MANAGEMENT OF

WAITING LINES
 Chapter Outlines
01. Why is there Waiting 06. Queuing Models: Infinite
Lines? Source

02. Managerial implications 07. Queuing Models: Finite

of Waiting Lines Source

03. Goal of Waiting Line

08. Constraint Management
Management

04. Characteristics of 09. The Psychology of

Waiting Lines Waiting

05. Measures of Waiting 10. Operations Strategy

Line Performance
 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
Waiting lines are non-value added occurrences
Why is there Waiting Line?
 Why is there Waiting Lines?

Waiting lines tend to form

even though a system is
basically underloaded
§ Variability
- Arrival and service
rates are variable
§ Services cannot be
completed ahead of time
and stored for later use
Managerial Implications
of Waiting Lines
Managerial Implications of
Waiting Lines
Managers have a number of very good reasons to be
concerned with waiting lines. Chief among those reasons are
the following:
q The cost to provide waiting space.
q A possible loss of business should customers leave
the line before being served or refuse to wait at all.
q A possible loss of goodwill.
q A possible reduction in customer satisfaction.
q The resulting congestion that may disrupt other
business operations and/or customers.
Goal of Waiting Line
Management
 Goal of Waiting Line Mangement

The goal of waiting-line

management is to
minimize the sum of two
costs: customer waiting
costs and service capacity
costs.

TC = Customer waiting cost + Capacity cost

Characteristics of
Waiting Lines
Characteristics of Waiting Lines

POPULATION SOURCE

NUMBER OF SERVERS
(CHANNELS)

ARRIVAL AND
SERVICE PATTERNS

QUEUE DISCIPLINE
(ORDER OF SERVICE)
 Population Source

There are two possibilities: infinite-source and finite-

source populations

§ Infinite-source situations exist whenever service is

unrestricted
§ When the potential number of customers is limited,
a finite-source situation exists
 Number of Servers (Channels)

The capacity of
queuing systems
is a function of
the capacity of
each server and
the number of
servers being
used Four common
variations of
queuing
systems
 Arrival and Service Patterns
Waiting lines are a direct result of arrival and service variability.
§ Arrival and service rates can be described by a Poisson distribution
or, equivalently, that the interarrival time and service time can be
described by a negative exponential distribution.

Poisson and negative exponential distributions

Behavior of Arrivals

Patient Reneging
customers

Jockeying Balking
 Queue Discipline

Queue discipline refers to the order in which customers

are processed

§ First-come, first-served
§ Priority
§ Preferred (loyalty programs/fee-based)
§ Reservation (appointment)
Measures of Waiting Line
Performance
Measures of Waiting Line Performance
The operations manager typically looks at five measures when evaluating existing or
proposed service systems. They relate to potential customer dissatisfaction and costs:

The average The average System The implied The implied

number of time utilization, cost of a cost of a
customers customers which refers given level of given level of
waiting, wait, either to the capacity and capacity and
either in line in line or in percentage its related its related
or in the the system. of capacity waiting line. waiting line.
system. utilized.
Queuing Models:
Infinite - sourcs
QUEUING MODELS: INFINITE-SOURCE

1. Single server, exponential service time.

2. Single server, constant service time.
3. Multiple servers, 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
Average number or steady (i.e., steady state)
timewaiting in line
Infinite-source symbols
 Basic Relationships of
Infinite source
The arrival and service rates, represented by λ and M, must be in
the same units (e.g., customers per hour)
§ System utilization: This reflects the ratio of demand (as measured
by the arrival rate) to supply or capacity (as measured by the
product of the number of servers M, and the service rate, μ).

§ The average number of customers being served:

 Basic Relationships of
Infinite source
Little’s law applied in waiting line:
Number of people waiting in line = (Average Customer Arrival Rate) x
(Average Time in the System)

§ The average number of customers:

§ The average time customers are:

Lq will usually be one of the first values you will want to determine in problem solving.
 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.
 Solution
a) The arrival rate is given in the problem: λ = Basic relationships
18 customers per hour. Change the service
time to a comparable hourly rate. Thus, 60
minutes per hour/3 minutes per customer =
μ = 20 customers per hour
Single Server, Exponential Service Time, M/M/1¹

The simplest model involves a system that has one server (or a single crew). 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.
 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:
a) System utilization.
b) Percentage of time the server (agent) will be idle.
c) The expected number of customers waiting to be
served.
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.
Solution
 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.
 Example
Wanda’s Car Wash & Dry is an automatic, five-minute 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.
Solution
 Multiple Servers, M/M/S

Two or more servers are working independently to provide service to

customer arrivals. Use of the model involves the following assumptions:

1) A Poisson arrival rate and exponential Multiple-server queuing formulas

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
Queuing Models:
Finite - source
QUEUING MODEL: FINITE-SOURCE

§ The finite-source model is appropriate for cases in which the

calling population is limited to a relatively small number of
potential calls.
§ There may be more than one server or channel
§ Arrival rates are required to be Poisson and service times
exponential.
§ Arrival rate of customers in a finite situation is affected by the
length of the waiting line
§ Arrival rate decreases as the length of the line increases
Finite-source queuing formulas
and notation
Constraint
Management
 Constraint Management
Managers may be able to reduce waiting times by actively managing one or
more system constraints. Typically, in the short term, the facility size and the
number of servers are fixed resources. However, some other options might
be considered:

Look
Shift
for a
Demand
bottleneck
Use
temporary Standardize
workers service
Psychology of Waiting Line
Management
The Psychology of Waiting
Occupied time Unexplained waits
feels shorter than are longer than
explained waits
unoccupied time.

Unfair waits are

People want to longer than
get started equitable waits

The more
Anxiety makes CEO valuable the service,
waits seem longer the longer the
customer will wait

Uncertain waits David H. Maister

Solo waits feel
are longer than
longer than group
known, finite waits
waits
 Disney’s approach to managing
waiting lines

1. Provide distractions
2. Provide alternatives for
those willing to pay a
premium
3. Keep customers informed
4. Exceed expectations
5. Comfortable waiting
environment
Operations Strategy
 Operations Strategy

Carefully Shift some

assess the arrivals to
costs and Increase the Standardi “off-times”
New
benefits of processing zation by using
processing
various rate (reduce
equipmen
alternative vs. variability § reservation
and systems
for number of in
methods § “early-bird”
capacity of servers processing)
specials
service
§ senior
systems. discounts
Thank You!