This document provides an overview of queuing (waiting line) theory. It discusses the basic components of a queuing model including arrival patterns, service rates, number of service facilities, system capacity, and queue discipline. It also describes common distributions for arrival and service times as well as different service arrangements. Several examples of queuing models and their applications are provided.

1. Operations Research
MBA-024

2. QUEUING (WAITING LINE) THEORY
UNIT IV

3. Basic Components of Queuing
Model
• The arrival pattern/arrival rate
• Service mechanism/service rate
• No. of service facilities
• Capacity of the system
• Queue discipline

4. The Input (or arrival) Pattern
• It represents the pattern in which customers arrive at the system.
• Arrivals may also be represented by the inter-arrival time, which is the
time period between two successive arrivals.
• Arrivals may be separated by equal intervals of time, or unequal but
definitely known intervals of time, or by unequal intervals of time whose
probabilities are known; these are called random arrivals.
• The rate at which customers arrive at the service station, that is, the
number of customers arriving per unit of time is called arrival rate.
• The assumption regarding the distribution of arrival rate has a great
impact on the mathematical model.
• If the number of customers is very large, the probability of an arrival in
the next interval of time does not depend upon the customers already in
the system.
• Hence, the arrival is completely random and it follows the Poisson process
with mean equals the average number of arrivals per unit time,
represented by λ.

5. The Service Mechanism (or service
pattern)
• The service pattern is similar to the arrival pattern but
there are some important differences.
• Service time may be a constant or an random variable.
• Distributions of service time which we are following are
‘negative exponential distribution’, which is characterised
by a single parameter, the mean rate µ or its mean service
time 1/µ.
• The servicing system in which the customers may be served
in batches of fixed size or of variable size by the same
server is termed as bulk service system.
• The system in which service depends on the number of
waiting customers is termed as state-dependent system.

6. Capacity of the System
• Some of the queuing processes admit physical
limitations to the amount of waiting room, so
that when the waiting line exceeds a fixed
length, no further customers are allowed to
enter until space becomes available by a
service completion.
• This type of situation is termed as finite source
queues.

7. Service Arrangements
• For providing service to the incoming customers, one or
more service points are established.
• The number depends in the number of customers, rate of
arrivals, time taken for providing service to a single
customer, and so on.
• Depending on these variables, a service channel is single or
multiple.
• When there are several service channels available to
provide service, much depends upon their arrangement.
They may be arranged in parallel or in series or a more
complex combination of both, depending on the design of
the system’s service mechanism.

8. Service Channels

9. Service Time

10. Queue Discipline

11. Customer Behaviour

12. Symbols

13. Single Channel Queuing Model
• Arrival rates follow Poisson distribution
• Service time follows exponential distribution
• Single server
• Capacity of system is infinite
• Queue discipline: FIFO

14. Formulae

15. Formulae

16. Formulae

17. A TV repairman finds that the time spent on his
job has an exponential distribution with mean 30
minutes. If he repairs sets in the order in which
they come in and if the arrival of sets is
approximately a Poisson with an average rate of
10 in an 8 hour day, what is the repairman’s
expected idle time each day? How many jobs are
ahead of the average set just brought in?

18. 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 the
customer.

19. Arrival of machinists at a tool crip is considered to be
Poisson distributed at an average rate of 6 per hour.
The length of the time the machinists must remain at
the tool crip is exponentially distributed with an
average time being 0.05 hours.
a.What is the probability that a machinist arriving at
the tool crip will have to wait?
b.What is the average no. of machinists at the tool
crip?
c.The co. will install a 2nd
tool crip when convinced that
a machinist would have to spend 6 minutes waiting
and being served at the tool crip. By how much the
flow of machinists to the tool crip should increase to
justify the addition of a 2nd
tool crip?

20. Customers arrive at a window drive in a bank
according to Poisson distribution with mean 10 per
hour. Service time per customer is exponential with
mean 5 minutes. The space in front of the window,
including that for the serviced car can accommodate a
maximum of 3 cars. Other cars can wait outside this
space.
a.What is the probability that an arriving customer car
drives directly to the space in front of the window?
b.What is the probability that an arriving customer car
will have to wait outside the indicated space?

21. A repairman is to be hired to repair machines
which break down at an average rate of 3 per
hour. The breakdown follows a Poisson
distribution. Non productive time of a machine is
considered to cost Rs. 10 per hour. Two repairmen
have been interviewed – one is slow but cheap
while the other is fast but expensive. The slow
repairman charges Rs. 5 per hour and he services
broken down machines at the rate of 4 per hour.
The fast repairman demands Rs. 7 per hour and he
services at an average rate of 6 per hour. Which
repairman should be hired?

22. Applications of Queue Model
• Scheduling of aircraft at landing and takeoff
from busy airports.
• Scheduling of issue and return of tools by
workmen from tool cribs in factories.
• Scheduling of mechanical transport fleets.
• Scheduling distribution of scarce war material.
• Scheduling of work and jobs in production
control.

23. Applications of Queue Model
• Minimisation of congestion due to traffic delay at
tool booths.
• Scheduling of parts and components to assembly
lines.
• Decisions regarding replacement of capital assets
taking into consideration mortality curves,
technological improvement and cost equations.
• Routing and scheduling of salesmen and sales
efforts.

24. Other Benefits of Queuing Theory
• Attempts to formulate, interpret and predict
for purposes of better understanding the
queues and for the scope to introduce
remedies such as adequate service with
tolerable waiting time.
• Provides models that are capable of
influencing arrival pattern of customers.

25. Other Benefits of Queuing Theory
• Determines the most appropriate amount of
service or number of service stations.
• Studies behaviours of waiting lines via
mathematical techniques utilising concept of
stochastic process.