Discrete-time stochastic processes

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Unformatted text preview: bution function X n Pr {Un ≤ u} = Pij−1 exp(−∫j u). j The epoch of the nth transition is Sn = U1 + · · · Un , which is also a rv (i.e., it it finite with probability 1, so limt→1 Pr {Sn ≤ t} = 1), i.e., for all n, the nth transition eventually occurs with probability 1. Now define Mi (t) as the number of transitions (given X0 = i) that occur by time t. Note that {Mi (t); t ≥ 0} is not a renewal process (neither delayed nor ordinary), but it still satisfies the set identity {Mi (t) ≥ n} = {Sn ≤ t}. This then implies that limt→1 Pr {Mi (t) ≥ n} = 1 for all n, or in other words,2 limt→1 Mi (t) = 1 with probability 1. Note that this result, which is stated in the following lemma, does not assume that the Markov process is irreducible or recurrent. Lemma 6.1. Let Mi (t) be the number of transitions in the interval (0, t] of a Markov process starting with X0 = i. Then lim Mi (t) = 1 t→1 6.2.2 with probability 1. (6.10) Renewals on successive entries to a state For an irreducible Markov process with X0 = i, let Mij (t) be the number of transitions into state j over the interval (0, t]. We want to find when this is a delayed renewal counting process. It is clear that the sequence of epochs at which state j is entered form renewal points, since they form renewal points in the embedded Markov chain and the time intervals between transitions depend only on the current state. The questions are whether the first 2 To spell this out, consider the sample function Mi (t, ω ) for any ω ∈ ≠. This is nondecreasing in t and thus either has a finite limit or goes to 1. The set of ω for which this limit is at most n is 0 since limt→1 Pr {Mi (t) ≥ n} = 1, and thus the limit is 1 with probability 1. 6.2. STEADY-STATE BEHAVIOR OF IRREDUCIBLE MARKOV PROCESSES 241 entry to state j must occur within some finite time, and then whether recurrences must occur within finite time. The following lemma answers these questions for the case where the embedded chain is recurrent (either positive recurrent or null recurrent). Lemma 6.2. Consider a Markov process with an irreducible recurrent embedded chain {Xn ; n ≥ 0}. Given X0 = i, let {Mij (t); t ≥ 0} be the number of transitions into a given state j in the interval (0, t]. Then {Mij (t); t ≥ 0} is a delayed renewal counting process (or, if j = i, is an ordinary renewal counting process). Proof: Given X0 = i, let Nij (n) be the number of transitions into state j that occur in the embedded Markov chain by the nth transition of the embedded chain. From Lemma 5.4, {Nij (n); n ≥ 0} is a delayed renewal process, and from Lemma 3.2, limn→1 Nij (n) = 1 with probability 1. Then Mij (t) = Nij (Mi (t)) where Mi (t) is the total number of state transitions (between all states) in the interval (0, t]. Thus, with probability 1, lim Mij (t) = lim Nij (Mi (t)) = lim Nij (n) = 1. t→1 t→1 n→1 where we have used Lemma 6.1, which asserts that limt→1 Mi (t) = 1 with probability 1. It follows that the time W1 at which the first transition into state j occurs, and the subsequent interval W2 until the next transition to state j , are both finite with probability 1. Subsequent intervals have the same distribution as W2 , and all intervals are independent, so {Mij (t); t ≥ 0} is a delayed renewal process with inter-renewal intervals {Wk ; k ≥ 1}. If i = j , then all Wk are identically distributed and we have an ordinary renewal process, completing the proof. The inter-renewal intervals W2 , W3 , . . . for {Mij (t); t ≥ 0} above are well-defined nonnegative iid rv’s whose distribution depends on j but not i. They either have an expectation as a finite number or can be regarded as having an infinite expectation. In either case, this expectation is denoted as E [W (j )] = W (j ). This is the mean time between successive entries to state j , and we will see later that in some cases this mean time can be infinite. In order to study the fraction of time spent in state j , we define a delayed renewal-reward process, based on {Mij (t); t ≥ 0}, for which unit reward is accumulated whenever the process is in state j (see Figure 6.4). If transition n − 1 of the embedded chain enters state j , then the interval Un until the nth transition is exponential with rate ∫j , so E [Un |Xn−1 =j ] = 1/∫j . Define pj as the limiting time-average fraction of time spent in state j (if such a limit exists). Then, since U (j ) = 1/∫j , Theorems 3.6 and 3.12, for ordinary and delayed renewal-reward processes respectively, state that pj = lim t→1 Rt 0 Rj (τ )dτ U (j ) 1 = = t W (j ) ∫j W (j ) with probability 1. (6.11) We can also investigate the limit, as t → 1, of the probability that X (t) = j . This is equal to limt→1 E [R(t)] for the renewal-reward process above. Because of the exponential holding 242 CHAPTER 6. MARKOV PROCESSES WITH COUNTABLE STATE SPACES t t ✛ Un Xn−1 =j ✛ ✲ t Rj (t) t Xn 6=j t Xn+1 6=j Xn+2 =j Wk ✲ t Xn+3 6=j Figure 6.4: The delayed renewal-reward process {Rj (t); t ≥ 0} for time in stat...
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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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