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take home exam - 18.445 Take Home Exam (Due May 12th) 1....

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Unformatted text preview: 18.445 Take Home Exam (Due May 12th) 1. Suppose is a continuous time Markov Chain taking non-negative integer values. Moreover, suppose (1) (2) ; has independent increments property; and such that when , (3) There exist . Let for all and non-negative integer . Find a sequence of differential equations for . Try to simplify (or solve) them as possible as you can. 2. Taxis looking for customers arrive at a taxi station as a Poisson process (rate 1 per minute), while customers looking for taxis arrive as a Poisson process (rate 1.25 per minute). Suppose taxis will wait, no matter how many taxis are in line before them. But customers who arrive to find 2 other customers in line go away immediately. Over the long run, what is average number of customers waiting at the station? 3. Suppose parameter times of and are two independent Poisson Processes with and , respectively. Suppose for are the waiting , and for are the waiting times of . What ( ) ( ). is the distribution of 4. Suppose that battery lifetimes are independent with the distribution, whose density at t > 0 is , for some and . In a system requiring one battery, the battery is replaced by a new one as soon as it dies. Let denote the total lifetime (that is current age plus remaining lifetime) of the battery in use at time t. Describe the limit distribution of this limit distribution. as , and calculate the mean of 5. (Coupling method) Suppose is a transition matrix on state space such that for all . Let be the stationary distribution of , i.e. and . Suppose and are two independent Markov Chain both with transition matrix . Assume and follows distribution . Let | . (a) Prove that (b) Show that (c) Let distribution . (d) Show that | (e) Show that | | | . . | , and . Show that . , where . is a Markov Chain with stationary ...
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This note was uploaded on 10/31/2011 for the course 18 18.445 taught by Professor Liewang during the Spring '11 term at MIT.

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