Question

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2. Customers arrive to a coffee cart according to a Poisson process with constant rate 12 per hour. Each customer is served by a single server and this takes an exponentially-distributed amount of time with mean 2 minutes irrespective of ev- erything else. When the coffee cart opens for service, there are already 7 people waiting. Denote by X = (X+,t> 0) the number of people waiting or in service at the coffee cart t hours after it opens. (a) Formulate a continuous-time Markov chain model for the problem for X, by specifying the state space E, 7), and Q. [1] (b) Draw the corresponding transition rate diagram. (c) What is the long run probability that n customers are waiting or in service, for n = 0,1,...? (d) What is the long run expected number customers waiting or in service? [1]

Answer 1

Solution a) This is a Mirli model (poission with Exp time] where aplival rate -d=121 hour Sesivice time u = 2/minite - 120 hour E = {0, 1,2 -9 since there can be infinite customers in the queue. Traffic intensity, etc. The to Assuming or coas the initial probability distribution for this morkor chain. TilO) -(-e) = 1- 1-10 = 9600 Assumving Q as the transition rate matrix Q = I u burd) de -(Mtd) M O- 120 - 132 12 120 - 132

b) Transition rate matrix u c) u u wet an be the limiting probability an= lim P(Qct)=n) - Cine) en - (in ) ( n = Cound) = (12)^(108) . 108 (120in on. t-o un กะ d) Average customers in system ECL) = npn = Average customers in Queue E (22) = E(L) - e e2 (1110) - The 9110 - 1190