The Output of Multiserver Queuing Systems

We derive a recurrence relation for ξ_n, the time until the nth departure measured from T₀, any well-defined instant in time. While the recurrence, valid for the G/G/s queue, is generally not easy to work with, in certain cases we can use it to obtain the distribution and moments of ξ_n. We obtain limited results for the M/M/s queue but investigate the M/D/s queue in detail. For the M/D/s queue, the results are used to determine the range of system parameters, p/s (the average utilization per server) and s, for which the distribution of ξ_s can be approximated by one of the gamma type. The quality of the approximation for a given p and s provides insight as to when the output of the M/D/s queue is “nearly Poisson.”