Simulation. Inverse Functions Actually, we’ve already done this with the normal distribution

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


0.1 = 1 -

ae

F(x)

x

F(a)

a

Inverse Exponential


0.1 = 1 -

ln(0.9) = -a

F(x)

x

F(a)

a

ae

Inverse Exponential


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


0 -- -- 0 -- --1 0.34723456

Avg = #DIV/0! #DIV/0!

M/M/1 Queue.347


0 -- -- 0 -- --1 0.347 0.34723456


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


0 -- -- 0 -- --1 0.347 0.3472 0.5393456




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


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


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



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


M/M/1 Queue

0.726


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



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


M/M/1 Queue


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



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


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


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



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


M/M/1 Queue1.349


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



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


M/M/1 Queue1.349

0.520


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


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



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


M/M/1 QueueM/M/1 Queue2.032 1.951 1.349 .795 .539 .347

0.3050.0740.0350.5201.5350.159


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



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


M/M/1 Performance Measures

Number in Queue

0

3

0 1 2 3 4

Time

Q(t

)


Number in System

0

3

0 1 2 3 4

Time

S(t

)


Busy/Idle

0

2

0 1 2 3 4

Time

B(t

)

Applications; Financial

Documents

Simulation. Inverse Functions Actually, we’ve already done this with the normal distribution