Mathematics

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12 pages, 1271 KiB

Open AccessArticle

COVID-19 Spatial Diffusion: A Markovian Agent-Based Model

by Marco Gribaudo, Mauro Iacono and Daniele Manini

Mathematics 2021, 9(5), 485; https://doi.org/10.3390/math9050485 - 26 Feb 2021

Cited by 11 | Viewed by 2276

We applied a flexible modeling technique capable of representing dynamics of large populations interacting in space and time, namely Markovian Agents, to study the evolution of COVID-19 in Italy. Our purpose was to show that this modeling approach, that is based on mean [...] Read more.

We applied a flexible modeling technique capable of representing dynamics of large populations interacting in space and time, namely Markovian Agents, to study the evolution of COVID-19 in Italy. Our purpose was to show that this modeling approach, that is based on mean field analysis models, provides good performances in describing the diffusion of phenomena, like COVID-19. The paper describes the application of this modeling approach to the Italian scenario and results are validated against real data available about the Italian official documentation of the diffusion of COVID-19. The model of each agent is organized similarly to what largely established in literature in the Susceptible-Infected-Recovered (SIR) family of approaches. Results match the main events taken by the Italian government and their effects. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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20 pages, 540 KiB

Open AccessArticle

Diffusion Limit of Multi-Server Retrial Queue with Setup Time

by Anatoly Nazarov, Alexander Moiseev, Tuan Phung-Duc and Svetlana Paul

Mathematics 2020, 8(12), 2232; https://doi.org/10.3390/math8122232 - 16 Dec 2020

Cited by 9 | Viewed by 2010

Abstract

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival [...] Read more.

In the paper, we consider a multi-server retrial queueing system with setup time which is motivated by applications in power-saving data centers with the ON-OFF policy, where an idle server is immediately turned off and an off server is set up upon arrival of a customer. Customers that find all the servers busy join the orbit and retry for service after an exponentially distributed time. For this model, we derive the stability condition which depends on the setup time and turns out to be more strict than that of the corresponding model with an infinite buffer which is independent of the setup time. We propose asymptotic methods to analyze the system under the condition that the delay in the orbit is extremely long. We show that the scaled-number of customers in the orbit converges to a diffusion process. Using this diffusion limit, we obtain approximations for the steady-state probability distribution of the number of busy servers and that of the number of customers in the orbit. We verify the accuracy of the approximations by simulations and numerical analysis. Numerical results show that the retrial system under the limiting condition consumes more energy than that with an infinite buffer in front of the servers. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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26 pages, 4700 KiB

Open AccessArticle

Short-Scale Stochastic Solar Energy Models: A Datacenter Use Case

by Sara Alouf and Alain Jean-Marie

Mathematics 2020, 8(12), 2127; https://doi.org/10.3390/math8122127 - 27 Nov 2020

Viewed by 1446

Abstract

Modeling the amount of solar energy received by a photovoltaic panel is an essential part of green IT research. The specific motivation of this work is the management of the energy consumption of large datacenters. We propose a new stochastic model for the [...] Read more.

Modeling the amount of solar energy received by a photovoltaic panel is an essential part of green IT research. The specific motivation of this work is the management of the energy consumption of large datacenters. We propose a new stochastic model for the solar irradiance that features minute-scale variations and is therefore suitable for short-term control of performances. Departing from previous models, we use a weather-oriented classification of days obtained from past observations to parameterize the solar source. We demonstrate through extensive simulations, using real workloads, that our model outperforms the existing ones in predicting performance metrics related to energy storage. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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16 pages, 362 KiB

Open AccessArticle

Performance Analysis of Hybrid MTS/MTO Systems with Stochastic Demand and Production

by Dieter Fiems, Eline De Cuypere, Koen De Turck and Dieter Claeys

Mathematics 2020, 8(11), 1925; https://doi.org/10.3390/math8111925 - 02 Nov 2020

Cited by 5 | Viewed by 2164

Abstract

We present a comprehensive numerical approach with reasonably light complexity in terms of implementation and computation for assessing the performance of hybrid make-to-stock (MTS)/make-to-order (MTO) systems. In such hybrid systems, semi-finished products are produced up front and stored in a decoupling inventory. When [...] Read more.

