Performance Measures in a Generalized Asymmetric Simple Inclusion Process
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) b...
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MDPI AG
2022-02-01
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| author | Yaron Yeger Uri Yechiali |
| author_facet | Yaron Yeger Uri Yechiali |
| author_sort | Yaron Yeger |
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| description | 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 <i>k</i> (<i>k</i> = 1, 2, …, <i>n</i>). 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 <i>j</i> opens with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>j</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow></semantics></math></inline-formula>, 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. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:30:52Z |
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| spelling | doaj.art-c65528370c7341358d8d719a342a72832023-11-23T20:57:10ZengMDPI AGMathematics2227-73902022-02-0110459410.3390/math10040594Performance Measures in a Generalized Asymmetric Simple Inclusion ProcessYaron Yeger0Uri Yechiali1School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, IsraelSchool of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, IsraelPerformance 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 <i>k</i> (<i>k</i> = 1, 2, …, <i>n</i>). 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 <i>j</i> opens with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi><mi>j</mi></msub><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac></mrow></semantics></math></inline-formula>, 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.https://www.mdpi.com/2227-7390/10/4/594asymmetric simple inclusion process (ASIP)generalized ASIP (G-ASIP)performance measureslimit laws |
| spellingShingle | Yaron Yeger Uri Yechiali Performance Measures in a Generalized Asymmetric Simple Inclusion Process Mathematics asymmetric simple inclusion process (ASIP) generalized ASIP (G-ASIP) performance measures limit laws |
| title | Performance Measures in a Generalized Asymmetric Simple Inclusion Process |
| title_full | Performance Measures in a Generalized Asymmetric Simple Inclusion Process |
| title_fullStr | Performance Measures in a Generalized Asymmetric Simple Inclusion Process |
| title_full_unstemmed | Performance Measures in a Generalized Asymmetric Simple Inclusion Process |
| title_short | Performance Measures in a Generalized Asymmetric Simple Inclusion Process |
| title_sort | performance measures in a generalized asymmetric simple inclusion process |
| topic | asymmetric simple inclusion process (ASIP) generalized ASIP (G-ASIP) performance measures limit laws |
| url | https://www.mdpi.com/2227-7390/10/4/594 |
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