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03faex2 - exactly one customer is lost during your service...

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632 Introduction to Stochastic Processes Fall 2003 Midterm Exam II Instructions: Justify non-obvious statements for full credit. Quote re- sults from class accurately. Points add up 100. 1. (a) (20 pts) Let N be a rate λ Poisson process. Find the conditional probabilities P [ N (5) = 4 | N (2) = 3] and P [ N (2) = 3 | N (5) = 4]. (b) Suppose male customers arrive in a store as a Poisson process with rates μ and female customers arrive at rate φ , and these arrival processes are independent of each other. (10 pts) Find an expression for the probability that the first 6 customers are divided evenly between males and females. (10 pts) Suppose during the first hour one male and one female customer arrived. What is the probability that the male customer arrived before the female? (c) (15 pts) Imagine a service system where the duration of service is exponentially distributed with rate μ . Customers who arrive while the server is busy turn away and are lost. Suppose you just arrived in service. If customers arrive as a Poisson process with rate λ , find the probability that
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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 continuous-time 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 discrete-time 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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