  In this paper, we provide a novel matrix-analytic approach for studying
doubly exponential solutions of randomized load balancing models (also known as
supermarket models) with Markovian arrival processes (MAPs) and phase-type (PH)
service times. We describe the supermarket model as a system of differential
vector equations by means of density dependent jump Markov processes, and
obtain a closed-form solution with a doubly exponential structure to the fixed
point of the system of differential vector equations. Based on this, we show
that the fixed point can be decomposed into the product of two factors
inflecting arrival information and service information, and further find that
the doubly exponential solution to the fixed point is not always unique for
more general supermarket models. Furthermore, we analyze the exponential
convergence of the current location of the supermarket model to its fixed
point, and apply the Kurtz Theorem to study density dependent jump Markov
process given in the supermarket model with MAPs and PH service times, which
leads to the Lipschitz condition under which the fraction measure of the
supermarket model weakly converges the system of differential vector equations.
This paper gains a new understanding of how workload probing can help in load
balancing jobs with non-Poisson arrivals and non-exponential service times.
