We consider a generalized switch model, which includes as special cases the model of multiuser data scheduling over a wireless medium, the input-queued cross-bar switch model and a discrete time version of a parallel server queueing system. Input flows $n=1,\ldots,N$ are served in discrete time by a switch. The switch state follows a finite state, discrete time Markov chain. In each state $m$, the switch chooses a scheduling decision $k$ from a finite set $K(m)$, which has the associated service rate vector $(\mu_1^m(k),\ldots,\mu_N^m(k))$.

We consider a heavy traffic regime, and assume a Resource Pooling (RP) condition. Associated with this condition is a notion of workload $X=\sum_n \zen Q_n$, where $\ze=(\ze_1,\ldots,\ze_N)$ is some fixed nonzero vector with nonnegative components, and $Q_1,\ldots,Q_N$ are the queue lengths. We study the MaxWeight discipline which always chooses a decision $k$ maximizing $\sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(k)$, that is, \[ k \in \mathop{\arg\max}_{i} \sum_n \gamma_n [Q_n]^{\beta} \mu_n^m(i), \] where $\beta>0$, $\gamma_1>0,\ldots,\gamma_N>0$ are arbitrary parameters. We prove that under MaxWeight scheduling and the RP condition, in the heavy traffic limit, the queue length process has the following properties: (a) The vector $(\gamma_1 Q_1^{\beta},\ldots,\gamma_N Q_N^{\beta})$ is always proportional to $\ze$ (this is "State Space Collapse"), (b) the workload process converges to a Reflected Brownian Motion, (c) MaxWeight minimizes the workload among all disciplines. As a corollary of these properties, MaxWeight asymptotically minimizes the holding cost rate \[ \sum_n \gamma_n Q_{n}^{\beta+1} \] at all times, and cumulative cost (with this rate) over finite intervals.

Let $X=\{(X_t, Y_t) \}_{t \in \mathbb{Z}}$ be a stationary time series where $X_t$ is binary valued and $Y_t$, the noisy observation of $X_t$, is real valued. Letting $\mathbf{P}$ denote the probability measure governing the joint process $\{(X_t, Y_t)\}$, we characterize $U(l, \mathbf{P})$, the optimal asymptotic average performance of a predictor allowed to base its prediction for $X_t$ on $Y_1, \ldots, Y_{t-1}$, where performance is evaluated using the loss function $l$. It is shown that the stationarity and ergodicity of $\mathbf{P}$, combined with an additional "conditional mixing" condition, suffice to establish $U(l, \mathbf{P})$ as the fundamental limit for the almost sure asymptotic performance. $U(l, \mathbf{P})$ can thus be thought of as a generalized notion of the Shannon entropy, which can capture the sensitivity of the underlying clean sequence to noise. For the case where $\mathbf{X}=\{ X_t \}$ is governed by $P$ and $Y_t$ given by $Y_t=g(X_t, N_t)$ where $g$ is any deterministic function and $\mathbf{N}=\{ N_t \}$, the noise, is any i.i.d. process independent of $\mathbf{X}$ (namely, the case where the "clean" process $\mathbf{X}$ is passed through a fixed memoryless channel), it is shown that, analogously to the noiseless case, there exist universal predictors which do not depend on $P$ yet attain $U(l, \mathbf{P})$. Furthermore, it is shown that in some special cases of interest [e.g., the binary symmetric channel (BSC) and the absolute loss function], there exist twofold universal predictors which do not depend on the noise distribution either. The existence of such universal predictors is established by means of an explicit construction which builds on recent advances in the theory of prediction of individual sequences in the presence of noise.

The behavior of a connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the $q$-calculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model.

We present some convergence results about the distortion $\mathcal{D}_{\mu,n,r}^{\nu}$ related to the Voronoï vector quantization of a $\mu$-distributed random variable using $n$ i.i.d. $\nu$-distributed codes. A weak law of large numbers for $n^{r/d}\mathcal{D}_{\mu,n,r}^{\nu}$ is derived essentially under a $\mu$-integrability condition on a negative power of a $\delta$-lower Radon--Nikodym derivative of $\nu$. Assuming in addition that the probability measure $\mu$ has a bounded $\varepsilon$-potential, we obtain a strong law of large numbers for $n^{r/d} \mathcal{D}_{\mu,n,r}^{\nu}$. In particular, we show that the random distortion and the optimal distortion vanish almost surely at the same rate. In the one-dimensional setting ($d=1$), we derive a central limit theorem for $n^{r}\mathcal{D}_{\mu,n,r}^{\nu}$. The related limiting variance is explicitly computed.

