  Marcel F. Neuts opened a key door in numerical computation of stochastic
models by means of phase-type (PH) distributions and Markovian arrival
processes (MAPs). To celebrate his 75th birthday, this paper reports a more
general framework of Markovian supermarket models, including a system of
differential equations for the fraction measure and a system of nonlinear
equations for the fixed point. To understand this framework heuristically, this
paper gives a detailed analysis for three important supermarket examples: M/G/1
type, GI/M/1 type and multiple choices, explains how to derive the system of
differential equations by means of density-dependent jump Markov processes, and
shows that the fixed point may be simply super-exponential through solving the
system of nonlinear equations. Note that supermarket models are a class of
complicated queueing systems and their analysis can not apply popular queueing
theory, it is necessary in the study of supermarket models to summarize such a
more general framework which enables us to focus on important research issues.
On this line, this paper develops matrix-analytical methods of Markovian
supermarket models. We hope this will be able to open a new avenue in
performance evaluation of supermarket models by means of matrix-analytical
methods.
