Managing Waiting Lines A big part of most people’s day and consequently, life is spent waiting— waiting for the stop light to turn green, waiting in a traffic jam, waiting at store checkouts, waiting for rides at amusement parks. . . the list goes on and on. In a life time, it is estimated that a person spends about five years waiting in lines. If the queue is long, people get restless, anxious and even bored. Sometimes people will not even join the queue if they see a long line (balking) or even walk away in disgust after having spent some time in the queue (reneging). Traditionally, a queue is pictured as a counter or a checkout with a server and a line of customers in front and one person being served at a time. But queues can take other forms, such as the following: • Bus transportation or rides, where the customers receive the service in batches or in bulk. • Fire or ambulance service, when the service provider goes to the customer to provide service. • Customer service, when the customers are in a queue over the phone instead of physically in a line. • Medical clinic or hospital, where the patients may join a network of queues as they move within the system to receive different types of service. For businesses, especially service businesses, waiting lines impact on customer satisfaction and the operating costs. If the number of servers is increased, the operating costs increase but customer waits decrease and the customer satisfaction is impacted positively. The reverse is equally true as well. Since services are more labor intensive in general compared to manufacturing, balancing the number of servers and the customer waiting time can have a significant impact on the health and the bottom line of the organization. It is not difficult to quantify the cost of providing the service since the wage and other expenses are known. If the customer is an internal employee, it is possible to quantify the cost of keeping the customer waiting in terms of the productive wage that is lost in waiting. What is really difficult to quantify is the cost of keeping the customers waiting when the customers are external. QSO 610 Module Nine    1  When the cost of keeping the customers waiting can be determined relatively accurately, as in case of internal customers, the minimization of the total cost approach can be used. When it is difficult to quantify the cost of keeping the customers waiting, as in case of external customers, the service level approach is frequently used. In this approach, instead of quantifying the cost of keeping the customers waiting, a business strives to achieve a reasonable level of service, such as not letting the average customer wait time exceed five minutes. Wherever possible, the wait times should be reduced such as by taking appointments or reservations. To the extent the wait is inevitable, customer’s reactions to waiting depends on his or her expectations ...

1. Managing Waiting Lines
A big part of most people’s day and consequently, life is spent
waiting— waiting for the stop
light to turn green, waiting in a traffic jam, waiting at store
checkouts, waiting for rides at
amusement parks. . . the list goes on and on. In a life time, it is
estimated that a person
spends about five years waiting in lines.
If the queue is long, people get restless, anxious and even
bored. Sometimes people will not
even join the queue if they see a long line (balking) or even
walk away in disgust after
having spent some time in the queue (reneging).
Traditionally, a queue is pictured as a counter or a checkout
with a server and a line of
customers in front and one person being served at a time. But
queues can take other forms,
such as the following:
• Bus transportation or rides, where the customers receive the
service in batches or in
bulk.
• Fire or ambulance service, when the service provider goes to
the customer to provide

2. service.
• Customer service, when the customers are in a queue over the
phone instead of
physically in a line.
• Medical clinic or hospital, where the patients may join a
network of queues as they
move within the system to receive different types of service.
For businesses, especially service businesses, waiting lines
impact on customer satisfaction
and the operating costs. If the number of servers is increased,
the operating costs increase
but customer waits decrease and the customer satisfaction is
impacted positively. The
reverse is equally true as well. Since services are more labor
intensive in general compared
to manufacturing, balancing the number of servers and the
customer waiting time can have
a significant impact on the health and the bottom line of the
organization.
It is not difficult to quantify the cost of providing the service
since the wage and other
expenses are known. If the customer is an internal employee, it
is possible to quantify the
cost of keeping the customer waiting in terms of the productive
wage that is lost in waiting.
What is really difficult to quantify is the cost of keeping the
customers waiting when the
customers are external.
QSO 610 Module Nine 1

