Simulation StatisticsSome StatisticsMore StatisticsSlide 4Traffic IntensityServer UtilizationStatistical ModelsTermsExamplesSlide 10Slide 11Random variableRandom Variable ExamplesDiscrete vs. Continuous RVDiscrete: Probability FunctionProbability Function ExampleCumulative Distribution FunctionCumulative Distribution Function - ExampleCumulative FunctionDiscrete vs. Continuous R.V.Discrete vs. Continuous Random Variables1Simulation StatisticsNumerous standard statistics of interestSome results calculated from parametersUsed to verify the simulationMost calculated by program2Some Statistics Average Wait time for a customer = total time customers wait in queue total number of customersAverage wait time of those who wait= total time of customers who wait in queue number of customers who wait3More StatisticsProportion of server busy time= number of time units server busy total time units of simulationAverage service Time= total service time number of customers serviced4More Statistics Average time customer spends in system= total time customers spend in system total number of customersProbability a customer has to wait in queue= number of customers who wait total number of customers5Traffic IntensityA measure of the ability of the server to keep up with the number of the arrivalsTI= (service mean)/(inter-arrival mean)If TI > 1 then system is unstable & queue grows without bound6Server Utilization% of time the server is busy serving customersIf there is 1 server SU = TI = (service mean)/(inter-arrival mean)If there are N serversSU = 1/N * (service mean)/(inter-arrival mean)7Statistical ModelsProbability: a quantitive measure of the chance or likelihood of an event occurring. Random: unable to be predicted exactlyIn 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 model8TermsEvent SpaceEvent Complement of an EventIntersectionUnionMutually Exclusive9ExamplesEvent Space: The set of all possible events that can occurex: {1,2,3,4,5,6}Event (E): Any single occurrenceex: E = {4,5} Complement of E: Set of all events except EEx: Complement of E = {1,2,3,6}10ExamplesUnion: Combination of any 2 event setsA= {1,2,3}B = {3,4}A U B = {1,2,3,4}11ExamplesIntersection: Overlap of common occurrence of 2 event setsA= {1,2,3} B = {3,4}A Π B = {3}Mutually Exclusive: 2 event sets that have no events in commonA= {1,2} B = {3,4}A Π B = { }12Random variablePractical Definitiona quantity whose value is determined by the outcome of a random experiment13Random Variable ExamplesX = the number of 4's that occur in 10 rollsY = the number of customers that arrive in 1 hourZ = the number of services that are completed in 5 minutes14Discrete vs. Continuous RVEXAMPLEDiscrete: X = number of customers that arrive in 1 hourContinuous: Y = gallons that flow into the pool in 1 hour????: Z = the average age of the customers that arrive in an hour15Discrete: Probability FunctionLet X be a discrete R.V. with possible values x1, x2,…xn. Let P be the probability functionP(xi) = (X = xi) such that(a) P(xi) >= 0 for i = 1,2,…n(b) Σ P(xi) = 116Probability FunctionExampleConsider 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 x17Cumulative Distribution FunctionCDF of a random variable X is F such that F(x) = P (X <= x) F(X) is continuousDiscrete: sum of probabilitiesContinuous: area under the curve18Cumulative Distribution Function - ExampleConsider the rolling of a fair die0 for x < 1 1/6 for x < 2F(X) = 2/6 for x <33/6 for x < 44/6 for x < 55/6 for x < 6 1 for x >= 619Cumulative Function1 2 3 4 5 611/21/620Discrete 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 piecewiseF(x) exists except in at most a finite number of points21Discrete vs. Continuous Random VariablesRandom variable: a function whose domain is the event space & whose range is some subset of real numbersIf 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
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