EGR 549
June 2005
Using Arena for Queuing Problems
Consider Example 5 on page 1079. The average interarrival time is 6 min (rate is 10
number per hour) and average service time is 5 min (rate is 12 number per hour). Both
times are exponentially distributed. Below is its Arena model.
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Case 1: Below is complete statistical output from Arena when simulation run length is
5000 min:
ARENA Simulation Results
sparisay - License: STUDENT
Summary for Replication 1 of 1
Project: Example 5 P1078
Run execution date : 6/ 1/2005
Analyst: sparisay
Model revision date: 6/ 1/2005
Replication ended at time
: 5000.0 Minutes
Base Time Units: Minutes
TALLY VARIABLES
Identifier
Average Half Width Minimum Maximum Observations
mechinist.VATime
4.9598 .31025 .00709 33.073
845
mechinist.NVATime
.00000 .00000 .00000 .00000
845
mechinist.WaitTime
19.111 5.7202 .00000 78.367
845
mechinist.TranTime
.00000 .00000 .00000 .00000
845
mechinist.OtherTime
.00000 .00000 .00000 .00000
845
mechinist.TotalTime
24.070 5.8260 .01551 84.347
845
tool center.Queue.WaitingTime
19.120 5.7202 .00000 78.367
846
2
DISCRETE-CHANGE VARIABLES
Identifier
Average Half Width Minimum Maximum Final Value
mechinist.WIP
4.0802 (Corr) .00000 18.000 2.0000
clerk.NumberBusy
.83946 .04583 .00000 1.0000 1.0000
clerk.NumberScheduled
1.0000 (Insuf) 1.0000 1.0000 1.0000
clerk.Utilization
.83946 .04583 .00000 1.0000 1.0000
tool center.Queue.NumberInQueue
3.2407 (Corr) .00000 17.000 1.0000
Identifier
mechinist.NumberIn
mechinist.NumberOut
clerk.NumberSeized
clerk.ScheduledUtilization
System.NumberOut
OUTPUTS
Value
847.00
845.00
846.00
.83946
845.00
Simulation run time: 0.02 minutes.
Simulation run complete.
Below is some important information from this statistical output:
mechinist.VATime
4.9598: Means ave. service time, close to Expo(5)
mechinist.TotalTime
24.070: Means W, average time in system
tool center.Queue.WaitingTime
19.120: Means Wq, average waiting time in line
tool center.Queue.NumberInQueue 3.2407: Means Lq, average number in line
clerk.Utilization
.83946: Means server’s utilization
Also it shows that during 5000 minutes of simulation (5000/60 hrs) 847 mechinist have
entered the system. Then the arrival rate is 847/(5000/60) = 10.16. It is close to arrival
rate of 10.
Case 2: Below is partial statistical output from Arena when simulation run length is
100000 min:
Replication ended at time
Base Time Units: Minutes
: 100000.0 Minutes
TALLY VARIABLES
Identifier
Average Half Width Minimum Maximum Observations
mechinist.VATime
5.0403 .07117 7.7405E-05 48.689
16538
mechinist.NVATime
.00000 .00000 .00000 .00000
16538
mechinist.WaitTime
24.225 4.7873 .00000 178.64 16538
mechinist.TranTime
.00000 .00000 .00000 .00000 16538
mechinist.OtherTime
.00000 .00000 .00000 .00000 16538
mechinist.TotalTime
29.265 4.8305 4.7005E-04 186.06
16538
tool center.Queue.WaitingTime
24.224 4.7873 .00000 178.64 16539
DISCRETE-CHANGE VARIABLES
Identifier
Average Half Width Minimum Maximum Final Value
mechinist.WIP
4.8403 .78626 .00000 37.000 7.0000
clerk.NumberBusy
.83361 .01697 .00000 1.0000 1.0000
clerk.NumberScheduled
1.0000 (Insuf) 1.0000 1.0000 1.0000
clerk.Utilization
.83361 .01697 .00000 1.0000 1.0000
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tool center.Queue.NumberInQueue
4.0067
.77416
.00000
36.000
6.0000
Please compare the two output and notice the change in total time in system and
utilization of the clerk for both run length and its difference with theoretical values.
Case 3: Now assume that the interarrival time is distributed as Uniform (5,7), which
means the average interarrival time is 6 min. (minimum is 5 and maximum is 7 min)
Moreover, assume service time is distributed as Normal (5, 0.5), which means the
average service time is 5 min. (standard deviation is 0.5) We include these information
in Arena model and run it. Below is its partial statistical output.