3. When the cost of keeping the customers waiting can be
determined relatively accurately, as
in case of internal customers, the minimization of the total cost
approach can be used. When
it is difficult to quantify the cost of keeping the customers
waiting, as in case of external
customers, the service level approach is frequently used. In this
approach, instead of
quantifying the cost of keeping the customers waiting, a
business strives to achieve a
reasonable level of service, such as not letting the average
customer wait time exceed five
minutes.
Wherever possible, the wait times should be reduced such as by
taking appointments or
reservations. To the extent the wait is inevitable, customer’s
reactions to waiting depends on
his or her expectations and perceptions. If the customer is
unhappy with the experience, the
customer is more likely share it with others than when the
customer is happy with the
experience. David Maister (1985) has presented a number of
interesting perspectives on the
psychology of waiting:
First impressions are important: The initial experience during
the service encounter
can color the customer’s attitude towards the rest of the service
experience. Pleasant
and comfortable physical surroundings and pleasant greetings

4. on arrival can make
the subsequent wait more tolerable.
Observing the First-Come-First-Served (FCFS) order of service
is important to
creating an impression of fairness. This can be achieved by
using a number system
or forming a single line before the service counters. Observing a
true FCFS system
also reduces the anxiety of the customer.
Also, important is to provide the customer with a feeling that
the service has started
as soon as possible after the customer has arrived. This can be
achieved by handing
out menus, asking the customer to fill out a medical history
form or having the nurse
take temperature and blood pressure readings. This makes the
subsequent wait for
the food or the doctor more tolerable.
Make the Wait More Tolerable: People detest “empty” time—
when they are doing
nothing or when they feel they have been forgotten. This feeling
can be avoided by
playing music while on hold during a phone call. In a
restaurant, waiting customers
can go to the bar or observe cooking in the kitchen through
glass windows. Wait time
can also be utilized productively by exposing customers to sales
commercials,
information about various services or by showing educational
films.
Some services are trying to make the waiting fun, so it is not
perceived as a wait. This

5. is achieved by entertaining displays along the line and creating
“pre-shows” for
customers waiting in the line. Thus, these services include the
waiting line in
designing the total service experience.
The following diagram from Fitzsimmons and Fitzsimmons
(2011) shows the five major
elements of the queuing system. The customers arrive from a
Calling Population (for
example, from the neighborhood for a neighborhood grocery
store). They arrive at a certain
2 QSO 610 Module Nine
rate (Arrival Process), which can vary over time. If no one is
waiting they receive immediate
service. If all servers are busy, the customers join a Queue. If
the queue is long, some
customers may not join the queue at all (balking). Even after
joining, some customers may
leave the queue before receiving service (reneging). The queues
may be of different types,
such as single or multiple lines (Queue Configuration).
Customers are selected from the
queue for service on the basis of FCFS or some other criterion
(Queue Discipline). The
customers then receive the service (Service Process) and exit
the system. The customers
may join the Calling Population to return again or may never

6. return for service.
Figure 9-1: Queuing System
Calling Population
The calling population may be homogeneous or not
homogeneous. When the customers are
of different types such as those with reservations or
appointments, walk-ins or emergency
patients at doctor’s office, we have a non-homogeneous
population.
The calling population may be large enough to be considered
Infinite or it may be small
enough to be considered Finite. If the population is finite, the
arrival rate would depend on
the number from the population that is receiving service.
Arrival Process
Time pattern of arrivals can be represented in two different
ways:
• Inter-arrival time: Time between successive arrivals
• Arrival rate: Number of customers arriving during each period
of time
The two are reciprocal of each other, i.e., Average arrival rate =
1/ Average inter-arrival time.
When the arrivals occur at random, the inter-arrival time

7. follows the exponential distribution
and the number of arrivals during a time period follows the
Poisson distribution. The
equivalence of the two can be seen in the following diagram
from Fitzsimmons and
Fitzsimmons (2011).
QSO 610 Module Nine 3
Figure 9-2: Poisson Distribution
Queue Configuration
The most queue configurations are one of two types shown in
the following diagram from
Fitzsimmons and Fitzsimmons (2011): Single Line or Multiple
Lines.
Single Line
Figure 9-3: Single Line
Multiple Lines

