TY - JOUR

T1 - Matrix Approach for Analyzing n-Site Generalized ASIP Systems

T2 - PGF and Site Occupancy Probabilities

AU - Yechiali, Uri

AU - Yeger, Yaron

N1 - Publisher Copyright:
© 2022 by the authors.

PY - 2022/12

Y1 - 2022/12

N2 - 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 (Formula presented.) linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of (Formula presented.) equations, and then further reduced to a set of (Formula presented.) equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an (Formula presented.) -site network can be derived from the corresponding PGF of an n-site system. Explicit results for networks with (Formula presented.) and (Formula presented.) sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site (Formula presented.) 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 (Formula presented.), the following interesting property holds: (Formula presented.). 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 (Formula presented.) sites are calculated. Numerical results are presented.

AB - 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 (Formula presented.) linear equations required to obtain the PGF of an n-site network can be first cut by half into a set of (Formula presented.) equations, and then further reduced to a set of (Formula presented.) equations. The latter set can be additionally split into several smaller triangular subsets. It is also shown how the PGF of an (Formula presented.) -site network can be derived from the corresponding PGF of an n-site system. Explicit results for networks with (Formula presented.) and (Formula presented.) sites are obtained. The matrix approach is utilized to explicitly calculate the probability that site (Formula presented.) 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 (Formula presented.), the following interesting property holds: (Formula presented.). 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 (Formula presented.) sites are calculated. Numerical results are presented.

KW - Asymmetric Simple Inclusion Process (ASIP)

KW - Generalized ASIP (G-ASIP)

KW - matrix approach

KW - multidimensional PGF

KW - site occupancies

UR - http://www.scopus.com/inward/record.url?scp=85143487811&partnerID=8YFLogxK

U2 - 10.3390/math10234624

DO - 10.3390/math10234624

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AN - SCOPUS:85143487811

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 23

M1 - 4624

ER -