Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes
In the paper, we consider the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>H</mi><mn>2</mn></msub></mrow></semantics></...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-10-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/13/10/716 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850205471392137216 |
|---|---|
| author | Youxin Liu Liwei Liu Tao Jiang Xudong Chai |
| author_facet | Youxin Liu Liwei Liu Tao Jiang Xudong Chai |
| author_sort | Youxin Liu |
| collection | DOAJ |
| description | In the paper, we consider the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>H</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>-distribution, which is a particular case of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>H</mi></mrow></semantics></math></inline-formula>-distribution. In other words, The first service phase is exponentially distributed, and the service rate is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>. After the first service phase, the customer can to go away with probability <i>p</i> or continue the service with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow></semantics></math></inline-formula> and service rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>μ</mi><mo>′</mo></msup></semantics></math></inline-formula>. We study an analysis of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>P</mi><msub><mi>H</mi><mn>2</mn></msub><mo>/</mo><mn>1</mn></mrow></semantics></math></inline-formula> queue model with catastrophes, which is regarded as a generalization of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>1</mn></mrow></semantics></math></inline-formula> queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given. |
| format | Article |
| id | doaj-art-a60385ee13de41cc8a02eaffecbc168b |
| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-a60385ee13de41cc8a02eaffecbc168b2025-08-20T02:11:04ZengMDPI AGAxioms2075-16802024-10-01131071610.3390/axioms13100716Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with CatastrophesYouxin Liu0Liwei Liu1Tao Jiang2Xudong Chai3School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210044, ChinaSchool of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210044, ChinaCollege of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, ChinaSchool of Mathematics-Physics and Finance, Anhui Polytechnic University, Wuhu 241000, ChinaIn the paper, we consider the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><msub><mi>H</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>-distribution, which is a particular case of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>H</mi></mrow></semantics></math></inline-formula>-distribution. In other words, The first service phase is exponentially distributed, and the service rate is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>μ</mi></semantics></math></inline-formula>. After the first service phase, the customer can to go away with probability <i>p</i> or continue the service with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo>)</mo></mrow></semantics></math></inline-formula> and service rate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>μ</mi><mo>′</mo></msup></semantics></math></inline-formula>. We study an analysis of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>P</mi><msub><mi>H</mi><mn>2</mn></msub><mo>/</mo><mn>1</mn></mrow></semantics></math></inline-formula> queue model with catastrophes, which is regarded as a generalization of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>M</mi><mo>/</mo><mi>M</mi><mo>/</mo><mn>1</mn></mrow></semantics></math></inline-formula> queue model with catastrophes. Whenever a catastrophe happens, all customers will be cleaned up immediately, and the queuing system is empty. The customers arrive at the queuing system based on a Poisson process, and the total service duration has two phases. Transient probabilities and steady-state probabilities of this queuing system are considered using practical applications of the modified Bessel function of the first kind, the Laplace transform, and probability-generating function techniques. Moreover, some important performance measures are obtained in the system. Finally, numerical illustrations are used to discuss the system’s behavior, and conclusions and future directions of the model are given.https://www.mdpi.com/2075-1680/13/10/716catastrophesBessel functionLaplace transformtransient and steady-state probabilitiesperformance measures |
| spellingShingle | Youxin Liu Liwei Liu Tao Jiang Xudong Chai Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes Axioms catastrophes Bessel function Laplace transform transient and steady-state probabilities performance measures |
| title | Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes |
| title_full | Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes |
| title_fullStr | Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes |
| title_full_unstemmed | Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes |
| title_short | Transient and Steady-State Analysis of an <i>M</i>/<i>PH</i><sub>2</sub>/1 Queue with Catastrophes |
| title_sort | transient and steady state analysis of an i m i i ph i sub 2 sub 1 queue with catastrophes |
| topic | catastrophes Bessel function Laplace transform transient and steady-state probabilities performance measures |
| url | https://www.mdpi.com/2075-1680/13/10/716 |
| work_keys_str_mv | AT youxinliu transientandsteadystateanalysisofanimiiphisub2sub1queuewithcatastrophes AT liweiliu transientandsteadystateanalysisofanimiiphisub2sub1queuewithcatastrophes AT taojiang transientandsteadystateanalysisofanimiiphisub2sub1queuewithcatastrophes AT xudongchai transientandsteadystateanalysisofanimiiphisub2sub1queuewithcatastrophes |