Replication ended at time
Base Time Units: Minutes
: 5000.0 Minutes
TALLY VARIABLES
Identifier
Average Half Width Minimum Maximum Observations
mechinist.VATime
4.9696 .03832 3.5993 6.4912
834
mechinist.NVATime
.00000 .00000 .00000 .00000
834
mechinist.WaitTime
.04671 .01559 .00000 1.5417
834
mechinist.TranTime
.00000 .00000 .00000 .00000
834
mechinist.OtherTime
.00000 .00000 .00000 .00000
834
mechinist.TotalTime
5.0163 .04881 3.5993 6.5680
834
tool center.Queue.WaitingTime
.04665 .01559 .00000 1.5417
835
DISCRETE-CHANGE VARIABLES
Identifier
Average Half Width Minimum Maximum Final Value
mechinist.WIP
.83689 .01035 .00000 2.0000 1.0000
clerk.NumberBusy
.82910 .00869 .00000 1.0000 1.0000
clerk.NumberScheduled
1.0000 (Insuf) 1.0000 1.0000 1.0000
clerk.Utilization
.82910 .00869 .00000 1.0000 1.0000
tool center.Queue.NumberInQueue
.00779 (Insuf) .00000 1.0000 .00000
Identifier
mechinist.NumberIn
mechinist.NumberOut
clerk.NumberSeized
clerk.ScheduledUtilization
System.NumberOut
OUTPUTS
Value
835.00
834.00
835.00
.82910
834.00
Please compare the output of Case 1 and Case 3 and notice the changes in total time in
system and utilization of the clerk (and other statistics). The difference is because of
different distributions that are used. In these cases the average values of distributions
used are the same however the standard deviations (variations) are different and it created
the difference in performance measures.
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Case 4: You can include break schedules (for lunch and ….) for the clerk. Assume in
Case 3 you give a break of 30 minutes for each 2.5 hours that the clerk is working.
Below is the partial statistical output.
Replication ended at time
Base Time Units: Minutes
: 5000.0 Minutes
TALLY VARIABLES
Identifier
Average Half Width Minimum Maximum Observations
mechinist.VATime
4.9836 .03898 3.3958 6.5010
832
mechinist.NVATime
.00000 .00000 .00000 .00000
832
mechinist.WaitTime
18.875 (Corr) .00000 44.612
832
mechinist.TranTime
.00000 .00000 .00000 .00000
832
mechinist.OtherTime
.00000 .00000 .00000 .00000
832
mechinist.TotalTime
23.859 (Corr) 4.1655 49.264
832
tool center.Queue.WaitingTime
18.880 (Corr) .00000 44.612
833
DISCRETE-CHANGE VARIABLES
Identifier
Average Half Width Minimum Maximum Final Value
mechinist.WIP
3.9819 (Corr) .00000 9.0000 5.0000
clerk.NumberBusy
.82932 (Corr) .00000 1.0000 1.0000
clerk.NumberScheduled
.82778 (Insuf) .00000 1.0000 1.0000
clerk.Utilization
.82932 (Corr) .00000 1.0000 1.0000
tool center.Queue.NumberInQueue
3.1526 (Corr) .00000 8.0000 4.0000
Please compare the output of Case 4 and Case 3 and notice the change in total time in
system and scheduled number of clerk (in red). These differences are because of the
breaks given to the clerk. When clerk has 30 minutes break each 2.5 hours (150 min) it
means the percentage of time that the clerk is available is 150/(150+30) = 0.83. This
value is close to the clerk.NumberScheduled
.82778.
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Consider problem 1 on page 1085. Below is its Arena model.
Below is its statistical output:
ARENA Simulation Results
sparisay - License: STUDENT
Summary for Replication 1 of 1
Project: Problem 1 page 1085
Run execution date : 6/ 1/2005
Analyst: sparisay
Model revision date: 6/ 1/2005
Replication ended at time
: 5000.0 Minutes
Base Time Units: Minutes
TALLY VARIABLES
Identifier
Average Half Width Minimum Maximum Observations
customer.VATime
20.413 (Insuf) .00000 182.95
227
customer.NVATime
.00000 (Insuf) .00000 .00000
227
customer.WaitTime
37.045 (Insuf) .00000 229.60
227
customer.TranTime
.00000 (Insuf) .00000 .00000
227
customer.OtherTime
.00000 (Insuf) .00000 .00000
227
customer.TotalTime
57.458 (Insuf) .00000 272.56
227
facility.Queue.WaitingTime
55.221 (Insuf) .00000 229.60
153
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DISCRETE-CHANGE VARIABLES
Identifier
Average Half Width Minimum Maximum Final Value
customer.WIP
2.6342 .29455 .00000 5.0000 3.0000
serverr.NumberBusy
.92903 (Insuf) .00000 1.0000 1.0000
serverr.NumberScheduled
1.0000 (Insuf) 1.0000 1.0000 1.0000
serverr.Utilization
.92903 (Insuf) .00000 1.0000 1.0000
facility.Queue.NumberInQueue
1.7052 (Insuf) .00000 3.0000 2.0000
Identifier
customerServed
customerBalk
Counters
Count Limit
152 Infinite
75 Infinite
The effective arrival rate per hour will be {(152 number served)/(5000 simulation
length)} * (60 minute per hour) = 1.82
customer.WaitTime
facility.Queue.WaitingTime
37.045
: This is average waiting time of all customers including
those who balked!
55.221 : This is average waiting time in line for those entered the
system
Please compare the average waiting time in line and server utilization with that of
WinQSB.
Important issues in simulation:
How is the field data collected?
How much details, from the original system, are incorporated into the model?
How well is the simulation experiment conducted? (transient period, run length, number
of replications)
How well are the statistical output analyzed and appropriate recommendations provided?
(making distinction between randomness effect and system design)
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