We study the stability of the optimal filter w.r.t. its initial condition and w.r.t. the model for the hidden state and the observations in a general hidden Markov model, using the Hilbert projective metric. These stability results are then used to prove, under some mixing assumption, the uniform convergence to the optimal filter of several particle filters, such as the interacting particle filter and some other original particle filters.

Bolker and Pacala recently introduced a model of an evolving population in which an individual's fecundity is reduced in proportion to the "local population density." We consider two versions of this model and prove complementary extinction/persistence results, one for each version. Roughly, if individuals in the population disperse sufficiently quickly relative to the range of the interaction induced by the density dependent regulation, then the population has positive chance of survival, whereas, if they do not, then the population will die out.

We consider spectrally negative Lévy process and determine the joint Laplace transform of the exit time and exit position from an interval containing the origin of the process reflected in its supremum. In the literature of fluid models, this stopping time can be identified as the time to buffer-overflow. The Laplace transform is determined in terms of the scale functions that appear in the two-sided exit problem of the given Lévy process. The obtained results together with existing results on two sided exit problems are applied to solving optimal stopping problems associated with the pricing of Russian options and their Canadized versions.

Choose $n$ independent random points on the boundary of a convex body $K \subset \R^d$. The intersection of the supporting halfspaces at these random points is a random convex polyhedron. The expectations of its volume, its surface area and its mean width are investigated. In the case that the boundary of $K$ is sufficiently smooth, asymptotic expansions as $n \to \infty$ are derived even in the case when the curvature is allowed to be zero. We compare our results to the analogous results for best approximating polytopes.

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall and Sinclair to generate random tilings of regions by lozenges. For an $\ell\times\ell$ region we bound the mixing time by $O(\ell^4\log\ell)$, which improves on the previous bound of $O(\ell^7)$, and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an $O(n^3\log n)$ upper bound on the mixing time of the Karzanov--Khachiyan Markov chain for linear extensions.

This article proves that the stability region of a two-station, five-class reentrant queueing network, operating under a nonpreemptive static buffer priority service policy, depends on the distributions of the interarrival and service times. In particular, our result shows that conditions on the mean interarrival and service times are not enough to determine the stability of a queueing network under a particular policy. We prove that when all distributions are exponential, the network is unstable in the sense that, with probability 1, the total number of jobs in the network goes to infinity with time. We show that the same network with all interarrival and service times being deterministic is stable. When all distributions are uniform with a given range, our simulation studies show that the stability of the network depends on the width of the uniform distribution. Finally, we show that the same network, with deterministic interarrival and service times, is unstable when it is operated under the preemptive version of the static buffer priority service policy. Thus, our examples also demonstrate that the stability region depends on the preemption mechanism used.

Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common $\ell_2$ metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or $m$-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.

Diffusion models arising in analysis of large biochemical models and other complex systems are typically far too complex for exact solution or even meaningful simulation. The purpose of this paper is to develop foundations for model reduction and new modeling techniques for diffusion models.

These foundations are all based upon the recent spectral theory of Markov processes. The main assumption imposed is $V$-uniform ergodicity of the process. This is equivalent to any common formulation of exponential ergodicity and is known to be far weaker than the Donsker--Varadahn conditions in large deviations theory. Under this assumption it is shown that the associated semigroup admits a spectral gap in a weighted $L_\infty$-norm and real eigenfunctions provide a decomposition of the state space into "almost"-absorbing subsets. It is shown that the process mixes rapidly in each of these subsets prior to exiting and that the conditional distributions of exit times are approximately exponential.

These results represent a significant expansion of the classical Wentzell--Freidlin theory. In particular, the results require no special structure beyond geometric ergodicity; reversibility is not assumed and meaningful conclusions can be drawn even for models with significant variability.

In this paper we propose a numerical scheme for a class of backward stochastic differential equations (BSDEs) with possible path-dependent terminal values. We prove that our scheme converges in the strong $L^2$ sense and derive its rate of convergence. As an intermediate step we prove an $L^2$-type regularity of the solution to such BSDEs. Such a notion of regularity, which can be thought of as the modulus of continuity of the paths in an $L^2$ sense, is new. Some other features of our scheme include the following: (i) both components of the solution are approximated by step processes (i.e., piecewise constant processes); (ii) the regularity requirements on the coefficients are practically "minimum"; (iii) the dimension of the integrals involved in the approximation is independent of the partition size.

Stochastic evolutionary systems of additive functional type, described by processes with locally independent increments, are considered with Markov switching in an asymptotic split state space having a stoppage state. The average and diffusion approximation limit theorems are established in both single and double merging. The proofs of these results are obtained using a singular perturbation approach of linear reducible--invertible operators and the tightness of processes. Particular cases of these systems including integral functionals, dynamic systems, storage processes and compound Poisson processes are also considered. The application of limit theorems in reliability and reward problems is discussed.