Sample Path Large Deviation Principle Research Articles

Sample path moderate deviations for the cumulative fluid produced by an increasing number of exponential on-off sources

We consider the superposition of the cumulative fluid generated by an increasing number of stationary iid on-off sources with exponential iid on- and off-time distributions. We establish a family of sample path large deviation principles when the fluid is centered and then scaled with a factor between the inverse of the number of sources and its square root. The common rate function in this family also appears in a large deviation principle for the tail probabilities of an integrated Ornstein---Uhlenbeck process. When the produced fluid is centered and scaled with the square root of the inverse of the number of sources it converges to this integrated Ornstein---Uhlenbeck process in distribution. We discuss several representations of the rate function. We apply the results to queueing systems loaded with on-off traffic and approaching critical loading.

Minimizing Large Deviation Paths for a Family of Long-Range Dependent Processes and Their Fractional Brownian Approximations

The long-range dependent cumulative traffic processes generated by the superposition of constant rate fluid sources having exponentially distributed inter-start times and Pareto-distributed durations with finite mean and infinite variance satisfy a large deviation principle when the session intensity is increased and the processes are centered and appropriately scaled. We show that close to its zero point the rate function in this large deviation principle behaves approximately like the one appearing in a sample path large deviation principle for fractional Brownian motion. This allows for an approximation of large deviation rates and minimizing paths of the traffic processes by those of fractional Brownian motions. We explicitly calculate certain minimizing paths representing the asymptotically most probable behavior of the traffic processes leading to a fixed positive or negative increase and compare them to the corresponding behavior of fractional Brownian motions. We develop analogous results for the minimizing paths of the queue length processes obtained by sending the traffic through a buffer with infinite capacity and constant service rate. These paths are approximated by minimizing paths of reflected fractional Brownian motion.

Sample path large deviations for multiclass feedforward queueing networks in critical loading

We consider multiclass feedforward queueing networks with first in first out and priority service disciplines at the nodes, and class dependent de terministic routing between nodes. The random behavior of the network is constructed from cumulative arrival and service time processes which are as sumed to satisfy an appropriate sample path large deviation principle. We establish logarithmic asymptotics of large deviations for waiting time, idle time, queue length, departure and sojourn-time processes in critical loading. This transfers similar results from Puhalskii about single class queueing net works with feedback to multiclass feedforward queueing networks, and com plements diffusion approximation results from Peterson. An example with renewal inter arrival and service time processes yields the rate function of a reflected Brownian motion. The model directly captures stationary situations.

Large deviations for risk models in which each main claim induces a delayed claim

In reality insurance claims may be delayed for several reasons and risk models with this feature have been discussed for some years. In this paper we present a sample path large deviation principle for the delayed claims risk model presented in (Yuen K.C., Guo J., and Ng K.W., 2005, On ultimate ruin in a delayed-claims model, Journal of Applied Probability, 42, 163–174). Roughly speaking each main claim induces another type of claim called by-claim; any by-claim occurs later than its main claim and the time of delay is random. Successively we use Gärtner Ellis Theorem to prove a large deviation principle for a more general version of this model, in which the following items depend on the evolution of an irreducible (continuous time) Markov chain with finite state space: the intensity of the Poisson claim number process, the distribution of the claim sizes and the distribution of the random times of delay. Finally we present the Lundberg's estimate for the ruin probabilities; in the fashion of large deviations this estimate provides the exponential decay of the ruin probability as the initial capital goes to infinity.

Sample path large deviations for a family of long-range dependent traffic and associated queue length processes

We consider the long-range dependent cumulative traffic generated by the superposition of constant rate fluid sources having exponentially distributed inter start times and Pareto distributed durations with finite mean and infinite variance. We prove a sample path large deviation principle when the session start time intensity is increased and the processes are centered and scaled appropriately. Properties of the rate function are investigated. We derive a sample path large deviation principle for a related family of stationary queue length processes. The large deviation approximation of the steady-state queue length distribution is compared with the corresponding empirical distribution obtained by a computer simulation.

Large deviations for processes with discontinuous statistics

This paper is devoted to the problem of sample path large deviations for the Markov processes on R_+^N having a constant but different transition mechanism on each boundary set {x:x_i=0 for i\notin\Lambda, x_i>0 for i\in\Lambda}. The global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates. Our results complete the proof of Dupuis and Ellis of the sample path large deviation principle for Markov processes describing a general class of queueing networks.

