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Gerold Alsmeyer - One of the best experts on this subject based on the ideXlab platform.

  • A Min-Type Stochastic Fixed-Point Equation Related to the Smoothing Transformation
    arXiv: Probability, 2009
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    This paper is devoted to the study of the stochastic xed-Point Equation X d = inf i 1:Ti>0 Xi=Ti

  • a stochastic Fixed Point Equation for weighted minima and maxima
    arXiv: Probability, 2008
    Co-Authors: Gerold Alsmeyer, Uwe Rösler
    Abstract:

    Given any finite or countable collection of real numbers $T_j,j\in J$, we find all solutions $F$ to the stochastic Fixed Point Equation \[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the $W_j,j\in J$, are independent real-valued random variables with distribution $F$ and $\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $|J|\geq 2$ and all $T_j$ are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation $T$ has a characteristic exponent $\alpha$ given by $\sum_{j\in J}T_j^{\alpha}=1$ and the set of solutions depends on the closed multiplicative subgroup of $\mathbb {R}^{>}=(0,\infty)$ generated by the $T_j$ which is either $\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in \mathbb {Z}\}$ for some $r>1$. The first case being trivial, the nontrivial Fixed Points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any Fixed Point is harmonic with respect to $\varLambda =\sum_{j\geq 1}\delta_{T_j}$, i.e. $\varGamma =\varGamma \star \varLambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem.

  • A stochastic Fixed Point Equation for weighted minima and maxima
    Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 2008
    Co-Authors: Gerold Alsmeyer, Uwe Rösler
    Abstract:

    Given any finite or countable collection of real numbers Tj, j 2 J, we find all solutions F to the stochastic Fixed Point Equation W d = inf j2J T jWj,

  • A stochastic maximin Fixed-Point Equation related to game tree evaluation
    Journal of Applied Probability, 2007
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    After suitable normalization the asymptotic root value W of a minimax game tree of order b ≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval of R appears to be a particular solution of the stochastic maximin Fixed-Point Equation W L = ξ max1≤i≤b min1≤j ≤b Wi,j ,

  • A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation
    Journal of Applied Probability, 2007
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    After suitable normalization the asymptotic root valueWof a minimax game tree of orderb≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval ofRappears to be a particular solution of the stochastic maximin Fixed-Point EquationWξ max1≤i≤bmin1≤j≤bWi,j, whereWi,jare independent copies ofWanddenotes equality in law. Moreover, ξ=g'(α) > 1, whereg(x) := (1 − (1 −x)b)band α denotes the unique Fixed Point ofgin (0, 1). This Equation, which takes the formF(t) =g(F(t/ξ)) in terms of the distribution functionFofW, is studied in the present paper for a reasonably extended class of functionsgso as to encompass more general stochastic maximin Equations as well. A complete description of the set of solutionsFis provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic Fixed-Point Equations for weighted minima and maxima is also discussed.

Matthias Meiners - One of the best experts on this subject based on the ideXlab platform.

  • A Min-Type Stochastic Fixed-Point Equation Related to the Smoothing Transformation
    arXiv: Probability, 2009
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    This paper is devoted to the study of the stochastic xed-Point Equation X d = inf i 1:Ti>0 Xi=Ti

  • A stochastic maximin Fixed-Point Equation related to game tree evaluation
    Journal of Applied Probability, 2007
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    After suitable normalization the asymptotic root value W of a minimax game tree of order b ≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval of R appears to be a particular solution of the stochastic maximin Fixed-Point Equation W L = ξ max1≤i≤b min1≤j ≤b Wi,j ,

  • A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation
    Journal of Applied Probability, 2007
    Co-Authors: Gerold Alsmeyer, Matthias Meiners
    Abstract:

    After suitable normalization the asymptotic root valueWof a minimax game tree of orderb≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval ofRappears to be a particular solution of the stochastic maximin Fixed-Point EquationWξ max1≤i≤bmin1≤j≤bWi,j, whereWi,jare independent copies ofWanddenotes equality in law. Moreover, ξ=g'(α) > 1, whereg(x) := (1 − (1 −x)b)band α denotes the unique Fixed Point ofgin (0, 1). This Equation, which takes the formF(t) =g(F(t/ξ)) in terms of the distribution functionFofW, is studied in the present paper for a reasonably extended class of functionsgso as to encompass more general stochastic maximin Equations as well. A complete description of the set of solutionsFis provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic Fixed-Point Equations for weighted minima and maxima is also discussed.

