The embedded random walk in the stationary M/M/1 queue (Q1851129)

In an M/M/1 queue the embedded Markov chain of the queue length process immediately after its jump times (the arrival or departure epochs) is a random walk with reflection at the origin. The author derives a simple formula (not containing any integral) for the distribution of this embedded random walk and studies the monotonicity properties of the corresponding expected values. The asymmetry between the number of arrivals and the number of departures until the \(n\)th jump is brought out in terms of an inequality involving their expected values.