On the spectrum of Markov semigroups via sample path large deviations

The essential spectral radius of a sub-Markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process. When the family of rescaled processes satisfies sample path large deviation principle, the spectral radius and the essential spectral radius are expressed in terms of the rate function. The paper is motivated by applications to reflected diffusions and jump Markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated.

Sample Path Large Deviations Principles for Poisson Shot Noise Processes and Applications

This paper concerns sample path large deviations for Poisson shot noise processes, and applications in queueing theory. We first show that, under an exponential tail condition, Poisson shot noise processes satisfy a sample path large deviations principle with respect to the topology of pointwise convergence. Under a stronger superexponential tail condition, we extend this result to the topology of uniform convergence. We also give applications of this result to determining the most likely path to overflow in a single server queue, and to finding tail asymptotics for the queue lengths at priority queues.

Sample Path Large Deviations for Diffusion Processes on Configuration Spaces over a Riemannian Manifold

In this paper, we establish a sample path large deviation principle for a class of diffusion processes on configuration spaces over a Riemannian manifold. The rate functional turns out to be the energy of the paths associated to the _L_2-Wasserstein distance.

Large deviations problems for star networks: The min policy

We are interested in analyzing the effect of bandwidth sharing for telecommunication networks. More precisely, we want to calculate which routes are bottlenecks by means of large deviations techniques. The method is illustrated in this paper on a star network, where the bandwidth is shared between customers according to the so-called min policy. We prove a sample path large deviation principle for a rescaled process n^{-1}Q_{nt}, where Q_t represents the joint number of connections at time t. The main result is to compute the rate function explicitly. The major step consists in deriving large deviation bounds for an empirical generator constructed from the join number of customers and arrivals on each route. The rest of the analysis relies on a suitable change of measure together with a localization procedure. An example shows how this can be used practically.

Large deviations for multi-dimensional reflected fractional Brownian motion

By proving the continuity of multi-dimensional Skorokhod maps in a quasi-linearly discounted uniform norm on the doubly infinite time interval R, and strengthening know sample path large deviation principles for fractional Brownian motion to this topology, we obtain large deviation principles for the image of multi-dimensional fractional Brownian motions under Skorokhod maps as an immediate consequence of the contraction principle. As an application, we explicitly calculate large deviation decay rates for steady-state tail probabilities of certain queueing systems in multi-dimensional heavy traffic models driven by fractional Brownian motions.

Analysis of G/D/1 Queueing Systems with Inputs Satisfying Large Deviation Principle under Weak * Topology

The large deviation principle (LDP) which has been effectively used in queueing analysis is the sample path LDP, the LDP in a function space endowed with the uniform topology. Chang [5] has shown that in the discrete-time G/D/1 queueing system under the FIFO discipline, the departure process satisfies the sample path LDP if so does the arrival process. In this paper, we consider arrival processes satisfying the LDP in a space of measures endowed with the weak* topology (Lynch and Sethuraman [12]) which holds under a weaker condition. It is shown that in the queueing system mentioned above, the departure processes still satisfies the sample path LDP. Our result thus covers arrival processes which can be ruled out in the work of Chang [5]. The result is then applied to obtain the exponential decay rate of the queue length probability in an intree network as was obtained by Chang [5], who considered the arrival process satisfying the sample path LDP.

Large Deviations Rate Function for Polling Systems

In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n−1Qnt, where Qt represents the joint number of clients at time t in a polling system with N nodes, one server and Markovian routing. By the way, the large deviation principle is proved and the rate function is shown to have the form conjectured by Dupuis and Ellis. We introduce a so called empirical generator consisting of Qt and of two empirical measures associated with St, the position of the server at time t. One of the main step is to derive large deviations bounds for a localized version of the empirical generator. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program. The method seems to have a wide range of application including the famous Jackson networks, as shown at the end of this study. An example illustrates how this technique can be used to estimate stationary probability decay rate.

Largest Weighted Delay First Scheduling: Large Deviations and Optimality

We consider a single server system with N input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight vector α=(α1,...,αN). We show that under the LWDF discipline the sequence of scaled stationary distributions of the delay \(\hat{w}_{i}\) of each flow satisfies a large deviation principle with the rate function given by a finite- dimensional optimization problem. We also prove that the LWDF discipline is optimal in the sense that it maximizes the quantity $$\min_{i= 1,\space ...,\space N}\left[α_i \lim_{n\to\infty}\frac{−1}{n}\log P(\hat{w}_i>n)\right],$$ within a large class of work conserving disciplines.

