Simulation of systems
Steps in Simulation
Problem formulation
Setting of objectives and overall project plan
Model conceptualization
Data collection
Model Translation
No
Verified ?
Yes
No
No
Validated ?
Yes
Experimental Design
Production run and analysis
Yes
More Run?
No
Yes
Documentation and reporting
02-08-2021
Implementation
Simulation example
Simulating single server queue
Queuing systems
There are many situations in daily life when a queue is formed.
Queuing theory is the mathematical study of waiting lines and it is very useful for analyzing the procedure of queuing of
daily life of human being.
Queuing theory applies not only in day to day life but also in sequence of computer programming, networks, medical field,
banking sectors etc.
The Process Flow in Queue
Calling
Population
Waiting
Line
Server
Arrival Event Processing
Arrival
Event
Server
Busy?
No
Unit Enters
Service
Yes
Unit Enters
Queue
Departure Event Processing
Departure Event
No
Server Idle Time
Begins
Yes
Remove one waiting
unit from the Queue
and start service
Anyone Waiting?
Simulating a Single Server Queuing System
To simulate the functioning of a single server queuing system Simulate Entity – An
object of Interest in a system
The assumptions are
The calling population is infinite
The time between arrivals is known and is uniformly distributed between limits (say 1 to 8
min)
Mean time between arrivals is 4.5 min
The service time is known and is uniformly distributed between limits (say 1 to 6)
Mean service time is 3.5 min
Mean service time is less than mean time between arrivals hence the system is stable
The system capacity is infinite
FIFO is the rule
02-08-2021
Modelling & Simulation ─ Application Areas
1
Random
Number
0-0.067
0.068- 0.133
0.134-0.333
0.334-0.600
0.601-0.867
0.868-0.9999
Modelling & Simulation ─ Application Areas
1
0
5
0
0
5
5
2
8
8
3
8
0
11
3
3
4
5
6
7
8
9
10
3
2
7
7
5
1
2
2
11
13
20
27
32
33
35
37
6
3
3
6
6
2
3
2
11
17
20
27
33
39
41
44
0
4
0
0
1
6
6
7
17
20
23
33
39
41
44
46
6
7
3
6
7
8
9
9
3
4
Solution………….
Customer
1
2
3
4
5
6
7
8
9
10
Inter Arrival
Time (min)
Arrival Time Service Time
(Clock)
(min)
0
8
11
13
20
27
32
33
35
37
8
3
2
7
7
5
1
2
2
02-08-2021
5
3
6
3
3
6
6
2
3
2
Time
Service
Begins
(clock)
0
8
11
17
20
27
33
39
41
44
Waiting Time
Time
Time Customer Server
(min)
Service Ends Spends in the Idle Time
(Clock)
System (min)
(min)
0
0
0
4
0
0
1
6
6
7
24
5
11
17
20
23
33
39
41
44
46
5
3
6
7
3
6
7
8
9
9
63
3
4
7
Metrics of the Queueing System
Average Waiting Time
Probability that the customer
has to wait
=
=
=
Server Utilisation
=
=
Mean Waiting Time
=
=
Mean time in system
=
=
=
Total time spent in queue / Total
Number of Customers
24/10 = 2.4 min
No. of Customers who wait / Total No. of
Customers
2/10 = 0.2
1-(Server Idle Time /Total Time)
1-(7/46) = 95.65 %
Total waiting time / No. of Customers
who wait
24/5 = 4.8 min
Total time in system / Total No of
Customers
63/10 = 6.3 min
Simulating single server queue
Simulating Single sever queuing systems
Let us consider an example of super market, who have a single billing counter, following are observations and
assumption of the market
• The calling population is infinite
• Mean time of arrival of a customer is uniformly distributed between 2 to 7min, with mean of 5 min
Inter arrival Time
Probability
2
3
4
5
6
7
0.10
0,15
0,20
0.25
0.20
0.10
Service time per customer is uniformly distributed between 1 to 6 min with mean of 4 min
Service Time
1
2
3
4
5
6
Probability
0.10
0,10
0,20
0.30
0.20
0.10
• Since the mean customer arrival time is more than service time of each customer, hence the system is stable
• First Come First Serve rule is applicable
02-08-2021
DR. ASHISH KUMAR SAXENA
Simulation Process flow
Problem statement: Simulate the billing process of the supermarket, and find out efficiency of the system over
finite event
Customer arrival
to billing
counter
Stand in Waiting
line
Billing
Process
There are two events, in which system stage changes with time.
• Arrival event
• Departure event
Between these two events the complete process take place …….
So we should discuss these two events in detail.