Uwe Rösler - One of the best experts on this subject based on the ideXlab platform.

  • All solutions of the stochastic Fixed Point Equation of the Quicksort process
    Advances in Applied Probability, 2018
    Co-Authors: S. Hallmann, Uwe Rösler, M. Wnuk
    Abstract:

    Abstract The Quicksort process R (Rösler (2018)) can be characterized as the unique endogenous solution of the inhomogeneous stochastic Fixed Point Equation R=D(UR1(1∧t∕U)+

  • a stochastic Fixed Point Equation for weighted minima and maxima
    arXiv: Probability, 2008
    Co-Authors: Gerold Alsmeyer, Uwe Rösler
    Abstract:

    Given any finite or countable collection of real numbers $T_j,j\in J$, we find all solutions $F$ to the stochastic Fixed Point Equation \[W\stackrel{\mathrm {d}}{=}\inf_{j\in J}T_jW_j,\] where $W$ and the $W_j,j\in J$, are independent real-valued random variables with distribution $F$ and $\stackrel{\mathrm {d}}{=}$ means equality in distribution. The bulk of the necessary analysis is spent on the case when $|J|\geq 2$ and all $T_j$ are (strictly) positive. Nontrivial solutions are then concentrated on either the positive or negative half line. In the most interesting (and difficult) situation $T$ has a characteristic exponent $\alpha$ given by $\sum_{j\in J}T_j^{\alpha}=1$ and the set of solutions depends on the closed multiplicative subgroup of $\mathbb {R}^{>}=(0,\infty)$ generated by the $T_j$ which is either $\{1\}$, $\mathbb {R}^{>}$ itself or $r^{\mathbb {Z}}=\{r^n\dvt n\in \mathbb {Z}\}$ for some $r>1$. The first case being trivial, the nontrivial Fixed Points in the second case are either Weibull distributions or their reciprocal reflections to the negative half line (when represented by random variables), while in the third case further periodic solutions arise. Our analysis builds on the observation that the logarithmic survival function of any Fixed Point is harmonic with respect to $\varLambda =\sum_{j\geq 1}\delta_{T_j}$, i.e. $\varGamma =\varGamma \star \varLambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet--Deny theorem.

  • A stochastic Fixed Point Equation for weighted minima and maxima
    Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 2008
    Co-Authors: Gerold Alsmeyer, Uwe Rösler
    Abstract:

    Given any finite or countable collection of real numbers Tj, j 2 J, we find all solutions F to the stochastic Fixed Point Equation W d = inf j2J T jWj,

  • A stochastic Fixed Point Equation related to weighted branching with deterministic weights.
    Electronic Journal of Probability, 2006
    Co-Authors: Gerold Alsmeyer, Uwe Rösler
    Abstract:

    For real numbers $C,T_{1},T_{2},...$ we find all solutions $\mu$ to the stochastic Fixed Point Equation $W \sim\sum_{j\ge 1}T_{j}W_{j}+C$, where $W,W_{1},W_{2},...$ are independent real-valued random variables with distribution $\mu$ and $\sim$ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of ${ R}_{*}={ R}\backslash\{0\}$ generated by the $T_{j}$. If this group is continuous, i.e. ${R}_{*}$ itself or the positive halfline ${R}_{+}$, then all nontrivial Fixed Points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Levy measure of any Fixed Point is harmonic with respect to $\Lambda=\sum_{j\ge 1}\delta_{T_{j}}$, i.e. $\Gamma=\Gamma\star\Lambda$, where $\star$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

  • Fixed Points with finite variance of a smoothing transformation
    Stochastic Processes and their Applications, 2003
    Co-Authors: Amke Caliebe, Uwe Rösler
    Abstract:

    Let T=(T1,T2,T3,...) be a sequence of real random variables. We investigate the following Fixed Point Equation for distributions [mu]: W[congruent with][summation operator]j=1[infinity] TjWj, where W,W1,W2,... have distribution [mu] and T,W1,W2,... are independent. The corresponding functional Equation is [phi](t)=E [product operator]j=1[infinity] [phi](tTj), where [phi] is a characteristic function. We consider solutions of the Fixed Point Equation with finite variance. Results about existence and uniqueness are derived. In the situation of solutions with zero expectation we give a representation of the characteristic functions of solutions and treat the question of moments and -Lebesgue densities. The article extends results on the case of non-negative T and non-negative solutions.

