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This paper describes the heavy-traffic behavior of an M/G/1 last-in-first-out preemptive resume queue. An appropriate framework for the analysis is provided by measure-valued processes. In particular, the paper exploits the setting of recent works by Le Gall and Le Jan.Their finite-measure-valued exploration process corresponds to our RES-measure (residual services measure) process, that captures all the relevant information about the evolution of the queue, while their height process corresponds to the queue-length process. The heavy-traffic “diffusion” approximations for the RES-measure and the queue-length processes are derived under the usual second moment assumptions on the service distributions. The tightness of queue lengths argument uses estimates for the total size and height of large Galton–Watson trees.
This paper introduces a new aspect of queueing theory, the study of systems that service customers with specific timing requirements (e.g., due dates or deadlines). Unlike standard queueing theory in which common performance measures are customer delay, queue length and server utilization, real-time queueing theory focuses on the ability of a queue discipline to meet customer timing requirements, for example, the fraction of customers who meet their requirements and the distribution of customer lateness. It also focuses on queue control policies to reduce or minimize lateness, although these control aspects are not explicitly addressed in this paper. To study these measures, we must keep track of the lead times (dead-line minus current time) of each customer; hence, the system state is of unbounded dimension. A heavy traffic analysis is presented for the earliest-deadline-first scheduling policy. This analysis decomposes the behavior of the real-time queue into two parts: the number in the system (which converges weakly to a re flected Brownian motion with drift) and the set of lead times given the queue length. The lead-time profile has a limit that is a nonrandom function of the limit of the scaled queue length process. Hence, in heavy traffic, the system can be characterized as a diffusion evolving on a one-dimensional manifold of lead-time profiles. Simulation results are presented that indicate that this characterization is surprisingly accurate. A discussion of open research questions is also presented.
This paper presents a large deviations principle for the average of real-valued processes indexed by the positive integers, one which is particularly suited to queueing systems with many traffic flows. Examples are given of how it may be applied to standard queues with finite and infinite buffers, to priority queues and to finding most likely paths to overflow.
We prove, using results for hydrodynamic limits,that an exclusion process starting from an ergodic initial distribution converges to product measure in one dimension. Our only assumption is the existence of a nonzero mean for the underlying random walk.
We consider stochastic analogs of classical billiard systems. A particle moves at unit speed with constant direction in the interior of a bounded, d-dimensional region with continuously differentiable boundary. The boundary need not be connected; that is, the “table” may have inte- rior “obstacles.” When the particle strikes the boundary, a new direction is chosen uniformly at random from the directions that point back into the interior of the region and the motion continues. Such chains are closely related to those that appear in shake-and-bake simulation algorithms. For the discrete time Markov chain that records the locations of successive hits on the boundary, we show that, uniformly in the starting point, there is exponentially fast total variation convergence to an invariant distribution. By analyzing an associated nonlinear, first-order PDE, we investigate which regions are such that this chain is reversible with respect to surface measure on the boundary. We also establish a result on uniform total variation Césaro convergence to equilibrium for the continuous time Markov process that tracks the position and direction of the particle. A key ingredient in our proof is a result on the geometry of $C^1$ regions that can be described loosely as follows:associated with any bounded $C^1$ region is an integer N such that it is always possible to pass a message between any two locations in the region using a relay of exactly N locations with the property that every location in the relay is directly visible from its predecessor. Moreover, the locations of the intermediaries can be chosen from a fixed, finite subset of positions on the boundary of the region. We also consider corresponding results for polygonal regions in the plane.
We develop an algorithm for simulating “perfect” random samples from the invariant measure of a Harris recurrent Markov chain.The method uses backward coupling of embedded regeneration times and works most effectively for stochastically monotone chains, where paths may be sandwiched between “upper” and “lower” processes. We give an approach to finding analytic bounds on the backward coupling times in the stochastically monotone case. An application to storage models is given.
It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a first-order asymptotic expansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.
We consider all two-times iterated Itô integrals obtained by pairing m independent standard Brownian motions. First we calculate the conditional joint characteristic function of these integrals, given the Brownian increments over the integration interval, and show that it has a form entirely similar to what is obtained in the univariate case. Then we propose an algorithm for the simultaneous simulation of the $m^2$ integrals conditioned on the Brownian increments that achieves a mean square error of order $1/n^2$, where n is the number of terms in a truncated sum. The algorithm is based on approximation of the tail-sum distribution, which is a multivariate normal variance mixture, by a multivariate normal distribution.
We study a specific particle system in which particles undergo random branchingand spatial motion. Such systems are best described, mathematically, via measure valued stochastic processes. As is now quite standard, we study the so-called superprocess limit of such a system as both the number of particles in the system and the branchingrate tend to infinity. What differentiates our system from the classical superprocess case, in which the particles move independently of each other, is that the motions of our particles are affected by the presence of a global stochastic flow. We establish weak convergence to the solution of a well-posed martingale problem. Usingthe particle picture formulation of the flow superprocess, we study some of its properties. We give formulas for its first two moments and consider two macroscopic quantities describing its average behavior, properties that have been studied in some detail previously in the pure flow situation, where branching was absent. Explicit formulas for these quantities are given and graphs are presented for a specific example of a linear flow of Ornstein–Uhlenbeck type.
Occupation time functionals for a diffusion process or a birth-and-death process on the edges of a graph $\Gamma$ depending only on the values of the process on a part $\Gamma' \subset \Gamma$ of $\Gamma$ are closely related to so-called eigenvalue depending boundary conditions for the resolvent of the process. Under the assumption that the connected components of $\Gamma\backslash\Gamma'$ are trees, we use the special structure of these boundary conditions to give a procedure that replaces each of the trees by only one edge and that associates this edge with a speed measure such that the respective functional for the appearing process on the simplified graph coincides with the original one.