Law of Large Number Limits of Limited Processor-Sharing Queues

Abstract

Motivated by applications in computer and communication systems, we consider a processor-sharing queue where the number of jobs served is not larger than K. We propose a measure-valued fluid model for this limited processor-sharing queue and show that there exists a unique associated fluid model solution. In addition, we show that this fluid model arises as the limit of a sequence of appropriately scaled processor-sharing queues.

Steady state approximations of limited processor sharing queues in heavy traffic

“…As in [25,26], we introduce a measure-valued state descriptor (Q(·), Z(·)), which is rich enough to describe the evolution of the system with given initial conditions and stochastic primitives. For any Borel set A ⊂ R + , Q(t)(A) denotes the total number of jobs in buffer whose job size belongs to A; and Z(t)(A) denotes the total number of jobs in service whose residual job size belongs to set A. Denote M the space of all non-negative finite Borel measures on [0, ∞).…”

“…Some stochastic ordering results are derived in Nuyens and van der Weij [19]. Recently, Zhang, Dai and Zwart [25,26] studied the stochastic processes underlying the LPS queue in the heavy traffic regime, an asymptotic regime where the traffic intensity converges to 1. The study was carried in a general setting, allowing the inter-arrival and service times to have general distributions.…”

Tail-robust scheduling via limited processor sharing

“…Some stochastic ordering results are derived in Nuyens & van der Weij [9]. Zhang, Dai & Zwart [10,11,12] develop fluid, diffusion and heavy traffic approximations. Finally, Gupta & Harchol-Balter [13] consider approximation methods and Markov decision techniques to determine the optimal level c when the system is not work-conserving.…”

Monotonicity in the Limited Processor-Sharing Queue

“…Very recently, Zhang et al [14,15] and Zhang and Zwart [16] have investigated the LPS queue, and described the behaviour of the queue in light and heavy traffic. They derived an approximation for the waiting probability in the limit c → ∞ [14,15], and for the steady-state queue length and response time in heavy traffic [16].…”

“…They derived an approximation for the waiting probability in the limit c → ∞ [14,15], and for the steady-state queue length and response time in heavy traffic [16].…”