Pub Date : 2025-07-23DOI: 10.1016/j.rinam.2025.100613
Can Li , Xin Wang , Yubin Yan , Zexin Hou
In this paper, we consider a time semi-discrete scheme for a tempered subdiffusion equation with nonsmooth data. Due to the low regularity of the solution, the optimal convergence rate cannot be achieved when the L1 time-stepping scheme is directly applied to discretize the tempered fractional derivative. By introducing a correction term at the initial time step, we propose a corrected L1 scheme which recover to the optimal convergence rate. Theoretical error estimates and numerical experiments validate the improvement.
{"title":"A corrected L1 scheme for solving a tempered subdiffusion equation with nonsmooth data","authors":"Can Li , Xin Wang , Yubin Yan , Zexin Hou","doi":"10.1016/j.rinam.2025.100613","DOIUrl":"10.1016/j.rinam.2025.100613","url":null,"abstract":"<div><div>In this paper, we consider a time semi-discrete scheme for a tempered subdiffusion equation with nonsmooth data. Due to the low regularity of the solution, the optimal convergence rate cannot be achieved when the L1 time-stepping scheme is directly applied to discretize the tempered fractional derivative. By introducing a correction term at the initial time step, we propose a corrected L1 scheme which recover to the optimal convergence rate. Theoretical error estimates and numerical experiments validate the improvement.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100613"},"PeriodicalIF":1.4,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144687563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-22DOI: 10.1016/j.rinam.2025.100617
Bing Tan, Yingzhe Fan
This paper investigates the global well-posedness of the Cauchy problem for the Vlasov–Fokker–Planck equation coupled with the incompressible Euler system around a normalized global Maxwellian in a periodic spatial domain. The system describes the interaction between a fluid governed by Euler equations and a particle distribution evolving under the VFP dynamics, with coupling through a drag force. We establish the existence and uniqueness of global mild solutions for small initial data in a low regularity function space by employing Fourier analysis.
Compare to the Navier–Stokes–Vlasov-Fokker–Planck system (Tan and Fan, 2023) where velocity dissipation estimates can be directly derived from the viscous term, the Vlasov–Euler–Fokker–Planck system lacks such direct accessibility to velocity dissipation due to its inherent structural differences. To overcome this obstacle, we need to exploit the macroscopic dissipation inherent in the macroscopic equation. Then the dissipation of velocity is indirectly captured by combining the macroscopic dissipation of and the linear dissipation of within the equation. Finally the uniform energy functionals of the solution can be obtained by utilizing the refined energy estimate.
本文研究了周期空间域上Vlasov-Fokker-Planck方程与不可压缩欧拉系统在规格化全局麦克斯韦方程组周围耦合的Cauchy问题的全局适定性。该系统描述了由欧拉方程控制的流体与在VFP动力学下演化的粒子分布之间的相互作用，并通过阻力进行耦合。利用傅里叶分析，建立了低正则性函数空间Lk1LT∞Lv2上小初始数据全局温和解的存在唯一性。与Navier-Stokes-Vlasov-Fokker-Planck系统（Tan and Fan, 2023）相比，Vlasov-Euler-Fokker-Planck系统由于其固有的结构差异，无法直接获得速度耗散估计。在Navier-Stokes-Vlasov-Fokker-Planck系统中，可以直接从粘性项中导出速度耗散估计。为了克服这个障碍，我们需要利用宏观方程中固有的宏观耗散b。然后结合方程中b的宏观耗散和u−b的线性耗散，间接捕捉速度耗散。最后利用精化的能量估计得到解的均匀能量泛函。
{"title":"Global existence for the Vlasov–Euler–Fokker–Planck system in low-regularity space","authors":"Bing Tan, Yingzhe Fan","doi":"10.1016/j.rinam.2025.100617","DOIUrl":"10.1016/j.rinam.2025.100617","url":null,"abstract":"<div><div>This paper investigates the global well-posedness of the Cauchy problem for the Vlasov–Fokker–Planck equation coupled with the incompressible Euler system around a normalized global Maxwellian in a periodic spatial domain. The system describes the interaction between a fluid governed by Euler equations and a particle distribution evolving under the VFP dynamics, with coupling through a drag force. We establish the existence and uniqueness of global mild solutions for small initial data in a low regularity function space <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>T</mi></mrow><mrow><mi>∞</mi></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math></span> by employing Fourier analysis.