The embedded random walk in the stationary M/M/1 queue

We consider the stationary queue length process of the standard M/M/1 queue just after its jump times (i.e., the times of arrivals or departures). Simple formulas for the distribution of this embedded random walk with immediate reflection at zero are derived. We study the monotonicity properties of the corresponding expected values and point out an asymmetry between the number of arrivals and the number of departures until the n-th jump time.