ENEE620. Final examination 2021
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ENEE620. Final examination, 5/18/2021.
Problem 1.
Let (N (t), t ≥ 0) be a Poisson process with rate λ = 2.
(a) Let Tk , k = 1, 2, . . . be the kth arrival time. Find P (T1 + T2 < T3 ).
(b) Let S … Unif[0, 1] be independent of N (t). Find E[N2 (S)], where N (S) is the number of arrivals at a random time S.
(c) Deine Xn = N (n2 ), n = 0, 1, 2, . . . . Find the probability P (Xn+1 = j|Xn = i,Xn-1 = in- 1 , . . . , X0 = i0 ) and argue whether the sequence (Xn , n ≥ 0) forms a Markov chain.
Problem 2.
Let (Xn )n be a sequence of RVs, not assumed to be identically distributed or independent, and letSn = ∑
Xn , n ≥ 1 be the partial sums.
Problem 3.
Let (Yn )n be a sequence of i.i.d. RVs such that P (Y = 
= 
, and set Xn = ∏ 1≤i≤n Yi , n ≥ 1. (a) Show that (Xn )n forms a martingale; ind EX.
(b) Show that the martingale converges and identify the limit. Does it converge in L1 ? If yes, explain which
suficient conditions you used; if not, explain which necessary conditions fail to hold.
Problem 4.
A queueing system has 2 servers with exponential service rates μ1 and μ2 . Customers enter the system at Poisson rate λ and join a single queue. If the arriving customer inds the system empty, he chooses one of the servers randomly with probability 1/2.
(a) Show that the system can be modeled by a Markov chain with states S = {0, a,b, 2, 3, . . . }, where a means that there is one customer at server 1 and b that there is one customer at server 2 (and no other customers in the system). Prove that this Markov chain is reversible,i.e., itsatisies the detailed balance condition πsQst = πt Qts for all t, s ∈ S.
(b) Find the limiting distribution π of the system.
Problem 5.
(a) Let Y … Unif( — 1, 1) and consider a random process X(t) = Y3t, t ≥ 0. Is the process X(t) stationary?
(b) Now let Y … Unif(0, 1) and consider X(t) = eY t, t ≥ 0. Are the increments of X(t) independent? Are they stationary (meaning that for any 0 < t1 < t2 the distribution of X(t2 + s) — X(t1 + s) does not depend on s) ?
2023-11-27