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Lecture9 - Stochastic Models EM-602 I W-710(NJ lntroductii...

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EM-602 I W-710 (NJ) Management Science Lecture 9 Stochastic Models Probability Review Discrete Case (part 1) l In the discrete case, Pi is a valid Probability Mass Function if and only if: Pi 2 0 Vi pi I 1 Vi c Pi = 1 Stochastic Models lntroductii l Deterministic models assumed that outcomes were known wlth certainty l Stochastic models do not make that assumption l Outcomes of a stochastic process are not known with certainty l Probabllltles can be associated wlth various outcomes Probability Review Discrete Case (part 2) l The mean (u) and variance (0’) for a discrete distribution are given by: Probability Review Probability Review Continuous Case (part 1) Continuous Case (part 2) l In the conttnuous case,/@) Is a valid Probabiltty Density Function if and only if: l The mean (F) and variance (a’) for a discrete distribution are given by: f(x) 2 0 VX f(x) s 1 VX r 5 f(x)tfx = 1 -r . . EM-602 / QM-710 (NJ) Lecture 9 Page 9-l
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Probability Review Common Dlstributions l Discrete Distributions Binomial Distribution Poisson Distribution l Continuous Distributions Normal Distribution Negative Exponential Distribution Probability Functions Poisson Dlstribution l Used in queueing systems to describe discrete arrival patterns l For example: Pk , the probability of having k arrivals in a given interval, if the mean rate of arrivals is i, arrivals per unit time, is given by: ihe-’ A-k’, i.>o . : Probabilitv Functions Exponential Distribution l Used to queueing to describe service patterns l Probability density function for a negative exponential distribution is given by: f(x)= ).lfz-u’ where u is the mean service rate Probability Functions Binomial Distribution l Used in Bernoulli trials with only 2 discrete outcomes such as success I fail l For example: Pk , the probability of having R successes in R trials, if the probability of a success is p is given by: pk= ]I /.?^(l-p)=’ 0 Probability Functions Nomal Dlstrbution l Used to describe numerous random events in nature - familiar bell-shaped Curve l Probability density function for a normal distribution is given by: -(x-W)* f(x)=-j=&=e’“. where u is the mean and a is the standard deviation Markov Process Introduction l Stochastic system in which future state depends only on the preceding State l Markovian (memoryless) property l States of system can be finite or countably infinite (discrete in number) EM-602 I QM-710 (NJ) Lecture 9 Page 9-2
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-- Markov Process Terminology l Unit of time in Markov Process is caked an Epoch l Pu is deffned as the probabilfty of the system undergoing transltlon from state i
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