Exponential Bounds for Queues with Markovian Arrivals

Exponential bounds P[queue ≥ b] ≤ φe^(-γb) are found for queues whose increments are described by Markovian Additive Processes. This is done application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant γ is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor φ

Local Correlation Functions for Mean-field Dynamical Semigroups on C*algebras

This paper concerns the dynamics of local correlation functions in dissipative mean-field systems. We extend the abstract notion of a mean-field dynamical semigroup on a C*-algebra given in [1], from an evolution on site-averaged observables, to one on a class of local observables. Conditions are established under which this generalized mean-field dynamics factorises, in the thermodynamic limit, into contributions from disjoint regions. Correspondingly, the nested correlation functions factorise into contributions for single site observables in this limit. We demonstrate that these conditions are satisfied for a large class of model systems

Rigorous Bounds for Loss Probabilities in Multiplexers of Discrete Heterogenous Markovian Sources

Exponential upper bounds of the form P[queue ≥ b] ≤ φy^(-b) are obtained for the distribution of the queue length in a model of a multiplexer in which the input is a heterogeneous superposition of discrete Markovian on-off sources. These bounds are valid at all queue lengths, rather than just asymptotic in the limit b→∞. The decay constant y is found by numerical solution of a single transcendental equation which determines the effective bandwidths of the sources in the limit b→∞. The prefactor φ is given explicitly in terms of y. The bound provides a means to determine rigorous corrections to effective bandwidths for multiplexers with finite buffers

Exponential Upper Bounds via Martingales for Multiplexers with Markovian Arrivals.

We obtain explicit upper bounds in closed form for the queue length in a slotted time FCFS queue in which the service requirement is a sum of independent Markov processes on the state space {O, 1}, with integral service rate. The bound is of the form P[queue length ≥ b] ≤ cy^(-b) for any b ≥ 1 where c 1 are given explicitly in terms of the parameters of the model. The model can be viewed as an approximation for the burst-level component of the queue in an ATM multiplexer. We obtain heavy traffic bounds for the mean queue length and show that for typical parameters this far exceeds the mean queue length for independent arrivals at the same load. We compare our results on the mean queue length with an analytic expression for the case of unit service rate, and compare our results on the full distribution with computer simulations

Large deviations and overflow probabilities for the general single-server queue, with applications

We consider from a thermodynamic viewpoint queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (a_t, v_t, t ϵ R_+) and a rate function I such that if (W_t, t ϵ R_+) denotes the workload process, then lim_(t→∞) (v_t)^(-1)logP(W_t/a_t > w) = -I(w) on the continuity set of I. In the case that a_t = v_t = t it has been argued heuristically, and recently proved in a fairly general context (for discrete time models) by Glynn and Whitt [8], that the queue-length distribution (that is, the distribution of supremum of the workload process Q = sup_(t≥0) W_t) decays exponentially: P(Q > b) ~ e^(-δb) and the decay rate δ is directly related to the rate function I. We establish conditions for a more general result to hold, where the scaling functions are not necessarily linear in t: we find that the queue-length distribution has an exponential tail only if lim_(t→∞) a_t/v_t is finite and strictly positive; otherwise, provided our conditions are satisfied, the tail probabilities decay like P(Q > b) ~ e^(-δv(a^(-1)(b))). We apply our results to a range of workload processes, including fractional Brownian motion (a model that has been proposed in the literature (see, for example, Leland et al [10] and Norros [13]) to account for self-similarity and long range dependence) and, more generally, Gaussian processes with stationary increments. We also show that the martingale upper bound estimates obtained by Daley and Duffield [5], when the workload is modelled as an Ornstein-Uhlenbeck position process, are asymptotically correct

Bounds and comparisons of the loss ratio in queues driven by an M/M/∞ source.

We obtain upper bounds for the loss probability in a queue driven by an M/M/∞ source. The bound is compared with exact numerical results, and with bounds for two related arrivals models: superposed two state Markov fluids, and the Ornstein—Uhlenbeck process. The bounds are shown to behave continuously through approximation procedures relating the models

Local dynamics of mean-field Quantum Systems

In this paper we extend the theory of mean-field-dynamical semigroups given in {DW1,Du1] to treat the irreversible mean-field dynamics of quasi-local mean-field observables. These are observables which are site averaged except within a region of tagged sites. In the thermodynamic limit the tagged sites absorb the whole lattice, hut also become negligible in proportion to the hulk. We develop the theory in detail for a class of interactions which contains the mean-field versions of quantum lattice interactions with infinite range. For this class we obtain an explicit form of the dynamics in the thermodynamic limit. We show that the evolution of the bulk is governed by a flow on the one-particle state space, whereas the evolution of local perturbations in the tagged region factorizes over sites, and is governed by a cocycle of completely positive maps. We obtain an H-theorem which suggests that local disturbances typically become completely delocalized for large times, and we show this to be true for a dense set of interactions. We characterize all limiting evolutions for certain subclasses of interactions, and also exhibit some possibilities beyond the class we study in detail: for example, the limiting evolution of the bulk may exist, while the local cocycle does not. In another case the hulk evolution is given by a diffusion rather than a flow, and the local evolution no longer factorizes over sites

Mean-Field Dynamical Semigroups on C*-Algebras

We study a notion of the mean-field limit of a sequence of dynamical semigroups on the n-fold tensor products of a C*-algebra A with itself. In analogy with the theory of semigroups on Banach spaces we give abstract conditions for the existence of these limits. These conditions are verified in the case of semigroups whose generators are determined by the successive resymmetrizations of a fixed operator, as well as generators which can be approximated by generators of this type. This includes the time evolutions of the mean-field versions of quantum lattice systems. In these cases the limiting dynamical semigroup is given by a continuous flow on the state space of A. For a class of such flows we show stability by constructing a Liapunov function. We also give examples where the limiting evolution is given by a diffusion, rather than a flow on the state space of A

Exact Solution of the Infinite-Range Quantum Mattis Model

We have solved the quantum version of the Mattis model with infinite-range interactions. A variational approach gives the exact solution for the infinite-range system, in spite of the non-commutative nature of the quantum spin components; this implies that quantum effects are not predominant in determining the macroscopic properties of the system. Nevertheless, the model has a surprisingly rich phase behaviour, exhibiting phase diagrams with tricritical, three-phase and critical end points.Comment: 14 pages, 11 figure