Stevenson
18
Management of
Waiting Lines
18-2
Learning Objectives
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Explain why waiting lines form in systems.
Implications of waiting line
Goal of waiting line management
Characteristics of waiting line
Measures of waiting line performance
Queuing models – Infinite Source and Finite
Source
Constraint management
Psychology of waiting
Operations strategy
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Why do waiting lines form?
 Waiting lines occur when there is a temporary
imbalance between supply (capacity) and demand.
 Waiting lines add to the cost of operation and they
reflect negatively on customer service,
 It is important to balance the cost of having
customers wait with the cost of providing service
capacity.
 From a managerial perspective, the key is to
determine the balance that will provide an
adequate level of service at a reasonable cost.
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Disney World
 Waiting in lines does not add enjoyment
 Waiting in lines does not generate
revenue
Waiting lines are non-value added occurrences
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Waiting lines
 Waiting Time: Operators and machines
waiting for parts or work to arrive from
suppliers or other operations. Customers
waiting in line.
 One of the “seven wastes”
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Waiting Lines
 Queuing theory: Mathematical approach to
the analysis of waiting lines.
 Goal of queuing analysis is to minimize the
sum of two costs
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Customer waiting costs
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Service capacity costs
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Goal of waiting line management
Minimize the sum of two costs: customer waiting costs and
service capacity costs
Cost
Total
cost
=
Customer
waiting cost
+
Total cost
Capacity
cost
Cost of
service
capacity
Cost of
customers
waiting
Service capacity
Optimum
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Implications of Waiting Lines
 Cost to provide waiting space
 Loss of business
 Customers leaving
 Customers refusing to wait
 Loss of goodwill
 Reduction in customer satisfaction
 Congestion may disrupt other business
operations
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System Characteristics
 Population Source
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Infinite source: customer arrivals are
unrestricted
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Finite source: number of potential
customers is limited
 Number of Servers (channels)
 Arrival and service patterns
 Queue discipline (order of service)
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System Characteristics:
Number of servers/phases
Multiple channel
Channel: A server in
a service system
Multiple phase
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System Characteristics:
Arrival and Service Patterns
The Poisson distribution often provides a reasonably
good description of customer arrivals per unit of time
(e.g., per hour).
 A discrete probability distribution
 probability of a given number of events occurring in a fixed
interval of time
 events occur with a known average rate and independently.
 The expected value and variance of (random) variable is equal to λ
P(r) = (e-λ λr ) / r! where
r = arrivals/time unit and
λ = mean arrivals/time unit
f(t) = μe – μt
where t = service time and
μ = mean service time
m = the average number
of customers who can be
served per time period.
Mean service time = 1/m
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System Characteristics:
Queue Discipline
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First-come, first-served
Priority
Preferred (loyalty programs/fee-based)
Reservation (appointment)
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Waiting line Models
 Patient
 Customers enter the waiting line and remain until
served
 Reneging
 Waiting customers grow impatient and leave the
line
 Jockeying
 Customers may switch to another line
 Balking
 Upon arriving, decide the line is too long and
decide not to enter the line
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Waiting Line Performance
 Waiting line performance relates to potential
customer dissatisfaction and cost:
 The average number of customers, either in line
or in the system
 The average time the customer waits, either in
line or in the system
 System utilization
 The implied cost of a given level of capacity and
its related waiting line
 The probability that an arrival would have to wait
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Waiting Time vs. Utilization
Average number on
time waiting in line
System utilization reflects the extent to which
servers are busy
0
System Utilization
100%
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Queuing Models: Infinite-Source
1.Single channel, exponential service time
2.Single channel, constant service time
3.Multiple channel, exponential service
time
4.Multiple priority service, exponential
service time
All of the above assume Poisson arrival rate,
and average arrival and service rates are steady
(i.e., steady state)
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Basic Relationships of Infinite
Source Queuing Model
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Infinite Source
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LEGENDS
 M/M/1 – Poisson Arrival Rate (the first M),
Poisson (or exponential) service rate (the
second M), and 1 server (the 1)
 M/D/1 – Poisson Arrival Rate (the first M),
deterministic (constant) service rate (the D),
and 1 server (the 1)
 M/M/S - Poisson Arrival Rate (the first M),
Poisson (or exponential) service rate (the
second M), and multiple servers (the S)
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Infinite source
Lq is a key value, and generally one of the first values we should calculate
Equation will be different for M/M/1, M/D/1 models, M/M/S models
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Example: Infinite Source
Customers arrive at a bakery at an average rate of 18 per hour on
weekday mornings. The arrival distribution can be described by a
Poisson distribution with a mean of 18. Each clerk can serve a
customer in an average of three minutes; this time can be described by
an exponential distribution with a mean of 3.0 minutes.
