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Management of Waiting Lines McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
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Management ofWaiting Lines
McGraw-Hill/Irwin Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved.
You should be able to:1. What imbalance does the existence of a waiting line
reveal?2. What causes waiting lines to form, and why is it
impossible to eliminate them completely?3. What metrics are used to help managers analyze
waiting lines?4. What are some psychological approaches to managing
lines, and why might a manager want to use them?5. What very important lesson does the constant service
time model provide for managers?
Instructor Slides 18-2
Waiting lines occur in all sorts of service systems
Wait time is non-value added Wait time ranges from the acceptable to the emergent
Short waits in a drive-thruSitting in an airport waiting for a delayed flightWaiting for emergency service personnel
Waiting time costsLower productivityReduced competitivenessWasted resourcesDiminished quality of life
Instructor Slides 18-3
Queuing theoryMathematical approach to the analysis of
waiting linesApplicable to many environments
Call centersBanksPost officesRestaurantsTheme parksTelecommunications systemsTraffic management
Instructor Slides 18-4
Waiting lines tend to form even when a system is not fully loadedVariability
Arrival and service rates are variableServices cannot be completed ahead of time
and stored for later use
Instructor Slides 18-5
Why waiting lines cause concern:1. The cost to provide waiting space2. A possible loss of business when customers leave the
line before being served or refuse to wait at all3. A possible loss of goodwill4. A possible reduction in customer satisfaction5. Resulting congestion may disrupt other business
operations and/or customers
Instructor Slides 18-6
The goal of waiting line management is to minimize total costs: Costs associated with customers waiting for service Capacity cost
Instructor Slides 18-7
The basic characteristics of waiting lines1. Population source2. Number of servers (channels)3. Arrival and service patterns4. Queue discipline
Instructor Slides 18-8
Calling population
Arrivals Waitingline
ExitService
System
Processing Order
Instructor Slides 18-9
Infinite sourceCustomer arrivals are unrestrictedThe number of potential customers greatly
exceeds system capacityFinite source
The number of potential customers is limited
Instructor Slides 18-10
ChannelA server in a service systemIt is assumed that each channel can handle one
customer at a timePhases
The number of steps in a queuing system
Instructor Slides 18-11
Instructor Slides 18-12
Arrival pattern Most commonly used models assume the arrival rate
can be described by the Poisson distributionArrivals per unit of time
Equivalently, interarrival times are assumed to follow the negative exponential distributionThe time between arrivals
Service pattern Service times are frequently assumed to follow a
negative exponential distribution
Instructor Slides 18-13
Instructor Slides 18-14
Queue disciplineThe order in which customers are processed
Most commonly encountered rule is that service is provided on a first-come, first-served (FCFS) basis
Non FCFS applications do not treat all customer waiting costs as the same
Instructor Slides 18-15
Managers typically consider five measures when evaluating waiting line performance:1. The average number of customers waiting (in line or
in the system)2. The average time customers wait (in line or in the
system)3. System utilization4. The implied cost of a given level of capacity and its
related waiting line5. The probability that an arrival will have to wait for
service
Instructor Slides 18-16
The average number waiting in line and the average time customers wait in line increase exponentially as the system utilization increases
Instructor Slides 18-17
Four basic infinite source modelsAll assume a Poisson arrival rate
1. Single server, exponential service time2. Single server, constant service time3. Multiple servers, exponential service time4. Multiple priority service, exponential service time
Instructor Slides 18-18
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Instructor Slides 18-19
System Utilization
Average number of customers being served
M
r
Instructor Slides 18-20
Little’s LawFor a stable system the average number of
customers in line or in the system is equal to the average customer arrival rate multiplied by the average time in the line or system
ss
WL
WL
Instructor Slides 18-21
The average number of customers Waiting in line for service:
In the system:
The average time customers are Waiting in line for service
In the system
]dependent. [Model qL
rLL qs
q
q
LW
s
qs
LWW
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Instructor Slides 18-22
M/M/1
n
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Instructor Slides 18-23
M/D/1 If a system can reduce variability, it can shorten waiting
lines noticeably For, example, by making service time constant, the average
number of customers waiting in line can be cut in half
Average time customers spend waiting in line is also cut by half.
Similar improvements can be made by smoothing arrival rates (such as by use of appointments)
)(2
2
qL
Instructor Slides 18-24
Assumptions:A Poisson arrival rate and exponential service
timeServers all work at the same average rateCustomers form a single waiting line (in order
to maintain FCFS processing)
Instructor Slides 18-25
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Instructor Slides 18-26
Service system design reflects the desire of management to balance the cost of capacity with the expected cost of customers waiting in the system
Optimal capacity is one that minimizes the sum of customer waiting costs and capacity or server costs
Instructor Slides 18-27
Instructor Slides 18-28
An issue that often arises in service system design is how much space should be allocated for waiting lines
The approximate line length, Lmax, that will not be exceeded a specified percentage of the time can be determined using the following:
1
percentage
specified1
where
ln
lnor
log
logmax
qLK
KKL
Instructor Slides 18-29
Multiple priority model Customers are processes according to some measure of
importance Customers are assigned to one of several priority classes
according to some predetermined assignment methodCustomers are then processed by class, highest class
firstWithin a class, customers are processed by FCFSExceptions occur only if a higher-priority customer
arrives That customer will be processed after the customer
currently being processed
Instructor Slides 18-30
Instructor Slides 18-31
Appropriate for cases in which the calling population is limited to a relatively small number of potential calls
Arrival rates are required to be Poisson Unlike the infinite-source models, the arrival rate is
affected by the length of the waiting lineThe arrival rate of customers decreases as the length of
the line increases because there is a decreasing proportion of the population left to generate calls for service
Service rates are required to be exponential
Instructor Slides 18-32
Procedure:1. Identify the values for
a. N, population sizeb. M, the number of servers/channelsc. T, average service timed. U, average time between calls for service
2. Compute the service factor, X=T/(T + U)3. Locate the section of the finite-queuing tables for N4. Using the value of X as the point of entry, find the values of
D and F that correspond to M5. Use the values of N, M, X, D, and F as needed to determine
the values of the desired measures of system performance
Instructor Slides 18-33
Instructor Slides 18-34
Managers may be able to reduce waiting lines by actively managing one or more system constraints: Fixed short-term constraints
Facility sizeNumber of servers
Short-term capacity optionsUse temporary workersShift demandStandardize the serviceLook for a bottleneck
Instructor Slides 18-35
If those waiting in line have nothing else to occupy their thoughts, they often tend to focus on the fact they are waiting in lineThey will usually perceive the waiting time to
be longer than the actual waiting timeSteps can be taken to make waiting more
acceptable to customersOccupy them while they wait
In-flight snackHave them fill out forms while they waitMake the waiting environment more comfortableProvide customers information concerning their wait
Instructor Slides 18-36
Managers must carefully weigh the costs and benefits of service system capacity alternatives
Options for reducing wait times: Work to increase processing rates, instead of increasing
the number of servers Use new processing equipment and/or methods Reduce processing time variability through
standardization Shift demand
Instructor Slides 18-37
