TY - JOUR

T1 - Performance Measures in a Generalized Asymmetric Simple Inclusion Process

AU - Yeger, Yaron

AU - Yechiali, Uri

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/2/1

Y1 - 2022/2/1

N2 - 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 pj =1, is optimal with regard to (i) minimizing mean traversal time, (ii) minimizing the system’s n 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.

AB - 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 pj =1, is optimal with regard to (i) minimizing mean traversal time, (ii) minimizing the system’s n 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.

KW - Asymmetric simple inclusion process (ASIP)

KW - Generalized ASIP (G-ASIP)

KW - Limit laws

KW - Performance measures

UR - http://www.scopus.com/inward/record.url?scp=85124984884&partnerID=8YFLogxK

U2 - 10.3390/math10040594

DO - 10.3390/math10040594

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AN - SCOPUS:85124984884

SN - 2227-7390

VL - 10

JO - Mathematics

JF - Mathematics

IS - 4

M1 - 594

ER -