Supplement
B Waiting Lines
1. Define “customers” in waiting lines.
2. Why do waiting lines form?
a. Temporary imbalance between demand for service and capacity of the system.
b. Can develop even if the time to process is constant due to variability in demand.
3. If both demand and service rates are constant, and service rate > than demand, no waiting line
forms.
4. The analysis of waiting lines is of concern to managers because it affects process design,
capacity planning, process performance, and ultimately, supply chain performance.
5. Applies to many service or manufacturing situations.
a. Relating customer arrival and service-system processing characteristics to service-system
output
6. Service is the act of processing a customer (or manufacturing job).
a. Hair cutting in a hair salon
b. Satisfying customer complaints
c. Processing production orders
d. Theatergoers waiting to purchase tickets
e. Trucks waiting to be unloaded at a warehouse
f. Patients waiting to be examined by a physician
1. Structure of Waiting-Line Problems
1. Customer population
a. The source of input to the service system.
b. Whether the input source is finite or infinite will have an effect on the waiting line
characteristics.
• When several customers from a finite source are already in the waiting line, the
chance of new customer arrivals is reduced.
• When the input source is infinite, customers already in the waiting line do not affect
probability of another arrival.
c. Whether the customers are patient or impatient also affects waiting line characteristics.
• Patient customers wait until served.
• Impatient customer arrivals either balk at long lines (leave immediately), or join the
line and renege (leave after becoming discouraged with slow progress).
2. The service system
a. Number of lines
• A single-line arrangement is favored when servers are capable of general service.
Keeps servers uniformly busy
Levels waiting times among customers, gives sense of fairness.
• A multiple-line arrangement is favored when servers provide a limited set of services.
Customers wait in the appropriate line for a particular service.
b. Arrangement of service facilities
• Channel: One or more facilities required to perform a given service
• Phase: A single step in providing a service
• Multiple-channel, single-phase
Used when demand is large enough to warrant providing the same service at
more than one facility
3. Priority rule
a. First-come, first-served (FCFS)—priority discipline is assumed.
b. Other rules
• Earliest due date (EDD)
• Shortest processing time (SPT)
• Discussed in Chapter 10: Operations Planning and Scheduling.
c. Preemptive discipline—allows a higher priority customer to be served ahead of another
who would have been served first according to the normal priority discipline (such as
FCFS).
4. Probability Distributions
a. The source of variation in waiting-line problems come from the random arrivals of
customers and the variation of service.
5. Arrival distribution
• Customer arrivals can often be described by the Poisson distribution with mean =
T
and variance also =
T .
• Arrival distribution: the probability of n arrivals in T time periods.
• Interarrival times, the average time between arrivals: the probability that the next
customer will arrive in the next T time periods.
( )
T
n
ne
n
T
P
−
=!
,2,1,0=nfor
where
=
n
P
Probability of
n
arrivals in
T
time periods
=
Average numbers of customer arrivals per period
=e
7183.2
6. Service time distribution can be described by an exponential distribution with mean = 1/
and variance = (1/
)2
• Service time distribution: The probability that the service time will be no more than
T time periods can be described by the exponential distribution.
( )
T
eTtP
−
−= 1
Where
=
Average number of customers completing service per period
=t
Service time of the customer
=T
Target service time
• The exponential distribution assumes that each service time is independent of those
that preceded it.
2. Using Waiting-Line Models to Analyze Operations
Balance costs against benefits of improving service system, but also consider the costs of not
making improvements.
1. Waiting line operating characteristics of the system
a. Line length— number of customers in line. Long lines indicate poor customer service,
inefficient service, or inadequate capacity.
b. Number of customers in system—customers in line and being served. A large number
causes congestion and dissatisfaction.
c. Waiting time in line—waiting for service to begin. Long waits are associated with poor
service.
2. Single-server model. Simplest waiting-line model
a. Assumptions
Number of servers
1
Number of phases
1
Customer population (input source)
Infinite and all customers are patient
Arrival distribution
Poisson; mean rate of
Service distribution
Exponential; mean rate of
, where the mean
service rate exceeds the mean arrival rate
Priority rule
FCFS
Waiting line
Single line; no length restrictions
b. Formulas
=
Average utilization of system
=
=
n
P
Probability that
n
customers are in the system
( )
n
−= 1
=L
Average number of customers in the service system
−
=
=
q
L
Average number of customers in the waiting line
L
=
=W
Average time spent in the system, including service
−
=1
=
q
W
Average waiting time in line
W
=
c. Use Application B.1: Single Server Model
Customers arrive at a checkout counter at an average 20 per hour, according to a Poisson
distribution. They are served at an average rate of 25 per hour, with exponential service
times. Use the single-server model to estimate the operating characteristic of this system.
20=
customer arrival rate per hour
25=
customer service rate per hour
1.
Average utilization of system
8.0
25
20 ===
2.
Average number of customers in the service system
4
2025
20 =
−
=
−
=
L
3.
Average number of customers in the waiting line
( )
2.348.0 === LLq
4.
Average time spent in the system, including service
2.0
2025
11 =
−
=
−
=
W
5.
