Unformatted text preview: 5. Customers enter an inﬁnite server system according to a Poisson process with rate λ . The r th arriving customer ﬁrst enters an (inﬁnite 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 λ 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 )] = λ Z t g ( u ) du and Var( X ( t )) = λ Z t g 2 ( u ) du. Hint: Condition on N ( t )....
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 Fall '09
 GLENTAKAHARA
 Probability, Probability theory, Stochastic process, Poisson process

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