ass4 - 5. Customers enter an innite server system according...

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STAT455/855 Fall 2009 Applied Stochastic Processes Assignment #4 Due Monday, Nov.23 Starred questions are for 855 students only. 1. Ross, Chapter 5, #44 (8th Edition: Chapter 5, #44). 2. Ross, Chapter 5, #53 (8th Edition: Chapter 5, #53). 3. Ross, Chapter 5, #59 (8th Edition: Chapter 5, #59). 4. Shocks to a system occur according to a Poisson process with rate λ = 0 . 2. The system will fail if it receives two consecutive shocks that are less than one time unit apart. Suppose at time 2 the system is observed to be still functioning. (a) If it is known that exactly one shock occurred in the interval [0 , 2), what is the probability that the next shock will cause the system to fail? (b) If it is known that exactly two shocks occurred in the interval [0 , 2), what is the probability that the next shock will cause the system to fail? Hint: You need to find the distribution of the time of the most recent shock before time 2 given that there were 2 shocks in [0 , 2) and they were at least 1 time unit apart.
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Unformatted text preview: 5. Customers enter an innite server system according to a Poisson process with rate . The r th arriving customer rst enters an (innite capacity) waiting room where he waits for a time X r , after which time he then receives service which takes a time Y r . Upon completion of service the customer leaves the system. Suppose that X r and Y r , r 1, are independent Uniform(0,1) random variables. Let W ( t ) be the number of customers waiting at time t and let S ( t ) be the number of customers in service at time t . Find the joint distribution of W ( t ) and S ( t ). ? 6. Let { N ( t ) : t } be a Poisson process with rate and let g : [0 , ) [0 , ) be a nonnegative integrable function. Let X ( t ) = N ( t ) X i =1 g ( S i ) , where S i is the time of the i th event (and X ( t ) = 0 if N ( t ) = 0). Show that E [ X ( t )] = Z t g ( u ) du and Var( X ( t )) = Z t g 2 ( u ) du. Hint: Condition on N ( t )....
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