Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs
Ophthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacture...
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MDPI AG
2024-11-01
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| author | Yingying Pan Jingchuan Zhang Zaiming Liu |
| author_facet | Yingying Pan Jingchuan Zhang Zaiming Liu |
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| description | Ophthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacturer (OEM) and results in a delay time between the server breakdown and the start of the repair. In this paper, we model this scenario as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi></msub></semantics></math></inline-formula>/M/1 queuing system with two types of breakdowns and delayed repairs due to complex breakdowns, where the delay time and repair times for simple and complex breakdowns are generally distributed. We obtain the steady-state probabilities and provide the recursive formulas for the Laplace–Stieltjes transforms (LSTs) of conditional residual delay time and repair times given the system state. For the fully observable case, we derive the equilibrium joining strategies of customers who decide to join or balk based on their observation of the system state. Moreover, two numerical experiments are conducted to explore the equilibrium joining probabilities. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2024-11-01 |
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| series | Mathematics |
| spelling | doaj-art-c0ed5efd23ed4c678f92a3ae4cd9dd6d2024-12-13T16:27:27ZengMDPI AGMathematics2227-73902024-11-011223369510.3390/math12233695Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed RepairsYingying Pan0Jingchuan Zhang1Zaiming Liu2School of Mathematics and Statistics, Central South University, Changsha 410083, ChinaAlibaba Business School, Hangzhou Normal University, Hangzhou 311121, ChinaSchool of Mathematics and Statistics, Central South University, Changsha 410083, ChinaOphthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacturer (OEM) and results in a delay time between the server breakdown and the start of the repair. In this paper, we model this scenario as an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi></msub></semantics></math></inline-formula>/M/1 queuing system with two types of breakdowns and delayed repairs due to complex breakdowns, where the delay time and repair times for simple and complex breakdowns are generally distributed. We obtain the steady-state probabilities and provide the recursive formulas for the Laplace–Stieltjes transforms (LSTs) of conditional residual delay time and repair times given the system state. For the fully observable case, we derive the equilibrium joining strategies of customers who decide to join or balk based on their observation of the system state. Moreover, two numerical experiments are conducted to explore the equilibrium joining probabilities.https://www.mdpi.com/2227-7390/12/23/3695queuing gameequilibrium strategybreakdownsdelayed repairsstate-dependent |
| spellingShingle | Yingying Pan Jingchuan Zhang Zaiming Liu Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs Mathematics queuing game equilibrium strategy breakdowns delayed repairs state-dependent |
| title | Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs |
| title_full | Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs |
| title_fullStr | Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs |
| title_full_unstemmed | Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs |
| title_short | Equilibrium Strategies in an M<sub>n</sub>/M/1 Queue with Server Breakdowns and Delayed Repairs |
| title_sort | equilibrium strategies in an m sub n sub m 1 queue with server breakdowns and delayed repairs |
| topic | queuing game equilibrium strategy breakdowns delayed repairs state-dependent |
| url | https://www.mdpi.com/2227-7390/12/23/3695 |
| work_keys_str_mv | AT yingyingpan equilibriumstrategiesinanmsubnsubm1queuewithserverbreakdownsanddelayedrepairs AT jingchuanzhang equilibriumstrategiesinanmsubnsubm1queuewithserverbreakdownsanddelayedrepairs AT zaimingliu equilibriumstrategiesinanmsubnsubm1queuewithserverbreakdownsanddelayedrepairs |