Discrete-time stochastic processes

The important thing here is for the reward function

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Unformatted text preview: robability of a renewal at time kd is given by lim Pr {N (kd) − N (kd − d) = 1} = k→1 3.4 d . X (3.20) Renewal-reward processes; time-averages There are many situations in which, along with a renewal counting process {N (t); t ≥ 0}, there is another randomly varying function of time, called a reward function {R(t); t ≥ 0}. 106 CHAPTER 3. RENEWAL PROCESSES R(t) models a rate at which the process is accumulating a reward. We shall illustrate many examples of such processes and see that a “reward” could also be a cost or any randomly varying quantity of interest. The important restriction on these reward functions is that R(t) at a given t depends only on the particular inter-renewal interval containing t. We start with several examples to illustrate the kinds of questions addressed by this type of process. Example 3.4.1. (time-average Residual Life) For a renewal counting process {N (t), t ≥ 0}, let Y (t) be the residual life at time t. The residual life is defined as the interval from t until the next renewal epoch, i.e., as SN (t)+1 − t. For example, if we arrive at a bus stop at time t and buses arrive according to a renewal process, Y (t) is the time we have to wait for a bus to arrive (see Figure 3.6). We interpret {Y (t); t ≥ R } as a reward function. The time0 6 (1/t) t Y (τ )dτ . We are interested in the average of Y (t), over the interval (0, t], is given by 0 limit of this average as t → 1 (assuming that it exists in some sense). Figure 3.6 illustrates a sample function of a renewal counting process {N (t); t ≥ 0} and shows the residual life Y (t) for R that sample function. Note that, for a given sample function {Y (t) = y (t)}, the t integral 0 y (τ ) dτ is simply a sum of isosceles right triangles, with part of a final triangle at the end. Thus it can be expressed as Z t 0 n(t) 1X 2 y (τ )dτ = xi + 2 i=1 Z t y (τ )dτ τ =sn(t) where {xi ; 0 < i < 1} is the set of sample values for the inter-renewal intervals. Since this relationship holds for every sample point, we see that the random variable Rt 0 Y (τ )dτ can b e expressed in terms of the inter-renewal random variables Xn as Z t Y (τ )dτ = τ =0 Zt N (t) 1X 2 Xn + Y (τ )dτ . 2 n=1 τ =SN (t) Although the final term above can be easily evaluated for a given SN (t) (t), it is more convenient to use the following bound: Z N (t) N (t)+1 1X 2 1 t 1X 2 X≤ Y (τ )dτ ≤ Xn . 2t n=1 n t τ =0 2t n=1 (3.21) The term on the left can now be evaluated in the limit t → 1 (for all sample functions except a set of probability zero) as follows: lim t→1 6 Rt PN (t) n=1 2t 2 Xn = lim t→1 £§ 2 E X2 Xn N (t) = . N (t) 2t 2E [X ] PN (t) n=1 (3.22) Y (τ )dτ is a random variable just like any other function of a set of variables. It has a sample 0 value for each sample function of {N (t); t ≥ 0}, and its distribution function could be calculated in a straightforward but tedious way. For arbitrary stochastic processes, integration and differentiation can require great mathematical sophistication, but none of those subtleties occur here. 3.4. RENEWAL-REWARD PROCESSES; TIME-AVERAGES 107 N (t) ✛ X2 ✲ X1 S1 S2 S3 S4 S5 S6 X5 ❅ ❅ ❅Y (t) X2 X4 ❅ ❅ ❅ X6 ❅ ❅ X1 ❅ ❅ ❅ ❅ ❅ X3 ❅ ❅ ❅❅ ❅ ❅❅ t ❅ ❅❅ ❅ ❅ S1 S2 S3 S4 S5 S6 Figure 3.6: Residual life at time t. For any given sample function of the renewal process, the sample function of residual life decreases linearly with a slope of −1 from the beginning to the end of each inter-renewal interval. The second equality above follows by applying the strong law of large numbers (Theorem P 2 1.5) to n≤N (t) Xn /N (t) as N (t) approaches infinity, and by applying the strong law for renewal processes (Theorem 3.1) to N (t)/t as t → 1. The right hand term of (3.21) is handled almost the same way: £§ PN (t)+1 2 PN (t)+1 2 E X2 Xn Xn N (t) + 1 N (t) n=1 n=1 lim = lim = . (3.23) t→1 t→1 2t N (t) + 1 N (t) 2t 2E [X ] Combining these two results, we see that, with probability 1 (abbreviated as W.P.1), the time-average residual life is given by £§ Rt E X2 τ =0 Y (τ ) dτ lim = W.P.1. (3.24) t→1 t 2E [X ] 2 2 Note that this time-average depends on the second moment of X ; this is X + σ 2 ≥ X , so the time-average residual life is at least half the expected inter-renewal interval (which is not surprising). On the other hand, the second moment of X can be arbitrarily large (even infinite) for any given value of E [X ], so that the time average residual life can be arbitrarily large relative to E [X ]. This can be explained intuitively by observing that large inter-renewal intervals are weighted more heavily in this time-average than small interrenewal intervals. As an example, consider an inter-renewal random variable X that takes on value ≤ with § probability 1 − ≤ and value 1/≤ with probability ≤. Then, for small ≤, £ E [X ] ∼ 1, E X 2 ∼ 1/≤, and the time average residual life is approximately 1/(2≤) (see Figure 3.7...
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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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