Stochastic Processes and Queueing Theory — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

2372574701 (NEP-UGCF) DSC: Stochastic Process and Queueing Theory 2. Attempt six questions in all. 4. Attempt five more questioLs from the Q. No. 2 to 7. (a) For a (M|M|1) with FCFS service discipline and infinite system capacity queuing system, show (in usual notations) P(queue length 2 n) = pn (b) Let X be a non-negative integral valued random variable with P(X = k) = pk; k = 0,1,2... and probability generating function P(s). Find the generating function of P(X 2 k). (c) (d) Let the probability of transition from a dry day to a wet day.be 0.55 and that of transition from a wet day to a dry day be 0.6. The initial probability distribution of a day being a dry day or a wet day is (0.7 0.3). Find the probability of the second day being a wet day. In the classical ruin problem find the expected gain for a player and interpret the result. (e) Show that the initial distribution together with transition matrix completely defines a Markov chain. ® Let X be a positive discrete random variable such that P[X=k] = pk, Obtain a relation between P(s) and Q(s). Show that E[X] = Q(1) = Zqk . (g) Let Yn= aoxn+ a[Xn_I (n= 1,2 ,.... ) where ao, a[ are constants and Xn, n = 0,1,2... are i.i.d. random variables with mean 0 and variance o2. Is {Yn, n > 1} covariance stationary? (3x5) (a) Let X be a random variable (r.v) denoting the number of tosses required to get two consecutive heads when a fair coin is tossed. Obtain.the p.g.f. of the r.v. X. (b) Let SN = X] + X2 + ... + XN, Where X£'s are i.i.d. random variables and N is a random variable independent of x£'s. Find expectation and variance of sN. ,7,8, (a) Consider the process X(t)=±(Arcos©rt+Btsin©rt) Where Ar, Br are r=1 uncorrelated random variables with mean 0 and variance o2 and Or are constants. Is the process {X(t), t € T} covariance stationary? (b) A doctor can either go by a car or take a cab to go to his clinic every day. He never goes two days in a row in car , but if he goes by car one day , then the next day he is equally likely to travel by cab as he is to go by car again .On the first day of the week he throws a dice and goes by car to his clinic iff "3 " appears. Find the probability that (i) he goes by car to his (ii) he takes a cab in the long run. clinic on the second day of the week. 4. (a) Consider the Markov Chain with three states, S = {0, 1,2) having the following transition matrix: (i) Is the chain irreducible? (i) Find the nature of the states 0, 1 and 2. (b) Consider a sequence {,, n≥2) of independent coin tossing trials with probability p for head() in a trial. Denote the states of X by states 1,2,3,4 according as the trial numbers (n-1) and n results in HH,HI,TH,TT respectively. Show that (X, n22) is a Markov chain and construct the (7,8) transition probability matrix. (b) Let {X,, n≥ 0} be a Markov chain with S = {0, 1, 2), transition probability matrix and the initial distribution P|, =i|=. = 1=0, 1, 2. Find the probabilities