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Simulating Single server queuing models

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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.

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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.

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Flowchart of an arrival event IdleBusy An Arrival Status of Server Customer joins queue Customer enters service More

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Flowchart of a Departure event NOYes A Departure Queue Empty ? Set system status to idle Remove customer from Queue and begin service More

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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

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Initialization Time = 0 system Server 0 0 0 0 0 00 0 0.4 999. System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D Statistical Counters

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Arrival Time = 0.4 system 01 0.4 100 0 1.6 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Arrival Time = 1.6 system 11 1.6 100 1.2 1.6 2.1 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 0.4 1.6 A1=0.4, A2=1.2, A3=0.5, A4=1.7, A5=0.2, A6=1.6, A7=0.2, A8=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Arrival Time = 2.1 21 2.1 1.6 2.1 100.5 1.7 2.1 3.8 2.4 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 0.4 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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 2.4 11 2.4 2.1 20.81.1 2.0 2.4 3.8 3.1 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 1.6 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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 3.1 01 3.1 31.8 2.7 3.1 3.8 3.3 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(t) A D 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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 3.1 00 3.3 31.8 2.9 3.3 3.8 999. System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 3.1 01 3.8 41.8 2.9 3.8 4.0 4.9 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 3.1 11 4.0 41.8 3.1 4.0 5.6 4.9 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Departure Time = 3.1 01 4.9 52.7 4.0 4.9 5.6 8.6 System state Server status # in que Times of Arrival Time Of Last event Clock Eventlist Number delayed Total delay Area Under Q(t) Area Under B(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=1.4 S1=2.0, S2=0.7, S3=0.2, S4=1.1, S5=3.7, S6=0.6 Statistical Counters

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Monte Carlo Simulation Solving deterministic problems using stochastic models. –Example: estimate It is efficient in solving multi dimensional integrals.

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Monte Carlo Simulation To illustrate, consider a known region R with area A and R 1 subset of R whose area A 1 in unknown. To estimate the area of R 1 we can through random points in the region R. The ratio of points in the region R 1 over the points in R approximately equals the ratio of A 1 /A. R R1R1

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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] ab R1R1 R M 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 R 1, say C 1 3. The estimate of I is then C 1 M(b-a)/n

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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

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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.

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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

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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

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