Boundary behavior of diffusion approximations to Markov jump processes

We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.     

Busy period distribution in state-dependent queues

We consider a state-dependent M/M/1 queue in which the arrival rate is a function of the instantaneous unfinished work (work backlog) in the system, and the customer's exponential service time distribution is allowed to depend on the unfinished work in the system at the instant that customer arrived. We obtain asymptotic approximations to both the busy period distributions as well as the residual busy period distribution. Our approximations are valid for systems with a rapid arrival rate and small mean service times.This research was supported in part by NSF Grants DMS-84-06110 and DMS-86-20267 and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences. C. Knessl was partially supported by an I.B.M. Graduate Fellowship.     

A Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

A Ray Approximation to a PdE that Arises in the Study of Tandem Queues

We use singular perturbation methods to analyze a diffusion equation that arose in studying two tandem queues. Denoting by p ( n ₁,  n ₂) the probability that there are n ₁ customers in the first queue and n ₂ customers in the second queue, we obtain the approximation p ( n ₁,  n ₂)∼ɛ² P ( X ,  Y )=ɛ² P (ɛ n ₁, ɛ n ₂), where ɛ is a small parameter. The diffusion approximation P satisfies an elliptic PDE with a nondiagonal diffusion matrix and boundary conditions that involve both normal and tangential derivatives. We analyze the boundary value problem using the ray method of geometrical optics and other singular perturbation techniques. This yields the asymptotic behavior of P ( X ,  Y ) for X and/or Y large.     

A singular perturbation approach to first passage times for Markov jump processes

We introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for Markov jump processes. Our methods are applied directly to the integrai equation for the mean first passage time and do not involve the use of diffusion approximations. An absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a Wiener-Hopf problem in the neighborhood of the boundary. A model of unimolecular dissociation is considered to illustrate our methods.     

A Markov-modulated M/G/1 queue I: Stationary distribution

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.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

Uniform expansion of the transition rate in Kramers' problem

Kramers' model of diffusion over potential barriers, e.g., chemical reactions, based on the noise activated escape of a particle from a potential well, is considered. Kramers derived escape rates valid for intermediate and large damping, and in a separate analysis, for small damping. In the small damping limit, Kramers' intermediate result reduces to the transition state rate which does not agree with the small damping result. A new escape rate is derived that is uniformly valid for all values of the damping coefficient. The new rate reduces to Kramers' results in the appropriate limits and, in particular, connects Kramers' intermediate and small damping results.This work was partially supported by the Air Force Office of Scientific Research under Grant No. AFOSR-83-0086, U.S. Department of Energy under Grant No. DE-AC02-78ERO-4650, and the National Science Foundation under Grant No. MCS-83-00562. One of us (BJM) gratefully acknowledges the support of a John Simon Guggenheim Memorial Foundation Fellowship.