8. Figure 9-4: Multiple Lines
The Multiple Lines offer a number of advantages over the
Single Line configuration. Multiple
Lines allow differentiation of service (express vs. normal
lanes), allow specialization
(commercial vs. consumer banking), allow customers to select a
server of choice and
4 QSO 610 Module Nine
reduction in the tendency to balk by the perception of short
multiple lines. At the same time,
the Multiple Lines configuration leads to the tendency to switch
lines (called jockeying)
Single Line has its advantages too. It allows for true FCFS
queue discipline, more orderly
movement of line, privacy as queue is separated from the
server, and reduction in the
average customer waiting time. Because of the significant
advantages that Single Line
arrangement offers over the Multiple Lines, its use has
increased over the years.
The Single Line configuration is sometimes practiced by
providing numbers to customers so
customers do not have to wait in line and can engage in other
activities such as browsing
various products in the stores. Single Line configuration can

9. also be a virtual as when a
number of phone calls queue up at a customer service phone
number.
Queues are considered to be infinite, when the space available
for queuing is sufficient at
most times. Queues are considered finite, when the space is so
limited that all customers
are not able to join the queue—such as when only so many
parking spots are available.
Queue Discipline
FCFS is the most common queue discipline as it is the “fair”
criterion. Sometimes other
queue disciplines are used. For example, department and
grocery stores have separate
lines for customers with fewer items. But within the separate
lines, the FCFS rule is used. In
case of emergency services, cases that are most urgent preempt
others.
Service Process
Different server arrangements are used in different situations. In
some cases, such as
supermarkets, customers can even serve themselves using the
self-checkout lanes. Many
services, such as grocery stores, use servers in parallel.
Multistage services may have
server stations in a series. If service times are occur randomly,
they follow exponential
distribution just as the customer inter-arrival times usually do.

10. Queuing Models
The capacity of a system is determined by the servers and other
resources available within
the organization. Demand for services on the other hand is
usually highly variable and
unpredictable. At the same time, the supply of services cannot
be inventoried like
manufactured goods, nor the satisfaction of demand be
postponed in most cases. Thus,
capacity planning or matching supply and demand in case of
services is not an easy job.
As discussed earlier, the number of servers in a service
organization affects the cost of
providing the service and at the same time, the cost associated
with keeping the customers
waiting. Determining the number of servers to have to meet the
demand at different times is
QSO 610 Module Nine 5
an important decision for a service. A number of analytical
models exist that can help in
making that decision.
The queuing models help in determining the steady state
operating characteristics of
different queuing systems. The system, when it is starting up,
such as in the morning of the
After-Thanksgiving Sale, is in a state called the transient state.

11. When the system settles
down and starts operating normally, it reaches the steady state.
In the steady state, the
distributions that the operating characteristics follow become
time independent.
Secondly, all waiting line models make a number of
assumptions. The formulas for the
various operating characteristics are therefore based on meeting
of those assumptions. If
any of those assumptions are violated, the formulas are not
valid. In such circumstances,
simulation models can be used where those assumptions can be
avoided.
All queuing systems contain two parts, as shown in the diagram
for the Single Server Model
shown below: the Servers and the Queue. The two parts together
are called the system. The
following symbols are used to describe a queuing system:
n = number of customers in the system
s = number of servers
λ = the average arrival rate
μ = the average service rate
ρ = utilization of server(s)
qL = average number of customers in the queue
sL = average number of customers in the system
qW = average time in the queue
sW = average time in the system

12. nP = Probability of n exactly n customers in the system
The single server model (also called the M/M/1 Model), is the
simplest model. It includes one
server with one line in front of the server as shown in the
diagram below. The formulas for
steady state operating system characteristics are based on the
following assumptions:
Calling population: Infinite (Large) in size.
Arrival process: Random arrivals with number of arrivals per
time period following the
Poisson distribution.
Queue configuration: Single waiting line of infinite queue
capacity (no restrictions on queue
length) with no balking or reneging.
Queue discipline: FCFS.
Service process: One server with service times that are
exponentially distributed
6 QSO 610 Module Nine
Figure 9-5: Single Server Model
The formulas for the Single Server Model are given below:
ρ = Average utilization of system
μ
λ

13. =
=nP Probability that customers are in the system n ( ) nρρ−= 1
)n(P ≥ = Probability that n or more customers are in the system
= nρ
=sL Average number of customers in the service system λμ
λ
−
=
=qL Average number of customers in the waiting line sLρ=
=Ws Average time spent in the system, including service λμ −
=
1
=qW Average waiting time in line sWρ=
Let us take up the following bank example to understand the
application of the queuing
formulas to a single server system:
Customers arrive at a small local bank every 2.5 minutes on an
average. Number of
customers arriving each minute is Poisson distributed. Most
times, there is only one teller
serving the customers. It takes an average of 2 minutes to serve
each customer. The
services times are exponentially distributed. Determine the
operating characteristics of the
small local bank.

