Limit laws for the asymmetric inclusion process

The Asymmetric Inclusion Process (ASIP) is a unidirectional lattice-gas flow model which was recently introduced as an exactly solvable 'Bosonic' counterpart of the 'Fermionic' asymmetric exclusion process. An iterative algorithm that allows the computation of the probability generating function (PGF) of the ASIP's steady state exists but practical considerations limit its applicability to small ASIP lattices. Large lattices, on the other hand, have been studied primarily via Monte Carlo simulations and were shown to display a wide spectrum of intriguing statistical phenomena. In this paper we bypass the need for direct computation of the PGF and explore the ASIP's asymptotic statistical behavior. We consider three different limiting regimes: heavy-traffic regime, large-system regime, and balanced-system regime. In each of these regimes we obtain - analytically and in closed form - stochastic limit laws for five key ASIP observables: traversal time, overall load, busy period, first occupied site, and draining time. The results obtained yield a detailed limit-laws perspective of the ASIP, numerical simulations demonstrate the applicability of these laws as useful approximations.