Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience
We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are wai...
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| Format: | Article |
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MDPI AG
2023-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/9/1979 |
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| author | Yacov Satin Rostislav Razumchik Ivan Kovalev Alexander Zeifman |
| author_facet | Yacov Satin Rostislav Razumchik Ivan Kovalev Alexander Zeifman |
| author_sort | Yacov Satin |
| collection | DOAJ |
| description | We consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are waiting. As a result, customers in the queue may become impatient and leave it. Under this setting and with certain restrictions on the intensity functions, the quantity of interest, the total number of customers in the system, is the level-dependent birth-and-death process (BPD). In this paper, for the first time in the literature, explicit upper bounds for the distance between two probability distributions of this BDP are obtained. Using the obtained ergodicity bounds in combination with the sensitivity bounds, we assess the stability of BDP under perturbations. Truncation bounds are also given, which allow for numerical solutions with guaranteed truncation errors. Finally, we provide numerical results to support the findings. |
| first_indexed | 2024-03-11T04:13:44Z |
| format | Article |
| id | doaj.art-fad3bff70cc6497cb25601bef20bdb33 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T04:13:44Z |
| publishDate | 2023-04-01 |
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| spelling | doaj.art-fad3bff70cc6497cb25601bef20bdb332023-11-17T23:18:24ZengMDPI AGMathematics2227-73902023-04-01119197910.3390/math11091979Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s ImpatienceYacov Satin0Rostislav Razumchik1Ivan Kovalev2Alexander Zeifman3Department of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, RussiaFederal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 44-2 Vavilov Str., 119133 Moscow, RussiaDepartment of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, RussiaDepartment of Applied Mathematics, Vologda State University, 15 Lenina Str., 160000 Vologda, RussiaWe consider a non-standard class of Markovian time: varying infinite capacity queues with possibly heterogeneous servers and impatience. We assume that during service time, a customer may switch to the faster server (with no delay), when such a server becomes available and no other customers are waiting. As a result, customers in the queue may become impatient and leave it. Under this setting and with certain restrictions on the intensity functions, the quantity of interest, the total number of customers in the system, is the level-dependent birth-and-death process (BPD). In this paper, for the first time in the literature, explicit upper bounds for the distance between two probability distributions of this BDP are obtained. Using the obtained ergodicity bounds in combination with the sensitivity bounds, we assess the stability of BDP under perturbations. Truncation bounds are also given, which allow for numerical solutions with guaranteed truncation errors. Finally, we provide numerical results to support the findings.https://www.mdpi.com/2227-7390/11/9/1979nonstationary queuing systemimpatiencebirth-death processergodicityboundslimiting characteristics |
| spellingShingle | Yacov Satin Rostislav Razumchik Ivan Kovalev Alexander Zeifman Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience Mathematics nonstationary queuing system impatience birth-death process ergodicity bounds limiting characteristics |
| title | Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience |
| title_full | Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience |
| title_fullStr | Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience |
| title_full_unstemmed | Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience |
| title_short | Ergodicity and Related Bounds for One Particular Class of Markovian Time—Varying Queues with Heterogeneous Servers and Customer’s Impatience |
| title_sort | ergodicity and related bounds for one particular class of markovian time varying queues with heterogeneous servers and customer s impatience |
| topic | nonstationary queuing system impatience birth-death process ergodicity bounds limiting characteristics |
| url | https://www.mdpi.com/2227-7390/11/9/1979 |
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