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