A generalized ASIP with arrivals to all sites and particle movements in all directions

A generalized n-site Asymmetric Simple Inclusion Process (ASIP) network is studied, where gate-opening instants are determined by a renewal process and arrivals occur to all sites. Various types of batch particle movements between sites are analyzed: (i) unidirectional probabilistic forward movements; (ii) probabilistic forward movements combined with feedback to the first site; and (iii) general probabilistic multidirectional movements. In contrast to the tedious successive substitution method used in previous ASIP studies, an efficient matrix approach is applied to derive the multidimensional probability generating function (PGF) of site occupancies right after gate opening instants. The complexity of the ASIP processes allows us to obtain explicit PGF results for small-size networks only, while for larger networks, a formula to calculate the mean site occupancies is derived for all types of movements. In movement case (i) the means are directly and explicitly calculated. For movement case (ii), where the network is homogeneous with equal probabilities of forward movements from site i to downstream sites j≥i, we show that the ratio between the mean occupancies of consecutive sites approaches a constant when the network becomes large, and calculate this ratio. Finally, we investigate an n-site network where at gate opening instants all gates open simultaneously, and particles move in all directions. Numerical examples are presented.