quantitative techniques

2. TODAY, WE SHALL FIRST FINISH THE
UNFINISHED AGENDA OF THE LAST SESSION.

3. We started with Queuing Theory in
the last session…

4. C – 4
Queueing Theory …
 It is a mathematical/analytical theory that
enable us to understand the behavior of a
queueing system and also, helps in solving
various problems associated with the
queueing system.
 It is study of waiting line systems in its all
possible dimensions.
 It is mathematical analysis of queues and
waiting times in stochastic systems.

5. C – 5
Queueing Theory studies – QUEUEING SYSTEM!!!
 QUEUEING SYSTEM:
 It is a system that takes CUSTOMERS who are coming for
taking services as input, asks them to wait in a QUEUE if the
server is busy, SERVER provides necessary services to the
customers; and finally, it outputs a SERVED CUSTOMER.
Dave’s
Car Wash
Enter Exit
Population of
dirty cars
Arrivals
from the
general
population …
Queue
(waiting line)
Service
facility
Exit the system
Arrivals to the system Exit the system
In the system

6. C – 6
QUEUEING SYSTEM…
 QUEUEING SYSTEM : is a system that has the
following inter-related components :
 INPUT: Every queueing system receives inputs of those
customers who calls for the services. And, the population
from which they come, is known as the Calling Population.
 QUEUE: A queue is very important part of a queueing
system. It consists of those customers who have come for
the services, but has to wait as the service center is busy in
giving services to other customers.
 SERVICE CENTER: A service center is a facility that
provides necessary services to the customers for which they
have entered into the queueing system.
 OUTPUT: Output of a queueing system consists of those
customers who have obtained the necessary services for
which they have entered into a queueing system.

7. To model a Queueing System, we have
to make assumptions about it!!!!

8. C – 8
The Model which we are going to study is …
Service
facility

9. ASSUMPTIONS ABOUT THE SIMPLE - ONE
CHANNEL & ONE SERVER MODEL
ASSUMPTIONS ABOUT THE INPUTS :
 The size of the Calling Population is infinite.
 The process of customers arrival to the queuing system is random and it
follows Poisson Probability Distribution.
 The customers are coming individually and independently.
 The mean arrival rate per unit of time is  which is assumed to be a constant.
 Once a customer has arrived to the Queueing System, he/she will not go back
without taking service; that is to say, no balking is allowed.

10. ASSUMPTIONS ABOUT THE SIMPLE - ONE
CHANNEL & ONE SERVER MODEL (CONTINUED..)
ASSUMPTIONS ABOUT THE QUEUE:
 There is only one queue.
 The size of the queue may be infinite.
 The queue discipline is first come, first serve ( FCFS).
 Once a customer has joined the queue for some time, he/she will not leave it
without taking services. That is to say that, no reneging is allowed.

11. ASSUMPTIONS ABOUT THE SIMPLE - ONE
CHANNEL & ONE SERVER MODEL (CONTINUED..)
ASSUMPTIONS ABOUT THE SERVICE CENTER:
 There is only one service center.
 The services are provided in one phase.
 The service time devoted to a customer is uncertain and is Exponentially
Distributed.
 The mean service rate per unit of the time is μ and μ > λ.
 There is no holiday/vacation/break etc. during the working hours of the
service center.

12. ASSUMPTIONS ABOUT THE SIMPLE - ONE
CHANNEL & ONE SERVER MODEL (CONTINUED..)
ASSUMPTIONS ABOUT THE OUTPUT:
 Every customer will be served.
 Each customer will be a satisfied customer in the sense that whatever services,
he/she wants to have obtains it.
ASSUMPTIONS ABOUT THE OVER ALL QUEUEING SYSTEM:
 The system will be in the equilibrium state or steady state.
 When the system opens or closes down, there should not be any queue.

13. C – 13
We are looking for…
 Measures related to Probability.
 Measures related to the System as Whole
 Measures related to the Queue
(Unconditional)
 Measures related to the Queue
(Conditional)

14. C – 14
Summary of Measures…
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15. Any question before we
proceed further.

