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Keywords = time-non-homogeneous birth-death processes

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15 pages, 312 KiB  
Article
A Matrix-Multiplicative Solution for Multi-Dimensional QBD Processes
by Valeriy Naumov
Mathematics 2024, 12(3), 444; https://doi.org/10.3390/math12030444 - 30 Jan 2024
Cited by 1 | Viewed by 1149
Abstract
We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=+M×J, where + is the set of non-negative integers and J={1,2,,n}. The [...] Read more.
We consider an irreducible positive-recurrent discrete-time Markov process on the state space X=+M×J, where + is the set of non-negative integers and J={1,2,,n}. The number of states in J may be either finite or infinite. We assume that the process is a homogeneous quasi-birth-and-death process (QBD). It means that the one-step transition probability between non-boundary states (k,i) and (n,j) may depend on i,j, and nk but not on the specific values of k and n. It is shown that the stationary probability vector of the process is expressed through square matrices of order n, which are the minimal non-negative solutions to nonlinear matrix equations. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory, Simulation and Applications)
21 pages, 786 KiB  
Article
On the Dynamics in Decoupling Buffers in Mass Manufacturing Lines: A Stochastic Approach
by Gilberto Pérez-Lechuga, Francisco Venegas-Martínez, Marco A. Montufar-Benítez and Jaime Mora-Vargas
Mathematics 2022, 10(10), 1686; https://doi.org/10.3390/math10101686 - 14 May 2022
Cited by 1 | Viewed by 2422
Abstract
This paper analyzes the flow of the contents of interleaved buffers with continuously operating machines in a mass production line. Under this framework, the products to be manufactured advance from station to station to receive a physical–chemical transformation that adds value as they [...] Read more.
This paper analyzes the flow of the contents of interleaved buffers with continuously operating machines in a mass production line. Under this framework, the products to be manufactured advance from station to station to receive a physical–chemical transformation that adds value as they progress in the process. The existence of decoupling buffers between operations (between two consecutive workstations) is a common practice in order to alleviate the pressure that is ahead due to the lack of synchronization between consecutive operations, which causes leisure and/or bottlenecks in the system. In this proposal, we analyze the dynamics of a mass manufacturing line with intermediate decoupling buffers. To do that, we use a regenerative stochastic process approach to build a model where the products stored in each buffer are taken all at once by the consecutive machine. In a second approach, we use a homogeneous birth–death process with constant input–output and assume that the products are taken one by one by the consecutive machine. Finally, we use a non-homogeneous birth–death process to analyze the dynamics of a system whose inputs and outputs depend on time. These proposals are accompanied by numerical examples that illustrate its practical utility. Full article
(This article belongs to the Special Issue Industrial Mathematics in Management and Engineering)
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23 pages, 646 KiB  
Article
A Time-Non-Homogeneous Double-Ended Queue with Failures and Repairs and Its Continuous Approximation
by Antonio Di Crescenzo, Virginia Giorno, Balasubramanian Krishna Kumar and Amelia G. Nobile
Mathematics 2018, 6(5), 81; https://doi.org/10.3390/math6050081 - 11 May 2018
Cited by 29 | Viewed by 4208
Abstract
We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by [...] Read more.
We consider a time-non-homogeneous double-ended queue subject to catastrophes and repairs. The catastrophes occur according to a non-homogeneous Poisson process and lead the system into a state of failure. Instantaneously, the system is put under repair, such that repair time is governed by a time-varying intensity function. We analyze the transient and the asymptotic behavior of the queueing system. Moreover, we derive a heavy-traffic approximation that allows approximating the state of the systems by a time-non-homogeneous Wiener process subject to jumps to a spurious state (due to catastrophes) and random returns to the zero state (due to repairs). Special attention is devoted to the case of periodic catastrophe and repair intensity functions. The first-passage-time problem through constant levels is also treated both for the queueing model and the approximating diffusion process. Finally, the goodness of the diffusive approximating procedure is discussed. Full article
(This article belongs to the Special Issue Stochastic Processes with Applications)
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