Matrix Exponential Distributions Research Articles

An Operational Calculus for Matrix-Exponential Distributions, with Applications to a Brownian (q, Q) Inventory Model

A distribution G on (0, ∞) is called matrix-exponential if the density has the form αeTz s where α is a row vector, T a square matrix and s a column vector. Equivalently, the Laplace transform is rational. For such distributions, we develop an operator calculus, where the key step is manipulation of analytic functions f(z) extended to matrix arguments. The technique is illustrated via an inventory model moving according to a reflected Brownian motion with negative drift, such that an order of size Q is placed when the stock process down-crosses some level q. Explicit formulas for the stationary density are found under the assumption that the leadtime Z has a matrix-exponential distribution, and involve expressions of the form f(T) where f(z) = √1 − 2z.

Representations for matrix-geometric and matrix-exponential steady-state distributions with applications to many-server queues

Neuts has shown that a Markov chain of GI/M/1 type has a matrix-geometric steady-state distribution, extending earlier work by Evans [13] and Wallace [31]. Various authors later observed that, under this steady-state distribution, the “level” component is phase type. We give a probabilistic interpretation of this result by constructing, from a sample path of the original process, a Markov chain with the correct distribution of time to absorption. We give a similar construction for the matrix-exponential steady-state distribution of Sengupta's continuous-level chain [27]. We extend Sengupta's analysis of the GI/PH/1 queue to the many-server case and thereby prove that the steady-state waiting time distribution in a GI/PH/c queue is phase type. This leads to a proof of a recent conjecture of Abate, Choudhury and Whitt [1] and gives an alternative approach to Neuts and Takahashi's [20] results on the exponential decay parameter of the waiting-time distribution

Delay analysis of a cellular mobile priority queueing system

A nonpreemptive priority queueing system incorporating a channel reservation policy and a hysteresis mechanism is considered for use in cellular mobile networks. The aim is to provide for priority for handover attempts over new call. Attempts in such a way as to avoid the forced termination of calls in progress without unduly affecting the performance seen by new call attempts. The system model is found to have a matrix-geometric solution with a phase-type distribution for the handover delay and a matrix-exponential distribution for the delay seen by new call attempts. It is also shown that the system can be represented by an M/G/1 queue with multiple vacations. A new result is derived for the M/G/1 queue with multiple vacations and impatient customers and this allows for the analysis to be extended so that new call arrivals are subject to a fixed timeout in queue. Results are provided which show how nonpreemptive priority in conjunction with channel reservation and hysteresis an provide for fast and efficient handover performance for cellular mobile networks. >

Performance analysis of a digital link with heterogeneous multislot traffic

Presents a unified model to compute various performance measures when Poissonian and non-Poissonian (renewal) multi-slot traffic streams are offered to a digital link in a (double) loss system. the authors represent the non-Poissonian arrival process by a matrix-exponential distribution, requiring only that the inter-arrival distribution has a rational Laplace transform. Several distributions are considered as the non-Poissonian traffic. The resulting model uses matrix algebraic techniques only, thus not requiring any complex and/or tedious transform techniques. The authors also incorporate various control policies in their modeling framework using acceptance functions. Through computational studies, they conclude that the second and the third moments of the non-Poissonian traffic have significant impact on various performance measures. >

The semi-markovian queue: theory and applications

In this paper, we study a first-come-first-served single server semi-Markovian queue in which both the arrival and service mechanisms are semi-Markov processes. The interarrival time and service times may depend on one another and the marginal distribution of the service times is assumed to be phase-type. For this queue, we show that the distributions of waiting time, time in system and virtual waiting time are matrix-exponential. Further, these matrix-exponential distributions have phase-type representations. For the special case when the interarrival times are independent of the service times, we show that the queue length distribution is matrix-geometric. For this special case, we prove that the queue length distribution problem is the dual of the waiting time distribution problem, i.e., finding the solution of one problem immediately gives the solution of the other. We show that our methods are computationally feasible and report our numerical experience. We give Examples where such queues arise naturally. In particular, we discuss an application in manufacturing, a periodic queue and a queue with Markov modulated arrivals and services.