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Performance Measures in a Generalized Asymmetric Simple Inclusion Process

Author

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  • Yaron Yeger

    (School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel)

  • Uri Yechiali

    (School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel)

Abstract

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 = 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.

Suggested Citation

  • Yaron Yeger & Uri Yechiali, 2022. "Performance Measures in a Generalized Asymmetric Simple Inclusion Process," Mathematics, MDPI, vol. 10(4), pages 1-25, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:4:p:594-:d:749397
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    References listed on IDEAS

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    1. Onno Boxma & Offer Kella & Uri Yechiali, 2016. "An ASIP model with general gate opening intervals," Queueing Systems: Theory and Applications, Springer, vol. 84(1), pages 1-20, October.
    2. James R. Jackson, 1957. "Networks of Waiting Lines," Operations Research, INFORMS, vol. 5(4), pages 518-521, August.
    3. James R. Jackson, 1963. "Jobshop-Like Queueing Systems," Management Science, INFORMS, vol. 10(1), pages 131-142, October.
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    Cited by:

    1. Uri Yechiali & Yaron Yeger, 2022. "Matrix Approach for Analyzing n -Site Generalized ASIP Systems: PGF and Site Occupancy Probabilities," Mathematics, MDPI, vol. 10(23), pages 1-33, December.

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