A Switching Fluid Limit of a Stochastic Network Under a State-Space-Collapse Inducing Control with Chattering

Routing mechanisms for stochastic networks are often designed to produce state space collapse (SSC) in a heavy-traffic limit, i.e., to confine the limiting process to a lower-dimensional subset of its full state space. In a fluid limit, a control producing asymptotic SSC corresponds to an ideal sliding mode control that forces the fluid trajectories to a lower-dimensional sliding manifold. Within deterministic dynamical systems theory, it is well known that sliding-mode controls can cause the system to chatter back and forth along the sliding manifold due to delays in activation of the control. For the prelimit stochastic system, chattering implies fluid-scaled fluctuations that are larger than typical stochastic fluctuations. In this paper we show that chattering can occur in the fluid limit of a controlled stochastic network when inappropriate control parameters are used. The model has two large service pools operating under the fixed-queue-ratio with activation and release thresholds (FQR-ART) overload control which we proposed in a recent paper. We now show that, if the control parameters are not chosen properly, then delays in activating and releasing the control can cause chattering with large oscillations in the fluid limit. In turn, these fluid-scaled fluctuations lead to severe congestion, even when the arrival rates are smaller than the potential total service rate in the system, a phenomenon referred to as congestion collapse. We show that the fluid limit can be a bi-stable switching system possessing a unique nontrivial periodic equilibrium, in addition to a unique stationary point

Heavy-traffic limits for waiting times in many-server queues with abandonment

We establish heavy-traffic stochastic-process limits for waiting times in many-server queues with customer abandonment. If the system is asymptotically critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a bounding argument shows that the abandonment does not affect waiting-time processes. If instead the system is overloaded, as in the efficiency-driven (ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh Annual Allerton Conference on Communication, Control and Computing (1999) 1095--1104], we treat customer abandonment by studying the limiting behavior of the queueing models with arrivals turned off at some time $t$. Then, the waiting time of an infinitely patient customer arriving at time $t$ is the additional time it takes for the queue to empty. To prove stochastic-process limits for virtual waiting times, we establish a two-parameter version of Puhalskii's invariance principle for first passage times. That, in turn, involves proving that two-parameter versions of the composition and inverse mappings appropriately preserve convergence.Comment: Published in at http://dx.doi.org/10.1214/09-AAP606 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Martingale proofs of many-server heavy-traffic limits for Markovian queues

This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.Comment: Published in at http://dx.doi.org/10.1214/06-PS091 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Queues with superposition arrival processes in heavy traffic

AbstractTo help provide a theoretical basis for approximating queues with superposition arrival processes, we prove limit theorems for the queue-length process in a Σ GIi/G/s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal processes each with rate n−1. The traffic intensity ρ is allowed to approach the critical value one as n increases. If n(1−ρ)2 → c, 0 < c < ∞, then a limit is obtained that depends on c. The two iterated limits involving ρ and n, which do not agree, are obtained as c → 0 and c → ∞

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

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-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 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 pre-specified fixed ratio. For large systems that fixed ratio is achieved approximately. For the fluid limit, or FWLLN, 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, remain stochastically bounded, and need to be considered without spatial scaling. Asymptotically, these queue-difference processes operate in a faster time scale than the fluid-scaled processes. In the limit, due to 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 (AP) allows us to replace the driving processes with the long-run average behavior of the FTSP.Comment: There are 55 pages, 46 references and 0 figure

Continuity of a queueing integral representation in the M1{M}_{\mathbf{1}} topology

We establish continuity of the integral representation $y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$.Comment: Published in at http://dx.doi.org/10.1214/09-AAP611 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org