A. What are the arrival and service rates?
B. Compute the average number of customers being served at any
time.
C. Suppose it has been determined that the average number of
customers waiting in line is 8.1. Compute the average number of
customers in the system (i.e., waiting in line or being served), the
average time customers wait in line, and the average time in the
system.
D. Determine the system utilization for M = 1, 2, and 3 servers.
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Example: Infinite Source
 λ=18 customers per hour
 We need to change the service time to comparable
hourly rate. 60 minutes per hour, and 3 minutes
per customer means 20 customers per hour. μ =
20 customers/per hour
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Different models within Infinite
Source
Single Server, Exponential Service Time,
M/M/1
The queue discipline is first-come, first-served, and it is
assumed that the customer arrival rate can be approximated
by a Poisson distribution and service time by a negative
exponential distribution. There is no limit on length of queue
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M/M/1 Example
An airline is planning to open a satellite ticket desk in a new shopping
plaza, staffed by one ticket agent. It is estimated that requests for tickets
and information will average 15 per hour, and requests will have a
Poisson distribution. Service time is assumed to be exponentially
distributed. Previous experience with similar satellite operations suggests
that mean service time should average about three minutes per request.
Determine each of the following:
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a. System utilization.
b. Percentage of time the server (agent) will be idle.
c. The expected number of customers waiting in the line.
d. The average time customers will spend in the system.
e. The probability of zero customers in the system and the probability
of four customers in the system.
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M/M/1 Example
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Different models within Infinite
Source
Single Server, Constant Service Time, M/D/1
Waiting lines are a consequence of random, highly variable
arrival and service rates. If a system can reduce or eliminate
the variability of either or both, it can shorten waiting lines
noticeably. A case in point is a system with constant service
time. The effect of a constant service time is to cut in half the
average number of customers waiting in line
The average time customers spend waiting in line is also cut in half.
Similar improvements can be realized by smoothing arrival times (e.g., by
use of appointments).
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M/D/1 Example
Wanda's Car Wash & Dry is an automatic, fiveminute operation with a single bay. On a typical
Saturday morning, cars arrive at a mean rate of
eight per hour, with arrivals tending to follow a
Poisson distribution. Find
 a. The average number of cars in line.
 b. The average time cars spend in line and
service
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M/D/1 Example
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Different models within Infinite
Source
Multiple Servers, M/M/S
 Two or more servers are working independently. Use
of the model involves the following assumptions:
 1. A Poisson arrival rate and exponential service time.
 2. Servers all work at the same average rate.
 3. Customers form a single waiting line (in order to
maintain first-come, first-served processing).
 The multiple-server formulas are more complex than
the single-server formulas, especially the formulas for
Lq and P0..
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M/M/S Equations
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Queuing Model: Finite Source
 Calling population is limited
 One person may be responsible for handling
breakdowns on 15 machines
 There may be more than one server or channel
 Arrival rates are required to be Poisson and
service times exponential.
 arrival rate of customers here is affected by the
length of the waiting line
 the arrival rate decreases as the length of the line
increases
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Finite Source Queuing
Formulas and Notations
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Constraint Management
Managers may be able to reduce waiting times
by actively managing one or more system
constraints.
 Use temporary workers
 Shift demand
 Standardize the service
 Look for a bottleneck
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Psychology of waiting
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Occupied time feels shorter than unoccupied time
People want to get started
Anxiety makes waits seem longer
Uncertain waits are longer than known, finite waits
Unexplained waits are longer than explained waits
Unfair waits are longer than equitable waits
The more valuable the service, the longer the
customer will wait
 Solo waits feel longer than group waits
Source: David H. Maister, “The Psychology of Waiting Lines,”
davidmaister.com, blog, September 8, 2008
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Two key items to remember
 Waiting lines forms because arrival rates and
services rates are different. In other words,
there is mismatch between supply and
demand, or waiting lines happen due to the
variability in arrival and service rates.
 Increase in system utilization results in an
exponential increase in customer waiting
time.
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Disney’s approach to
managing waiting lines
 Provide distractions
 Provide alternatives for those willing to
pay a premium
 Keep customers informed
 Exceed expectations
 Comfortable waiting environment and
other distractions
 A form of reservations (Fast Pass)
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Operations Strategy
 Carefully assess the costs and benefits of various
alternatives for capacity of service systems.
 Increase the processing rate vs. number of servers.
 New processing equipment/methods
 Standardization (reduce variability in processing)
 Shift some arrivals to “off-times” by using
reservations systems
 “early-bird” specials
 senior discounts
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