The matrix M/M/∞ system: retrial models and Markov Modulated sources

Abstract

The matrix-geometric work of Neuts could be viewed as a matrix variant of M/M/ 1. A 2 × 2 matrix counterpart of Neuts for M/M/ ∞ is introduced, the stability conditions are identified, and the ergodic solution is solved analytically in terms of the ten parameters that define it. For several cases of interest, system properties can be found from simple analytical expressions or after easy numerical evaluation of Kummer functions. When the matrix of service rates is singular, a qualitatively different solution is derived. Applications to telecommunications include some retrial models and an M/M/ ∞ queue with Markov-modulated input.