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Simulating Single server queuing models
Simulating Single server queuing models
• Consider the following sequence of activities that each customer undergoes:1. Customer arrives
2. Customer waits for service if the server is busy.
3. Customer receives service.
4. Customer departs the system.
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.
Flowchart of an arrival event
IdleBusy
An Arrival
Status of Server
Customer joins queueCustomer enters service
More
Flowchart of a Departure event
NO Yes
A Departure
Queue Empty ?
Set system status to idle
Remove customer from Queue and begin service
More
An example of a hand simulation
• Consider the following IAT’s and ST’s:
• A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4, A9=1.9, …
• S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
• Want: Average delay in queue • Utilization
InitializationTime = 0
system
Server
00 0
0 0 0 0
00.4
999.
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
Statistical Counters
ArrivalTime = 0.4
system
01 0.4
1 0 0 0
0.41.6
2.4
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
0.4
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
ArrivalTime = 1.6
system
11 1.6
1.6
1 0 0 1.2
1.62.1
2.4
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D0.4
1.6
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
ArrivalTime = 2.1
21 2.1
1.6
2.1
1 0 0.5 1.7
2.13.8
2.4
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D0.4
1.6
System
2.1
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 2.4
11 2.4
2.1
2 0.8 1.1 2.0
2.43.8
3.1
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D1.6
2.1
System
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 3.1
01 3.1
3 1.8 1.8 2.7
3.13.8
3.3
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D2.1
System
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 3.1
00 3.3
3 1.8 1.8 2.9
3.33.8
999.
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
System
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 3.1
01 3.8
4 1.8 1.8 2.9
3.84.0
4.9
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
System
3.8
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 3.1
11 4.0
4.0
4 1.8 1.8 3.1
4.05.6
4.9
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
System
3.8
4.0
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
DepartureTime = 3.1
01 4.9
5 2.7 2.7 4.0
4.95.6
8.6
System state
Serverstatus
# in que
Times of Arrival
TimeOf Lastevent
Clock
Eventlist
Numberdelayed
Totaldelay
AreaUnderQ(t)
AreaUnderB(t)
A
D
System
4.0
A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6
Statistical Counters
Monte Carlo Simulation
• Solving deterministic problems using stochastic models. – Example: estimate
• It is efficient in solving multi dimensional integrals.
b
adxxgI )(
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
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]
a b
R1
RM
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
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
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.
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
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