Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

摘要

In Internet environment, traffic flow to a link is typically modeled bysuperposition of ON/OFF based sources. During each ON-period for a particularsource, packets arrive according to a Poisson process and packet sizes (henceservice times) can be generally distributed. In this paper, we establish heavytraffic limit theorems to provide suitable approximations for the system underfirst-in first-out (FIFO) and work conserving service discipline, which statethat, when the lengths of both ON- and OFF-periods are lightly tailed, thesequences of the scaled queue length and workload processes converge weakly toshort-range dependent reflecting Gaussian processes, and when the lengths ofON- and/or OFF periods are heavily tailed with infinite variance, the sequencesconverge weakly to either reflecting fractional Brownian motions (FBMs) orcertain type of long-range dependent reflecting Gaussian processes depending onthe choice of scaling as the number of superposed sources tends to infinity.Moreover, the sequences exhibit a state space collapse-like property when thenumber of sources is large enough, which is a kind of extension of thewell-known Little's law for M/M/1 queueing system. Theory to justify theapproximations is based on appropriate heavy traffic conditions whichessentially mean that the service rate closely approaches the arrival rate whenthe number of input sources tends to infinity.