 
Summary: Performance Evaluation xxx (2005) xxxxxx
Solving the ME/ME/1 queue with statespace methods
and the matrix sign function
Nail Akar
Electrical and Electronics Engineering Department, Bilkent University, Bilkent, 06800 Ankara, Turkey
Received 18 September 2003; received in revised form 12 December 2004
Abstract
Matrix exponential (ME) distributions not only include the wellknown class of phasetype distributions but also
can be used to approximate more general distributions (e.g., deterministic, heavytailed, etc.). In this paper, a novel
mathematical framework and a numerical algorithm are proposed to calculate the matrix exponential representation
for the steadystate waiting time in an ME/ME/1 queue. Using statespace algebra, the waiting time calculation
problem is shown to reduce to finding the solution of an ordinary differential equation in statespace form with order
being the sum of the dimensionalities of the interarrival 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.
© 2004 Elsevier B.V. All rights reserved.
Keywords: GI/GI/1 queue; Lindley's equation; Matrix exponential distribution; Realization theory; Matrix sign function
1. Introduction
