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

Abstract

We prove a many-server heavy-traffic fluid limit 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-withthresholds (FQR-T) control, which we proposed in a recent paper 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. After the control is activated, it aims to keep the two queues at a prespecified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN (functional weak law of large numbers), we consider a sequence of properly scaled X models in overload operating under FQR-T. Our proof of the FWLLN follows the compactness approach, i.e., we show that the sequence of scaled processes is tight and then show that all converging subsequences have the specified limit. The characterization step is complicated because the queue-difference processes, which determine the customer-server assignments, need to be considered without spatial scaling. Asymptotically, these queue-difference processes operate on a faster time scale than the fluid-scaled processes. In the limit, because of a separation of time scales, the driving processes converge to a time-dependent steady state (or local average) of a time-varying fast-time-scale process (FTSP). This averaging principle allows us to replace the driving processes with the long-run average behavior of the FTSP.1. 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.) The X model has two classes of customers and two service pools, one for each class, but with both pools capable of handling customers from either class. The service-time distributions depend on both the class and the pool. The FQR-T control was suggested in Perry and Whitt [35] as a way to automatically initiate sharing (i.e., sending customers from one class to the other service pool) when the system encounters an unexpected overload, while ensuring that sharing does not take place when it is not needed.

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

“…If class 2 is overloaded, then an optimal ratio r 2,1 should hold between the queues. In [24] we showed that the FQR-T control achieves the target ratios asymptotically as the scale increases (in the fluid limit), for the time-dependent transient performance as well as in steady state. Moreover, the FQR-T control produces a tractable fluid limit.…”

“…By Lemma 3.1 in [24], π 1,2 (γ) is well defined for all γ ∈ S, but D(γ, ·) is positive recurrent if and only if 0 < π 1,2 (γ) < 1 and γ ∈ S b . By Theorem 6.1 of [23], Both the FWLLN and the FCLT depend critically on distributional and topological characteristics of the FTSP's.…”

“…Observe that Theorem 1 concludes that if x(0) ∈ A, then x(t) ∈ A for all t over some interval [0, δ) (that part of the theorem follows from Theorem 4.5 in [24]), so that we have SSC in the sense that the original six-dimensional process is a deterministic function of a two-dimensional process. More importantly for the FCLT, we also have that Q n…”

“…where N a i , N s i,j and N u i for i = 1, 2; j = 1, 2 are eight mutually independent rate-1 Poisson processes. Theorem 5.1 of [24] gives a representation of the CTMC in terms of these processes. Corollaries 6.1-6.3 plus Theorem 6.4 of [24] then establish state space collapse (SSC) results yielding an asymptotically equivalent three-dimensional representation of X n 6 involving these mutually independent rate-1 Poisson processes plus two others.…”

“…In particular, the limiting fluid approximation is a deterministic function characterized by an ODE (and an initial condition), which is driven by the time-varying instantaneous average behavior of a family of fast-time-scale stochastic processes (FTSP's), which produces the AP. See §1.3 of [24] for a discussion of the literature on AP's; notable contributions in the queueing literature are by Coffman et al [4] and Hunt and Kurtz [15]. See [6] for a (quite different) FCLT involving an AP, building on [15].…”