Invariant states and rates of convergence for a critical fluid model of a processor sharing queue

Abstract

This paper contains an asymptotic analysis of a fluid model for a heavily loaded processor sharing queue. Specifically, we consider the behavior of solutions of critical fluid models as time approaches ∞. The main theorems of the paper provide sufficient conditions for a fluid model solution to converge to an invariant state and, under slightly more restrictive assumptions, provide a rate of convergence. These results are used in a related work by Gromoll for establishing a heavy traffic diffusion approximation for a processor sharing queue.

Diffusion limits of limited processor sharing queues

“…)We are interested in the limit of the diffusion scaled process 4 J. ZHANG, J. G. DAI AND B. ZWART state. The counterpart of this study for standard PS queues has been carried out in [22]. Standard PS queues are relatively tractable since their fluid models can be related (by means of a time-change) to a renewal equation.…”

Heavy traffic limit for a processor sharing queue with soft deadlines

“…The state of the processor sharing queue with deadlines will be tracked using a measure valued process in the right half-plane. This idea builds on previous work on the GI/GI/1 processor sharing queue [8,9,10,20,21]. The setup requires detailed information about how arriving jobs affect the system state.…”

“…Note that projection of Z(·) onto the first coordinate yields the state descriptor used in [8,9,10,20,21]. For each r in a sequence R of positive real numbers tending to infinity, define a diffusion scaled version of the state descriptor bŷ Let α be the limiting arrival rate (as r → ∞) for jobs entering the system and let ϑ ∈ M be the limiting joint distribution (as r → ∞) of service times and initial lead times (the precise form of these assumptions is given in Section 2.3 below).…”

Diffusion approximation for a processor sharing queue in heavy traffic

“…The process µ(·) has previously been used by Grishechkin [5], along with other measure valued descriptors, in his heavy traffic analysis of the steady state distribution of a processor sharing queue. It has also been used recently by Gromoll, Puha and Williams [6], Puha and Williams [16] and Puha, Stolyar and Williams [15] for obtaining fluid limit results. Similar measure valued descriptors have also been used recently to describe other queueing systems.…”