Average waiting time in line
( )
16.02.08.0 === WWq
d. Active Model B.1 in MyLab Operations Management provides additional insight on the
single-server model and its uses.
e. Tutor Model B.1 in MyLab Operations Management provides a new example to practice
the single-server model.
f. Analyzing the Service Rate. Use Application B.2:
In the checkout counter example, what service rate is required to have customers average
only 10 minutes in the system?
( )
17.0
1=
−
=
W
hr (or 10 minutes)
( )
117.0 =−
, where
20=
customers arrival rate per hour
( )
88.25
17.0
2017.01 =
+
=
or about 26 customers per hour
3. Multiple-server model. Service system has only one phase, multiple-channels.
a. Assumptions (in addition to single-server model)
• There are
s
identical servers
• The service distribution for each server is exponential, with a mean service time of
1
b. Use Application B.3: Multiple-Server Model
Suppose the manager of the checkout system decides to add another counter. The arrival rate
is still 20 customers per hour, but now each checkout counter will be designed to service
customers at the rate of 12.5 per hour. What is the waiting time in line of the new system?
2=s
,
5.12=
customers per hour,
20=
customers per hour
Solve for the operating characteristics of this model by using either POM for Windows or OM
Explorer. The solution below uses POM for Windows.
c. Tutor B.2 in MyLab Operations Management provides a new example to practice the
multiple-server model.
d. Active Model B.2 in MyLab Operations Management provides additional insight on the
multiple-server model and its uses for this problem.
4. Little’s Law
a. Relates the number of customers in a waiting line system to the waiting time of
customers.
b. Using the notation from the single-server and multiple-server models: expressed as
qq WLorWL
==
c. Holds for a wide variety of arrival processes, service time distributions, and numbers of
servers.
d. You need only know two of the parameters to estimate the third.
e. Can be used for service and manufacturing applications.
• Service
The manager can estimate
W
, the average time each customer spent in the
facility
If the time a customer spends at the facility is unreasonable, the manager can
focus on either adding capacity or improving the work methods to reduce the
time spent serving the customers
• Manufacturing
f. Provides basis for measuring the effects of process improvements.
g. Is not applicable to situations where the customer population is finite.
5. Finite-source model
a. Assumptions
• Follows the assumption of the single-server, except that the customer population is
finite
• Having only
N
potential customers
• If
30N
, then the single-server model with the assumption of infinite customer
population is adequate
b. Finite-Source Model. Use Application B.4:
DBT Bank has 8 copy machines located in various offices throughout the building. Each
machine is used continuously and has an average time between failures of 50 hours. Once
failed, it takes 4 hours for the service company to send a repair person to have it fixed. What
is the average number of copy machines in repair or waiting to be repaired?
02.0501==
copiers per hour
25.041 ==
copiers per hour
Solve for the operating characteristics of this model by using either POM for Windows or OM
Explorer. The solution below uses POM for Windows.
c. Use Application B.5: Hilltop Produce (an elaborate example.)
The Hilltop Produce store is staffed by one checkout clerk. The average checkout time is
exponentially distributed around an average of two minutes per customer. An average of
20 customers arrive per hour.
What is the average utilization rate?
667.0
30
20 ===
What is the probability that three or more customers will be in the checkout area?
First calculate 0, 1, and 2 customers will be in the checkout area:
( ) ( )( )
( ) ( )( )
( ) ( )( )
0
0
0
1
1
1
2
2
2
1 0.333 0.667 0.333
1 0.333 0.667 0.222
1 0.333 0.667 0.148
P
P
P
= − = =
= − = =
= − = =
Then calculate 3 or more customers will be in the checkout area:
( ) ( )
0 1 2
1 1 0.333 0.222 0.148 0.297P P P− + + = − + + =
What is the average number of customers in the waiting line?
( )
333.1
2030
20
667.0 =
−
=
−
==
LLq
What is the average time customers spend in the store?
min6min601.0
2030
11 ==
−
=
−
=hrhrW
d. Tutor B.3 in MyLab Operations Management provides a new example to practice the
finite-source model.
e. Active Model B.3 in MyLab Operations Management provides additional insight on the
finite-source model and its uses for this problem.
3. Waiting Lines and Simulation
• Often, the nature of the customer population, the constraints on the waiting line, the priority
rule used, the service-time distribution, and the arrangement of the facilities are such that
waiting-line theory is no longer useful. In these cases, simulation often is used.
• Supplement E: “Simulation”, discusses simulation.
1. Waiting-line theory is no longer useful and simulations are often used under conditions
related to:
a. Nature of the customer population
b. Constraints on the line
c. Priority rule
d. Service time distribution
e. Arrangement of facilities
2. SimQuick is an easy-to-use package provided in MyLab Operations Management.
The steps to use SimQuick include:
a. draw a flowchart of the process using SimQuick’s building blocks
b. enter information describing each building block into SimQuick tables
i. when arrivals occur
c. run the model and interpret results
4. Decision Areas for Management
1. Arrival rates—Adjust through advertising, promotions, pricing, appointments.
2. Number of service facilities—Adjust service system capacity.