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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)= ).lfzu’ 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 EM602 I QM710 (NJ) Lecture 9
Page 92  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
Onestep Transltlon Matrtx  a square
stochastic matrix P1 ] whose elements
consist of individual onestep
transition probabilities Pu l l Markov Process Markov Process Onestep Transition Matrix
l OneStep Transition Diagram Onestep Transition Matrix l OneStep 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 revisits 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 onestep transition probabilities
are given in a transition matrix EM602 / QM710 (NJ) Lecture 9
Page 93 Example Example Taxi Company Taxi Company
Transition Diagram Onestep 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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 Spring '94
 DonaldC.Johnson
 Probability theory, ProbabiliTy Review

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