Question: Consider anM/M/1 queue with arrival rateand the service ratewith Note:This could be a partial proof for the famous result saying that the 'departure process' of

Consider anM/M/1 queue with arrival rateand the service ratewith

Note:This could be a partial proof for the famous result saying that the 'departure process' of anM/M/1 queue in equilibrium is also a Poisson process with rate.

Hints:LetDbe the inter-departure time andNbe the number of customers in the system. Find the distribution ofD(e.g., using CDF or Laplace transform) via conditioning on{N1}or{N= 0}, i.e., conditioning on whether the system is empty or not. Under each of these cases, you should be able to expressDin terms of other known random variables (e.g., exponential with rateor). Then, remove the conditioning.

Consider anM/M/1 queue with arrival rateand the service ratewith Note:This could be

3. (15 points): (from past exam) Consider an M /M / 1 queue with arrival rate A and the service rate u with A

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