We present a comprehensive numerical approach with reasonably light complexity in terms of implementation and computation for assessing the performance of hybrid make-to-stock (MTS)/make-to-order (MTO) systems. In such hybrid systems, semi-finished products are produced up front and stored in a decoupling inventory. When an order arrives, the products are completed and possibly customised. We study this system in a stochastic setting: demand and production are modelled by random processes. In particular, our model includes two coupled Markovian queues: one queue represents the decoupling inventory and the other the order backlog. These queues are coupled as order processing can only occur when both queues are non-empty. We rely on matrix analytic techniques to study the performance of the MTO/MTS system under non-restrictive stochastic assumptions. In particular, we allow for arrival correlation and non-exponential setup and MTS and MTO processing times, while the hybrid MTS/MTO system is managed by an

(s, S)

-type threshold policy that governs switching from MTO to MTS and back. By some numerical examples, we assess the impact of inventory control, irregular order arrivals, setup and order processing times on inventory levels and lead times. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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16 pages, 342 KiB

Open AccessArticle

Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

by Bara Kim, Jeongsim Kim and Jerim Kim

Mathematics 2020, 8(11), 1893; https://doi.org/10.3390/math8111893 - 31 Oct 2020

Cited by 1 | Viewed by 1532

Abstract

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability [...] Read more.

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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25 pages, 768 KiB

Open AccessArticle

On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

by Antonio Gómez-Corral, Martín López-García, Maria Jesus Lopez-Herrero and Diana Taipe

Mathematics 2020, 8(10), 1718; https://doi.org/10.3390/math8101718 - 07 Oct 2020

Cited by 7 | Viewed by 1986

Abstract

In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary [...] Read more.

In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary probabilities and moments of first-passage times to higher and lower levels. We provide a matrix-analytic scheme for numerically computing hitting probabilities, the number of upcrossings, sojourn time analysis, and the random area under the level trajectory. Our algorithmic solution is inspired from Gaussian elimination, which is applicable in all our descriptors since the underlying rate matrices have a block-structured form. Using the results obtained, numerical examples are given in the context of varicella-zoster virus infections. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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23 pages, 751 KiB

Open AccessFeature PaperArticle

The Effect of Setting a Warning Vaccination Level on a Stochastic SIVS Model with Imperfect Vaccine

by Maria Gamboa and Maria Jesus Lopez-Herrero

Mathematics 2020, 8(7), 1136; https://doi.org/10.3390/math8071136 - 11 Jul 2020

Cited by 4 | Viewed by 2237

Abstract

This paper deals with a stochastic Susceptible-Infective-Vaccinated-Susceptible (SIVS) model with infection reintroduction. Health policies depend on vaccine coverage,

v_{0}

, that guarantees herd immunity levels in the population. Vaccine failures occur when an organism develops a disease despite of being [...] Read more.

This paper deals with a stochastic Susceptible-Infective-Vaccinated-Susceptible (SIVS) model with infection reintroduction. Health policies depend on vaccine coverage,

v_{0}

, that guarantees herd immunity levels in the population. Vaccine failures occur when an organism develops a disease despite of being vaccinated against it. After vaccination, a proportion of healthy individuals unsuccessfully tries to increase antibody levels and, consequently these individuals are not immune to the vaccine preventable disease. When an infectious process is in progress, the initial vaccine coverage drops down and herd immunity will be lost. Our objective was to introduce a warning vaccination level and define random measures quantifying the time until the number of vaccinated descends to a warning vaccination level (i.e., the so-called sleeping period), and the epidemic size. A sensitivity analysis was performed to assess the influence of the model parameters on the variation and robustness of the sleeping period and the number of infections observed within it. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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12 pages, 427 KiB

Open AccessArticle

Antagonistic One-To-N Stochastic Duel Game

by Song-Kyoo (Amang) Kim

Mathematics 2020, 8(7), 1114; https://doi.org/10.3390/math8071114 - 06 Jul 2020

Cited by 7 | Viewed by 2046

Abstract

This paper is dealing with a multiple person game model under the antagonistic duel type setup. The unique multiple person duel game with the one-shooting-to-kill-all condition is analytically solved and the explicit formulas are obtained to determine the time dependent duel game model [...] Read more.

