Unformatted text preview: exactly one customer is lost during your service. 2. (10 pts) Batch arrivals in an M/M/1 queue. Consider an M/M/1 queueing system with arrival rate λ and service rate μ . Each arrival brings a batch of customers, and each batch contains k customers with probability p k . The numbers { p k : k ≥ 1 } are given parameters. Let the Markov chain X t be the number of customers in the system at time t . Describe the rates q ( i,j ), i 6 = j , for this Markov chain. 3. Consider a continuoustime Markov chain X t on the state space S = { 1 , 2 , 3 , 4 } . The rates are q (1 , 2) = α , q (2 , 1) = β , and q (1 , 3) = q (2 , 4) = 1. All other rates q ( i,j ) for i 6 = j are zero, so 3 and 4 are absorbing states. (a) (15 pts) Give the transition matrix of a discretetime chain that records the states in the order in which X t visits them. (b) (20 pts) Start the chain X t at state 1. Find the probability that the chain is eventually absorbed in state 3....
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 Spring '07
 Seppalainen
 Probability, Probability theory, Markov chain, Queueing theory

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