Pub Date : 2024-03-09DOI: 10.61102/1024-2953-mprf.2023.29.5.006
G. Fayolle, P. Muhlethaler
The purpose of this paper is to analyze the so-called backoff tech- nique of the IEEE 802.11 protocol with buffers. This protocol rules the trans- missions on a radio channel between nodes (or stations) of a network exchanging packets of information. In contrast to existing models, packets arriving at a sta- tion which in the backoff state are not discarded, but are stored in a buffer of in nite capacity. The backoff state corresponds to the number of time-intervals (mini-slot) that a node must wait before its packet is actually transmitted. As in previous studies, the key point of our analysis hinges on the assumption that the time on the radio channel is viewed as a random succession of trans- mission slots (whose duration corresponds to a packet transmission time) and mini-slots, which stand for the time intervals during which the backoff of the station is decremented. During these mini-slots the channel is idle, which im- plies that there is no packet transmission. These events occur independently with given probabilities, and the external arrivals of messages follow a Pois- son process. The state of a node is represented by a three-dimensional Markov chain in discrete-time, formed by the triple (backoff counter, number of packets at the station, number of transmission attempts). Stability (ergodicity) condi- tions are obtained for an arbitrary station and interpreted in terms of maximum throughput. Several approximations related to these models are also discussed.
{"title":"A Markovian Analysis of an IEEE-802.11 Station with Buffering","authors":"G. Fayolle, P. Muhlethaler","doi":"10.61102/1024-2953-mprf.2023.29.5.006","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.5.006","url":null,"abstract":"The purpose of this paper is to analyze the so-called backoff tech- nique of the IEEE 802.11 protocol with buffers. This protocol rules the trans- missions on a radio channel between nodes (or stations) of a network exchanging packets of information. In contrast to existing models, packets arriving at a sta- tion which in the backoff state are not discarded, but are stored in a buffer of in nite capacity. The backoff state corresponds to the number of time-intervals (mini-slot) that a node must wait before its packet is actually transmitted. As in previous studies, the key point of our analysis hinges on the assumption that the time on the radio channel is viewed as a random succession of trans- mission slots (whose duration corresponds to a packet transmission time) and mini-slots, which stand for the time intervals during which the backoff of the station is decremented. During these mini-slots the channel is idle, which im- plies that there is no packet transmission. These events occur independently with given probabilities, and the external arrivals of messages follow a Pois- son process. The state of a node is represented by a three-dimensional Markov chain in discrete-time, formed by the triple (backoff counter, number of packets at the station, number of transmission attempts). Stability (ergodicity) condi- tions are obtained for an arbitrary station and interpreted in terms of maximum throughput. Several approximations related to these models are also discussed.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140256061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.61102/1024-2953-mprf.2023.29.4.004
V. Chulaevsky, Y. Suhov
We prove a uniform exponential localization of eigenfunctions and simplicity of spectrum for a class of limit-periodic lattice Schr¨odinger operators. An important ingredient of the proof is a generalized variant of the well-known Minami estimates (correlation inequalities for the eigenvalues) to the case where the spectral intervals can be arbitrarily placed in the real line. The new corre- lation inequalities allow us to substantially simplify and make more transparent the application of the KAM (Kolmogorov-Arnold-Moser) techniques.
{"title":"Uniform Anderson Localization and Non-local Minami-type Estimates in Limit-periodic Media","authors":"V. Chulaevsky, Y. Suhov","doi":"10.61102/1024-2953-mprf.2023.29.4.004","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.004","url":null,"abstract":"We prove a uniform exponential localization of eigenfunctions and simplicity of spectrum for a class of limit-periodic lattice Schr¨odinger operators. An important ingredient of the proof is a generalized variant of the well-known Minami estimates (correlation inequalities for the eigenvalues) to the case where the spectral intervals can be arbitrarily placed in the real line. The new corre- lation inequalities allow us to substantially simplify and make more transparent the application of the KAM (Kolmogorov-Arnold-Moser) techniques.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139389766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.61102/1024-2953-mprf.2023.29.4.003
P. Sergey, Z. Elena
Can a local disaster lead to extinction? We answer this question in this work. In the paper [19] we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated suf- ficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in [19] was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the bal- ance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.