</div><div>Compare to the Navier–Stokes–Vlasov-Fokker–Planck system (Tan and Fan, 2023) where velocity dissipation estimates can be directly derived from the viscous term, the Vlasov–Euler–Fokker–Planck system lacks such direct accessibility to velocity dissipation due to its inherent structural differences. To overcome this obstacle, we need to exploit the macroscopic dissipation <span><math><mi>b</mi></math></span> inherent in the macroscopic equation. Then the dissipation of velocity is indirectly captured by combining the macroscopic dissipation of <span><math><mi>b</mi></math></span> and the linear dissipation of <span><math><mrow><mi>u</mi><mo>−</mo><mi>b</mi></mrow></math></span> within the equation. Finally the uniform energy functionals of the solution can be obtained by utilizing the refined energy estimate.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100617"},"PeriodicalIF":1.4,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144680600","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-22DOI: 10.1016/j.rinam.2025.100608
Yujie Yun, Tieqiang Gang, Lijie Chen
In this study, we employ the iterative Linear Quadratic Gaussian (ILQG) method, discretized based on the high-order exponential Runge–Kutta methods, to numerically solve stochastic optimal control problems. In the sense of weak convergence, we derive a mean-square third-order scheme with an additive noise, and provide corresponding order conditions. As the analysis of order conditions is local, the analysis is transformed into a error estimate of the discrete problem with control constraints. Finally, the global control law is approximated by computing the node control via the ILQG method. The numerical experiment further demonstrates the significant stability of ILQG in dealing with stochastic semilinear control problems. The proposed approach presents the advantages of simplicity and efficiency.
{"title":"Discrete ILQG method based on high-order exponential Runge–Kutta discretization","authors":"Yujie Yun, Tieqiang Gang, Lijie Chen","doi":"10.1016/j.rinam.2025.100608","DOIUrl":"10.1016/j.rinam.2025.100608","url":null,"abstract":"<div><div>In this study, we employ the iterative Linear Quadratic Gaussian (ILQG) method, discretized based on the high-order exponential Runge–Kutta methods, to numerically solve stochastic optimal control problems. In the sense of weak convergence, we derive a mean-square third-order scheme with an additive noise, and provide corresponding order conditions. As the analysis of order conditions is local, the analysis is transformed into a <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> error estimate of the discrete problem with control constraints. Finally, the global control law is approximated by computing the node control via the ILQG method. The numerical experiment further demonstrates the significant stability of ILQG in dealing with stochastic semilinear control problems. The proposed approach presents the advantages of simplicity and efficiency.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100608"},"PeriodicalIF":1.4,"publicationDate":"2025-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144679589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-18DOI: 10.1016/j.rinam.2025.100615
Xiaofeng Li
This study presents the first analytical solution for wave propagation over composite seabeds integrating sinusoidal sandbars with truncated semi-elliptical topographies, overcoming limitations of conventional mild-slope equations in handling elliptical curvature effects, coupled Bragg scattering, and singularities at truncated boundaries. Utilizing Frobenius series expansion and multi-region field matching, we systematically quantify how geometric parameters— ratio, , and —govern wave reflection coefficients (). Key discoveries reveal that the ratio controls resonance peak frequencies (inducing 12% shifts per 0.1 change), the radius parameter triggers complete reflection () at a critical value of 0.5, and optimal expands reflection bandwidth by up to 22%. This work transcends classical studies on singular seabed types, establishes a theoretical foundation for designing wave-control metamaterials via multiscale resonances, and bridges classical potential flow theory with modern coastal engineering applications in wave energy harvesting, coastal protection, and offshore structure design.
