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, 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 rho = ES/ET, then sufficient conditions for EW(i)   inverted right perpendicular rho inverted left perpendicular, where for any real x, inverted right perpendicularxinverted left perpendicular denotes the smallest integer greater than or equal to x. Explicitly, for i  = 1. Furthermore, components with indices lower than inverted right perpendicular rho inverted left perpendicular all share the same finite moment conditions. This is not true for i > inverted right perpendicular rho inverted left perpendicular; these components have individual finite moment conditions: EW(i)(alpha)  = 1. Finally, for S in a large class of service distributions, these conditions are also necessary.
necessary and sufficient conditions workload components multiserver queues finite moment conditions