MDPI - Publisher of Open Access Journals

15 pages, 312 KiB

Open AccessArticle

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 1152

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, n}

. The [...] Read more.

We consider an irreducible positive-recurrent discrete-time Markov process on the state space

X = ℤ_{+}^{M} \times J

, where

ℤ_{+}

is the set of non-negative integers and

J = {1, 2, \dots, 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

n - k

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

Open AccessArticle

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 2427

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 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)

► Show Figures

Figure 1

23 pages, 646 KiB

Open AccessArticle

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 4210

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 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)

► Show Figures

Figure 1

Search Results (3)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (3)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI