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ps4 - Urban OR Fall 2006 Problem Set 4(Due Wednesday...

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Urban OR Fall 2006 Problem Set 4 (Due: Wednesday, November 8, 2006) Problem 1 [This is a review problem on M/G/1 queues. We have already done the first four parts in class, as an example.] Consider a single-server queueing system with infinite queue capacity. Arrivals of customers to this system occur in a Poisson manner at the rate of 36 per hour. The service times, S , of customers are mutually independent and their duration is uniformly distributed between 1 and 2 minutes, in other words, f S ( t ) = U [1, 2] in units of minutes. (1) Compute the expected waiting time in queue, W q , and the expected number of customers in the queue, L q , when this queueing system is in steady state. [Numerical answers are expected.] (2) Assume now that service times have a negative exponential probability density 2 function with f S ( t ) = 2 e 3 t for t 0 , in units of minutes. Repeat part (1), i.e., 3 compute numerically the values of W q and of L q for this case. (3) Assume now that service times are constant and all have duration equal to 1.5 minutes. Repeat part (1), i.e., compute numerically the values of W q and of L q for this case. (4) Using the G/G/1 bounds discussed in class, compute numerically upper and lower bounds for W q and for L q for the case of Part (1). (5) Return to the original case (uniform service times) from here until the end of the problem. Suppose that immediately after the completion of a service, exactly 2 customers are left in the system. Carefully write an expression for the probability that after the end of the next service, exactly 4 customers will be left in the system. Do NOT evaluate numerically this expression. (6) Repeat part (4) under the supposition that after the completion of a service, exactly 0 customers are left in the system. Carefully write an expression for the probability that after the end of the next service, exactly 4 customers will be left in the system. Do NOT evaluate numerically this expression. (7) Consider again part (5) where after the completion of a service, exactly 2 customers are left in the system. Carefully write an expression for the probability that TWO service completions later, exactly 4 customers will be left in the system. Do NOT evaluate numerically this expression.

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ps4 - Urban OR Fall 2006 Problem Set 4(Due Wednesday...

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