510 research outputs found
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 . Then, the
waiting time of an infinitely patient customer arriving at time 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 , 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 Ξ£ GIi/G/s model, in which the arrival process is the superposition of n independent and identically distributed stationary renewal processes each with rate nβ1. The traffic intensity Ο is allowed to approach the critical value one as n increases. If n(1βΟ)2 β c, 0 < c < β, then a limit is obtained that depends on c. The two iterated limits involving Ο and n, which do not agree, are obtained as c β 0 and c β β
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 topology
We establish continuity of the integral representation
, , mapping a function into a function
when the underlying function space is endowed with the Skorohod
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 -continuity is based on a new characterization of the
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 .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
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