14. The average arrival and the average service rates can be
calculated as follows:
λ = the average arrival rate = 1 / 2.5 = 0.4 customers per minute
μ = the average service rate = 1/ 2 = 0.5 customers per minute
Operating Characteristics of the small local bank with one teller
can be determined as
follows:
QSO 610 Module Nine 7
=ρ Average utilization of teller 80
50
40
.
.
.
===
μ
λ
= 80%
=sL
Average number of customers in the
service bank

15. 4
4050
40
=
−
=
−
=
..
.
λμ
λ
customers
=qL
Average number of customers in the
waiting line
23480 .*.Ls === ρ customers
=sW
Average time spent in the bank,
including service
10
4050
11
=

16. −
=
−
=
..λμ
minutes
=qW Average waiting time in line 81080 === *.Wsρ minutes
=0P Probability that no customer is in the bank ( )
20120808011 0 .*..).(n ==−=−= ρρ
=1P Probability that there is one customer is in the
bank ( ) 1608020808011 1 ..*..).(n ==−=−= ρρ
=2P
=
Probability that there is one customer is in the
bank ( ) 128064020808011 2 ..*..).(n ==−=− ρρ
=3P
=
Probability that there is one customer is in the
bank and so on. ( ) 10240512020808011 3 ..*..).(n ==−=− ρρ
)(P 4≥ = Probability that there 4 or more customers are in the
bank = 4096080 4 .. =
Are you satisfied with the operation of the bank? Hint: Look at

17. the average time in the line.
The multiples lines system is like the system in a grocery or
department store where one
sees multiple lines with one server for each line. The multiple
lines systems can be
considered to be a composite of s single server systems,
assuming that there are s lines
with s servers. If we make the following assumptions in
addition to all the assumptions for
the single server model, we can use the single server model for
this case:
1. The customers split equally into the multiple lines.
2. All the servers work at the same pace.
3. Customers do not switch lines.
Based on assumption 1, if λ overall is the average arrival rate
for the entire system, then the
average arrival rate for each line would be s/overallλλ = , where
s is the number of servers
(lines). The average service rate would be the same for each
server, equal to μ for each of
the lines based on assumption 2. Each line can now be treated as
a single server system
with λ as the average arrival rate and μ as the average service
rate and the formulas for the
single server system can be used.
For an example, assume that the small local bank feels that an
average wait time of 8
minutes in the queue is too long for the customers and is afraid
that it would lose business.
The bank thinks that the average wait time in the queue should
not exceed 5 minutes at the
most. The bank is considering having two tellers with a separate

18. line in front of each teller.
8 QSO 610 Module Nine
The average arrival rate for each of the two lines will now be
half of the overall rate.
Therefore, λ = the average arrival rate = 0.4 / 2 = 0.2 customers
per minute
= Average utilization of teller 40
50
20
.
.
.
===
μ
λ
= 40%
=
ρ
sL
Average number of customers in the
service bank

19. 670
2050
20
.
..
.
=
−
=
−
=
λμ
λ
customers
=qL
Average number of customers in the
waiting line
267067040 ..*.Ls === ρ customers
=sW
Average time spent in the bank,
including service
333
2050
11

20. .
..
=
−
=
−
=
λμ
minutes
=qW Average waiting time in line 33133340 ..*.Ws === ρ
minutes
The average waiting time in the line would now be only 1.33
minutes, much less than the 5
minutes that the bank was striving for. Of course, the operating
cost in terms of labor cost is
going to double. Some of it might be recovered by the increased
business the better service
might generate.
Single line with multiple servers (called the multiple server
system) is becoming very
common because of its many advantages. It is a true FCFS
system and results in smaller
wait time for the customers. The formulas for this case are more
complex as given below:
QSO 610 Module Nine 9

