Discrete-time stochastic processes

# 116 chapter 3 renewal processes p p a function rz is

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Unformatted text preview: 3) As t → 1, N (t) → 1, and thus, by the strong law of large numbers, the ﬁrst term on the right side of (3.33) approaches E [Rn ] with probability 1. Also the second term approaches 1/X by the strong law for renewal processes. Thus the product of the two terms approaches the limit E [Rn ] /X . The right hand term of (3.32) is handled almost the same way, PN (t)+1 n=1 t Rn = PN (t)+1 Rn N (t) + 1 N (t) n=1 . N (t) + 1 N (t) t (3.34) It is seen that the terms on the right side of (3.34) approach limits as before and thus the term on the left approaches E [Rn ] /X withRprobability 1. Since the upper and lower τ bound in (3.32) approach the same limit, (1/t) 0 R(τ ) dτ approaches the same limit and the theorem is proved. The restriction to non-negative renewal-reward functions in Theorem 3.6 is slightly artiﬁcial. The same result holds for non-positive reward functions simply by changing the directions of the inequalities in (3.32). Assuming that E [Rn ] exists (i.e., that both its positive and negative parts are ﬁnite), the same result applies in general by splitting an arbitrary reward function into a positive and negative part. This gives us the corollary: Corollary 3.1. Let {R(t); t > 0} be a renewal-reward function for a renewal process with expected inter-renewal time E [X ] = X . If E [Rn ] exists, then with probability 1 Z 1t E [Rn ] lim R(τ ) dτ = . (3.35) t→1 t τ =0 X Example 3.4.4. (Distribution of Residual Life) Example 3.4.1 treated the time-average value of the residual life Y (t). Suppose, however, that we would like to ﬁnd the time-average distribution function of Y (t), i.e., the fraction of time that Y (t) ≤ y as a function of y . The 3.4. RENEWAL-REWARD PROCESSES; TIME-AVERAGES 111 approach, which applies to a wide variety of applications, is to use an indicator function (for a given value of y ) as a reward function. That is, deﬁne R(t) to have the value 1 for all t such that Y (t) ≤ y and to have the value 0 otherwise. Figure 3.10 illustrates this function for a given sample path. Expressing this reward function in terms of Z (t) and X (t), we have ( 1 ; X (t) − Z (t) ≤ y R(t) = R(Z (t), X (t)) = . 0 ; otherwise ✛y✲ 0 ✛y✲ ✲ X3 ✛ S1 S2 S3 ✛y✲ S4 Figure 3.10: Reward function to ﬁnd the time-average fraction of time that {Y (t) ≤ y }. For the sample function in the ﬁgure, X1 > y , X2 > y , and X4 > y , but X3 < y Note that if an inter-renewal interval is smaller than y (such as the third interval in Figure 3.10), then R(t) has the value one over the entire interval, whereas if the interval is greater than y , then R(t) has the value one only over the ﬁnal y units of the interval. Thus Rn = min[y , Xn ]. Note that the random variable min[y , Xn ] is equal to Xn for Xn ≤ y , and thus has the same distribution function as Xn in the range 0 to y . Figure 3.11 illustrates this in terms of the complementary distribution function. From the ﬁgure, we see that Z1 Zy E [Rn ] = E [min(X, y )] = Pr {min(X, y ) > x} dx = Pr {X > x} dx. (3.36) x=0 x=0 ✛ ✒ ° °✲ Pr {min(X, y ) > x} ° ✠ Pr {X > x} x y Figure 3.11: Rn for distribution of residual life. Rt Let FY (y ) = limt→1 (1/t) 0 R(τ ) dτ denote the time-average fraction of time that the residual life is less than or equal to y . From Theorem 3.6 and Eq.(3.36), we then have Z E [Rn ] 1y FY (y ) = = Pr {X > x} dx. (3.37) X X x=0 As a check, note that this integral is increasing in y and approaches 1 as y → 1. In the development so far, the reward function R(t) has been a function solely of the age and duration intervals. In more general situations, where the renewal process is embedded in some more complex process, it is often desirable to deﬁne R(t) to depend on other aspects of the process as well. The important thing here is for the reward function to be independent of the renewal process outside the given inter-renewal interval so that the accumulated rewards 112 CHAPTER 3. RENEWAL PROCESSES over successive inter-renewal intervals are IID random variables. Under this circumstance, Theorem 3.6 clearly remains valid. The subsequent examples of Little’s theorem and the M/G/1 expected queueing delay both use this more general type of renewal-reward function. The above time-average is sometimes visualized by the following type of experiment. For some given large time t, let T be a uniformly distributed random variable over (0, t]; T is Rt independent of the renewal-reward process under consideration. Then (1/t) 0 R(τ ) dτ is the expected value (over T ) of R(T ) for a given sample point of {R(τ ); τ >0}. Theorem 3.6 states that in the limit t → 1, all sample points (except a set of probability 0) yield the same expected value over T . This approach of viewing a time-average as a random choice of time is referred to as random incidence. Random incidence is awkward mathematically, since the random variable T changes with the overall time t and has n...
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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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