Simulating%20Single%20server%20queuing%20models

1
Simulating Single server queuing models
2
Simulating Single server queuing models

• Consider the following sequence of activities
  that each customer undergoes
• Customer arrives
• Customer waits for service if the server is busy.
• Customer receives service.
• Customer departs the system.

3
Analytical Solutions

• Analytical solutions for W, L, Wq, Lq exist
  However, analytical solution exist at infinity
  which cannot be reached.
• Therefore, Simulation is a most.

4
Flowchart of an arrival event
Idle Busy
An Arrival
Status of Server
Customer enters service
Customer joins queue
More
5
Flowchart of a Departure event

• NO Yes

A Departure
Queue Empty ?
Remove customer from Queue and begin service
Set system status to idle
More
6
An example of a hand simulation

• Consider the following IATs and STs
• A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
  A70.2, A81.4, A91.9,
• S12.0, S20.7, S30.2, S41.1, S53.7, S60.6
• Want Average delay in queue
• Utilization

7
System state
Initialization Time 0 system
0.4
999.
A
Server




0
D
Clock
Eventlist
0
0
0
in que
Time Of Last event
Server status
0
0
0
0
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
Statistical Counters
8
Arrival Time 0.4 system
System state
1.6
2.4
A




0.4
D
Clock
0.4
Eventlist
0
1
0.4
in que
Time Of Last event
Server status
0
1
0
0
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
9
Arrival Time 1.6 system
System state
2.1
2.4
A
1.6



1.6
D
Clock
0.4
Eventlist
1
1
1.6
in que
Time Of Last event
1.6
Server status
1.2
1
0
0
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
10
Arrival Time 2.1
System state
3.8
2.4
A
1.6
2.1


2.1
D
Clock
0.4
Eventlist
2
1
2.1
in que
Time Of Last event
1.6
Server status
1.7
1
0
0.5
2.1
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
11
Departure Time 2.4
System state
3.8
3.1
A
2.1



2.4
D
Clock
1.6
Eventlist
1
1
2.4
in que
Time Of Last event
2.1
Server status
2.0
2
0.8
1.1
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
12
Departure Time 3.1
System state
3.8
3.3
A




3.1
D
Clock
2.1
Eventlist
0
1
3.1
in que
Time Of Last event
Server status
2.7
3
1.8
1.8
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
13
Departure Time 3.1
System state
3.8
999.
A




3.3
D
Clock
Eventlist
0
0
3.3
in que
Time Of Last event
Server status
2.9
3
1.8
1.8
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
14
Departure Time 3.1
System state
4.0
4.9
A




3.8
D
Clock
3.8
Eventlist
0
1
3.8
in que
Time Of Last event
Server status
2.9
4
1.8
1.8
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
15
Departure Time 3.1
System state
5.6
4.9
A
4.0



4.0
D
Clock
3.8
Eventlist
1
1
4.0
in que
Time Of Last event
Server status
3.1
4.0
4
1.8
1.8
Area Under B(t)
Total delay
Area Under Q(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
16
Departure Time 3.1
System state
5.6
8.6
A




4.9
D
Clock
4.0
Eventlist
0
1
4.9
in que
Time Of Last event
Server status
4.0
5
2.7
2.7
Total delay
Area Under Q(t)
Area Under B(t)
Number delayed
Times of Arrival
System
Statistical Counters
A10.4, A21.2, A30.5, A41.7, A50.2, A61.6,
A70.2, A81.4 S12.0, S20.7, S30.2, S41.1,
S53.7, S60.6
17
Monte Carlo Simulation

• Solving deterministic problems using stochastic
  models.
• Example estimate
• It is efficient in solving multi dimensional
  integrals.

18
Monte Carlo Simulation

• To illustrate, consider a known region R with
  area A and R1 subset of R whose area A1 in
  unknown.
• To estimate the area of R1 we can through random
  points in the region R. The ratio of points in
  the region R1 over the points in R approximately
  equals the ratio of A1/A.

R
R1
19
Monte Carlo Simulation

• To estimate the integral I. one can estimate the
  area under the curve of g.
• Suppose that M max g(x) on a,b

1. Select random numbers X1, X2, ,Xn in
a,b And Y1, Y2, ,Yn in 0,M 2. Count how
many points (Xi,Yi) in R1, say C1 3. The estimate
of I is then C1M(b-a)/n
M
R
R1
a
b
20
Advantages of Simulation

• Most complex, real-world systems with stochastic
  elements that cannot be described by mathematical
  models. Simulation is often the only
  investigation possible
• Simulation allow us to estimate the performance
  of an existing system under proposed operating
  conditions.
• Alternative proposed system designs can be
  compared with the existing system
• We can maintain much better control over the
  experiments than with the system itself
• Study the system with a long time frame

21
Disadvantages of Simulation

• Simulation produces only estimates of performance
  under a particular set of parameters
• Expensive and time consuming to develop
• The Large volume of numbers and the impact of the
  realistic animation often create high level of
  confidence than is justified.

22
Pitfalls of Simulation

• Failure to have a well defined set of objectives
  at the beginning of the study
• Inappropriate level of model details
• Failure to communicate with manager during the
  course of simulation
• Treating a simulation study as if it is a
  complicated exercise in computer programming
• Failure to have well trained people familiar with
  operations research and statistical analysis
• Using commercial software that may contain errors

23
Pitfalls of Simulation cont.

• Reliance on simulator that make simulation
  accessible to anyone
• Misuse of animation
• Failure to account correctly for sources of
  randomness in the actual system
• Using arbitrary probability distributions as
  input of the simulation
• Do output analysis un correctly
• Making a single replication and treating the
  output as true answers
• Comparing alternative designs based on one
  replication of each design
• Using wrong measure of performance