Discrete-time stochastic processes

Let j be a non negative integer valued random

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Unformatted text preview: process, with IID inter-renewal times, is delayed until after the epoch of the first renewal. What we shall discover is intuitive — both the time average behavior and, in essence, the limiting ensemble behavior are not affected by the 11 This is often called the PASTA property, standing for Poisson arrivals see time-averages. This holds with great generality, requiring only that time-averages exist and that the state of the system at a given time t is independent of future arrivals. 3.7. DELAYED RENEWAL PROCESSES 123 distribution of the first renewal epoch. It might be somewhat surprising, however, to find that this irrelevance of the distribution of the first renewal epoch holds even when the mean of the first renewal epoch is infinite. To be more precise, we let {Xi ; i≥1} be a set of independent non-negative random variables. X1 has some distribution function G(x), whereas {Xi ; i ≥ 2} are identically distributed with some distribution function F (x). Typically, G(P 6= F (x), since if equality held, we would x) have an ordinary renewal process. Let Sn = n Xi be the nth renewal epoch and let i=1 N (t) be the number of renewal epochs up to and including time t (i.e., N (t) ≥ n if and only if Sn ≤ t). {N (t); t ≥ 0} is then called a delayed renewal counting process. The following simple lemma follows from lemma 3.1. Lemma 3.2. Let {N (t); t ≥ 0} be a delayed renewal counting process. Then limt→1 N (t) = 1 with probability 1 and limt→1 E [N (t)] = 1. Proof: Conditioning on X1 = x, we can write N (t) = 1 + N 0 (t − x) where N 0 {t; t ≥ 0} is the ordinary renewal counting process with inter-renewal intervals X2 , X3 , . . . . From Lemma 3.1, limt→1 N 0 (t − x) = 1 with probability 1, and limt→1 E [N 0 (t − x)] = 1. Since this is true for every finite x > 0, and X1 is finite with probability 1, the lemma is proven. Theorem 3.9 (Strong Law for Delayed Renewal Processes). Let a delayed renewal R1 process have mean inter-renewal interval X 2 = x=0 [1 − F (x)] dx. Then lim t→1 N (t) 1 = t X2 with probability 1. (3.68) Proof: As in the proof of Theorem 3.1, we have SN (t) SN (t)+1 t ≤ ≤ . N (t) N (t) N (t) PN (t) SN (t) X1 n=2 Xn lim = lim + lim t→1 N (t) t→1 N (t) t→1 N (t) − 1 (3.69) N (t) − 1 . N (t) (3.70) From Lemma 3.2, N (t) approaches 1 as t → 1. Thus for any finite sample value of X1 , the first limit on the right side of (3.70) approaches 0. Since X1 is a random variable, it takes on a finite value with probability 1, so this first term is 0 with probability 1 (note that this does not require X1 to have a finite mean). The second term in (3.70) approaches X 2 with probability 1 by the strong law of large numbers. The same argument applies to the right side of (3.69), so that limt→1 N t t) = X 2 with probabilty 1. Eq. (3.68) then follows. ( A truncation argument, as in Exercise 3.3, shows that the theorem is still valid if X 2 = 1. Next we look at the elementary renewal theorem and Blackwell’s theorem for delayed renewal processes. To do this, we view a delayed renewal counting process {N (t); t ≥ 0} as an ordinary renewal counting process that starts at a random non-negative epoch X1 with some distribution function G(t). Define No (t − X1 ) as the number of renewals that occur in the interval (X1 , t]. Conditional on any given sample value x for X1 , {No (t−x); t−x≥0} is an ordinary renewal counting process and thus, given X1 = x, limt→1 E [No (t − x)] /(t−x) = 1/X 2 . 124 CHAPTER 3. RENEWAL PROCESSES Since N (t) = 1 + No (t − X1 ) for t > X1 , we see that, conditional on X1 = x, lim t→1 E [N (t) | X1 =x] E [No (t − x)] t − x 1 = lim = . t→1 t t−x t X2 (3.71) Since this is true for every finite sample value x for X1 , we can take the expected value over X1 to get the following theorem: Theorem 3.10 (Elementary Delayed Renewal Theorem). For a delayed renewal process with E [Xi ] = X 2 for al l i ≥ 2, lim t→1 E [N (t)] 1 = .. t X2 (3.72) The same approach gives us Blackwell’s theorem. Specifically, if {Xi ; i≥2} are non-arithmetic, then, using Blackwell’s theorem for ordinary renewal processes, for any δ > 0, lim t→1 E [No (t − x + δ ) − No (t − x)] 1 = . δ X2 (3.73) Thus, conditional on any sample value X1 = x, limt→1 E [N (t+δ ) − N (t) | X1 =x] = δ /X 2 . Taking the expected value over X1 gives us limt→1 E [N (t + δ ) − N (t)] = δ /X 2 . The case in which {Xi ; i≥2} are arithmetic with span d is somewhat more complicated. If X1 is arithmetic with span d (or a multiple of d), then the first renewal epoch must be at some multiple of d and d/X 2 gives the expected number of arrivals at time id in the limit as i → 1. If X1 is non-arithmetic or arithmetic with a span other than a multiple of d, then the effect of the first renewal epoch never dies out, since all subsequent renewals occur at multiples of d from this first epoch. This gives us the theorem: Theorem 3.11 (Blackwell for Delayed Renewal). If {Xi ; i≥2} are non-arithmetic, then, for al l δ > 0, lim t→1 E [N (t + ...
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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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