Discrete-time stochastic processes

R a1 r d3 a4 r right moving forward mm1 process d3 r

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Unformatted text preview: Since k=1 πi /∫i is finite for any integer k, it means that i the probability of states larger than k is increasing with time. In the limit t → 1, the state disperses over an infinite set of possibilities with increasingly small probabilities for each. To look at this in another way, the expected time to the next transition, starting in steady state for the embedded chain, is 1. This is probably not a phenomenon that can be understood intuitively. Perhaps the best one can hope for is to see that it does not violate the things we know. 6.2.4 The equations for the steady state process probabilities P Let us now come back to the case where i πi /∫i < 1, which is the case of virtually all applications. We have seen that a Markov process can be specified in terms of the timetransitions qij = ∫i Pij , and it is useful to express the steady state equations for pj directly in terms of qij rather than indirectly in terms of the embedded chain. First note from P P (6.18) that if i πi /∫i < 1, then j pj = 1, so that the limiting time averages behave as P we would expect in all but the peculiar case where i πi /∫i = 1. We also note that (6.18) specifies the embedded steady state probabilities πi in terms of the pi . Since πi = pi ∫i α, P where α is independent of i, we can use the normalization i πi = 1 to obtain pi ∫i . k pk ∫k πi = P We can substitute πi as given by (6.19) into (6.6), obtaining pj ∫j = state j . Since ∫i Pij = qij , X X pj ∫j = pi qij ; pi = 1. i (6.19) P i pi ∫i Pij for each (6.20) i This set of equations is known as the steady state equations for the Markov process. The P condition i pi = 1 has been added as a normalization condition. Equation (6.20) has a nice interpretation in that the term on the left is the steady state rate in time at which transitions occur out of state j and the term on the right is the rate in time at which transitions occur into state j . Since the total number of entries to j must differ by at most 1 from the exits from j for each sample path, this equation is not surprising. We know that (6.6) has a unique solution with all πi > 0 if the embedded chain is positive recurrent, and thus (6.20) also has a unique solution with all pi > 0 under the conditions P that the embedded chain is positive recurrent and i πi /∫i < 1. This final condition is not quite what we want, since we would like to solve (6.20) directly without worrying about the embedded chain. 6.2. STEADY-STATE BEHAVIOR OF IRREDUCIBLE MARKOV PROCESSES 245 P If we find a solution to (6.20), however, and if i pi ∫i < 1 in that solution, then the corresponding set of πi from (6.19) must satisfy (6.6) and be the unique steady state solution for the embedded chain. Thus the solution for pi must be the corresponding steady state solution for the Markov process. This is summarized in the following theorem. TheoremP 6.2. Assume an irreducible Markov process and let {pi ; i ≥ 0} be a solution to (6.20). If i pi ∫i < 1, then, first, that solution is unique, second, each pi is positive, and third, the embedded Markov chain is positive recurrent with the steady state πi satisfying P (6.19). Also, if the embedded chain is positive recurrent, and i πi /∫i < 1 then the set of pi satisfying (6.18) is the unique solution to (6.20). 6.2.5 The sampled-time approximation again For an alternative view of the probabilities {pi }, consider the special case (but the typical case) where the transition rates {∫i } are bounded. Consider the sampled-time approximation to the process for a given increment size δ ≤ [maxi ∫i ]−1 (see Figure 6.3). Let {wi ; i ≥ 0} be the set of steady state probabilities for the sampled-time chain, assuming that they exist. These steady state probabilities satisfy wj = X i6=j wi qij δ + wj (1 − ∫j δ ); wj ≥ 0; X wj = 1. (6.21) j P The first equation simplifies to wj ∫j = i6=j wi qij , which is the same as (6.20). It follows that the steady state probabilities {pi ; i ≥ 0} for the process are the same as the steady state probabilities {wi ; i ≥ 0} for the sampled-time approximation. Note that this is not an approximation; wi is exactly equal to pi for all values of δ ≤ 1/ supi ∫i . We shall see later that the dynamics of a Markov process are not quite so well modeled by the sampled time approximation except in the limit δ → 0. 6.2.6 Pathological cases The example in Figure 6.5 gives some insight into the case of positive recurrent embedded P chains with i πi /∫i = 1. It models a variation of an M/M/1 queue in which the server becomes increasingly rattled and slow as the queue builds up, and the custormers become almost equally discouraged about entering. The downward drift in the transitions is more than overcome by the slow down in higher states. Transitions continue to occur, but the number of transitions per unit time goes to 0 with increasing time. Exercise 6.1 gives some added insight into this type of situation. P P It is also possible for (6.20) to have...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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