STAT 333 tutorial10

# STAT 333 tutorial10 - t(a Find the generator of the process...

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Stat 333 - Tutorial 10 1. A small computer network consists of two computers, A and B. The computers can access data from each other but must do so via requests made to a mainframe computer. We will focus on the mainframe computer and the number of requests that it handles. For simplicity, assume that A or B will not generate requests for data while a previous request is pending. As such, at any time t , the mainframe computer is handling 0, 1 or 2 requests. Additional assumptions: The computers A and B operate independently and if a particular computer has no requests at time t , the conditional probability that it will do so in ( t,t + dt ) is λdt + o ( dt ) . If the mainframe computer is handling i polling requests at time t , the conditional probability that it will complete processing of these requests in ( t,t + dt ) is μ i dt + o ( dt ) , i = 1 , 2 . Let X t represent the number of requests the mainframe computer is handling at time
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Unformatted text preview: t . (a) Find the generator of the process { X t } t ≥ . (b) Draw the state transition graph. Determine the classes of this Markov process. Are they transient or recurrent? Is the MP irreducible? (c) Determine the equilibrium distribution for this MP. 2. Potential customers arrive at a full-service, one-pump gas station at a Poisson rate of 20 cars per hour. However, customers will only enter the station for gas if there are no more than two cars (including the one currently being attended to) at the pump. Suppose the amount of time required to service a car is exponentially distributed with a mean of ﬁve minutes. (a) What fraction of the attendant’s time will be spent servicing cars? (b) What fraction of potential customers are lost? 1...
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