Conditional inter‐departure times from the M/G/s queue

We study the mean and the distribution of the time elapsing between two consecutive departures from the stationary M/G/s queue given the number of customers left behind by the first departure is equal to n. It is conjectured that if the failure rate of the service time distribution is increasing (decreasing), then (i) the limit of the mean conditional inter‐departure time as n tends to infinity is less (greater) than the mean service time divided by the number of servers s, and (ii) the conditional inter‐departure times are stochastically decreasing (increasing) in n for all n≥s.