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Consider an M/M/1 queue with arrival rate λ and the service rate μ with λ < μ. Prove that, in the the

steady-state, the inter-departure time (the time between two consecutive departure instants) is exponentially distributed with rate λ.

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

Hints: Let D be the inter-departure time and N be the number of customers in the system. Find the distribution of D (e.g., using CDF or Laplace transform) via conditioning on {N ≥ 1} or {N = 0}, i.e., conditioning on whether the system is empty or not. Under each of these cases, you should be able to express D in terms of other known random variables (e.g., exponential with rate λ or μ). Then, remove the conditioning.

Screen Shot 2019-10-15 at 12.11.03 PM.png

Screen Shot 2019-10-15 at 12.11.03 PM.png

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