{"title":"Persistence in Perturbed Contact Models in Continuum","authors":"P. Sergey, Z. Elena","doi":"10.61102/1024-2953-mprf.2023.29.4.003","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.003","url":null,"abstract":"Can a local disaster lead to extinction? We answer this question in this work. In the paper [19] we considered contact processes on locally compact metric spaces with state dependent birth and death rates and formulated suf- ficient conditions on the rates that ensure the existence of invariant measures. One of the crucial conditions in [19] was the critical regime condition, which meant the existence of a balance between birth and death rates in average. In the present work, we reject the criticality condition and suppose that the bal- ance condition is violated. This implies that the evolution of the correlation functions of the contact model under consideration is determined by a nonlocal convolution type operator perturbed by a (negative) potential. We show that local peaks in mortality do not typically lead to extinction. We prove that a family of invariant measures exists even without the criticality condition and these measures can be described using the Feynman-Kac formula.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139391619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.61102/1024-2953-mprf.2023.29.4.007
A. Logachov, O. Logachova, E. Pechersky, E. Presman, A. Yambartsev
The symmetric birth and death stochastic process on the non-negative integers x ∈ Z + with polynomial rates x α , α ∈ [1, 2], x 6= 0, is studied. The process moves slowly and spends more time in the neighborhood of the state 0. We prove the convergence of the scaled process to a solution of stochastic differential equation without drift. Sticking phenomenon appears at the limiting process: trajectories, starting from any state, take finite time to reach 0 and remain there indefinitely.
本文研究了在非负整数 x∈Z + 上以多项式速率 x α , α∈ [1, 2], x 6= 0 的对称出生和死亡随机过程。该过程移动缓慢，在状态 0 附近停留的时间较长。我们证明了缩放过程对无漂移随机二阶方程解的收敛性。在极限过程中会出现粘滞现象：从任何状态出发的轨迹都需要花费有限的时间到达 0，并无限地停留在那里。
{"title":"Diffusion Approximation for Symmetric Birth-and-Death Processes with Polynomial Rates","authors":"A. Logachov, O. Logachova, E. Pechersky, E. Presman, A. Yambartsev","doi":"10.61102/1024-2953-mprf.2023.29.4.007","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.007","url":null,"abstract":"The symmetric birth and death stochastic process on the non-negative integers x ∈ Z + with polynomial rates x α , α ∈ [1, 2], x 6= 0, is studied. The process moves slowly and spends more time in the neighborhood of the state 0. We prove the convergence of the scaled process to a solution of stochastic differential equation without drift. Sticking phenomenon appears at the limiting process: trajectories, starting from any state, take finite time to reach 0 and remain there indefinitely.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139391264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.61102/1024-2953-mprf.2023.29.4.005
V.V. Vedenyapin, A.A. Bay, V. I. Parenkina, A.G. Petrov
The relativistic equations of gravitation and electromagnetism in the form of Vlasov – Einstein – Maxwell equations are proposed and analyzed. For weakly relativistic equations we get an analog of Mealn – McCree solution. We also study Lagrange points in non-relativistic case with Einstein lambda- term.
{"title":"Minimal Action Principle for Gravity and Electrodynamics, Einstein Lambda, and Lagrange Points","authors":"V.V. Vedenyapin, A.A. Bay, V. I. Parenkina, A.G. Petrov","doi":"10.61102/1024-2953-mprf.2023.29.4.005","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.005","url":null,"abstract":"The relativistic equations of gravitation and electromagnetism in the form of Vlasov – Einstein – Maxwell equations are proposed and analyzed. For weakly relativistic equations we get an analog of Mealn – McCree solution. We also study Lagrange points in non-relativistic case with Einstein lambda- term.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-01-02DOI: 10.61102/1024-2953-mprf.2023.29.4.001
E. Chernousova, S. Molchanov, A. Shiryaev
This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.
本文介绍赫米特多项式及其在风险理论和站点渗流中的应用。
{"title":"Wick–Fourier–Hermite Series in the Theory of Linear and Nonlinear Transformations of Gaussian Distributions","authors":"E. Chernousova, S. Molchanov, A. Shiryaev","doi":"10.61102/1024-2953-mprf.2023.29.4.001","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.001","url":null,"abstract":"This article provides information on Hermite polynomials and its application to some problems in risk theory and site percolation.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-22DOI: 10.61102/1024-2953-mprf.2023.29.4.002
A. Komech, A. Merzon
We describe Malyshev’s method of automorphic functions in ap- plication to boundary value problems in angles and to diffraction by wedges. We give a concise survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller, G.E. Shilov and others.
