Fundamentals of queueing theory (2nd ed.).
Fundamentals of queueing theory (2nd ed.).
Performance of Computer Communication Systems: A Model-Based Approach
Performance of Computer Communication Systems: A Model-Based Approach
Linear System Theory and Design
Linear System Theory and Design
Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues
Queueing Systems: Theory and Applications
The PH/PH/1 queue at epochs of queue size change
Queueing Systems: Theory and Applications
Modeling and analysis of power-tail distributions via classical teletraffic methods
Queueing Systems: Theory and Applications
Approximation models of feed-forward G/G/1/N queueing networks with correlated arrivals
Performance Evaluation
System-theoretical algorithmic solution to waiting times in semi-Markov queues
Performance Evaluation
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Matrix exponential (ME) distributions not only include the well-known class of phase-type distributions but also can be used to approximate more general distributions (e.g., deterministic, heavy-tailed, etc.). In this paper, a novel mathematical framework and a numerical algorithm are proposed to calculate the matrix exponential representation for the steady-state waiting time in an ME/ME/1 queue. Using state-space algebra, the waiting time calculation problem is shown to reduce to finding the solution of an ordinary differential equation in state-space form with order being the sum of the dimensionalities of the inter-arrival and service time distribution representations. A numerically efficient algorithm with quadratic convergence rates based on the matrix sign function iterations is proposed to find the boundary conditions of the differential equation. The overall algorithm does not involve any transform domain calculations such as root finding or polynomial factorization, which are known to have potential numerical stability problems. Numerical examples are provided to demonstrate the effectiveness of the proposed approach.