Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

Abstract

This paper is concerned with a bivariate Markov process {Xt, Nt ; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt ; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Tackling Latency via Replication in Distributed Systems

“…To determine the waiting-time distribution, we observe the queues only during the all-busy periods and define an age process, following [2,19]. Different from [2,19], which consider queues with service times independent of the system state, the replication and canceling mechanism introduces dependencies among the servers that cannot be analyzed by existing models.…”

“…Different from [2,19], which consider queues with service times independent of the system state, the replication and canceling mechanism introduces dependencies among the servers that cannot be analyzed by existing models. We thus define a bivariate Markov process {X(t), J(t)|t≥0}, where the age X(t) is the total time-insystem of the youngest job in service at time t. The age X(t) thus takes values in [0, ∞), increasing linearly with rate 1 as long as no new jobs start service.…”

“…To determine the PH representation (swait, Swait) of the waiting-time distribution, we rely on the stationary distribution π(x) of the (X(t), J(t)) process, which has a matrix exponential representation [19] π(x)=π(0) exp(T x), for x>0. The m×m matrix T satisfies the non-linear integral equation…”

Sojourn times in fluid queues with independent and dependent input and output processes

“…A lot of results are available for such systems in case of discrete jobs: elegant and numerically efficient procedures to obtain the matric-geometric distribution of the queue length [3] and the matrix-exponential distribution of the sojourn time [4]. The introduction of the age process based analysis of discrete queues [5,6,7] made it possible to obtain a lower order matrix-exponential representation of the sojourn time distribution if the arrival and service processes are independent.…”

The BMAP/G/1 queue: A tutorial

“…Sengupta [63] showed that the functional equation arising in the solution to the GI/PH/I queue could be written in a matrix exponential form. It was immediately clear that a similar situation occurred in the PH/G/1 queue and, consequently, also for the BMAP/G/1 queue for the matrix G. Neuts [64] proved the result for the MMPP/G/1 queue and the result for the BMAP/G/1 queue was derived simultaneously by Lucantoni [7] and Ramaswami [65].…”