Theory of queues

Module 4
Queuing Theory
MBA SEMESTER 2
Quantitative Analysis (QA)-II
PREPAREDBY : JIGNESH J KARIYA
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Theory of Queues
2By: Jignesh Kariya

Introduction
3By: Jignesh Kariya
A common situation that occurs in everyday life is that of waiting in a line
either at bus stops, petrol pumps, restaurants, ticket booths, bank,
hospital and so on.
Queues (waiting Lines) are also found in workshops where the machines
wait to be repaired ; at a tool crib(cheat) where the mechanics wait to
receive tools; in a warehouse where items wait to be used . Incoming
calls wait to mature in the telephone exchange, trucks wait to be
unloaded, airplanes wait either to take off or land and so on.
Queuing theory can be applied to a variety of situations where it is not
possible to accurately predict the arrival rate (or time) of customers and
service rate (or time) of service facility or facilities.
Queuing theory can be used to determine the level of services that
balances the following two conflicting costs :

Introduction
4By: Jignesh Kariya
1. Cost of offering the service
2. Cost incurred due to delay in offering service.
The first cost is associated with the service facilities and their operation,
and the second represents the cost of customers waiting for service.
Obviously an increase in the existing service facilities would reduce the
customer’s waiting time and decreasing the level of service would result
in long queue. This means in the level of service increases the cost of
operating service facilities but the decreases the cost of customers
waiting for service.
Since customer waiting cost for service is difficult to estimate, it is usually
measured in terms of loss of sales or goodwill when the customer is a
human being and has no sympathy with the service system. But if the
customer is machine waiting for repair then cost of waiting is measured
in terms of cost production.

Introduction
5By: Jignesh Kariya

The Structure Of A Queuing System
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The major components of any waiting line(queuing) system are :
1. Calling population (or input score)
2. Queuing Process
3. Queue discipline
4. Service Process (or Mechanism)

7By: Jignesh Kariya
 Potential customers who arrive to the queuing system is referred as ‘Calling
Population’ also known as ‘customer (input) source’.
The manner in which customers arrive at the service facility, individually, or in
batches, at scheduled or unscheduled time is called the arrival process. The
customer's entry into the queuing system depends upon the queue conditions.
Customers, from a queue, are selected for service according to certain rules
known as queue discipline.
 A service facility may be without server (self service), or may include one or
more servers operating either in a series (as a team) or in parallel (multiple
service channels). The rate (constant or random) at which service is rendered is
known as the service process. After the service is rendered, the customer leaves
the system.
 If the server is idle at the time of the customer's arrival, then the customer is
served immediately, otherwise the customer is asked to join a queue or wailing
line, which may have single, multiple or even priority lines.

8By: Jignesh Kariya
Calling population Characteristics
The arrivals or inputs to the system are characterized by:
• Size of calling population
• behavior of the arrivals
• Pattern of arrivals at the system
The calling population need not be homogeneous and may consist of
several subpopulations. For example, patients arriving at the OPD of a
hospital are usually of three categories: walk-in patients, patients with
appointments and emergency patients. Each patient class places different
demands on service facility, but the waiting expectations of each category
differ significantly.
Size of calling population
The size of calling population, whether it is homogeneous or consists of
several subpopulations, is considered to be either finite (limited) or infinite
(unlimited).

9By: Jignesh Kariya
Calling population Characteristics cont..
If customer's arrival depends on the number of customers already in the
system (in service plus in queue), the calling population is called limited or
finite.
An example of a finite calling population is a factory only has four
machines, which often require repair/service and two of them (say) are in
working condition. Then at any point in time, there are only two machines
that could possibly require service.
Alternately, if new customer's arrival is independent of the number of
customers already in the system, the calling population is called unlimited
or infinite.
Examples of infinite population include customers arriving at a bank or
super market, students arriving to get admission at a university, cars
arriving at a highway petrol pump, etc.

10By: Jignesh Kariya
• behavior of the arrivals
If a customer, on arriving at the service system waits in the queue until
served and docs not switch between waiting lines. He is called a patient
customer.
Ex: Machines arrived at the maintenance shop are examples of patient
customers.
Whereas the customer, who waits for a certain time in the queue and
leaves the service system without getting service due to certain reasons is
called an impatient customer.
EX : a customer who has just arrived at a grocery store and finds that the
salesmen arc busy in serving the customers already in the system, will
either wait for service till his patience is exhausted or estimates that his
waiting time may be excessive and so leaves immediately to seek service
elsewhere.

The behavior of the arrivals at any queuing system is categorized as :
• Balking Customers do not join the queue either by seeing the
number of customer already in service system or by estimating the
excessive waiting time for the desired service.
• Reneging Customers, after joining the queue, wait for sometime in
the queue but leave before being served on account of certain
reasons.
• Jockeying Customers move from one queue to another hoping to
receive service more quickly(a common scene at a railway booking
window).

