1. Simulation
imitation of some real thing, or a
Simulation is
process.
The act of simulating something generally involves
representation of certain
key characteristics or
behaviours
of a selected physical or abstract system.
Simulation involves the use of models to represent real
life situation.
2. Simulation Model
A simulation model is a mathematical model that
calculates the impact of uncertain inputs and
decisions we make on outcomes that we care about,
such as profit and loss, investment returns, etc.
A simulation model will include:
Model inputs that are uncertain numbers/
uncertain variables
Intermediate calculations as required
Model outputs that depend on the inputs -- These
are uncertain functions
3. Simulation techniques
Simulation techniques can be used to assist
management decision-making, where analytical
methods are either not available or inappropriate.
Typical business problems where simulation could be
used to aid management decision-making are
Inventory control.
Queuing problems.
Production planning.
4. Simulation and Queuing problems.
A major application of simulation has been in the
analysis of waiting line, or queuing systems.
Since the time spent by people and things waiting in
line is a valuable resource, the reduction of waiting
time is an important aspect of operations
management.
Waiting time has also become more important
because of the increased emphasis on quality.
Customers equate quality service with quick service
and providing quick service has become an important
aspect of quality service
5. Queuing problems.
For queuing systems, it is usually not possible to
develop analytical formulas, and simulation is often
the only means of analysis.
Simulation can hence be used to investigate
problems that are common in any situation involving
customers, items or orders arriving at a given point,
and being processed in a specified order.
For ex:
Customers arrive in a bank and form a single queue,
which feeds a number of service desks. The arrival
rate of the customers will determine the number of
service desks to have open at any specific point in time
6. Components of queuing systems
A queue system can be divided into four components
Arrivals: Concerned with how items (people, cars etc)
arrive in the system.
Queue or waiting line: Concerned with what happens
between the arrival of an item requiring service and the
time when service is carried out.
Service: Concerned with the time taken to serve a
customer.
Outlet or departure: The exit from the system.
A queuing problem involves striking a balance
between the cost of making reductions in service time
and the benefits gained from such a reduction
7. Structures of queuing system
There are a number of structures of queuing systems
in practice.
We will study only one i.e. single queue – single
service point.
Single queue – single service point
Queue discipline is first come – first served.
Arrivals* are random and for simulation this
randomness must be taken into account.
Service times** are random and for simulation this
randomness must be taken into account
successive
*Inter-arrival time: Is the time between the arrival of
customers in a queuing situation.
**Service time: Is the length of time taken to serve customers
8. Random Numbers
What is the purpose of random numbers?
There is randomness in the way customers are
likely to arrive.
The service time in most of the cases is also
variable.
The purpose of the random numbers is to allow
you to randomly select an arrival or service time
from the appropriate distribution.
To account for randomness, random numbers are
used.
9. Random Numbers
Such numbers can be computer generated, and are
often listed in published statistical tables.
Here we have a set of random numbers.
89 07 37 29 28 08 75 01 21 63
34 65 11 80 34 14 92 48 83 91
52 49 98 44 80 04 42 37 87 96
The random numbers are displaced as two-digit
numbers in the range between 00 and 99.
Every number is equally likely to occur and there is
no pattern, and thus no way of predicting what
number will be next in the sequence.
10. Example Problem
The arrival time of a customer at a retail sales depot is according to the
following distribution
Simulate the process for 10 arrivals and estimate the average waiting
time for the customer and percentage idle time for the server.
Use the following random numbers:
For IAT: 25, 19, 64, 82, 62, 74, 29, 92, 24, 23, 68, 96.
For ST: 92, 41,66,07,44,29,52,43,87,55,47,83
Assume that the shop opens at 9:00 am in the morning.
Service time
(in minutes)
Probability
3 0.3
4 0.5
5 0.1
6 0.1
Inter-arrival time
(in minutes)
Probability
3 0.1
4 0.2
5 0.5
6 0.1
7 0.1
11. Solution
Inter arrival time Probability
Cumulative
Probability
Basis of random allocation
3 0.1 0.1 0.0 -- 0.09
4 0.2 0.3 0.1 -- 0.29
5 0.5 0.8 0.3 -- 0.79
6 0.1 0.9 0.9 -- 0.89
7 0.1 1 0.9 -- 0.99
Service time Probability
Cumulative
Probability
Basis of random allocation
3 0.3 0.3 0.0 -- 0.29
4 0.5 0.8 0.3 -- 0.79
5 0.1 0.9 0.8 -- 0.89
6 0.1 1 0.9 -- 0.99
Calculation of Basis of random allocation
12. Solution
Inter arrival
time
Basis of random
allocation
3 0.0 -- 0.09
4 0.1 -- 0.29
5 0.3 -- 0.79
6 0.9 -- 0.89
7 0.9 -- 0.99
Service time
Basis of random
allocation
3 0.0 -- 0.29
4 0.3 -- 0.79
5 0.8 -- 0.89
6 0.9 -- 0.99
Customer
Random
Number
Inter arrival
time
Random
number
Service
time
Time of
arrival
Service
starts at
Service
ends at
Waiting
customer
Idle time
1 25 4 92 6 9.04 9.04 9.10 -- 4
2 19 4 41 4 9.08 9.10 9.14 2 --
3 64 5 66 4 9.13 9.14 9.18 1 --
4 82 6 07 3 9.19 9.19 9.22 -- 1
5 62 5 44 4 9.24 9.24 9.28 -- 2
9.29 9.29 9.32 -- 1
9.33 9.33 9.37 -- 1
9.40 9.40 9.44 -- 3
9.44 9.44 9.49 -- --
9.48 9.49 9.53 1 --
Total time 53 mins
Total waiting time 4 mins
Total idle time 12 mins
6 74 5 29 3
7 29 4 52 4
8 92 7 43 4
9 24 4 87 5
10 23 4 55 4
13. Solution
Average waiting time per customer is
4/10 = .4 minutes
Percentage average for the server is
(12/53)*100 = 22.64%
