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

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. 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 so that EWi < ∞, where Wi is the i’th smallest component of the ordered workload vector, depend crucially on the traffic intensity relative to i — specifically, on whether i ≤ dρe or i > dρe, where for any real x, dxe denotes the smallest interger greater than or equal to x. Explicitly, for i ≤ dρe, EW i < ∞, provided that ES1 < ∞, where β1(i) = (s− bρc+ α)/(s− bρc), for α ≥ 1. Furthermore, components with indices lower than dρe all share the same finite moment conditions. This is not true for i > dρe; these components have individual finite moment conditions: EW i < ∞ provided that ES2 < ∞, where β2(i) = (s − i+ α)/(s − i), for α ≥ 1. Finally, for S in the class L βj(i) 1 , j = 1, 2, defined in [9], these conditions are also necessary.