16. C – 16
Time to Practice … whatever we have learnt
so far!!!

17. C – 17
Practice Problem #1:
 Currently, a store has a separate checkout counter
for senior citizens. On average, 30 senior citizens per
hour arrive at the counter, according to a Poisson
distribution, and are served at an average rate of 35
customers per hour, with exponential service times.
Find the following operating characteristics:
a. Probability of zero customers in the system
b. Average utilization of the checkout clerk
c. Average number of customers in the system
d. Average number of customers in line
e. Average time spent in the system
f. Average waiting time in line

18. C – 18
Practice Problem #2:
 A clerk services 12 customers per hour who arrive for
services at a rate of 8 per hour.
Answer the following:-
(i) What is the average number of customers waiting for the service
of the clerk?
(ii) What is the average time a customer has to wait before getting
services?
(iii) The management is contemplating to have another clerk and
this will reduce service time to 3 minutes per customer and will
mean additional cost of Rs 500/- per day. If the waiting time of
customer costs Rs. 1.20 per minute, should the management
have another clerk? Assume 8 hours working day.

19. C – 19
Practice Problem #3:
 At a telephone booth, consumers come at the rate of
8 per hour and take on the average 5 minutes for its
use. The telephone company has a policy to install
another telephone booth if waiting time exceeds 20
minutes. Find the arrival rate at which another
telephone booth can be justified.

20. C – 20
Practice Problem #3:
 Arrivals at a telephone booth are considered to be
Poisson with an average time of 10 minutes between
one arrival and the next. The length of a phone call is
assumed to be exponentially distributed with mean 3
minutes. What is the probability that a person
arriving at the booth will have to wait? The second
booth can be started when an arrival would expect to
wait for at least 3 minutes. By how much should the
flow of arrivals increase in order to justify a second
booth?

21. C – 21
Practice Problem #4:
 To decide which repairman - A or B should be
selected for a company's workshop, the management
provides the following information:
 Machine break down rate = 4 machines per hour.
If machine opportunity cost = Rs 20/- per M.H., then
take a suitable decision.
A B
Wages per hour Rs. 20 Rs. 30
Service capacity (p.h.) 6 8
Spare parts & other items consumption (p.h) Rs. 10 Rs. 12
Workshop expenses (p.h) Rs. 25 Rs. 28

22. C – 22
Where we see applications of Queuing
Theory?

25. WHAT NEXT?

26. Meet Sardar Manjit Singh…
He has a truck.
He wants to know should he replace
the truck by a new truck – after one
year? After two years? After three
years?
When should he replace?

27. Why Sardar Manjit Singh is facing
this problem?
He can replace the truck after the
life of the truck!!!!
Do you think that he should do
that?

28. Should this man change his MACHINE?
THINK!!!

29. What should I do –
Individual Replacement
or Group Replacement?
This is Mr. How Jones.
He owns a cinema hall
in the city of Turino,
Italy. The hall is having
around 5,000 LED Light
Bulbs. If one such bulb
is failed and he gets it
replaced, it costs him €
4.75; but if he gets
them replaced in group,
it costs € 0.65 per LED
light bulb.

30. If you wish to solve their problems in an
OPTIMUM manner, then we have to study
REPLACEMENT MODELS

31. Let’s have a quick look at what reasons may
compel a firm to replace their assets.

32. Reasons for replacement…
• Impairment of an asset
• Change in Technology or the asset has become obsolete
• The company has decided to change its business/production line
• To expand the capacity
• To have a balance between the asset’s capital cost and
operating/maintenance costs
• End of the life of the assets

33. When to replace an asset?
• Run an asset till failed! Go on using an asset till it lasts and get it
repaired when it failed. In this case, we need to manage spare parts
inventory.
• Replace an asset when it is the most economical to replace – Go
on using an asset till it is economical to use! The Day one finds
that the asset is not-economical, replace the same. But,
before that time of replacement, the asset has to be properly
maintained. It is determining the length of time before replacing an
existing asset.
• For Bulk assets -Time based maintenance. Based on a fixed time
interval, irrespective of the condition of the equipment, replace all the
assets whether good or bad! But, before that fixed time interval,
assets have be replaced on individual basis.

