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SimulationSimulation
Inverse Functions
• Actually, we’ve already done this with the normal distribution.
Inverse Normal
• Actually, we’ve already done this with the normal distribution.
x
3.0
0.1
x = + z
= 3.0 + 0.3 x 1.282
= 3.3846
XZ
0
Inverse ExponentialExponential Life
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 0.5 1 1.5 2 2.5 3
Time to Fail
Den
sit
y
a
f x e x( )
f(x)F a X a( ) Pr{ }
e dxxa
0
e x a
1 e a
Inverse Exponential
xF Xe
1 -)(F(x)
x
Inverse Exponential
aF ae
= 1 - )(F(x)
x
F(a)
a
Inverse Exponential
Suppose we wish to find a such that the probability of a failure is limited to 0.1.
F(x)
x
F(a)
a
Inverse Exponential
Suppose we wish to find a such that the probability of a failure is limited to 0.1.
0.1 = 1 -
ae
F(x)
x
F(a)
a
Inverse Exponential
Suppose we wish to find a such that the probability of a failure is limited to 0.1.
0.1 = 1 -
ln(0.9) = -a
F(x)
x
F(a)
a
ae
Inverse Exponential
Suppose we wish to find a such that the probability of a failure is limited to 0.1.
0.1 = 1 -
ln(0.9) = -a
ae
F(x)
x
F(a)
a
a = - ln(0.9)/
Inverse Exponential
Suppose a car battery is governed by an exponential distribution with = 0.005. We wish to determine a warranty period such that the probability of a failure is limited to 0.1.
a = - ln(0.9)/
= - (-2.3026)/0.005
= 21.07 hrs.
F(x)
x
F(a)
a
Model
Customers arrive randomly in accordance withsome arrival time distribution. One server services customers in order of arrival. The service time is random following someservice time distribution.
M/M/1 Queue
M/M/1 Queue assumes exponential interarrival times and exponential service times
A eiAi
S eiSi
M/M/1 Queue
M/M/1 Queue assumes exponential interarrival times and exponential service times
A eiAi
S eiSi
South Dakota
School of Mines & Technology
Expectations for ExponentialExpectations for Exponential
Exponential Review
Expectations
Introduction toProbability & Statistics
Exponential ReviewExponential Review
M/M/1 Queue
2.032 1.951 1.349 .795 .539 .347
0.3050.0740.0350.5201.5350.159
M/M/1 Queue
2.032 1.951 1.349 .795 .539 .347
0.3050.0740.0350.5201.5350.159
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.0354 1.349 1.349 1.869 0.000 0.5205 1.951 1.951 3.486 0.000 1.5356 2.032 3.486 3.646 1.454 1.613
Avg = 0.261 0.699
M/M/1 Queue
.347
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.34723456
Avg = #DIV/0! #DIV/0!
M/M/1 Queue.347
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.34723456
Avg = #DIV/0! #DIV/0!
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 0
Sum =TimeAvg
M/M/1 Queue.539 .347
0.305
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.3472 0.5393456
Avg = #DIV/0! #DIV/0!
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.6520.7260.7950.8301.3491.8691.9512.0323.4863.646
Sum = 0.192TimeAvg 0.053
M/M/1 Queue.539 0.652
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.5393456
Avg = 0.000 0.305
.795
M/M/1 Queue.539
0.074
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.6523456
Avg = 0.000 0.305
0.652.795
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.7260.7950.8301.3491.8691.9512.0323.4863.646
Sum = 0.305TimeAvg 0.084
M/M/1 Queue
0.726
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873456
Avg = 0.056 0.246
.795
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.7950.8301.3491.8691.9512.0323.4863.646
Sum = 0.379TimeAvg 0.104
M/M/1 Queue
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795456
Avg = 0.056 0.246
.795
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.795 3 arrive 0 1 1 0.000
Sum = 0.379TimeAvg #DIV/0!
M/M/1 Queue.795
0.035
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795456
Avg = 0.056 0.246
0.8301.349
M/M/1 Queue
0.830
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.035456
Avg = 0.038 0.176
1.349
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.795 3 arrive 0 1 1 0.0000.830 3 depart 0 0 0 0.035
Sum = 0.414TimeAvg #DIV/0!
M/M/1 Queue1.349
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.0354 1.34956
Avg = 0.038 0.176
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.795 3 arrive 0 1 1 0.0000.830 3 depart 0 0 0 0.0351.349 4 arrive 0 1 1 0.000
Sum = 0.414TimeAvg #DIV/0!
M/M/1 Queue1.349
0.520
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.0354 1.34956
Avg = 0.038 0.176
1.8691.951
M/M/1 Queue1.8691.951
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.0354 1.349 1.349 1.869 0.000 0.52056
Avg = 0.028 0.262
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.795 3 arrive 0 1 1 0.0000.830 3 depart 0 0 0 0.0351.349 4 arrive 0 1 1 0.0001.869 4 depart 0 0 0 0.520
Sum = 0.934TimeAvg #DIV/0!
M/M/1 QueueM/M/1 Queue2.032 1.951 1.349 .795 .539 .347
0.3050.0740.0350.5201.5350.159
Time of Start Depart Time in Time inPart No. Arrival Service Time Queue System
0 -- -- 0 -- --1 0.347 0.347 0.652 0.000 0.3052 0.539 0.652 0.726 0.113 0.1873 0.795 0.795 0.830 0.000 0.0354 1.349 1.349 1.869 0.000 0.5205 1.951 1.951 3.486 0.000 1.5356 2.032 3.486 3.646 1.454 1.613
Avg = 0.261 0.699
M/M/1 Event Calendar
Event Event BusyTime Part No. Type Q(t) S(t) B(t) Time
0 -- start 0 0 0 00.347 1 arrive 0 1 1 00.539 2 arrive 1 2 1 0.1920.652 1 depart 0 1 1 0.1130.726 2 depart 0 0 0 0.0740.795 3 arrive 0 1 1 0.0000.830 3 depart 0 0 0 0.0351.349 4 arrive 0 1 1 0.0001.869 4 depart 0 0 0 0.5201.951 5 arrive 0 1 1 0.0002.032 6 arrive 1 2 1 0.0813.486 5 depart 0 1 1 1.4543.646 6 depart 0 0 0 0.160
Sum = 2.629TimeAvg 0.721
M/M/1 Performance Measures
Number in Queue
0
3
0 1 2 3 4
Time
Q(t
)
M/M/1 Performance Measures
Number in System
0
3
0 1 2 3 4
Time
S(t
)
M/M/1 Performance Measures
Busy/Idle
0
2
0 1 2 3 4
Time
B(t
)
Applications; Financial