• Pattern of arrivals at the system
Customers may arrive in batches or individually. These customers may
arrive at a service facility either on scheduled time or on unscheduled time.
The arrival process of customers to the service system is classified into two
categories : Static and Dynamic
In static arrival pattern, the control depends on the nature of arrival rate
(random or constant): In random (or unscheduled) arrivals the times are
random variable and therefore we need to understand the average and
frequency distribution of the times.
In both the cases, the arrival process can be described either by the
average arrival time or by the average inter-arrival time(average time
between two consecutives arrival).

The number of unscheduled arrivals to a service facility, in some fixed
period of time, can be studied by a statistical probability distribution such a
Poisson Distribution.
The dynamic arrival process is controlled by both the service facility and the
customers. The service facility adjusts its capacity to match changes in the
service intensity.
The arrival time distribution can be approximated by one of the following
probability distribution.
 Poisson Distribution
 Exponential Distribution
 Erlang Distribution

The poisson distribution, a discrete probability distribution, describes the
arrival rate variability i.e. number of random arrivals at a service facility in a
fixed period of time.
Another probability distribution that describes the average time between
arrivals (inter-arrival time) when arrival rate is poisson is called Exponential
Probability distribution.
Let n customers arrive in a time 0 to t. If λ is the expected number of
arrivals in a given time interval 0 to t will be λ *t
P(x=n) = ((λt)n e- λt) / n! For n=0,1,2..
P(x=0) = ((λt)0 e- λt) / 0! = e- λt

2. Queuing Process
The queuing process refers to the number of queues and their respective
lengths. The number of queues single, multiple or priority queues depend
upon the layout of a service system.
The length (or size) of a queue depends upon operational situations such as
physical space, legal restrictions, and attitude of the customers.
In certain cases, a service system is unable to accommodate more than the
required number of customers at a time. No further customers are allowed
to enter until more space is made available to accommodate new
customers. Such type of situations are referred to as finite (or limited)
source queue.
Examples of finite source queues are cinema halls, restaurants, etc.

Queuing Process cont..
On the other hand, if a service system is able to accommodate any number
of customers at a time, then it is referred to as infinite (or unlimited) source
queue. For example, in a sales department where the customer orders are
received, there is no restriction on the number of orders that can come in,
a queue of any size can be formed.
In many other situations, when arriving customers find long queue(s) in
front of a service facility, they often do not enter the service system even
though additional waiting space is available. The queue length in such cases
depends upon the attitude of the customers.
For example, when a motorist finds that there are many vehicles waiting at
the petrol station, in most of the cases, he does not stop at this station and
seeks service elsewhere.

Queuing Process cont..
In some finite source queuing systems, the maximum permissible queue is
of zero length, i.e. no queue is allowed to form. For example, when a
parking space (service facility) cannot accommodate additional incoming
vehicles (customers), the motorists are diverted elsewhere.
Multiple queues at a service facility can also be finite or infinite. But this
has certain advantages such as
• Division of manpower is possible.
• Customer has the option of joining any queue and can also switch to the
end of any other queue.
• Balking behavior of the customers can be controlled.

3. Queue Discipline
Queue discipline refers to the selections of customers from a queue for
service.
The queue discipline is the order or manner in which customers from the
queue are selected for service.
There are number of ways in which customers in the queue are served.
Some of these are :
a. Static Queue Discipline :
These are based on the individual customer’s status in the queue. Few of
such discipline are :
 FCFS : If the customers are served in the order of their arrival, then this
is known as the ‘First Come First Serve’ service discipline.
 LCFS : ‘Last Come First Serve’

Queue Discipline Cont..
b. Dynamic Queue Discipline :
These are based on the individual customer’s attributes in the queue. Few
of such discipline are :
 Service In Random Order (SIRO) : under this rule customers are selected
for service at random, irrespective of their arrivals in the service system.
 Priority Service : Under this rule customers are grouped in priority
classes on the basis of some attributes such as service time or urgency.
The FCFS rule is used within each class to provide service. The payment
of telephone or electricity bills by cheque or cash are examples of this
discipline.
 Pre-emptive priority (Emergency) : Under this rule, an important
customer is allowed to enter into the service immediately after entering
into the system, even if a customer with a lower priority is already in
service.

