Multiplexing on-off sources with subexponential on periods

Abstract

Consider an aggregate arrival process A/sup N/ obtained by multiplexing N on-off sources with exponential off periods of rate /spl lambda/ and subexponential on periods /spl tau//sup on/. For this process its activity period I/sup N/ satisfies P[I/sup N/>t]/spl sim/(1+/spl lambda/E/spl tau//sup on/)/sup N-1/P[/spl tau//sup on/>t] as t/spl rarr//spl infin/ for all sufficiently small /spl lambda/. When N goes to infinity, with /spl lambda/N/spl rarr//spl Lambda/, A/sup N/ approaches an M/G//spl infin/ type process, for which the activity period I/sup /spl infin//, or equivalently a busy period of an M/G//spl infin/ queue with subexponential service requirement /spl tau//sup on/, satisfies P[I/sup /spl infin//>t]/spl sim/e/sup /spl Lambda/Er(on)/P[/spl tau//sup on/>t] as t/spl rarr//spl infin/. For a simple subexponential on-off fluid flow queue we establish a precise asymptotic relation between the Palm queue distribution and the time average queue distribution. Further, a queueing system in which one on-off source, whose on period belongs to a subclass of subexponential distributions, is multiplexed with independent exponential sources with aggregate expected rate Ee/sub t/, is shown to be asymptotically equivalent to the same queueing system with the exponential arrival processes being replaced by their total mean value Ee/sub t/. For a fluid queue with the limiting M/G//spl infin/ arrivals we obtain a tight asymptotic lower bound for large buffer probabilities. Based on this bound, we suggest a computationally efficient approximation for the case of finitely many subexponential on-off sources. The accuracy of this approximation is verified with extensive simulation experiments.