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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.