simu3-12

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Building Simulation Model
Input Data Analysis
Random Number Generator True System
Simulation Model
validate
Output Data Analysis
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No Done
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Definition
Inter-arrival time exponentially distributed
Constant Rate
T
Number of arrival in a given time duration T is Poisson distribution
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Memoryless property
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Examples of Littles Law
4
people, 4 machines, 4 operations/part, one part arrival/day Version2 : Everyone work on the same operation on all ten parts. 4 parts come out every 4 days. T1=4, T2 =4+1, T3 =4+2 and T4 =4+3 Average Cycle time T=5.5 N=5.5
E[ x] m
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Var[ x] s
2
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Standard Normal Distribution
N(0,1)
1 ( x ) 2 / 2 ( x) e 2s
Z
Z ~ N (0,1)
X ~ N (m , s )
X m
s
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Example
What
is the probability of X3, when X is a normal distribution with mean = 1, and sigma=2?
Pr( X 3)
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Bernoulli
Bernoulli(p)
0 if P( x) 1 p if 1 if
What
is the probability for X >1 when X is exponentially distributed with =2?
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Normal Distribution
N(m,s2)
1 ( x m ) 2 / 2s 2 f ( x) e 2s
x0 0 x 1 x 1
1 p if x 0 p( x) p if x 1 0 otherwise
E[ x] p
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Var[ x] p(1 p)
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Binomial
Bin(t,p)
t x tx p ( 1 p ) if x {0,1,..., t} p( x) x 0 otherwise
e-t e-(t-t0)
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Application of Memoryless Triangular
Combination
of two Poisson Processes
1
+
2
12
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Application of Memoryless Triangular
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Littles Law
N q W
Long
lead time of production occurs uncertainties planning higher work in process longer lead time
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M/M/1
1 f ( x) b a 0 if a xb otherwise
0 if x a F ( x) if b a 1 if
ab E[ x] 2
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(b a) Var ( X ) 12
2
5
Exponential Distribution
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Little’s Law
1
2 3 Total arrival rate
N T
departures
}

Average # of customers in this enclosed subsystem N
T
Average flow time= cycle time
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notation
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Queueing Notations
of parallel servers m: 1,2…., System limit K : 1,2…., Queue discipline (Z)
No
Symbol FCFS LCFS RSS PR GD
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Random
Variable Sample Statistics Parameter Uncertainty Mean Variance
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Uniform Distribution
Uniform
Distribution U(a,b)
xa a xb xb
m
( / m 1)
Idle Probability
p0 1
Utilization = 1-p0 =
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M/M/1
Number of Customer in System N


1

m
Cycle Time
Output?
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Burke’s Theorem
A
Poisson process supplying arrivals to a server with exponentially distributed service times results in a Poisson departure process with the exact same rate
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Queueing Notations
Interarrival-time
distribution(A)
Explanation Exponential Deterministic General
Symbol M D G
Service
time distribution(B) shares the same
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Examples of Littles Law
4
people, 4 machines, 4 operations/part, one part arrival/day Version1 : Everyone work on the different parts resulting one part produced everyday. Cycle time T=4 N=4

Transient
Behaviour of a system in a short period Depend on the initial state of the system Example, number of customers arrived in the first operation hour of a bank.
Splitting
of Poisson Process

p
(1-p)
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Queueing Theory
Queue A(t) Customer Arrival m Servers B(t) Customer Departure
K : Capacity of the system Z: Queue discipline
e x if x { 0 , ,...} p( x) x! 0 otherwise
0 if x 0 x i F ( x) e if x 0 i 0 i!
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E[ x] Var[ x]
0 x t i t i F ( x) p ( 1 p ) i i 0 1
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if if if
x0 0 xt xt
E[ x] tp Var[ x] tp (1 p)
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Poisson
Poisson()
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An Important Law
Central
Limit Theorem Let X1, X2,… be a sequence of independent identically distributed random variables with finite means m and finite non-zero variance s2, and let Sn = X1 + X2 + … + Xn Then
IE3110E Simulation Part III Basic Statistics and Probability
Department of Industrial & Systems Engineering A/P Lee Loo Hay