{"title":"On Malyshev’s Method of Automorphic Functions in Diffraction by Wedges","authors":"A. Komech, A. Merzon","doi":"10.61102/1024-2953-mprf.2023.29.4.002","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.4.002","url":null,"abstract":"We describe Malyshev’s method of automorphic functions in ap- plication to boundary value problems in angles and to diffraction by wedges. We give a concise survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller, G.E. Shilov and others.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139161726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.61102/1024-2953-mprf.2023.29.2.001
A. Piatnitski, E. Zhizhina
The paper deals with periodic homogenization problem for a para- bolic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic functions both in spatial and temporal variables and that the scal- ing is diffusive, that is, the scaling factor of the temporal variable is equal to the square of the scaling factor of the spatial variable. Under the assumption that the convolution kernel has a nite second moment and that the operator is symmetric in spatial variables we show that the equation under study ad- mits homogenization, and we prove that the limit operator is a second order differential parabolic operator with constant coefficients.
{"title":"Homogenization of Non-Autonomous Operators of Convolution Type in Periodic Media","authors":"A. Piatnitski, E. Zhizhina","doi":"10.61102/1024-2953-mprf.2023.29.2.001","DOIUrl":"https://doi.org/10.61102/1024-2953-mprf.2023.29.2.001","url":null,"abstract":"The paper deals with periodic homogenization problem for a para- bolic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic functions both in spatial and temporal variables and that the scal- ing is diffusive, that is, the scaling factor of the temporal variable is equal to the square of the scaling factor of the spatial variable. Under the assumption that the convolution kernel has a nite second moment and that the operator is symmetric in spatial variables we show that the equation under study ad- mits homogenization, and we prove that the limit operator is a second order differential parabolic operator with constant coefficients.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136112787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Charles McCarthy, Gavin Nop, Reza Rastegar, Alexander Roitershtein
We consider a discrete-time random motion, Markov chain on the Poincaré disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random Möbius transformations. We exploit an isomorphism between the underlying group of Möbius transformations and to study the random motion through its relation to a one-dimensional random walk. More specifically, we show that key geometric characteristics of the random motion, such as Busemann functions and bipolar coordinates evaluated at its location, and hyperbolic distance from the origin, can be either explicitly computed or approximated in terms of the random walk. We also consider a variant of the model where the motion is not confined to a single arc, but rather the particle switches between arcs of a parabolic pencil of circles at random times.
{"title":"Random walk on the Poincaré disk induced by a group of Möbius transformations.","authors":"Charles McCarthy, Gavin Nop, Reza Rastegar, Alexander Roitershtein","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>We consider a discrete-time random motion, Markov chain on the Poincaré disk. In the basic variant of the model a particle moves along certain circular arcs within the disk, its location is determined by a composition of random Möbius transformations. We exploit an isomorphism between the underlying group of Möbius transformations and <math><mi>ℝ</mi></math> to study the random motion through its relation to a one-dimensional random walk. More specifically, we show that key geometric characteristics of the random motion, such as Busemann functions and bipolar coordinates evaluated at its location, and hyperbolic distance from the origin, can be either explicitly computed or approximated in terms of the random walk. We also consider a variant of the model where the motion is not confined to a single arc, but rather the particle switches between arcs of a parabolic pencil of circles at random times.</p>","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6999048/pdf/nihms-1037382.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37612704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-01-01DOI: 10.15496/PUBLIKATION-9137
M. Möhle
Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length Ln of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.
{"title":"Absorption Time and Tree Length of the Kingman Coalescent and the Gumbel Distribution","authors":"M. Möhle","doi":"10.15496/PUBLIKATION-9137","DOIUrl":"https://doi.org/10.15496/PUBLIKATION-9137","url":null,"abstract":"Formulas are provided for the cumulants and the moments of the time T back to the most recent common ancestor of the Kingman coalescent. It is shown that both the jth cumulant and the jth moment of T are linear combinations of the values ζ(2m), m ∈ {0, . . . , bj/2c}, of the Riemann zeta function ζ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length Ln of the Kingman coalescent restricted to a sample of size n is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.","PeriodicalId":48890,"journal":{"name":"Markov Processes and Related Fields","volume":null,"pages":null},"PeriodicalIF":0.2,"publicationDate":"2015-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67157736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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