An ODE for an overloaded X model involving a stochastic averaging principle

Abstract

We study an ordinary differential equation (ODE) arising as the many-server heavy-traffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the X model in the call-center literature, operates under the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unanticipated overload. Each pool serves only its own class until a threshold is exceeded; then one-way sharing is activated with all customer-server assignments then driving the two queues toward a fixed ratio. For large systems, that fixed ratio is achieved approximately. The ODE describes system performance during an overload. The control is driven by a queue-difference stochastic process, which operates in a faster time scale than the queueing processes themselves, thus achieving a time-dependent steady state instantaneously in the limit. As a result, for the ODE, the driving process is replaced by its long-run average behavior at each instant of time; i.e., the ODE involves a heavy-traffic averaging principle (AP).1. Introduction. We study an ordinary differential equation (ODE) that arises as the many-server heavy-traffic (MS-HT) fluid limit of a sequence of overloaded Markovian X queueing models under the fixed-queue-ratiowith-thresholds (FQR-T) control. The ODE is especially interesting, because it involves a heavy-traffic averaging principle (AP).The system consists of two large service pools that are designed to operate independently, but can help each other when one of the pools, or both, encounter an unexpected overload, manifested by an instantaneous shift in the arrival rates. We assume that the time that the arrival rates shift and the values of the new arrival rates are not known when the overload

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

“…The downward rates are 1 + ≡ 1 1 m 1 + 1 2 z 1 2 + 2 2 m 2 − z 1 2 + 1 q 1 and r + ≡ 2 (16) As in §7.1 of Perry and Whitt [37], we identify important subsets of the state space ≡ 0 Let D be a random variable that has the steady-state limiting distribution of the FTSP D s as s → and let 1 2 ≡ P D > 0 (18) That is, 1 2 is the probability that the stationary FTSP associated with ∈ is strictly positive. It turns out that D and 1 2 are well defined throughout .…”

“…The AP takes place only in the set . In Theorem 6.1 of Perry and Whitt [37] we showed that positive recurrence of the FTSP, and thus the set , depends only on the constant drift rates in the two regions:…”

“…For the derivation of the quantities in (8) see Theorem 2.3 in Whitt [48], especially equation (2.19), and §5.1 in Perry and Whitt [35]. See also §6 in Perry and Whitt [37].…”

“…3. We have studied the FTSP and the ODE in Perry and Whitt [37], to which we refer for more details. In §3.…”

“…Introduction. In this paper we prove that the deterministic fluid approximation for the overloaded X call-center model, suggested in Perry and Whitt [36] and analyzed in Perry and Whitt [37], arises as the manyserver heavy-traffic fluid limit of a properly scaled sequence of overloaded Markovian X models under the fixed-queue-ratio-with-thresholds (FQR-T) control. (A list of all the acronyms appears in §F in the appendix.)…”