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Where u is the mean and a is the standard deviation

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Unformatted text preview: tandard deviation : Probabilitv Functions l l Markov Process Exponential Distribution Introduction Used to queueing to describe service patterns Probability density function for a negative exponential distribution is given by: l l l f(x)= ).lfz-u’ Stochastic system in which future state depends only on the preceding State Markovian (memoryless) property States of system can be finite or countably infinite (discrete in number) where u is the mean service rate EM-602 I QM-710 (NJ) Lecture 9 Page 9-2 -- Markov Process Markov Process Terminology l l l Stochastic Matrices Unit of time in Markov Process is caked an Epoch Pu is deffned as the probabilfty of the system undergoing transltlon from state i to state j in one epoch Principle of Stationarity - Transition probabilities do not change within the scope of the analysis Stochastic Matrix - a matrix in which the sum of the elements of each row equals exactly 1 One-step Transltlon Matrtx - a square stochastic matrix P-1 ] whose elements consist of individual one-step transition probabilities Pu l l Markov Process Markov Process One-step Transition Matrix l One-Step Transition Diagram One-step Transition Matrix l One-Step Transitton Diagram - graphical representation of transttton probaMlttks State at end of Epoch Pertinent Questions l l l l Example Taxi Company Taxi Company If it is known that the taxi is now at the airport, what Is the probability of it being at the hotel in 2 epochs, 3 epochs, etc.? How many epochs elapse (on average) before the taxi re-visits the airport? Over the long run, what proportion of the taxi’s time is spent in each state? If the fare for each route is known, what is the expected revenue for the taxi owner? l l l A taxi company provides shuttle transportation service Service is provided between the airport, the hotel, and downtown The one-step transition probabilities are given in a transition matrix EM-602 / QM-710 (NJ) Lecture 9 Page 9-3 Example Example Taxi Company Taxi Company Transition Diagram One-step Transition Matrix 1R 113 l/2 ” Tow” Ah I3 l/3 l/3 2n @ Terminology Terminology Absolute Probability l l l nth step Probability Absolute Probability n:n) is deflned as the probability the system is in state j after n epochs (after the nfh transition) Absolute Probability Vector (state ve...
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