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Keywords = multiple vacation policy

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18 pages, 1048 KiB  
Article
Reliability Analysis and Numerical Simulation of Industrial Robot Drive System with Vacation
by Yanling Li, Genqi Xu and Yihui Wang
Axioms 2025, 14(4), 275; https://doi.org/10.3390/axioms14040275 - 4 Apr 2025
Viewed by 483
Abstract
With the advancement of science and technology, industrial robots have become indispensable equipment in advanced manufacturing and a critical benchmark for assessing a nation’s manufacturing and technological capabilities. Enhancing the reliability of industrial robots is therefore a pressing priority. This paper investigates the [...] Read more.
With the advancement of science and technology, industrial robots have become indispensable equipment in advanced manufacturing and a critical benchmark for assessing a nation’s manufacturing and technological capabilities. Enhancing the reliability of industrial robots is therefore a pressing priority. This paper investigates the drive system of industrial robots, modeling it as a series system comprising multiple components (n) with a repairman who operates under a single vacation policy. The system assumes that each component’s lifespan follows an exponential distribution, while the repairman’s repair and vacation times adhere to general distributions. Notably, the repairman initiates a vacation at the system’s outset. Using the supplementary variable method, a mathematical model of the system is constructed and formulated within an appropriate Banach space, leading to the derivation of the system’s abstract development equation. Leveraging functional analysis and the C0-semigroup theory of bounded operators, the study examines the system’s adaptability, stability, and key reliability indices. Furthermore, numerical simulations are employed to analyze how system reliability indices vary with parameter values. This work contributes to the field of industrial robot reliability analysis by introducing a novel methodological framework that integrates vacation policies and general distribution assumptions, offering new insights into system behavior and reliability optimization. The findings have significant implications for improving the design and maintenance strategies of industrial robots in real-world applications. Full article
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20 pages, 2256 KiB  
Article
Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation
by S. P. Niranjan, S. Devi Latha, Miroslav Mahdal and Krishnasamy Karthik
Mathematics 2024, 12(1), 75; https://doi.org/10.3390/math12010075 - 25 Dec 2023
Cited by 5 | Viewed by 1558
Abstract
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 a and [...] Read more.
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 a and maximum bulk size b. In the first essential service (FES) completion epoch, if the server fails, with probability δ, then the renewal of the service station is considered. On the other hand, if there is no server failure, with a probability 1δ, 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 β. On the contrary, a customer who does not require feedback leaves the system with a probability 1β. If the queue length is less than a after SES, the server may leave for a single vacation with probability 1β. 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 a. Even after setup time completion, the service process begins only with a queue length ‘N’ (N > b). 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. Full article
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29 pages, 432 KiB  
Article
Analysis of Stochastic State-Dependent Arrivals in a Queueing-Inventory System with Multiple Server Vacation and Retrial Facility
by M. Nithya, Gyanendra Prasad Joshi, C. Sugapriya, S. Selvakumar, N. Anbazhagan, Eunmok Yang and Ill Chul Doo
Mathematics 2022, 10(17), 3041; https://doi.org/10.3390/math10173041 - 23 Aug 2022
Cited by 7 | Viewed by 2592
Abstract
This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. [...] Read more.
This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of λ1(θ1). If the server is not on vacation, the primary (retrial) arrival occurs with a rate of λ2(θ2). Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of μ, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under (s,Q) ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced. Full article
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25 pages, 876 KiB  
Article
Analysis of Stock-Dependent Arrival Process in a Retrial Stochastic Inventory System with Server Vacation
by C. Sugapriya, M. Nithya, K. Jeganathan, N. Anbazhagan, Gyanendra Prasad Joshi, Eunmok Yang and Suseok Seo
Processes 2022, 10(1), 176; https://doi.org/10.3390/pr10010176 - 17 Jan 2022
Cited by 10 | Viewed by 3009
Abstract
The present study deals with the stock-dependent Markovian demand of a retrial queueing system with a single server and multiple server vacation. The items are restocked under a continuous review (s,Q) ordering policy. When there is no item in [...] Read more.
