Question

# 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.

### Recently Asked Questions

- In a clinical test with 1400 subjects, 420 showed improvement from the treatment. Find the margin of error for the 99% confidence interval used to estimate the

- The table shows, for the years 1997-2012, the mean hourly wage for residents of the 4) town of Pity Me and the mean weekly rent paid by the residents. Year

- can you show how it works