or random, often a Poisson

distribution

3. Behavior of arrivals Wait in the queue and do not switch lines No balking or reneging

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Waiting-Line Characteristics

Limited or unlimited queue length Queue discipline - first-in, first-out

(FIFO) is most common Other priority rules may be used in

special circumstances

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Service Characteristics

Queuing system designs Single-channel system, multiple-

channel system Single-phase system, multiphase

system

Service time distribution Constant service time Random service times, usually a

negative exponential distribution

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Queuing System Designs

Figure D.3

Departuresafter service

Single-channel, single-phase system

Queue

Arrivals

Single-channel, multiphase system

ArrivalsDeparturesafter service

Phase 1 service facility

Phase 2 service facility

Service facility

Queue

A family dentist’s office

A McDonald’s dual window drive-through

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Queuing System Designs

Figure D.3

Multi-channel, single-phase system

Arrivals

Queue

Most bank and post office service windows

Departuresafter service

Service facility

Channel 1

Service facility

Channel 2

Service facility

Channel 3

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Queuing System Designs

Figure D.3

Multi-channel, multiphase system

Arrivals

Queue

Some college registrations

Departuresafter service

Phase 2 service facility

Channel 1

Phase 2 service facility

Channel 2

Phase 1 service facility

Channel 1

Phase 1 service facility

Channel 2

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Measuring Queue Performance

1. Average time that each customer or object spends in the queue

2. Average queue length

3. Average time each customer spends in the system

4. Average number of customers in the system

5. Probability that the service facility will be idle

6. Utilization factor for the system

7. Probability of a specific number of customers in the system

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Queuing Costs

Figure D.5

Total expected cost

Cost of providing service

Cost

Low levelof service

High levelof service

Cost of waiting time

MinimumTotalcost

Optimalservice level

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Queuing Models

Table D.2

Model Name Example

A Single-channel Information counter system at department store(M/M/1)

Number Number Arrival Serviceof of Rate Time Population Queue

Channels Phases Pattern Pattern Size Discipline

Single Single Poisson Exponential Unlimited FIFO

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Model A – Single-Channel

1. Arrivals are served on a FIFO basis and every arrival waits to be served

2. Arrivals are independent of preceding arrivals

3. Arrivals are random and come from an infinite population

4. Service times are variable

5. The service rate is faster than the arrival rate

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Model A – Single-Channel

Table D.3

= Mean number of arrivals per time periodµ = Mean number of units served per time period

Ls = Average number of units (customers) in the system (waiting and being served)

=

Ws = Average time a unit spends in the system (waiting time plus service time)

=

µ –

1µ –

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Model A – Single-Channel

Table D.3

Lq = Average number of units waiting in the queue

=

Wq = Average time a unit spends waiting in the queue

=

= Utilization factor for the system

=

2

µ(µ – )

µ(µ – )

µ

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Model A – Single-Channel

Table D.3

P0 = Probability of 0 units in the system (that is, the service unit is idle)

= 1 –

Pn > k = Probability of more than k units in the system, where n is the number of units in the system

=

µ

µ

k + 1

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Single-Channel Example

= 2 cars arriving/hourµ = 3 cars

serviced/hour

Ls = = = 2 cars in the system on average

Ws = = = 1 hour average waiting time in the system

Lq = = = 1.33 cars waiting in line

2

µ(µ – )

µ –

1µ –

2

3 - 2

1

3 - 2

22

3(3 - 2)

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Single-Channel Example

Wq = = = 2/3 hour = 40 minute average waiting time

= /µ = 2/3 = 66.6% of time mechanic is busy

µ(µ – )

23(3 - 2)

µ

P0 = 1 - = .33 probability there are 0 cars in the system

= 2 cars arriving/hour µ = 3 cars serviced/hour

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Single-Channel ExampleProbability of more than k Cars in the System

k Pn > k = (2/3)k + 1

0 .667 Note that this is equal to 1 - P0 = 1 - .33

1 .4442 .2963 .198 Implies that there is a 19.8%

chance that more than 3 cars are in the system

4 .1325 .0886 .0587 .039

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Single-Channel EconomicsCustomer dissatisfaction

and lost goodwill = $10 per hour

Wq = 2/3 hourTotal arrivals = 16 per day

Mechanic’s salary = $56 per day

Total hours customers spend waiting per day

= (16) = 10 hours23

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Customer waiting-time cost = $10 10 = $106.6723

Total expected costs = $106.67 + $56 = $162.67

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In-Class Problems from the Lecture Guide Practice Problems

Problem 1:A new shopping mall is considering setting up an information desk manned by one employee. Based upon information obtained from similar information desks, it is believed that people will arrive at the desk at a rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that the arrivals follow a Poisson distribution and answer times are exponentially distributed.(a) Find the probability that the employee is idle.(b) Find the proportion of the time that the employee is busy.(c) Find the average number of people receiving and waiting to receive some information.(d) Find the average number of people waiting in line to get some information.(e) Find the average time a person seeking information spends in the system.(f) Find the expected time a person spends just waiting in line to have a question

answered (time in the queue).

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In-Class Problems from the Lecture Guide Practice Problems

Problem 2:Assume that the information desk employee in Problem 1 earns $10 per hour. The cost of waiting time, in terms of customer unhappiness with the mall, is $12 per hour of time spent waiting in line. Find the total expected costs over an 8-hour day.