34. Let’s talk something
about the FINANCE
Dimension of
Replacement Problems
before we proceed
further.

35. We need FINANCE to replace assets!!!
• Can we use DEPRECIATION RESERVE for replacement?

36. We need FINANCE to replace assets!!!
• Do we have something called Depreciation Reserve?
• What are the sources of financing an asset which we are going
to replace?

37. Let’s consider a Replacement Model …
Where the asset is deteriorating with time and
time value of money is ZERO.

38. Replacement
Model #1…
Assets that deteriorates over time
and time value of money is zero!!!!

39. Assumptions of Model #1
1. The given asset has a fixed life and the value of the asset is deteriorating
with the passage of time. It means that in no case the value of asset will
increase with the passage of time.
2. An asset will be replaced only at the end or in the beginning of a time
period.
3. Time value of money is zero.
4. The new asset that will replace the existing asset will have …
• Same capacity to serve
• Same capital and operating/maintenance costs
5. We are assuming that there is no inflation.

40. Assumptions of Model #1 (continued)
6. We are assuming that there WILL NOT be any impairment of the asset
during its life.
7. The objective will be MINIMIZATION of the total cost of using the machine.
8. We are assuming that there is complete certainty.

41. Let’s state the problem…
• Find the OPTIMAL TIME PERIOD after which an asset
should be replaced so that the total of the following costs is
MINIMUM.
1. Operating and Maintenance Costs
2. Capital cost in used to built the Asset

42. Let’s appreciate the behaviour of these costs.
AGE OF THE MACHINE
COST IN Rs.
PURCHASE PRICE
SALVAGE VALUE
TOTAL COST MAINTENANCE COST

43. Let’s a take a problem and see how one can solve
the same
Q.No.1 Decide optimum replacement period for an asset whose cost is Rs
2,00,000 and other information is as follows :-
Years 1 2 3 4 5 6
Operating Cost(Rs)10,000 12,000 14,000 18,000 20,000 25,000
Resale Value (Rs)1,80,000 1,50,000 1,25,000 1,00,000 80,000 75,000
Think!!! How can
we find the same?

44. Let’s see how can we get the optimum solution for the
problem
So, the optimum solution is – Replace the
asset after every one year and it will result
into a cost of Rs. 30,000 every year.

45. Let’s take another problem…

46. Answer:

47. Let’s a take a problem and see how one can solve
the same
Think!!! How can
we find the same?
Q.No.2. Consider the following data about machine A :
Purchase price = Rs 2,00,000
Years 1 2 3 4 5 6
Operating Cost (Rs.) 5000 7000 9000 11000 13000 15000
Resale Value (Rs.) 150000 130000 120000 115000 100000 90000
(i) Determine its economic life. (ii) If a new machine B whose
capacity is 150% of that of A and its optimum life is 5 years with AC =
Rs 40,000 has entered into the market, should the management replace A
by B ? If yes, when ? Assume that it has at present 3 machines of type A
which are two years old.

48. Solution:
Machine A has an average cost of Rs. 28, 333 with a replacement
period of 6 years while Machine B has an average cost of Rs. 80,
000 for a replacement period of 5 years (Note its capacity is 150%
times of that of Machine A)
What should be our decision? (A – Rs. 85,000 and B – Rs. 80,000)

49. Let’s a take a problem and see how one can solve
the same
Q.No.2. Consider the following data about machine A :
Purchase price = Rs 2,00,000
Years 1 2 3 4 5 6
Operating Cost (Rs.) 5000 7000 9000 11000 13000 15000
Resale Value (Rs.) 150000 130000 120000 115000 100000 90000
(i) Determine its economic life. (ii) If a new machine B whose
capacity is 150% of that of A and its optimum life is 5 years with AC =
Rs 40,000 has entered into the market, should the management replace A
by B ? If yes, when ? Assume that it has at present 3 machines of type A
which are two years old.

50. Nothing more today in this session!
ENJOY AND HAVE
A NICE DAY!