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Exit Probabilities and Balayage of Constrained Random Walks

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 Added by Ali Devin Sezer Dr.
 Publication date 2015
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and research's language is English




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Let $X$ be the constrained random walk on ${mathbb Z}_+^d$ representing the queue lengths of a stable Jackson network and $x$ its initial position. Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. $p_n doteq P_x(tau_n < tau_0)$ is a key performance measure for the queueing system represented by $X$, stability implies $p_nrightarrow 0$ exponentially. Currently the only analytic method available to approximate $p_n$ is large deviations analysis, which gives the exponential decay rate of $p_n$. Finer results are available via rare event simulation. The present article develops a new method to approximate $p_n$ and related expectations. The method has two steps: 1) with an affine transformation, move the origin onto the exit boundary of $tau_n$, take limits to remove some of the constraints on the dynamics, this yields a limit unstable constrained walk $Y$ 2) Construct a basis of harmonic functions of $Y$ and use them to apply the classical superposition principle of linear analysis. The basis functions are linear combinations of $log$-linear functions and come from solutions of harmonic systems, which are graphs whose vertices represent points on the characteristic surface of $Y$, the edges between the vertices represent conjugacy relations between the points, the loops represent membership in the boundary characteristic surfaces. Using our method we derive explicit, simple and almost exact formulas for $P_x(tau_n < tau_0)$ for $d$-tandem queues, similar to the product form formulas for the stationary distribution of $X$. The same method allows us to approximate the Balayage operator mapping $f$ to $x rightarrow {mathbb E}_x left[ f(X_{tau_n}) 1_{{tau_n < tau_0}} right]$ for a range of stable constrained random walks in $2$ dimensions. We indicate how the ideas of the paper relate to more general processes and exit boundaries.