02-08-2021
Simulating single server queue
Arrival Event Processing
Customer arrival
at
billing counter
No
Billing Counter busy ?
Yes
Customer
Billing
Process start
Customer need
to stand in
queue
Departure Event Processing
Customer
billing
Complete
No
Anyone waiting
in queue?
Yes
Server Idle time
start
1 Customer start
billing, 1 reduced
from queue
Details of Queue system
Random
Number
Random
Numbe
0-0.01
0-0.01
0.11-0.25
0.11-0.20
0.26-0.45
0.21-0.40
0.46-0.7
0.41-0.7
0.71-0.90
0.71-0.90
0.91-0.99
0.91-0.99
Customer Inter Arrival
Time (min)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Random
Number
Arrival
Time
(Clock)
Random
Service
Time
Waiting
Time
Number Time (min) Service Time (min) Service
Begins
Ends
(clock)
(Clock)
Time
Server
Customer
Idle
Spends in
Time
the System (min)
(min)
Probability that the customer has to wait =
𝑁𝑜 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑤ℎ𝑜 ℎ𝑎𝑠 𝑡𝑜 𝑤𝑎𝑖𝑡
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
=
Service utilization probability =1 −
𝑆𝑒𝑟𝑣𝑒𝑟 𝐼𝑑𝑙𝑒 𝑡𝑖𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒
Average time per customer spend in system
𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟 𝑠𝑝𝑒𝑛𝑑 𝑖𝑛 𝑠𝑦𝑠𝑡𝑒𝑚
=
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
Mean service time =
𝑇𝑜𝑡𝑎𝑙 𝑠𝑒𝑟𝑣𝑖𝑐𝑒 𝑡𝑖𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
Average time between arrivals =
Mean waiting time =
𝑇𝑜𝑡𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑎𝑟𝑟𝑖𝑣𝑎𝑙 𝑡𝑖𝑚𝑒
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
𝑇𝑜𝑡𝑎𝑙 𝑤𝑎𝑖𝑡𝑖𝑛𝑔 𝑡𝑖𝑚𝑒 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑤𝑎𝑖𝑡𝑒𝑑 𝑖𝑛 𝑠𝑦𝑠𝑡𝑒𝑚
Example 2: Call Center Problem
Multi Server Queueing Problem
• Consider a Call Center where technical staff take calls and provide service
• Two technical support people (server) exists
•Able more experienced, provides service faster
•Baker new, provides service slower
• Current rule: Able gets call if both people are idle
• Other possible rules:
•Baker gets call if both are idle
•Call is assigned randomly to Able and Baker
• Goal of study: Find out how well the current rule works!
Caller
Nr.
Rando Interarr Arrival
m
ival
Time
Numbe Time
r
When
Able
Avail..
When
Baker
Avail.
Server Rando Service
Chosen m
Time
Numbe
r
Time
Service
Begins
Able‘s
Service
Compl.
Time
Caller
Time in
Delay
System
/waitin
gTime
in
System
1
-
-
0
0
-
A
0.36
3
0
3
0
3
2
0.567
2
2
3
2
B
0.72
5
2
7
0
5
Example 3: Simulation of Inventory System
A dealer of electrical appliances has a certain product for which the probability distribution of demand per day and the
probability distribution of the lead-time, developed by past records are as shown in the following tables.
Probability distribution of lead demand
Probability distribution of lead time
The various costs involved are,
Ordering Cost = Rs. 50 per order
Holding Cost = Rs.1 per unit per day
Shortage Cost = Rs. 20 per unit per day
The dealer is interested in having an inventory policy with two parameters, the reorder point and the order quantity, i.e.,
at what level of existing inventory should an order be placed and the number of units to be ordered. Evaluate a simulation
plan for 15 days, which calls for a reorder quantity of 35 units and a re-order level of 20 units, with a beginning inventory
balance of 45 units.
Simulation of Inventory System
Day
Random
Number
Demand
Random
Number
(Lead
Time)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
58
45
43
36
46
46
70
32
12
40
51
59
54
16
68
7
6
6
6
6
6
7
5
4
6
6
7
6
4
7
73
21
-
Lead Time Inventory Quantity
(Days)
at the end received
of the day
3
2
-
45
38
32
26
20
14
8
1
31
27
21
15
8
2
33
26
35
35
-
Ordering
Cost
Holding
Cost
Shortage
Cost
50
50
100
38
32
26
20
14
8
36
31
27
21
15
8
37
33
26
-
02-08-2021
02-08-2021
Any Questions ?
02-08-2021
DR. ASHISH KUMAR SAXENA