This paper is dealing with a multiple person game model under the antagonistic duel type setup. The unique multiple person duel game with the one-shooting-to-kill-all condition is analytically solved and the explicit formulas are obtained to determine the time dependent duel game model by using the first exceed theory. The model could be directly applied into real-world situations and an analogue of the theory in the paper is designed for solving the best shooting time for hitting all other players at once which optimizes the payoff function under random time conditions. It also mathematically explains to build the marketing strategies for the entry timing for both blue and red ocean markets. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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13 pages, 478 KiB

Open AccessArticle

A Versatile Stochastic Duel Game

by Song-Kyoo (Amang) Kim

Mathematics 2020, 8(5), 678; https://doi.org/10.3390/math8050678 - 01 May 2020

Cited by 9 | Viewed by 2576

Abstract

This paper deals with a standard stochastic game model with a continuum of states under the duel-type setup. It newly proposes a hybrid model of game theory and the fluctuation process, which could be applied for various practical decision making situations. The unique [...] Read more.

This paper deals with a standard stochastic game model with a continuum of states under the duel-type setup. It newly proposes a hybrid model of game theory and the fluctuation process, which could be applied for various practical decision making situations. The unique theoretical stochastic game model is targeted to analyze a two-person duel-type game in the time domain. The parameters for strategic decisions including the moments of crossings, prior crossings, and the optimal number of iterations to get the highest winning chance are obtained by the compact closed joint functional. This paper also demonstrates the usage of a new time based stochastic game model by analyzing a conventional duel game model in the distance domain and briefly explains how to build strategies for an atypical business case to show how this theoretical model works. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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18 pages, 462 KiB

Open AccessArticle

Numerical Inverse Transformation Methods for Z-Transform

by Illés Horváth, András Mészáros and Miklós Telek

Mathematics 2020, 8(4), 556; https://doi.org/10.3390/math8040556 - 10 Apr 2020

Cited by 2 | Viewed by 3438

Abstract

Numerical inverse Z-transformation (NIZT) methods have been efficiently used in engineering practice for a long time. In this paper, we compare the abilities of the most widely used NIZT methods, and propose a new variant of a classic NIZT method based on contour [...] Read more.

Numerical inverse Z-transformation (NIZT) methods have been efficiently used in engineering practice for a long time. In this paper, we compare the abilities of the most widely used NIZT methods, and propose a new variant of a classic NIZT method based on contour integral approximation, which is efficient when the point of interest (at which the value of the function is needed) is smaller than the order of the NIZT method. We also introduce a vastly different NIZT method based on concentrated matrix geometric (CMG) distributions that tackles the limitations of many of the classic methods when the point of interest is larger than the order of the NIZT method. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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14 pages, 571 KiB

Open AccessArticle

A Versatile Queuing System For Sharing Economy Platform Operations

by Song-Kyoo Kim and Chan Yeob Yeun

Mathematics 2019, 7(11), 1005; https://doi.org/10.3390/math7111005 - 23 Oct 2019

Cited by 3 | Viewed by 2725

Abstract

The paper deals with a sharing economy system with various management factors by using a bulk input G/M/1 type queuing model. The effective management of operating costs is vital for controlling the sharing economy platform and this research builds the theoretical background to [...] Read more.

The paper deals with a sharing economy system with various management factors by using a bulk input G/M/1 type queuing model. The effective management of operating costs is vital for controlling the sharing economy platform and this research builds the theoretical background to understand the sharing economy business model. Analytically, the techniques include a classical Markov process of the single channel queueing system, semi-Markov process and semi-regenerative process. It uses the stochastic congruent properties to find the probability distribution of the number of contractors in the sharing economy platform. The obtained explicit formulas demonstrate the usage of functional for the main stochastic characteristics including sharing expenses due to over contracted resources and optimization of their objective function. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research)

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