MARKOV CHAINS:FIRST PASSAGE TIMES

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FIRST PASSAGE TIMES

Section 29.3 dealt with finding n-step transition probabilities from state i to state j. It is often desirable to also make probability statements about the number of transitions made by the process in going from state i to state j for the first time. This length of time is called the first passage time in going from state i to state j. When j = i, this first passage time is just the number of transitions until the process returns to the initial state i. In this case, the first passage time is called the recurrence time for state i.

To illustrate these definitions, reconsider the inventory example introduced in Sec. 29.1, where Xt is the number of cameras on hand at the end of week t, where we start with X0 = 3. Suppose that it turns out that

INTRODUCTION TO OPERATIONS RESEARCH-0730

In this case, the first passage time in going from state 3 to state 1 is 2 weeks, the first passage time in going from state 3 to state 0 is 3 weeks, and the recurrence time for state 3 is 4 weeks.

In general, the first passage times are random variables. The probability distributions associated with them depend upon the transition probabilities of the process. In particular, let f ij denote the probability that the first passage time from state i to j is equal to n.

For n > 1, this first passage time is n if the first transition is from state i to some state k (k * j) and then the first passage time from state k to state j is n - 1. Therefore, these probabilities satisfy the following recursive relationships:

INTRODUCTION TO OPERATIONS RESEARCH-0731

Thus, the probability of a first passage time from state i to state j in n steps can be computed recursively from the one-step transition probabilities.

In the inventory example, the probability distribution of the first passage time in going from state 3 to state 0 is obtained from these recursive relationships as follows:

INTRODUCTION TO OPERATIONS RESEARCH-0732

This equation recognizes that the first transition from state i can be to either state j or to some other state k. If it is to state j, the first passage time is 1. Given that the first transition is to some state k (k * j) instead, which occurs with probability pik, the con- ditional expected first passage time from state i to state j is 1 + kj. Combining these facts, and summing over all the possibilities for the first transition, leads directly to this equation.

For the inventory example, these equations for the ij can be used to compute the expected time until the cameras are out of stock, given that the process is started when three cameras are available. This expected time is just the expected first passage time 30. Since all the states are recurrent, the system of equations leads to the expressions

INTRODUCTION TO OPERATIONS RESEARCH-0733

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