Diffusion approximation for an overloaded X model via a stochastic averaging principle

Abstract

In previous papers we developed a deterministic fluid approximation for an overloaded Markovian queueing system having two customer classes and two service pools, known in the call-center literature as the X model. The system uses the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed as a way for one service system to help another in face of an unexpected overload. Under FQR-T, customers are served by their own service pool until a threshold is exceeded. Then, one-way sharing is activated with customers from one class allowed to be served in both pools. The control aims to keep the two queues at a pre-specified fixed ratio. We supported the fluid approximation by establishing a functional weak law of large numbers (FWLLN) involving a stochastic averaging principle. In this paper we develop a refined diffusion approximation for the same model based on a many-server heavy-traffic functional central limit theorem (FCLT).Since we are considering the case when sharing is taking place with class-1 customers receiving help, we essentially need only consider r 1,2 , which we henceforth denote by r, i.e., r ≡ r 1,2 .

On the instability of matching queues

“…for π in (29). By Lemma 2 and the FWLLN for the Poisson process, the second argument to the right of the equality in (59) converges weakly to λ i 0 e in D[0, δ), where e denotes the identity function e(t) = t. Since 1l N i (Q(s−)) is identically equal to 1 for all n large enough, again by Lemma 2, and M n i 0 /n ⇒ 0e in D[0, δ) as n → ∞, by virtue of Doob's martingale inequality, the limit (32) follows from (60) and Lemma 7.3 in [32] (a simple extension to the continuous mapping theorem).…”

“…In the limit, the effect of the "fast" (i.e., null) queues on the evolution of the positive fluid queues is averaged-out instantaneously, a phenomena known as a stochastic averaging principle (AP) in the literature. See [31,32] and the references therein, as well as [27,42] for recent examples of fast averaging in queueing networks.…”

A Fluid Limit for an Overloaded X Model via a Stochastic Averaging Principle

“…Hence, the six-dimensional Markov chain describing the system during overloads in the prelimit is replaced by a simplified lower dimensional deterministic function, which is easier to analyze. In addition, these SSC results are crucial to proving the FCLT for the system (Perry and Whitt [38]). …”

“…Because w x T is a continuous function of x for each fixed and T , we can apply this bound with the inequalities in the proof of Theorem 5.2 to deduce (38).…”

Dynamic overload balancing in server farms

“…The study of network systems in overload has received significant attention recently. Fluid limits [4], [5] and throughput optimization problems [6], [7] are investigated. Work [8] shows that the most-balanced queue growth rate vector exists in a single-commodity network in overload; this work uses deterministic fluid models for performance analysis and does not provide a stochastic control policy to achieve the most-balanced vector.…”