Steady-state Probability Vector Research Articles

The M/M/1 Queue with Randomly Varying Arrival and Service Rates: A Phase Substitution Solution

This paper presents an alternative procedure for computing the steady state probability vector of an M/M/1 queue with randomly varying arrival and service rates. By exploiting the structure of the infinitesimal generator of the underlying continuous-time Markov chain, the approach represents an efficient adaptation of the state reduction method introduced by Grassmann for solving problems involving M/M/1 queues under a random environment. We compare computational requirements of the proposed approach with the method of Neuts and block elimination under different rush-hour congestion patterns while keeping the overall traffic intensity constant as well as under different traffic intensities. We demonstrate that the proposed method requires minimal computing time to reach convergence and moreover the time requirement does not change much when traffic intensity increases.

Two units connected in series with general bulk service and random breakdown in unit 2

We study a queueing model consisting of two units 1 and 2 connected in series with a finite intermediate waiting room. The customers in the buffer are served according to a general bulk service rule with exponential times. Unit 2 is in the up state for a random interval of time following an exponential distribution with parameter α and the distribution of time spent in the down state is exponential with parameter β. Here we obtain the steady state probability vector for the number of customers in the queue.

A stable recursion for the steady state vector in markov chains of m/g/1 type

For the matrix analogues of Markov chains of the M/G/1 type, we derive a stable recursive scheme to compute the steady state probability vector. This scheme, which is the natural generalization of a clever device attributed to P.J. Burke in the M/G/1 case, is substantially superior to the Gauss-Seidel iterative scheme.

Technical Note—A Markov Chain Partitioning Algorithm for Computing Steady State Probabilities

This paper presents a partitioning algorithm for recursively computing the steady state probabilities for a finite, irreducible Markov chain or a Markov process. The algorithm contains a matrix reduction routine, followed by a vector enlargement routine. The matrix reduction routine repeatedly partitions the transition matrix for the Markov chain, creating a sequence of smaller, reduced transition matrices. The vector enlargement routine computes the components of the steady state probability vector by starting with the smallest reduced matrix and working sequentially toward the original transition matrix. This procedure produces an exact solution for the steady state probabilities. No special structure is required for the Markov chain. In theory, the procedure imposes no limit on the size of the largest Markov chain to which the partitioning algorithm can be applied. In practice, roundoff errors may require modifications to the method.

An algorithmic approach for a special class of Markov chains

In this paper, we develop an algorithmic method for the evaluation of the steady state probability vector of a special class of finite state Markov chains. For the class of Markov chains considered here, it is assumed that the matrix associated with the set of linear equations for the steady state probabilities possess a special structure, such that it can be rearranged and decomposed as a sum of two matrices, one lower triangular with nonzero diagonal elements, and the other an upper triangular matrix with only very few nonzero columns. Almost all Markov chain models of queueing systems with finite source and/or finite capacity and first-come-first-served or head of the line nonpreemptive priority service discipline belongs to this special class.

Explicit steady state solution for a Markovian inventory system*

Abstract The explicit steady state probability vector for the inventory level is presented for a Markovian system.

Efficient Algorithmic Solutions to Exponential Tandem Queues with Blocking

Stable queuing systems consisting of two groups of servers, having exponential service times, placed in tandem and separated by a finite buffer, are shown to have a steady-state probability vector of matrix-geometric form. The queue is stable as long as the Poisson arrival rate does not exceed a critical value, which depends in a complicated manner on the service rates, the numbers of servers in each group, the size of the intermediate buffer and the unblocking rule followed when system becomes blocked. The critical input rate is determined in a unified manner. For stable queues, it is shown how the stationary probability vector and other important features of the queue may be computed. The essential step in the algorithm is the evaluation of the unique positive solution of a quadratic matrix equation.