{"title":"Multiscale wave resonance in composite sinusoidal-elliptical topographies: Critical transitions and analytical control","authors":"Xiaofeng Li","doi":"10.1016/j.rinam.2025.100615","DOIUrl":"10.1016/j.rinam.2025.100615","url":null,"abstract":"<div><div>This study presents the first analytical solution for wave propagation over composite seabeds integrating sinusoidal sandbars with truncated semi-elliptical topographies, overcoming limitations of conventional mild-slope equations in handling elliptical curvature effects, coupled Bragg scattering, and singularities at truncated boundaries. Utilizing Frobenius series expansion and multi-region field matching, we systematically quantify how geometric parameters—<span><math><mrow><mi>a</mi><mo>/</mo><mi>b</mi></mrow></math></span> ratio, <span><math><mrow><mi>δ</mi><mo>/</mo><mi>a</mi></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>h</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>/</mo><mi>b</mi></mrow></math></span>—govern wave reflection coefficients (<span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>R</mi></mrow></msub></math></span>). Key discoveries reveal that the <span><math><mrow><mi>a</mi><mo>/</mo><mi>b</mi></mrow></math></span> ratio controls resonance peak frequencies (inducing 12% shifts per 0.1 change), the radius parameter <span><math><mrow><mi>r</mi><mo>=</mo><mrow><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>/</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></math></span> triggers complete reflection (<span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>R</mi></mrow></msub><mo>→</mo><mn>1</mn></mrow></math></span>) at a critical value of 0.5, and optimal <span><math><mrow><mi>δ</mi><mo>/</mo><mi>a</mi></mrow></math></span> expands reflection bandwidth by up to 22%. This work transcends classical studies on singular seabed types, establishes a theoretical foundation for designing wave-control metamaterials via multiscale resonances, and bridges classical potential flow theory with modern coastal engineering applications in wave energy harvesting, coastal protection, and offshore structure design.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100615"},"PeriodicalIF":1.4,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654180","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work deals with the offset fractional Fourier transform (OFrFT), which is a more general version of the fractional Fourier transform (FrFT). We demonstrate the basic properties such as translation, modulation and parity. The results are generalization of the FrFT properties. We study a relation of the OFrFT with the FrFT and the Fourier transform. Based on the relation, the key properties such as Parseval’s identity and inversion formula are derived. Applying the properties and the relation allow us to establish several versions of the uncertainty inequalities for the OFrFT. In addition, we discuss the comparison of the OFrFT with the FrFT in terms of properties and uncertainty principles. Finally, we perform an illustrative example to demonstrate that the value of Heisenberg uncertainty inequality for the OFrFT is bigger than that of Heisenberg uncertainty inequality for the FrFT and effect of the offset parameter in minimizing the Heisenberg uncertainty principle associated with the OFrFT.
{"title":"A comparative study on properties and uncertainty principles of fractional Fourier transform and offset fractional Fourier transform","authors":"Mawardi Bahri , Airien Nabilla B.A. , Nasrullah Bachtiar , Muhammad Zakir","doi":"10.1016/j.rinam.2025.100616","DOIUrl":"10.1016/j.rinam.2025.100616","url":null,"abstract":"<div><div>This work deals with the offset fractional Fourier transform (OFrFT), which is a more general version of the fractional Fourier transform (FrFT). We demonstrate the basic properties such as translation, modulation and parity. The results are generalization of the FrFT properties. We study a relation of the OFrFT with the FrFT and the Fourier transform. Based on the relation, the key properties such as Parseval’s identity and inversion formula are derived. Applying the properties and the relation allow us to establish several versions of the uncertainty inequalities for the OFrFT. In addition, we discuss the comparison of the OFrFT with the FrFT in terms of properties and uncertainty principles. Finally, we perform an illustrative example to demonstrate that the value of Heisenberg uncertainty inequality for the OFrFT is bigger than that of Heisenberg uncertainty inequality for the FrFT and effect of the offset parameter in minimizing the Heisenberg uncertainty principle associated with the OFrFT.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100616"},"PeriodicalIF":1.4,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-17DOI: 10.1016/j.rinam.2025.100595
Mohammad Adm , Jürgen Garloff
A sign regular matrix is a matrix having the property that its non-zero minors of all orders have, for each order, an identical sign. Such matrices arise in a wide range of applications. In this paper, intervals of real matrices with respect to the usual entry-wise partial ordering are considered. Using variation diminution, it is shown that all matrices in such an interval are sign-regular with the same signature of their minors if a specified finite set of element matrices in the interval has this property.
