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1. Prepared by Dr Ajay Parulekar

2. Topic Objectives
Prepared by Dr Ajay Parulekar
 To get acquainted with the basic terminology of queuing
theory
 To understand its use in day today business
 To learn applications of queuing theory through real life
business problems

3. Introduction
Prepared by Dr Ajay Parulekar
 A queuing model is a suitable model to represent a
service oriented problem where customers arrive
randomly to receive some service, the service time being
also a random variable
 The objective of a queuing model is to find out the
optimum service rate and the number of servers
(resources) so that the average cost of being in queuing
system and the cost of service are minimized
 Elements of a Queuing System:
I. Arrival Pattern: It is characterized by,
a. Number of customers arriving can be finite or infinite

4. Introduction
Prepared by Dr Ajay Parulekar
b. Time period between the arriving customers can be
constant or random. Most commonly used is Poisson
Exponential Distribution with mean arrival rate (average
number of arrivals per unit period – sec, min, hour etc)
denoted by λ. Hence mean time between successive
arrivals is (1 / λ)
II. Customer Behaviour
III. Queue (Waiting Line): It refers to customers who are
waiting to be served
a. Waiting time is the time spent by a customer in the queue
before being served.
b. Service time is the time spent by server in providing
service to a customer

5. Introduction
Prepared by Dr Ajay Parulekar
c. Waiting time in the system is the total time (waiting
time in the queue + service time)
d. Queue length is number of customers waiting in the
queue
e. System length is total customers in the system
(customers in the queue + customers being served)

6. Introduction
Prepared by Dr Ajay Parulekar
IV. Servers:
Servers can be a single one or can be multiple ones.
Arrangement of servers can be done in series or in
parallel.
The service time is the actual time for which service is
rendered to the customer. It can be constant or random.
Most of the times the service time follows exponential
continuous probability distribution with mean service
time as (1 / μ) where μ is the mean service rate
(average number of customers served per unit period of
time)
Service discipline is usually FIFO or FCFS provided
server is busy

7. Introduction
Prepared by Dr Ajay Parulekar
Service Discipline:
Service discipline is usually FIFO (First In First Out) or
FCFS (First Come First Served) provided server is busy.
It can also be LCFS (Last Come First Served) under
certain scenarios. Under SIRO (Service In Random
Order), customers are selected at random and time of
arrival of customer is inconsequential.
Under ‘Priority Service’, customers in a queue are
selected on a priority basis (for e.g. tenure of service,
VIPs etc.)
We shall be restricting ourselves to FCFS

8. Introduction
Prepared by Dr Ajay Parulekar
Customer Behaviour:
Customers who are patient by nature, will join an existing
queue and will wait for service. Whereas certain ‘impatient’
customers might defect from queue. These will not choose a
queue randomly (in case of multiple queues) and will keep
on switching to fast moving or shortest queue. This
behaviour is called ‘jockeying’. Under extreme condition,
customer might wait in the queue for sometime and then
leave the system because he feels system is working too
slowly. This is called as reneging. A certain type of customer
might, upon arrival not join the queue for some reason and
decide to return after sometime. Such behaviour is known
as‘balking’.
We shall be assuming there is no jockeying or balking or
reneging and customers leave the system only after
receiving service and not before.

9. Kendall’s Notation
Prepared by Dr Ajay Parulekar
A queuing model is symbolically represented in terms
of some important characteristics known as Kendall’s
notation given by (a/b/c) : (d/e)
where,
a: Probability distribution of the number of arrivals per
unit time.
b: Probability distribution of service time
c: Number of servers
d: Capacity of the system
e: Service discipline

10. Single server (Channel) Queuing Model (M/M/1) :
(∞ / FIFO)
Prepared by Dr Ajay Parulekar
Customers arrive through a single queue & are served by
single server on FIFO basis. Other assumptions,
i)Arrivals - as well departures follow Poisson Distribution
ii) Queue discipline is FIFO or FCFS
iii) Only one queue & one server
iv) Infinite potential customers, infinite system capacity
v) Mean arrival rate (λ) < Mean Service rate (μ)

11. IMP FORMULAE - Single server (Channel) Queuing
Model (M/M/1) : (∞ / FIFO)
Prepared by Dr Ajay Parulekar
MeanArrival Rate = λ
Mean Service Rate = μ
Utilisation Parameter /Traffic Intensity = (λ / μ)
Probability of having exactly‘n’ customers in the system,
Pn = ρn (1 – ρ)
Expected number of customers in the system Ls = ρ / (1 – ρ)
Expected number of customers in the queue Lq = ρ2 / (1 – ρ)
Mean waiting time in the systemWs = 1 / (µ – λ)
Mean waiting time in the queueWq = ρ / (µ – λ)

12. Multiple Servers (Channel) Queuing Model (M/M/k)
: (∞ / FIFO)
Prepared by Dr Ajay Parulekar
Customers arrive through a single queue & are served by
two or more servers which are arranged in parallel, on
FIFO basis. Other assumptions,
i)Arrivals as well departures follow Poisson Distribution
ii) Queue discipline is FIFO or FCFS
iii) Parallel servers (say k in numbers) serve a single queue
iv) All service facilities /servers provide the same service
at the same rate (μ)
v) Infinite potential customers
vi) Mean arrival rate (λ) < Combined Mean Service rate of
all the servers(kμ)

13. Example 1
Prepared by Dr Ajay Parulekar
A repairman is to be hired to repair machines which
breakdown at an average rate of 6 per hour. The
breakdown follow Poisson distribution. The productive
time of a machine is considered to cost INR 20 per hour.
Two repairman Mr. X & Mr. Y have been interviewed for
this purpose Mr. X charges INR 100 per hour and he
services breakdown machines at rate of 8 per hour. Mr. Y
demands INR 140 per hour and he services at an average
rate of 12 per hour. Which repairmen should be hired?
Why?(Assumes 8 hours shift per day).

14. Example 2
Prepared by Dr Ajay Parulekar
The mechanic at CarPoint is able to install new mufflers at
an average of three per hour while customers arrive at an
average rate of 2 per hour. Assuming that the conditions for
a single server infinite population model are all satisfied,
find out
a)The utilisation parameter
b)Average number of customers in the system
c)Average time a customer spends in the system
d)Average time a customer spends in the queue
e) Probability that there are more than 3 customers in the
system

15. Example 3
Prepared by Dr Ajay Parulekar
Customers arrive to a typist, known for quality typing, according to
Poisson probability distribution with an average inter arrival time of
20 minutes. If typist is not free, the customer will wait. The typist
completes the typing jobs at an average time of 15 minutes (with
exponential distribution). Find out,
a)What fraction of time is the typist busy?
b) Probability of having less than 3 customers at any time
c) Expected number of customers with typist
d) Expected number of customers waiting in the queue
e)Time spent by a customer in the queue
f) Average elapsed time between customer reaching and leaving the
typist
g) Expected time customer spends in the system
h) Probability that a customer has to wait for more than 10 min in
queue
i) Probability that a customer shall be in system for more than 10
min