Performance measures are studied for a generalized n-site asymmetric simple inclusion process (G-ASIP), where a general process controls intervals between gate-opening instants. General formulae are obtained for the Laplace–Stieltjes transform, as well as the means, of the (i) traversal time, (ii) busy period, and (iii) draining time. The PGF and mean of (iv) the system’s overall load are calculated, as well as the probability of an empty system, along with (v) the probability that the first occupied site is site k (k = 1, 2, …, n). Explicit results are derived for the wide family of gamma-distributed gate inter-opening intervals (which span the range between the exponential and the deterministic probability distributions), as well as for the uniform distribution. It is further shown that a homogeneous system, where at gate-opening instants gate j opens with probability $p_j=\frac{1}{n}$, is optimal with regard to (i) minimizing mean traversal time, (ii) minimizing the system’s load, (iii) maximizing the probability of an empty system, (iv) minimizing the mean draining time, and (v) minimizing the load variance. Furthermore, results for these performance measures are derived for a homogeneous G-ASIP in the asymptotic cases of (i) heavy traffic, (ii) large systems, and (iii) balanced systems.

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 $\left(\begin{array}{c}2n\\ n\end{array}\right)$ linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of $\left(\begin{array}{c}2n-1\\ n\end{array}\right)$ equations, and then further reduced to a set of $2^n-\left(n+1\right)$ equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an $\left(n+1\right)$-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 $k \left(k=1,2,\dots ,n\right)$ 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 $\lambda$, the following interesting property holds: $P\left(sitekisoccupied|\lambda =1\right)=P\left(sitek+1isoccupied|\lambda \to \infty \right)$. 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.

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 $$j \ge i$$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.

1. In many serial production environments, a product's quality deteriorates during its traversal time through the network. This phenomenon is common in industries such as fresh food (Bortolini et a...