{"title":"Variation diminution and intervals of sign regular matrices","authors":"Mohammad Adm , Jürgen Garloff","doi":"10.1016/j.rinam.2025.100595","DOIUrl":"10.1016/j.rinam.2025.100595","url":null,"abstract":"<div><div>A sign regular matrix is a matrix having the property that its non-zero minors of all orders have, for each order, an identical sign. Such matrices arise in a wide range of applications. In this paper, intervals of real matrices with respect to the usual entry-wise partial ordering are considered. Using variation diminution, it is shown that all matrices in such an interval are sign-regular with the same signature of their minors if a specified finite set of element matrices in the interval has this property.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100595"},"PeriodicalIF":1.4,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654176","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-12DOI: 10.1016/j.rinam.2025.100614
Fredrik Armerin
We derive an explicit formula for the moment generating function of a Brownian motion with drift reflected from above in one barrier. Some other properties of this stochastic process are also reported.
我们导出了一个带有漂移的布朗运动的力矩生成函数的显式公式。本文还报道了这一随机过程的其他一些性质。
{"title":"The moment generating function of a reflected Brownian motion with drift","authors":"Fredrik Armerin","doi":"10.1016/j.rinam.2025.100614","DOIUrl":"10.1016/j.rinam.2025.100614","url":null,"abstract":"<div><div>We derive an explicit formula for the moment generating function of a Brownian motion with drift reflected from above in one barrier. Some other properties of this stochastic process are also reported.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100614"},"PeriodicalIF":1.4,"publicationDate":"2025-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144604876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-09DOI: 10.1016/j.rinam.2025.100612
S. Zangoei Zadeh , M. Azizian , M. Sarvari
The Black–Scholes model, a powerful tool for valuation of equity options specially European equity options, is based on assumptions that are violated in some situations due to market realities. One of these cases is the instability of risk-free interest rates and the volatility of stock prices in the Black–Scholes model.
In this paper, in order to make the Black–Scholes model more in line with market realities, fixed parameters in the model, such as risk-free interest rates and stock price volatility, are considered with uncertainty. The obtained interval model is solved using discretization method and converting it into a minimization problem. Finally, The accuracy and efficiency of the method is tested by some numerical examples.
{"title":"An interval version of Black–Scholes European option pricing model and its numerical solution","authors":"S. Zangoei Zadeh , M. Azizian , M. Sarvari","doi":"10.1016/j.rinam.2025.100612","DOIUrl":"10.1016/j.rinam.2025.100612","url":null,"abstract":"<div><div>The Black–Scholes model, a powerful tool for valuation of equity options specially European equity options, is based on assumptions that are violated in some situations due to market realities. One of these cases is the instability of risk-free interest rates and the volatility of stock prices in the Black–Scholes model.</div><div>In this paper, in order to make the Black–Scholes model more in line with market realities, fixed parameters in the model, such as risk-free interest rates and stock price volatility, are considered with uncertainty. The obtained interval model is solved using discretization method and converting it into a minimization problem. Finally, The accuracy and efficiency of the method is tested by some numerical examples.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100612"},"PeriodicalIF":1.4,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144580081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-01DOI: 10.1016/j.rinam.2025.100609
Soumia EL OMARI, Said MELLIANI
This paper addresses proving that solutions exist for nonlinear elliptic problems characterized by boundary conditions of non-local type, as well as their uniqueness, within the framework of weighted Sobolev spaces. These problems are motivated by applications in petroleum engineering, where non-local boundary conditions model complex interactions in stratified reservoirs with three-dimensional geometries. Using the properties of Leray–Lions type operators, compactness arguments, and a priori estimates, we establish a fundamental theorem guaranteeing the existence of weak solutions under suitable assumptions. A rigorous proof of the uniqueness of solutions is also provided by exploiting the strict monotonicity of the operator. This work expands the modeling capabilities for contexts where non-local interactions play a key role, offering relevant mathematical tools for simulating oil well performance and other similar applications.
