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Graduate Program in Engineering and Technology Management

Graduate Program in Engineering and Technology Management. Simulation of Discrete Event Systems Aslı Sencer. Dynamic Simulation: Queueing System. Arrivals. Departures. Service. is identified by: Arrival rate, interarrival time distribution Service rate, service time distribution

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Graduate Program in Engineering and Technology Management

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  1. Graduate Program in Engineering and Technology Management Simulation of Discrete Event Systems Aslı Sencer

  2. Dynamic Simulation:Queueing System Arrivals Departures Service • is identified by: • Arrival rate, interarrival time distribution • Service rate, service time distribution • # servers • # queues • Queue capacities • Queue disciplines, FIFO, LIFO, etc. Aslı Sencer

  3. M/M/1 Queueing System Arrivals Departures Service M: interarrival time is exponentially distributed M: service time is exponentially distributed 1: There is a single server Aslı Sencer

  4. Ex3: Model Specifics • Initially (time 0) empty and idle • Base time units: minutes • Input data (assume given for now …), in minutes: Part Number Arrival Time Interarrival Time Service Time 1 0.00 1.73 2.90 2 1.73 1.35 1.76 3 3.08 0.71 3.39 4 3.79 0.62 4.52 5 4.41 14.28 4.46 6 18.69 0.70 4.36 7 19.39 15.52 2.07 8 34.91 3.15 3.36 9 38.06 1.76 2.37 10 39.82 1.00 5.38 11 40.82 . . . . . . . . . . • Stop when 20 minutes of (simulated) time have passed Aslı Sencer

  5. Queuing Simulation • Random variables: • Time between arrivals • Service time represented by probability distributions. • Events: • Arrival of a customer to the system • Departure from the system. • State variables: • # customers in the queue • Worker status {busy, idle} • Output measures: • Average waiting time in the queue • % utilization of the server • Average time spent in the system Aslı Sencer

  6. Output Performance Measures • Total production of parts over the run (P) • Average waiting time of parts in queue: • Maximum waiting time of parts in queue: N = no. of parts completing queue wait WQi = waiting time in queue of ith part Know: WQ1 = 0 (why?) N> 1 (why?) Aslı Sencer

  7. Output Performance Measures (cont’d.) • Time-average number of parts in queue: • Maximum number of parts in queue: • Average and maximum total time in system of parts: Q(t) = number of parts in queueat time t TSi = time in system of part i Aslı Sencer

  8. Output Performance Measures (cont’d.) • Utilization of the machine (proportion of time busy) • Many others possible (information overload?) Aslı Sencer

  9. Simulation by Hand • Manually track state variables, statistical accumulators • Use “given” interarrival, service times • Keep track of event calendar • “Lurch” clock from one event to the next • Will omit times in system, “max” computations here (see text for complete details) Aslı Sencer

  10. Simulation by Hand: Setup Aslı Sencer

  11. Simulation by Hand:t = 0.00, Initialize Aslı Sencer

  12. Simulation by Hand:t = 0.00, Arrival of Part 1 1 Aslı Sencer

  13. Simulation by Hand:t = 1.73, Arrival of Part 2 2 1 Aslı Sencer

  14. Simulation by Hand: t = 2.90, Departure of Part 1 2 Aslı Sencer

  15. Simulation by Hand:t = 3.08, Arrival of Part 3 3 2 Aslı Sencer

  16. Simulation by Hand:t = 3.79, Arrival of Part 4 4 3 2 Aslı Sencer

  17. Simulation by Hand:t = 4.41, Arrival of Part 5 5 4 3 2 Aslı Sencer

  18. Simulation by Hand:t = 4.66, Departure of Part 2 5 4 3 Aslı Sencer

  19. Simulation by Hand: t = 8.05, Departure of Part 3 5 4 Aslı Sencer

  20. Simulation by Hand:t = 12.57, Departure of Part 4 5 Aslı Sencer

  21. Simulation by Hand:t = 17.03, Departure of Part 5 Aslı Sencer

  22. Simulation by Hand:t = 18.69, Arrival of Part 6 6 Aslı Sencer

  23. Simulation by Hand: t = 19.39, Arrival of Part 7 7 6 Aslı Sencer

  24. Simulation by Hand:t = 20.00, The End 7 6 Aslı Sencer

  25. Ex3:Complete Record of the Hand Simulation Aslı Sencer

  26. Ex3: Simulation by Hand:Finishing Up • Average waiting time in queue: • Time-average number in queue: • Utilization of drill press: Aslı Sencer

  27. Entity Based-Simulation Since simulation ends at 20th minute, 6th item’s process will not be completed! The last event that occurs in a simulation will be the arrival of 7th item! Average wait time=(0+1.17+1.58+4.26+8.16+0)/6=2.53min Average time in system=(2.9+2.93+4.97+8.78+12.62+4.36)/6=36.56/6=6.09min Aslı Sencer

  28. Randomness in Simulation • The above was just one “replication” — a sample of size one (not worth much) • Made a total of five replications: • Confidence intervals for expected values: • In general, • For expected total production, Note substantial variability across replications Aslı Sencer

  29. Comparing Alternatives • Usually, simulation is used for more than just a single model “configuration” • Often want to compare alternatives, select or search for the best (via some criterion) • Simple processing system: What would happen if the arrival rate were to double? • Cut interarrival times in half • Rerun the model for double-time arrivals • Make five replications Aslı Sencer

  30. Results: Original vs. Double-Time Arrivals • Original – circles • Double-time – triangles • Replication 1 – filled in • Replications 2-5 – hollow • Note variability • Danger of making decisions based on one (first) replication • Hard to see if there are really differences • Need: Statistical analysis of simulation output data Aslı Sencer

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