1. Graduate Program in Engineering and Technology Management
Simulation of Discrete Event Systems
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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.
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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
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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
Stop when 20 minutes of (simulated) time have passed
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5. Queuing Simulation Random variables: Events: State 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
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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?)
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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 queue at time t
TSi = time in system of part i
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8. Output Performance Measures (cont’d.)
Utilization of the machine (proportion of time busy)
Many others possible (information overload?)
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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)
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10. Simulation by Hand: Setup
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11. Simulation by Hand: t = 0.00, Initialize
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12. Simulation by Hand: t = 0.00, Arrival of Part 1
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13. Simulation by Hand: t = 1.73, Arrival of Part 2
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14. Simulation by Hand: t = 2.90, Departure of Part 1
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15. Simulation by Hand: t = 3.08, Arrival of Part 32
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16. Simulation by Hand: t = 3.79, Arrival of Part 42
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17. Simulation by Hand: t = 4.41, Arrival of Part 53 2
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18. Simulation by Hand: t = 4.66, Departure of Part 25 4 3
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19. Simulation by Hand: t = 8.05, Departure of Part 34
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20. Simulation by Hand: t = 12.57, Departure of Part 4
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21. Simulation by Hand: t = 17.03, Departure of Part 5
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22. Simulation by Hand: t = 18.69, Arrival of Part 6
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23. Simulation by Hand: t = 19.39, Arrival of Part 76
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24. Simulation by Hand: t = 20.00, The End7 6
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25. Ex3:Complete Record of the Hand Simulation
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26. Ex3: Simulation by Hand: Finishing Up
Average waiting time in queue:
Time-average number in queue:
Utilization of drill press:
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27. Entity Based-Simulation
ITEM #
INTERARRIVAL TIME
ARRIVAL TIME
SERVICE TIME
START SERVICE TIME
END SERVICE TIME
WAIT TIME
TIME IN SYSTEM 1
1,73
0,00
2,90 2
1,35
1,76
4,66
1,17
2,93 3
0,71
3,08
3,39
8,05
1,58
4,97 4
0,62
3,79
4,52
12,57
4,26
8,78 5
14,28
4,41
4,46
17,03
8,16
12,62 6
0,70
18,69
4,36
23,05 7
15,52
19,39
2,07 8
3,15
34,91
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=( )/6=2.53min
Average time in system=( )/6=36.56/6=6.09min
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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
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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
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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
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