TY - JOUR

T1 - Sink or swim together

T2 - Necessary and sufficient conditions for finite moments of workload components in FIFO multiserver queues

AU - Scheller-Wolf, Alan

AU - Vesilo, Rein

PY - 2011

Y1 - 2011

N2 - 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 EWi<∞, where Wi 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⌉, EWα i < ∞, provided that ESβ 1(i) < ∞, where β1(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: EWα i < ∞ provided that ESβ 2(i) < ∞, where β2(i)=(s-i+α)/(s-i), for α≥1. Finally, for S in a large class of service distributions, these conditions are also necessary.

AB - 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 EWi<∞, where Wi 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⌉, EWα i < ∞, provided that ESβ 1(i) < ∞, where β1(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: EWα i < ∞ provided that ESβ 2(i) < ∞, where β2(i)=(s-i+α)/(s-i), for α≥1. Finally, for S in a large class of service distributions, these conditions are also necessary.

UR - http://www.scopus.com/inward/record.url?scp=78650412830&partnerID=8YFLogxK

U2 - 10.1007/s11134-010-9198-6

DO - 10.1007/s11134-010-9198-6

M3 - Article

AN - SCOPUS:78650412830

SN - 0257-0130

VL - 67

SP - 47

EP - 61

JO - Queueing Systems

JF - Queueing Systems

IS - 1

ER -