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Simulating Single server queuing models

Simulating Single server queuing models. 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.

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Simulating Single server queuing models

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  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 • IdleBusy 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 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

  7. System state Initialization Time = 0 system 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 A 0.4 D Clock 0.4 Eventlist 1 0 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 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

  9. Arrival Time = 1.6 system System state A 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 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

  10. Arrival Time = 2.1 System state A 2.1 D Clock 0.4 Eventlist 1 2 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 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

  11. Departure Time = 2.4 System state A 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 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

  12. Departure Time = 3.1 System state A 3.1 D Clock 2.1 Eventlist 1 0 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 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

  13. Departure Time = 3.1 System state 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 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

  14. Departure Time = 3.1 System state A 3.8 D Clock 3.8 Eventlist 1 0 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 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

  15. Departure Time = 3.1 System state A 4.0 D Clock 3.8 Eventlist 1 1 4.0 # in que Time Of Last event Server status 4.0 3.1 4 1.8 1.8 Area Under B(t) Total delay Area Under Q(t) Number delayed Times of Arrival System Statistical Counters 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

  16. Departure Time = 3.1 System state A 4.9 D Clock 4.0 Eventlist 1 0 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 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

  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

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