Operations Research Models and Methods / Computation

Markov Process

The Rate Matrix

The data describing a Markov Process is the Rate Matrix.

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Selecting the MPMatrix option from the OR_MS menu constructs the worksheet that holds this data. Names are entered for each state in the State Name array. Transition rates are entered in the rate matrix.

For the example, we left use the default names: State 0, State 1, etc. to represent the number of persons in the system. The mean time between arrivals is 0.5 minutes, so using minutes as the time interval, we determine an arrival rate of 2 per minute. When there are n persons using or waiting for the ATM (n < 5), an arrival adds one more to that number, so we place the number 2 as the rate of transition from state n to state n + 1 (when n is not equal to 5). We assume that arrivals balk when the foyer is full, so the row for state 5 does not show the arrival rate.

Service transitions are handled similarly. With a single ATM, the rate of departures is 2.5 per minute, so we use this rate for transitions from state n to state n -1 (when n is not equal to zero). When the foyer is empty, no departures can occur so this rate does not appear in row 0. All other entries in the rate matrix are 0, because no other single event causes a transition in the ATM example.

Operations Research Models and Methods
by Paul A. Jensen and Jon Bard, University of Texas, Copyright by the Authors