Queue Discipline Cont..
That is lower priority customer’s service is interrupted to start the service
for such a customer. This interrupted service is resumed after the priority
customer is served.
 Non- pre-emptive priority : In this case an important customer is
allowed to go ahead in the queue but the service his started
immediately on completion of the current service.
4. Service Process (or Mechanism)
The Service process is concerned with the manner in which customers are
serviced and leave the system.
It is characterized by:
 The arrangement of service facilities
 The distribution of service time
 Server’s behavior
 Management Policies

Service Process Cont..
 The arrangement of service facilities
The capacity of the service facility is measured in terms of customers who
can be served simultaneously and/or effectively.
The service facilities (or servers) commonly known as Service channels may
be in series, in parallel or mixed.
Single Queue, Single Service Facility

Single Queue, Multiple Service
facilities in Parallel

Multiple Queues, Multiple Service
facilities in Parallel

Service Process cont..
 The distribution of service times
Service time is the elapsed time from the beginning to the end of a
customer’s service.
The time taken by the server from the commencement of service to the
completion of service for a customer is known as the “service time”.
A random service time may be described in two ways :
a. Average Service Rate : µ * t
a. Average Length of Service Time : E(T) = 1 / µ

Performance Measure of a Queuing System
The performance measure of any queuing system, which are of a general
interest, for the evaluation of the performance of an existing queuing
system, and for designing a new system in terms of level of service a
customer receives as well as proper utilization of the service facilities are
listed as follows:
1. Time-related questions for the customers
(average (or expected) time spent by a customer in the queue and system)
2. Quantitative questions related to the number of customers
average (or expected) number of customers in the queue and system)
3. Questions Involving value of time both for customers and servers
Value of time both for customers and servers
4. Cost- related questions
Average cost required to operate the queuing system

1. Time-related questions for the customers
(a) Wq : What is the average (or expected) time an arriving customer has to
wait in a queue (denoted by (Wq) before being served.
(b) Ws: What is the average (or expected) time an arriving customer
spends in the system (denoted by Ws), including waiting and service. This
data can be used to make economic comparison of alternative queuing
systems.
2. Quantitative questions related to the number of customers
(a) Lq : The expected number of customers who are in the queue (queue
length) for service. This is denoted by Lq
(b)Ls : The expected number of customers who are in the system either
waiting in the queue or being serviced (denoted by Ls,). This data can be
used for finding the mean customer time spent in the system.

3. Questions Involving value of time both for customers and servers
(a) Pw : What is the probability that an arriving customer has to wait before
being served (denoted by Pw)? This is also called blocking probability.
(b) p = λ / μ : What is the probability that a server is busy at any particular
point in time (denoted by p )? This is the proportion of the time that a
server actually spends with the customer, i.e. the fraction of the time a
server is busy.
p = λ / μ
(c)Pn : What is the probability of n customers being in the queuing system
when it is in steady state condition? This is denoted by Pn, n = 0, 1....
(d) Pd : What is the probability of service denial when an arriving customer
cannot enter the system because the queue is full? This is denoted by Pd

4. Cost-related questions
(a) What is the average cost needed to operate the system per unit of time?
(b) How many servers (service centers) are needed to achieve cost
effectiveness?

Transient State and Steady States
At the beginning of service operations, a queuing system is influenced by
the initial conditions, such as number of customers in the system for
service and percentage of time servers are busy serving customers.
This period of transition is termed as transient-state. However, after
sufficient time has passed. the system becomes independent of
the initial conditions and and enters a steady state condition.
In the development of queuing theory models it is assumed that the system
has entered a steady-state.

Transient State and Steady States cont..

Relationship among Performance Measure

Relationship among Performance Measure cont..

Classification of Queuing Models
Different models in queuing theory are classified by using special
notations described initially by D.G.Kendall in 1953 in the form (a/b/c).
Later A.M. lee in 1966 added the symbols d and c to the kendallb
notation.

Classification of Queuing Models

Example - 1

Example – 2
Customers arrive at a box office window, being manned by a single individual,
according to a Poisson input process with a mean rate of 30 per hour. The time
required to serve a customer has an exponential distribution with a mean of 90
seconds. Find the average waiting time of a customer. Also, determine the average
number of customers in the system and the average queue length.

Example – 3 A self service store employs one cashier at its counter. Nine
customers arrive on an average every five minutes but the cashier can serve 10
customers in 5 minutes. Assuming Poisson distribution for arrival rate and
exponential distribution for service time, find
i. Average number of customers in the system.
ii. Average number of customers in the queue or average queue length.
iii. Average time a customer spends in the system.
iv. Average time a customer waits before being served.

Example - 4

Example – 5
Customers arrive at a clinic at the rate of 8 per hour (Poisson arrival) and the doctor
can serve at the rate of 9 per hour (exponential).
(1) What is the probability that a customer does not join the queue and walks into
the doctor’s room?
(2) What is the probability that there is no queue?
(3) What is the probability that there are 10 customers in the system?
(4) What is the expected number in the system?
(5) What is the expected waiting time in the queue?

Questions ???

Theory of queues

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