The present study deals with the stock-dependent Markovian demand of a retrial queueing system with a single server and multiple server vacation. The items are restocked under a continuous review (s,Q) ordering policy. When there is no item in the system, the server goes on vacation. Further, any arrival demand permits entry into an infinite orbit whenever the server is on vacation. In the Matrix geometric approach with the Neuts-Rao truncation technique, the steady-state joint distribution of the number of customers in orbit, the server status, and the inventory level is obtained. Under the steady-state conditions, some significant system performance measures, including the long-run total cost rate, are derived, and the Laplace-Stieltjes transform is also used to investigate the waiting time distribution. According to various considerations of uncontrollable parameters and costs, the merits of the proposed model, especially the important characteristics of the system with stock dependency over non-stock dependency, are explored. Ultimately, the important facts and ideas behind this model are given in conclusion. Full article
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16 pages, 452 KiB  
Article
Study on Transient Queue-Size Distribution in the Finite-Buffer Model with Batch Arrivals and Multiple Vacation Policy
by Wojciech M. Kempa and Rafał Marjasz
Entropy 2021, 23(11), 1410; https://doi.org/10.3390/e23111410 - 27 Oct 2021
Cited by 1 | Viewed by 1905
Abstract
The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring [...] Read more.
The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well. Full article
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29 pages, 7604 KiB  
Article
Optimizing a Multi-State Cold-Standby System with Multiple Vacations in the Repair and Loss of Units
by Juan Eloy Ruiz-Castro
Mathematics 2021, 9(8), 913; https://doi.org/10.3390/math9080913 - 20 Apr 2021
Cited by 10 | Viewed by 2399
Abstract
A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels [...] Read more.
A complex multi-state redundant system with preventive maintenance subject to multiple events is considered. The online unit can undergo several types of failure: both internal and those provoked by external shocks. Multiple degradation levels are assumed as both internal and external. Degradation levels are observed by random inspections and, if they are major, the unit goes to a repair facility where preventive maintenance is carried out. This repair facility is composed of a single repairperson governed by a multiple vacation policy. This policy is set up according to the operational number of units. Two types of task can be performed by the repairperson, corrective repair and preventive maintenance. The times embedded in the system are phase type distributed and the model is built by using Markovian Arrival Processes with marked arrivals. Multiple performance measures besides the transient and stationary distribution are worked out through matrix-analytic methods. This methodology enables us to express the main results and the global development in a matrix-algorithmic form. To optimize the model, costs and rewards are included. A numerical example shows the versatility of the model. Full article
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15 pages, 10792 KiB  
Article
Analytical Model of a Wireless Sensor Network (WSN) Node Operation with a Modified Threshold-Type Energy Saving Mechanism
by Wojciech M. Kempa
Sensors 2019, 19(14), 3114; https://doi.org/10.3390/s19143114 - 14 Jul 2019
Cited by 16 | Viewed by 3673
Abstract
In this article, a model of the operation of a wireless sensor network (WSN) node with an energy saving mechanism based on a threshold-controlled multiple vacation policy is considered. When the queue of packets directed to the node becomes empty, a multiple vacation [...] Read more.
In this article, a model of the operation of a wireless sensor network (WSN) node with an energy saving mechanism based on a threshold-controlled multiple vacation policy is considered. When the queue of packets directed to the node becomes empty, a multiple vacation period is started during which the receiving/transmitting of packets is blocked. In such a period, successive vacations of a fixed constant duration are taken until a predetermined number of N packets accumulated in the queue is detected. Then, at the completion epoch of this vacation, the processing restarts normally. The analytic approach is based on the conception of an embedded Markov chain; integral equations and renewal theory are applied to study the queue-size transient behaviour. The representations for the Laplace transforms of the queue-size distribution at an arbitrary fixed time t and on the idle and processing periods are obtained. The compact-form formulae for the distributions of the idle and processing period duration are derived. Numerical examples are attached as well. Full article
(This article belongs to the Special Issue Energy Harvesting and Energy-Neutral IoT Devices and Systems)
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18 pages, 357 KiB  
Article
An M[X]/G(a,b)/1 Queueing System with Breakdown and Repair, Stand-By Server, Multiple Vacation and Control Policy on Request for Re-Service
by G. Ayyappan and S. Karpagam
Mathematics 2018, 6(6), 101; https://doi.org/10.3390/math6060101 - 14 Jun 2018
Cited by 16 | Viewed by 6357
Abstract
In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server’s regular service time, re-service time, vacation time and stand-by server’s service time are [...] Read more.
In this paper, we discuss a non-Markovian batch arrival general bulk service single-server queueing system with server breakdown and repair, a stand-by server, multiple vacation and re-service. The main server’s regular service time, re-service time, vacation time and stand-by server’s service time are followed by general distributions and breakdown and repair times of the main server with exponential distributions. There is a stand-by server which is employed during the period in which the regular server remains under repair. The probability generating function of the queue size at an arbitrary time and some performance measures of the system are derived. Extensive numerical results are also illustrated. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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