Maaike Verloop - One of the best experts on this subject based on the ideXlab platform.

  • NET-COOP - Stability Properties of Networks with Interacting TCP Flows
    Lecture Notes in Computer Science, 2009
    Co-Authors: Carl Graham, Philippe Robert, Maaike Verloop
    Abstract:

    The asymptotic behavior of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. For this model, each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. In this situation Graham and Robert [6] has shown that the invariant distributions are in a one-to-one correspondence with the solutions of a Fixed Point Equation in a finite dimensional space. The purpose of this paper is to investigate the problem of uniqueness of the equilibrium of these networks, i.e., the uniqueness of the solutions of the associated Fixed Point Equation. Uniqueness results of such solutions are proved for different topologies: rings, trees and a linear network and with various configurations for routes through nodes.

  • Stability Properties of Networks with Interacting TCP Flows
    2009
    Co-Authors: Carl Graham, Philippe Robert, Maaike Verloop
    Abstract:

    The equilibrium distributions of a Markovian model describing the interaction of several classes of permanent connections in a network are analyzed. It has been introduced by Graham and Robert. For this model each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. It has been shown that the invariant distributions are determined by the solutions of a Fixed Point Equation in a finite dimensional space. In this paper, several examples of these Fixed Point Equations are studied. The topologies investigated are rings, trees and a linear network, with various sets of routes through the nodes.

Philippe Robert - One of the best experts on this subject based on the ideXlab platform.

  • Impatience in mobile networks and its application to data pricing
    2015
    Co-Authors: Salah Eddine Elayoubi, Philippe Robert, Christine Fricker, Fabrice Guillemin, Bruno Sericola
    Abstract:

    We consider in this paper an important Quality of Experience (QoE) indicator in mobile networks that is reneging of users due to impatience. We specifically consider a cell under heavy load conditions and compute the reneging probability by using a fluid limit analysis. By solving the Fixed Point Equation, we obtain a new QoE perturbation metric quantifying the impact of reneging on the performance of the system. This metric is then used to devise a new pricing scheme accounting of reneging. We specifically propose several flavors of this pricing around the idea of having a flat rate for accessing the network and an elastic price related to the level of QoE perturbation induced by communications.

  • NET-COOP - Stability Properties of Networks with Interacting TCP Flows
    Lecture Notes in Computer Science, 2009
    Co-Authors: Carl Graham, Philippe Robert, Maaike Verloop
    Abstract:

    The asymptotic behavior of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. For this model, each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. In this situation Graham and Robert [6] has shown that the invariant distributions are in a one-to-one correspondence with the solutions of a Fixed Point Equation in a finite dimensional space. The purpose of this paper is to investigate the problem of uniqueness of the equilibrium of these networks, i.e., the uniqueness of the solutions of the associated Fixed Point Equation. Uniqueness results of such solutions are proved for different topologies: rings, trees and a linear network and with various configurations for routes through nodes.

  • Stability Properties of Networks with Interacting TCP Flows
    2009
    Co-Authors: Carl Graham, Philippe Robert, Maaike Verloop
    Abstract:

    The equilibrium distributions of a Markovian model describing the interaction of several classes of permanent connections in a network are analyzed. It has been introduced by Graham and Robert. For this model each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. It has been shown that the invariant distributions are determined by the solutions of a Fixed Point Equation in a finite dimensional space. In this paper, several examples of these Fixed Point Equations are studied. The topologies investigated are rings, trees and a linear network, with various sets of routes through the nodes.

  • Analysis of Loss Networks with Routing
    Annals of Applied Probability, 2006
    Co-Authors: Nelson Antunes, Philippe Robert, Christine Fricker, Danielle Tibi
    Abstract:

    This paper analyzes stochastic networks consisting of finite capacity nodes with different classes of requests which move according to some routing policy. The Markov processes describing these networks do not have, in general, reversibility properties so that the explicit expression of their invariant distribution is not known. A heavy traffic limit regime is considered: the arrival rates of calls as well as the capacities of the nodes are proportional to a factor going to infinity. It is proved that, in the limit, the associated rescaled Markov process converges to a deterministic dynamical system with a unique equilibrium Point characterized by a non-standard Fixed Point Equation.