EE226: Random Processes in Systems Fall’06Problem Set 9 — Due December, 7Lecturer: Jean C. Walrand GSI: Assane GueyeThis problem set essentially reviews Convergence and Renewal processes. Not all exercisesare to be turned in. Only those with the sign F are due on Thursday, December 7that thebeginning of the class. Although the remaining exercises are not graded, you are encouragedto go through them.We will discuss some of the exercises during discussion sections.Please feel free to point out errors and notions that need to be clarified.Exercise 9.1. FAssume that Xnconverges in probability to X and f (·) is a continuous bounded function.Show that f(Xn) converges in probability to f(X)Exercise 9.2. Bonus FFAssume that Xnconverges in distribution to some random variable X.Show that we can find (Yn, Y ), where Ynis a sequence of random variables such that Ynhassame distribution as Xn, Y has the same distribution as X, and Ynconverges almost surely(a.s) to Y .Exercise 9.3. FIn the notes we have shown that if Xnconverges in probability to X, then it converges indistribution to X.Show that, conversely, if Xnconverges in distribution to a constant C, then it converges inprobability to C.Exercise 9.4. FProblem 21.2 of the course notes.Exercise 9.5. FProblem 21.9 of the course notes.Exercise 9.6. FRecall that in the study of renewal process we defined the inter-arrival times T = Ti+1−Ti, i ≥1 to be iid with distribution F (t).Letf(t) =1(1 + t)2to be the corresponding pdf.Find E[τ] where τ is the time until the next jump for the stationary process, and λ the rateof jumps.This exercise misses the point it was supposed to make. Answer the the next question:Find a distribution for which 0 < λ < ∞ and E[τ ] = ∞.9-1EE226 Problem Set 9 — Due December, 7 Fall’06Exercise 9.7. FIn the derivation of E[τ] in class we wroteE[τ] =Z∞0λt(1 − F (t))dt=λ2Z∞0λ(1 − F (t))dt2=λ2£t2(1 − F (t))¤∞0+λ2Z∞0t2f(t)dt (9.1)We claimed that the first term in the RHS of equation 9.1 vanishes because the mean ofT = T2− T1(inter-arrival time) should be finite.This argument is not quite correct; show the correct argument that is:t2(1 − F (t)) → 0 as t → ∞ if and on if E[T2] < ∞.Exercise 9.8. Bonus FFConsidering again the renewal process setting given in class, show that if the inter-arrivaltimes are iid uniform in [0, 1], then ²-coupling occurs in finite time.Exercise 9.9. FIn class we have shown that for a positive recurrent continuous-time Markov chain with ratematrix Q, and invariant distribution π, we have1TZT01[xt=i]dtas−→Xj1[j=i]πj= πiLetf : X → Rbe a bounded function from the state space X to the real line R.Show that1TZT0f(xt)dtas−→Xjf(j)πjExercise 9.10. FIn Prof. X’s group, John Lazy, a very daydreaming network manager has set up a printerwithout queue. Any request that finds the printer busy (i.e. already printing) is just lost.Assume that requests arrive at the printer according to a Poisson process with rate λ, andthe amount of time needed to print a request is a random variable having distribution Gwith mean µGand independent for each request.(a) What is the rate at which requests are accepted (i.e. requests get printed)?(b) What is the proportion of satisfied requests?Compute it for λ = 2 requests per second and µG= 2