An efficient computational method for solving nonlinear matrix equation and its application in queuing analysis

Abstract

The matrix analytic analysis of queues with complex arrival, vacation and service characteristics requires the solution of nonlinear matrix equation. The complexity and large dimensionality of the model require an efficient and smart algorithm for the solution. In this paper, we propose an efficient Adaptive Newton-Kantorovich (ANK) method for speeding up the algorithm solving the nonlinear matrix equation which is an inevitable step in the analysis of the queue with embedded Markov chain such as BMAP/SMSP/1/∞ queue or its discrete version. BMAP/SMSP/1/∞ is a queuing model with a Semi Markov Service time Process (SMSP) and a Batch Markovian Arrival Process (BMAP). The numerical result is presented for the discrete case of N-MMBP/D/1 queue which arises in analyzing traffic aspect of computer communication network, where MMBP is Markov Modulated Bermoulli Process. The comparisons of Adaptive Newton-Kantorovich (ANK) with Modified Newton-Kantorovich (MNK) show that ANK saves 30% of CPU time when the number of userN is 50.