Charles Tier

AN ASYMPTOTIC THEORY OF LARGE DEVIATIONS FOR MARKOV JUMP PROCESSES

We present new asymptotic methods for the analysis of Markov jump processes. The methods, based on the WKB and other singular perturbation techniques, are applied directly to the Kolmogorov equations and not to approximate equations that come e.g. from diffusion approximations. For time homogeneous processes, we construct approximations to the stationary density function and the mean first passage time from a given domain. Examples involving a random walk and a problem in queueing theory are presented to illustrate our methods. For a class of time inhomogeneous processes, we construct long time approximations to the transition probability density function and the probability of large deviations from a stable state. The law of large numbers is obtained as a special case.

Persistence in density dependent stochastic populations

Abstract Persistence, as measured by time to extinction, is studied in a density dependent population that is subject to small environmental and demographic randomness. Diffusion processes are formally derived from branching processes in constant and random environments. The moment generating function of the extinction time, which satisfies a second order ordinary differential equation, is found asymptotically in the limit of small diffusion and is related to the diffusion limit of the Galton-Watson process and the Ornstein-Uhlenbeck process. Extinction occurs with probability one, though the mean and variance of the extinction time are found to be exponentially large and suggest the extinction time is exponentially distributed. The notion of persistence is compared with other qualitative measures of stability. Four examples are studied and compared.

A subjective Bayesian approach to the theory of queues II—Inference and information in M/M/1 queues

This is a sequel to Part I of “A Subjective Bayesian Approach to the Theory of Queues”. The focus here is on inference and a use of Shannons measure of information for assessing the amount of information conveyed by the various types of data from queues. The notation and terminology used here is established in Part I.We consider an M/G/1 queueing system in which the arrival rate and service time density are functions of a two-state stochastic process. We describe the system by the total unfinished work present and allow the arrival and service rate processes to depend on the current value of the unfinished work. We employ singular perturbation methods to compute asymptotic approximations to the stationary distribution of unfinished work and in particular, compute the stationary probability of an empty queue.

Solution of Kramers–Moyal equations for problems in chemical physics

We derive asymptotic solutions of Kramers–Moyal equations (KMEs) that arise from master equations (MEs) for stochastic processes. We consider both one step processes, in which the system jumps from x to x+e or x−e with given probabilities, and general transitions, in which the system moves from x to x+eξ, where ξ is a random variable with a given probability distribution. Our method exploits the smallness of a parameter e, typically the ratio of the jump size to the system size. We employ the full KME to derive asymptotic expansions for the stationary density of fluctuations, as well as for the mean lifetime of stable equilibria. Thus we treat fluctuations of arbitrary size, including large fluctuations. In addition we present a criterion for the validity of diffusion approximations to master equations. We show that diffusion theory can not always be used to study large deviations. When diffusion theory is valid our results reduce to those of diffusion theory. Examples from macroscopic chemical kinetics a...

Two Parallel Queues with Dynamic Routing

We consider two parallel M/M/1 queueing systems where a new arrival (customer, job, message) joins the shorter of the two queues. Such problems arise naturally in computer communications and packet switched data networks. An asymptotic approach is developed to obtain approximations to the steady-state joint distribution of the number of customers in the two systems. We first analyze the case where the two queueing systems are identical and then consider the case when the two servers work at different rates. Our results are shown to agree with the expansions of known exact solutions, when such solutions are available, and to yield new approximations when such solutions are not available.

A QUEUEING SYSTEM WITH QUEUE LENGTH DEPENDENT SERVICE TIMES, WITH APPLICATIONS TO CELL DISCARDING IN ATM NETWORKS

A queueing system (M/G1,G2/1/K) is considered in which the service time of a customer entering service depends on whether the queue length, N(t), is above or below a threshold L. The arrival process is Poisson, and the general service times S1 and S2 depend on whether the queue length at the time service is initiated is <L or ≥L, respectively. Balance equations are given for the stationary probabilities of the Markov process (N(t),X(t)), where X(t) is the remaining service time of the customer currently in service. Exact solutions for the stationary probabilities are constructed for both infinite and finite capacity systems. Asymptotic approximations of the solutions are given, which yield simple formulas for performance measures such as loss rates and tail probabilities. The numerical accuracy of the asymptotic results is tested.

Asymptotic expansion for large closed queuing networks

In this paper, a new asymptotic method is developed for analyzing closed BCMP queuing networks with a single class (chain) consisting of a large number of customers, a single infinite server queue, and a large number of single server queues with fixed (state-independent) service rates. Asymptotic approximations are computed for the normalization constant (partition function) starting directly from a recursion relation of Buzen. The approach of the authors employs the ray method of geometrical optics and the method of matched asymptotic expansions. The method is applicable when the servers have nearly equal relative utilizations or can be divided into classes with nearly equal relative utilizations. Numerical comparisons are given that illustrate the accuracy of the asymptotic approximations.

Two Parallel M/G/1 Queues where Arrivals Join the System with the Smaller Buffer Content

We consider two parallel, infinite capacity, M/G/1 queues characterized by ( U_{1}(t), U_{2}(t) ) with U_{j}(t) denoting the unfinished work (buffer content) in queue j . A new arrival is assigned to the queue with the smaller buffer content. We construct formal (as opposed to rigorous) asymptotic approximations to the Joint stationary distribution of the Markov process ( U_{1}(t), U_{2}(t) ), treating separately the asymptotic limits of heavy traffic, light traffic, and large buffer contents. In heavy traffic, the stochastic processes U_{1}(t) + U_{2}(t) and U_{2}(t) - U_{1}(t) become independent, with the distribution of U_{1}(t) + U_{2}(t) identical to the heavy traffic waiting time distribution in the standard M/G/2 queue, and the distribution of U_{2}(t) - U_{1}(t) closely related to the tail of the service time density. In light traffic, we obtain a formal expansion of the stationary distribution in powers of the arrival rate.

Asymptotic expansions for large closed queueing networks with multiple job classes

A closed BCMP queuing network consisting of R job classes (chains), K+1 single-server, fixed-rate nodes, and M/sub j/ class j jobs (j=1, 2, . . ., R) is considered. Asymptotic expansions are constructed for the partition function under assumptions (1) K>>1, (2) M/sub j/>>1 for each j, and (3) K/M/sub j/=O(1). Analytic expressions for performance measures such as the mean queue length are also given. The approach employs the ray method and the method of matched asymptotic expansions. Numerical comparisons illustrate the accuracy of the approximations. >

An Asymptotic Solution of the First Passage Problem for Singular Diffusion in Population Biology

The first passage problem is studied for a singular diffusion process arising in population biology with the deterministic part having a stable equilibrium point and small diffusion. The Laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions. A formal asymptotic solution to this problem is constructed when the singular boundary point is absorbing and the regular boundary point is either absorbing or reflecting. The solution depends upon the initial value z of the process. For z near the equilibrium point the density is approximated by the Ornstein–Uhlenbeck equation, for z near the singular boundary point it is approximated by the diffusion limit of the linear birth-death process, and elsewhere it is approximated by the WKB solution. These different solutions are then connected together. Using these results, the process is shown to exit with probability one in the limit of small diffusion. The mean and varianc...