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Simulation Statistics Numerous standard statistics of

interest Some results calculated from

parameters Used to verify the simulation

Most calculated by program

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Some Statistics Average Wait time for a customer = total time customers wait in queue total number of customers

Average wait time of those who wait= total time of customers who wait in

queue number of customers who wait

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More StatisticsProportion of server busy time= number of time units server busy total time units of simulation

Average service Time= total service time number of customers serviced

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More Statistics Average time customer spends in system= total time customers spend in system total number of customers

Probability a customer has to wait in queue= number of customers who wait total number of customers

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Traffic Intensity A measure of the ability of the

server to keep up with the number of the arrivals

TI= (service mean)/(inter-arrival mean)

If TI > 1 then system is unstable & queue grows without bound

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Server Utilization % of time the server is busy serving

customers If there is 1 server

SU = TI = (service mean)/(inter-arrival mean)

If there are N servers SU = 1/N * (service mean)/(inter-

arrival mean)

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Statistical Models Probability: a quantitive measure of

the chance or likelihood of an event occurring.

Random: unable to be predicted exactly

In an experiment where events randomly occur but in which we have assigned to each possible outcome a probability, we have determined a probability or stochastic model

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Terms Event Space Event Complement of an Event Intersection Union Mutually Exclusive

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Examples Event Space: The set of all possible

events that can occur ex: {1,2,3,4,5,6}

Event (E): Any single occurrence ex: E = {4,5}

Complement of E: Set of all events except E Ex: Complement of E = {1,2,3,6}

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Examples Union: Combination of any 2

event sets A= {1,2,3} B = {3,4} A U B = {1,2,3,4}

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Examples Intersection: Overlap of common

occurrence of 2 event sets A= {1,2,3} B = {3,4} A Π B = {3}

Mutually Exclusive: 2 event sets that have no events in common A= {1,2} B = {3,4} A Π B = { }

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Random variablePractical Definition a quantity whose value is

determined by the outcome of a random experiment

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Random Variable Examples

X = the number of 4's that occur in 10 rolls

Y = the number of customers that arrive in 1 hour

Z = the number of services that are completed in 5 minutes

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Discrete vs. Continuous RVEXAMPLE Discrete: X = number of

customers that arrive in 1 hour Continuous: Y = gallons that flow

into the pool in 1 hour ????: Z = the average age of the

customers that arrive in an hour

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Discrete: Probability Function

Let X be a discrete R.V. with possible values x1, x2,…xn. Let P be the probability function

P(xi) = (X = xi) such that(a) P(xi) >= 0 for i = 1,2,…n(b) Σ P(xi) = 1

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Probability FunctionExample

Consider the rolling of a fair die1/6 for x = 1

P(x) = 1/6 for x = 21/6 for x = 31/6 for x = 41/6 for x = 51/6 for x = 60 for all other x

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Cumulative Distribution Function

CDF of a random variable X is F such that F(x) = P (X <= x)

F(X) is continuous Discrete: sum of probabilities Continuous: area under the curve

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Cumulative Distribution Function - Example

Consider the rolling of a fair die0 for x < 1 1/6 for x < 2

F(X) = 2/6 for x <33/6 for x < 44/6 for x < 55/6 for x < 6 1 for x >= 6

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Cumulative Function

1 2 3 4 5 6

1

1/2

1/6

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Discrete vs. Continuous R.V.

Cumulative Distribution Function (CDF) The CDF of a discrete R.V. X is F such

that F(x)= P (X<= x) Continuous: The CDF of a continuous

RV has the properties: F(x) is continuous, at least piecewise F(x) exists except in at most a finite

number of points

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Discrete vs. Continuous Random Variables

Random variable: a function whose domain is the event space & whose range is some subset of real numbers

If a random variable assumes a discrete (finite or countably infinite) number of values, it is called a discrete random variable. Otherwise, it is called a continuous random variable.