Matrix Approach for Analyzing n -Site Generalized ASIP Systems: PGF and Site Occupancy Probabilities

The Asymmetric Simple Inclusion Process (ASIP) is an n -site tandem stochastic network with a Poisson arrival influx into the first site. Each site has an unlimited buffer with a gate in front of it. Each gate opens, independently of all other gates, following a site-dependent Exponential inter-opening time. When a site’s gate opens, all particles occupying the site move simultaneously to the next site. In this paper, a Generalized ASIP network is analyzed where the influx is to all sites, while gate openings are determined by a general renewal process. A compact matrix approach—instead of the conventional (and tedious) successive substitution method—is constructed for the derivation of the multidimensional probability-generating function (PGF) of the site occupancies. It is shown that the set of ( 2 n n ) linear equations required to obtain the PGF of an n -site network can be first cut by half into a set of 2 n − 1 n equations, and then further reduced to a set of 2 n − n + 1 equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an n + 1 -site network can be derived from the corresponding PGF of an n -site system. Explicit results for networks with n = 3 and n = 4 sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site k k = 1 , 2 , … , n is occupied. We show that, in the case where arrivals occur to the first site only, these probabilities are functions of both the site’s index and the arrival flux and not solely of the site’s index. Consequently, refined formulas for the latter probabilities and for the mean conditional site occupancies are derived. We further show that in the case where the arrival process to the first site is Poisson with rate λ , the following interesting property holds: P s i t e k i s o c c u p i e d | λ = 1 = P s i t e k + 1 i s o c c u p i e d | λ → ∞ . The case where the inter-gate opening intervals are Gamma distributed is investigated and explicit formulas are obtained. Mean site occupancy and mean total load of the first k sites are calculated. Numerical results are presented.