21. Average utilization of system μ
λ
s
=
ρ =
=0P Probability that zero customers are in the system
( ) ( )
1
1
0 1
1
!!
−
−
=
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠

22. ⎞
⎜⎜
⎝
⎛
−
+= ∑
s
n
sn
sn ρ
μλμλ
=nP n Probability that customers are in the system
( )
( )
⎪
⎪
⎩
⎪ ⎪
⎨
⎧
≥
<<

23. =
−
snP
ss
snP
n
sn
n
n
0
0
!
0
!
μλ
μλ
=qL Average number of customers in the waiting line
( )
( )2
0

24. 1! ρ
ρμλ
−
=
s
P s
=Wq Average waiting time of customers in line λ
qL=
=Ws Average time spent in the system, including service μ
1
+= qW
=sL Average number of customers in the service system s
Wλ=
Let us revert to the bank example and study what would be
happen if the bank went in for
two tellers but with only one line instead of multiple lines.
Since the customers do not split
into two lines, the average arrival rate would remain λ = 1 / 2.5
= 0.4 customers per minute.
Let us calculate the new operating characteristics.
=ρ Average utilization of system %.

25. .*
.
s
4040
502
40
====
μ
λ
(same as Multiple Lines)
10 QSO 610 Module Nine
=0P Probability that zero customers are in the system
( ) ( )
1
1
0 1
1
!!
−
−

26. =
⎥
⎦
⎤
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
+= ∑
s
n
sn
sn ρ
μλμλ
( ) ( )
1
12

27. 0
2
401
1
2
50405040
−
−
= ⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∑ ⎟
⎠
⎞
⎜
⎝
⎛
−
+=
n

28. n
.!
..
!n
..
( ) ( ) ( )
1210
401
1
2
5040
1
5040
0
5040
−
⎥
⎥
⎦
⎤
⎢
⎢
⎣

29. ⎡
⎟
⎠
⎞
⎜
⎝
⎛
−
++=
.!
..
!
..
!
..
1
60
1
320801
−
⎥
⎦
⎤
⎢
⎣

30. ⎡
⎟
⎠
⎞
⎜
⎝
⎛
++=
.
.. =0.4286
=qL Average number of customers in the waiting line
( )
( )2
0
1! ρ
ρμλ
−
=
s
P s ( )
( )
15240
3602

31. 4064042860
4012
408042860
2
2
.
.*
.*.*.
.!
).(..
==
−
= customers
=qW Average waiting time of customers in line 3810040
15240
.
.
.Lq
===
λ
minute
Now, the customers will have to spend only 0.3810 minutes in
the queue!
Which arrangement do you prefer? Multiple Lines or Multiple

32. Server?
QSO 610 Module Nine 11
12 QSO 610 Module Nine
References
Fitzsimmons, J. A., & Fitzsimmons, M. J. (2011). Service
management: Operations, strategy,
information technology. (7th ed.). New York, NY: McGraw
Hill.
Exercise 16-2
A business school is considering replacing its copy machine
with a faster model. Past records show that the average student
arrival rate is 24 per hour, Poisson distributed, and that the
service times are distributed exponentially. The selection
committee has been instructed to consider only machines that
will yield an average turnaround time (i.e., expected time in the
system) of 5 minutes or less. What is the smallest processing
rate per hour that can be considered?
Exercise 16-4
On average, 4 customers per hour use the public telephone in

33. the sheriff’s detention area, and this use has a Poisson
distribution. The length of a phone call varies according to a
negative exponential distribution, with a mean of 5 minutes.
The sheriff will install a second telephone booth when an
arrival can expect to wait 3 minutes or longer for the phone.
a. By how much must the arrival rate per hour increase to
justify a second telephone booth?
b. Suppose the criterion for justifying a second booth is changed
to the following: install a second booth when the probability of
having to wait at all exceeds 0.6. Under this criterion, by how
much must the arrival rate per hour increase to justify a second
booth?