Sink or swim together: necessary and sufficient conditions for finite moments of workload components in FIFO multiserver queues

Abstract

Previously established necessary and sufficient conditions for finite stationary moments in stable FIFO GI/GI/s queues exist only for the first component of the workload vector, the delay, and the final component, which behaves as the total work in the system. In this paper, we derive moment results for all the components of the stationary workload vector in stable FIFO GI/GI/s queues. As in the case of stationary delay, the moment conditions for workload components incorporate the interaction between service-time distribution, traffic intensity and the number of servers in the queue.

If we denote a generic service-time random variable by S, a generic interarrival time by T, and define the traffic intensity as ρ=ES/ET, then sufficient conditions for EW _i <∞, where W _i is the ith smallest component of the ordered workload vector, depend crucially on the traffic intensity relative to i—specifically, on whether i≤⌈ρ⌉ or i>⌈ρ⌉, where for any real x, ⌈x⌉ denotes the smallest integer greater than or equal to x. Explicitly, for i≤⌈ρ⌉, $\mbox {E}W_{i}^{\alpha}<\infty$, provided that $\mbox {E}S^{\beta_{1}(i)}<\infty$, where β ₁(i)=(s−⌊ρ⌋+α)/(s−⌊ρ⌋), for α≥1. Furthermore, components with indices lower than ⌈ρ⌉ all share the same finite moment conditions. This is not true for i>⌈ρ⌉; these components have individual finite moment conditions: $\mbox {E}W_{i}^{\alpha}<\infty$ provided that $\mbox {E}S^{\beta_{2}(i)}<\infty$, where β ₂(i)=(s−i+α)/(s−i), for α≥1. Finally, for S in a large class of service distributions, these conditions are also necessary.