Large deviations and concentration properties for ∇ϕ interface models

We consider the massless field with zero boundary conditions outside D N ≡D∩ (ℤ d /N) (N∈ℤ+), D a suitable subset of ℝ d , i.e. the continuous spin Gibbs measure ℙ N on ℝ ℤd/N with Hamiltonian given by H(ϕ) = ∑ x,y:|x−y|=1 V(ϕ(x) −ϕ(y)) and ϕ(x) = 0 for x∈D N C . The interaction V is taken to be strictly convex and with bounded second derivative. This is a standard effective model for a (d + 1)-dimensional interface: ϕ represents the height of the interface over the base D N . Due to the choice of scaling of the base, we scale the height with the same factor by setting ξ N = ϕ/N. We study various concentration and relaxation properties of the family of random surfaces {ξ N } and of the induced family of gradient fields ∇ N ξ N as the discretization step 1/N tends to zero (N→∞). In particular, we prove a large deviation principle for {ξ N } and show that the corresponding rate function is given by ∫ D σ(∇u(x))dx, where σ is the surface tension of the model. This is a multidimensional version of the sample path large deviation principle. We use this result to study the concentration properties of ℙ N under the volume constraint, i.e. the constraint that (1/N d ) ∑ x∈DN ξ N (x) stays in a neighborhood of a fixed volume v > 0, and the hard–wall constraint, i.e. ξ N (x) ≥ 0 for all x. This is therefore a model for a droplet of volume v lying above a hard wall. We prove that under these constraints the field {ξ N of rescaled heights concentrates around the solution of a variational problem involving the surface tension, as it would be predicted by the phenomenological theory of phase boundaries. Our principal result, however, asserts local relaxation properties of the gradient field {∇ N ξ N (·)} to the corresponding extremal Gibbs states. Thus, our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments. The proofs have both probabilistic and analytic aspects. Essential analytic tools are ? p estimates for elliptic equations and the theory of Young measures. On the side of probability tools, a central role is played by the Helffer–Sjöstrand [31] PDE representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of T. Funaki and H. Spohn [25] on the structure of gradient fields.

Sample path large deviations in finer topologies*

In this paper we present sufficient conditions for sample path large deviation principles to be extended to finer topologies. We consider extensions of the uniform topology by Orlicz functional and we consider Lipschitz spaces: the former are concerned with cumulative path behavior while the latter are more sensitive to extremes in local variation. We also consider sample paths indexed by the half line, where the usual projective limit topologies are not strong enough for many applications. We introduce and apply a new technique extending large deviation principles to finer topologies. We show how to apply the results to obtain large deviations for weighted statistics, to improve Schilder's theorem as well as to obtain large deviations in queueing theory

A sample path large deviation principle for [formula omitted]-martingale measure processes

Gaussian White Noise, super-Brownian motion and the diffusion-limit Fleming–Viot process are examples of such infinite-dimensional Markov processes with continuous paths and L 2 -martingale measures we study in this work as regards to their sample path large deviation probabilities and their associated large deviation rate functions in the limit of small perturbations. We present a unified approach based on Girsanov transform techniques. We derive the rate function as a Lagrangian functional and, as an alternative representation, via some generalized derivatives in a `Cameron–Martin space'.

The linear geodesic property is not generally preserved by a FIFO queue

If a FIFO queue is fed by several input streams that jointly satisfy a sample path large deviation principle (LDP) with "linear geodesics," then the cumulative departures (up to a large time) also satisfy the LDP with a rate function which depends in a relatively simple way on the rate function corresponding to the inputs: this was demonstrated in a recent paper by the second author. It suggests the possibility of an iterative scheme which would allow one to determine the large deviation behavior of more complicated networks. To do this, however, one would require that the linear geodesic property be preserved: in this paper we demonstrate that in general it is not preserved. This is true even in the case of a single input stream.

Large deviations and the generalized processor sharing scheduling for a multiple-queue system

We establish asymptotic upper and lower bounds on the asymptotic decay rate of per-session queue length tail distributions for a multiple-queue system where a single constant rate server services the queues using the generalized processor sharing (GPS) scheduling discipline. In the special case where there are only two queues, the upper and lower bounds match, yielding the optimal bound proved in l15r. The dynamics of bandwidth sharing of a multiple-queue GPS system is captured using the notion of partial feasible sets, and the bounds are obtained using the sample-path large deviation principle. The results have implications in call admission control for high-speed communication networks.