Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation
In this paper, a bulk arrival and two-phase bulk service with active Bernoulli feedback, vacation, and breakdown is considered. The server provides service in two phases as mandatory according to the general bulk service rule, with minimum bulk size <inline-formula><math xmlns="http://...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/1/75 |
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| author | S. P. Niranjan S. Devi Latha Miroslav Mahdal Krishnasamy Karthik |
| author_facet | S. P. Niranjan S. Devi Latha Miroslav Mahdal Krishnasamy Karthik |
| author_sort | S. P. Niranjan |
| collection | DOAJ |
| description | In this paper, a bulk arrival and two-phase bulk service with active Bernoulli feedback, vacation, and breakdown is considered. The server provides service in two phases as mandatory according to the general bulk service rule, with minimum bulk size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> and maximum bulk size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>b</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. In the first essential service (FES) completion epoch, if the server fails, with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi mathvariant="sans-serif">δ</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, then the renewal of the service station is considered. On the other hand, if there is no server failure, with a probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mi>δ</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, then the server switches to a second essential service (SES) in succession. A customer who requires further service as feedback is given priority, and they join the head of the queue with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi></mrow></semantics></math></inline-formula>. On the contrary, a customer who does not require feedback leaves the system with a probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mi>β</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. If the queue length is less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> after SES, the server may leave for a single vacation with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula>. When the server finds an inadequate number of customers in the queue after vacation completion, the server becomes dormant. After vacation completion, the server requires some time to start service, which is attained by including setup time. The setup time is initiated only when the queue length is at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. Even after setup time completion, the service process begins only with a queue length ‘<i>N</i>’ (<i>N</i> > <i>b</i>). The novelty of this paper is that it introduces an essential two-phase bulk service, immediate Bernoulli feedback for customers, and renewal service time of the first essential service for the bulk arrival and bulk service queueing model. We aim to develop a model that investigates the probability-generating function of the queue size at any time. Additionally, we analyzed various performance characteristics using numerical examples to demonstrate the model’s effectiveness. An optimum cost analysis was also carried out to minimize the total average cost with appropriate practical applications in existing data transmission and data processing in LTE-A networks using the DRX mechanism. |
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| format | Article |
| id | doaj.art-f88d9b00a6724eb2ae24ef7383cffba0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T15:02:57Z |
| publishDate | 2023-12-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-f88d9b00a6724eb2ae24ef7383cffba02024-01-10T15:03:31ZengMDPI AGMathematics2227-73902023-12-011217510.3390/math12010075Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and VacationS. P. Niranjan0S. Devi Latha1Miroslav Mahdal2Krishnasamy Karthik3Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaDepartment of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaDepartment of Control Systems and Instrumentation, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 17, Listopadu 2172/15, 70800 Ostrava, Czech RepublicDepartment of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, IndiaIn this paper, a bulk arrival and two-phase bulk service with active Bernoulli feedback, vacation, and breakdown is considered. The server provides service in two phases as mandatory according to the general bulk service rule, with minimum bulk size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> and maximum bulk size <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>b</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. In the first essential service (FES) completion epoch, if the server fails, with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi mathvariant="sans-serif">δ</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, then the renewal of the service station is considered. On the other hand, if there is no server failure, with a probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mi>δ</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>, then the server switches to a second essential service (SES) in succession. A customer who requires further service as feedback is given priority, and they join the head of the queue with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>β</mi></mrow></semantics></math></inline-formula>. On the contrary, a customer who does not require feedback leaves the system with a probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mi>β</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. If the queue length is less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula> after SES, the server may leave for a single vacation with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><mn>1</mn><mo>−</mo><msup><mrow><mi>β</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></semantics></math></inline-formula>. When the server finds an inadequate number of customers in the queue after vacation completion, the server becomes dormant. After vacation completion, the server requires some time to start service, which is attained by including setup time. The setup time is initiated only when the queue length is at least <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow></mrow><mo>′</mo></msup><msup><mi>a</mi><mo>′</mo></msup></mrow></semantics></math></inline-formula>. Even after setup time completion, the service process begins only with a queue length ‘<i>N</i>’ (<i>N</i> > <i>b</i>). The novelty of this paper is that it introduces an essential two-phase bulk service, immediate Bernoulli feedback for customers, and renewal service time of the first essential service for the bulk arrival and bulk service queueing model. We aim to develop a model that investigates the probability-generating function of the queue size at any time. Additionally, we analyzed various performance characteristics using numerical examples to demonstrate the model’s effectiveness. An optimum cost analysis was also carried out to minimize the total average cost with appropriate practical applications in existing data transmission and data processing in LTE-A networks using the DRX mechanism.https://www.mdpi.com/2227-7390/12/1/75multiple control policyrenewal timebreakdownBernoulli feedback |
| spellingShingle | S. P. Niranjan S. Devi Latha Miroslav Mahdal Krishnasamy Karthik Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation Mathematics multiple control policy renewal time breakdown Bernoulli feedback |
| title | Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation |
| title_full | Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation |
| title_fullStr | Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation |
| title_full_unstemmed | Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation |
| title_short | Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation |
| title_sort | multiple control policy in unreliable two phase bulk queueing system with active bernoulli feedback and vacation |
| topic | multiple control policy renewal time breakdown Bernoulli feedback |
| url | https://www.mdpi.com/2227-7390/12/1/75 |
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