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273 - Ali Devin Sezer 2021
Let $X$ be the constrained random walk on $mathbb{Z}_+^d$ $d >2$, having increments $e_1$, $-e_i+e_{i+1}$ $i=1,2,3,...,d-1$ and $-e_d$ with probabilities $lambda$, $mu_1$, $mu_2$,...,$mu_d$, where ${e_1,e_2,..,e_d}$ are the standard basis vectors. The process $X$ is assumed stable, i.e., $lambda < mu_i$ for all $i=1,2,3,...,d.$ Let $tau_n$ be the first time the sum of the components of $X$ equals $n$. We derive approximation formulas for the probability ${mathbb P}_x(tau_n < tau_0)$. For $x in bigcup_{i=1}^d Big{x in {mathbb R}^d_+: sum_{j=1}^{i} x(j)$ $> left(1 - frac{log lambda/min mu_i}{log lambda/mu_i}right) Big}$ and a sequence of initial points $x_n/n rightarrow x$ we show that the relative error of the approximation decays exponentially in $n$. The approximation formula is of the form ${mathbb P}_y(tau < infty)$ where $tau$ is the first time the sum of the components of a limit process $Y$ is $0$; $Y$ is the process $X$ as observed from a point on the exit boundary except that it is unconstrained in its first component (in particular $Y$ is an unstable process); $Y$ and ${mathbb P}_y(tau< infty)$ arise naturally as the limit of an affine transformation of $X$ and the probability ${mathbb P}_x(tau_n < tau_0).$ The analysis of the relative error is based on a new construction of supermartingales. We derive an explicit formula for ${mathbb P}_y(tau < infty)$ in terms of the ratios $lambda/mu_i$ which is based on the concepts of harmonic systems and their solutions and conjugate points on a characteristic surface associated with the process $Y$; the derivation of the formula assumes $mu_i eq mu_j$ for $i eq j.$
Let $X$ be the constrained random walk on ${mathbb Z}_+^2$ having increments $(1,0)$, $(-1,1)$, $(0,-1)$ with jump probabilities $lambda(M_k)$, $mu_1(M_k)$, and $mu_2(M_k)$ where $M$ is an irreducible aperiodic finite state Markov chain. The process $X$ represents the lengths of two tandem queues with arrival rate $lambda(M_k)$, and service rates $mu_1(M_k)$, and $mu_2(M_k)$. We assume that the average arrival rate with respect to the stationary measure of $M$ is less than the average service rates, i.e., $X$ is assumed stable. Let $tau_n$ be the first time when the sum of the components of $X$ equals $n$ for the first time. Let $Y$ be the random walk on ${mathbb Z} times {mathbb Z}_+$ having increments $(-1,0)$, $(1,1)$, $(0,-1)$ with probabilities $lambda(M_k)$, $mu_1(M_k)$, and $mu_2(M_k)$. Let $tau$ be the first time the components of $Y$ are equal. For $x in {mathbb R}_+^2$, $x(1) + x(2) < 1$, $x(1) > 0$, and $x_n = lfloor nx rfloor$, we show that $P_{(n-x_n(1),x_n(2)),m)}( tau < infty)$ approximates $P_{(x_n,m)}(tau_n < tau_0)$ with exponentially vanishing relative error as $nrightarrow infty$. For the analysis we define a characteristic matrix in terms of the jump probabilities of $(X,M).$ The $0$-level set of the characteristic polynomial of this matrix defines the characteristic surface; conjugate points on this surface and the associated eigenvectors of the characteristic matrix are used to define (sub/super) harmonic functions which play a fundamental role both in our analysis and the computation / approximation of $P_{(y,m)}(tau < infty).$
Tight estimates of exit/containment probabilities are of particular importance in many control problems. Yet, estimating the exit/containment probabilities is non-trivial: even for linear systems (Ornstein-Uhlenbeck processes), the containment probability can be computed exactly for only some particular values of the system parameters. In this paper, we derive tight bounds on the containment probability for a class of nonlinear stochastic systems. The core idea is to compare the pull strength (how hard the deterministic part of the system dynamics pulls towards the origin) experienced by the nonlinear system at hand with that of a well-chosen process for which tight estimates of the containment probability are known or can be numerically obtained (e.g. an Ornstein-Uhlenbeck process). Specifically, the main technical contribution of this paper is to define a suitable dominance relationship between the pull strengths of two systems and to prove that this dominance relationship implies an order relationship between their containment probabilities. We also discuss the link with contraction theory and highlight some examples of applications.
The reproduction speed of a continuous-time branching random walk is proportional to a positive parameter $lambda$. There is a threshold for $lambda$, which is called $lambda_w$, that separates almost sure global extinction from global survival. Analogously, there exists another threshold $lambda_s$ below which any site is visited almost surely a finite number of times (i.e.~local extinction) while above it there is a positive probability of visiting every site infinitely many times. The local critical parameter $lambda_s$ is completely understood and can be computed as a function of the reproduction rates. On the other hand, only for some classes of branching random walks it is known that the global critical parameter $lambda_w$ is the inverse of a certain function of the reproduction rates, which we denote by $K_w$. We provide here new sufficient conditions which guarantee that the global critical parameter equals $1/K_w$. This result extends previously known results for branching random walks on multigraphs and general branching random walks. We show that these sufficient conditions are satisfied by periodic tree-like branching random walks. We also discuss the critical parameter and the critical behaviour of continuous-time branching processes in varying environment. So far, only examples where $lambda_w=1/K_w$ were known; here we provide an example where $lambda_w>1/K_w$.
148 - Chuang Xu , Arno Berger 2017
This paper studies best finitely supported approximations of one-dimensional probability measures with respect to the $L^r$-Kantorovich (or transport) distance, where either the locations or the weights of the approximations atoms are prescribed. Necessary and sufficient optimality conditions are established, and the rate of convergence (as the number of atoms goes to infinity) is discussed. In view of emerging mathematical and statistical applications, special attention is given to the case of best uniform approximations (i.e., all atoms having equal weight). The approach developed in this paper is elementary; it is based on best approximations of (monotone) $L^r$-functions by step functions, and thus different from, yet naturally complementary to, the classical Voronoi partition approach.
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