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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. Some Statistics. Average Wait time for a customer = total time customers wait in queue total number of customers - PowerPoint PPT Presentation
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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.