{"title":"Boundary conditions of nonlocal type in weighted Sobolev spaces for nonlinear elliptic problems","authors":"Soumia EL OMARI, Said MELLIANI","doi":"10.1016/j.rinam.2025.100609","DOIUrl":"10.1016/j.rinam.2025.100609","url":null,"abstract":"<div><div>This paper addresses proving that solutions exist for nonlinear elliptic problems characterized by boundary conditions of non-local type, as well as their uniqueness, within the framework of weighted Sobolev spaces. These problems are motivated by applications in petroleum engineering, where non-local boundary conditions model complex interactions in stratified reservoirs with three-dimensional geometries. Using the properties of Leray–Lions type operators, compactness arguments, and a priori estimates, we establish a fundamental theorem guaranteeing the existence of weak solutions under suitable assumptions. A rigorous proof of the uniqueness of solutions is also provided by exploiting the strict monotonicity of the operator. This work expands the modeling capabilities for contexts where non-local interactions play a key role, offering relevant mathematical tools for simulating oil well performance and other similar applications.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100609"},"PeriodicalIF":1.4,"publicationDate":"2025-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144517410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-06-27DOI: 10.1016/j.rinam.2025.100603
Yutaka Sakuma, Yan Linn Aung
In this paper, we consider an queue, where arriving customers decide whether to join the queue or not join based on the queue length at arrival instants. Kerner (2008, Stochastic Models) studies the queue, and derives a recursive formula for the Laplace-Stieltjes transform (LST, for short) of the conditional distribution of the server’s residual service time, given the queue length at arrival instants. This paper aims to analyze the queue in a much simpler way than the previous studies, and to show that our LST of the conditional distribution of the server’s residual service time is given in a more numerically stable form than that of the previous studies, specifically by avoiding the indeterminate form such as . We then use the formula to compute the customers joining probabilities in Nash equilibrium.
{"title":"A numerically stable formula for the conditional distribution of the residual service time in the Mn/PH/1 queue","authors":"Yutaka Sakuma, Yan Linn Aung","doi":"10.1016/j.rinam.2025.100603","DOIUrl":"10.1016/j.rinam.2025.100603","url":null,"abstract":"<div><div>In this paper, we consider an <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>P</mi><mi>H</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue, where arriving customers decide whether to join the queue or not join based on the queue length at arrival instants. Kerner (2008, <em>Stochastic Models</em>) studies the <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>G</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue, and derives a recursive formula for the Laplace-Stieltjes transform (LST, for short) of the conditional distribution of the server’s residual service time, given the queue length at arrival instants. This paper aims to analyze the <span><math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>P</mi><mi>H</mi><mo>/</mo><mn>1</mn></mrow></math></span> queue in a much simpler way than the previous studies, and to show that our LST of the conditional distribution of the server’s residual service time is given in a more numerically stable form than that of the previous studies, specifically by avoiding the indeterminate form such as <span><math><mrow><mn>0</mn><mo>/</mo><mn>0</mn></mrow></math></span>. We then use the formula to compute the customers joining probabilities in Nash equilibrium.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"27 ","pages":"Article 100603"},"PeriodicalIF":1.4,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144500879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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