Using theorem 38 we can calculate the limiting

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Unformatted text preview: E si 1/6 and W = WQ + Esi = 47. (c) What is the average queue length (counting the customer in service)? By Little’s formula, L = W = 47/6. 3.3 Age and Residual Life* Let t1 , t2 , . . . be i.i.d. interarrival times, let Tn = t1 + · · · + tn be the time of the nth renewal, and let N (t) = max{n : Tn t} be the number of renewals by time t. Let A(t) = t TN (t) and Z (t) = TN (t)+1 t A(t) gives the age of the item in use at time t, while Z (t) gives its residual lifetime. To explain the interest in Z (t) note that the interrarival times after TN (t)+1 will be independent of Z (t) and i.i.d. with distribution F , so if we can show that Z (t) converges in distribution, then the renewal process after time t will converge to an equilibrium. 110 CHAPTER 3. RENEWAL PROCESSES A(t) ⇥ T1 0 ⇥ T2 ⇥ Z (t) ⇥ TN (t) t ⇥ TN (t)+1 Figure 3.1: Age and residual life. 3.3.1 Discrete case The situation in which all the interarrival times are positive integers is very simple but also important because visits of a Markov chain to a fixed state, Example 3.1, are a special case. Let ( 1 if m 2 {T0 , T1 , T2 , ....
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