Oscillating random walk with a moving boundary

1994 ◽  
Vol 88 (1-3) ◽  
pp. 333-365
Author(s):  
Neal Madras ◽  
David Tanny
Keyword(s):  
Random Walk ◽  
Moving Boundary ◽  
Oscillating Random Walk

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

scholarly journals Recurrency of an Oscillating Random Walk

1978 ◽  
Vol 23 (1) ◽  
pp. 155-162 ◽  
Author(s):  
B. A. Rogozin ◽  
S. G. Foss
Keyword(s):  
Random Walk ◽  
Oscillating Random Walk

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

1986 ◽  
Vol 23 (03) ◽  
pp. 790-802 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
State Space ◽  
Complex Plane ◽  
General Class ◽  
Stochastic Behavior ◽  
Vacation Models ◽  
Server Vacations ◽  
Fixed Probability ◽  
Oscillating Random Walk

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules

1986 ◽  
Vol 23 (3) ◽  
pp. 790-802 ◽  
Author(s):  
J. Keilson ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
State Space ◽  
Complex Plane ◽  
General Class ◽  
Stochastic Behavior ◽  
Vacation Models ◽  
Server Vacations ◽  
Fixed Probability ◽  
Oscillating Random Walk

Processors handling multi-class traffic typically alternate between serving a particular class of traffic and performing other tasks, e.g., secondary service tasks or routine maintenance. The stochastic behavior of such systems is modeled by a newly introduced class of Bernoulli GI/G/1 vacation models. For this model, when a vacation is completed and customers are present, a customer is served. When a customer has just been served and other customers are present, the server accepts a customer with fixed probability p or commences a vacation of prespecified random duration with probability 1 – p. Whenever no customers are present, a vacation is taken. When p = 0 or p = 1 this schedule reduces to the previously introduced single service schedule and the exhaustive service schedule, respectively. An analysis of all three schedules on a state space incorporating server vacations is presented using simple methods in the complex plane. It is shown that the recent decomposition results for exhaustive service extend to the more general class of Bernoulli schedules.

scholarly journals First-passage time asymptotics over moving boundaries for random walk bridges

2018 ◽  
Vol 55 (2) ◽  
pp. 627-651 ◽  
Author(s):  
Fiona Sloothaak ◽  
Vitali Wachtel ◽  
Bert Zwart
Keyword(s):  
Random Walk ◽  
Moving Boundary ◽  
First Passage Time ◽  
Passage Time ◽  
Finite Variance ◽  
Tail Behavior ◽  
First Passage ◽  
The Impact ◽  
Return Point ◽  
Asymptotic Tail

Abstract We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

Heavy traffic limit theorems in fluctuation theory

1978 ◽  
Vol 10 (4) ◽  
pp. 852-866
Author(s):  
A. J. Stam
Keyword(s):  
Random Walk ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Central Limit ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
The Central Limit Theorem ◽  
Heavy Traffic Limit ◽  
Oscillating Random Walk

Let be a family of random walks with For ε↓0 under certain conditions the random walk U(∊)n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M∊ = max {U(∊)n, n ≧ 0}, v0 = min {n : U(∊)n = M∊}, v1 = max {n : U(∊)n = M∊}. The joint limiting distribution of ∊2σ∊–2v0 and ∊σ∊–2M∊ is determined. It is the same as for ∊2σ∊–2v1 and ∊σ–2∊M∊. The marginal ∊σ–2∊M∊ gives Kingman's heavy traffic theorem. Also lim ∊–1P(M∊ = 0) and lim ∊–1P(M∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U(∊)n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.

Wald's identity and random walk models for neuron firing

1976 ◽  
Vol 8 (02) ◽  
pp. 257-277 ◽  
Author(s):  
V. I. Kryukov
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Moving Boundary ◽  
First Passage Time ◽  
Model Parameters ◽  
First Passage ◽  
Fundamental Identity ◽  
Main Technique ◽  
Special Cases ◽  
Wide Range

A class of probability models for the firing of neurons is introduced and treated analytically. The cell membrane potential is assumed to be a one-dimensional random walk on the continuum; the first passage of the moving boundary triggers a nerve spike, an all-or-none event. The interspike interval distribution of first-passage time is shown to satisfy a Volterra integral equation suitable for numerical evaluation. Explicit solutions as well as their Laplace (Mellin) transforms are obtained for some special cases. The main technique used in this paper is the extension of Wald's fundamental identity of sequential analysis to a wide range of additive stochastic processes with an (eventually) linear boundary function. This identity is also useful in evaluating model parameters in terms of observed firing times, as well as providing a unified exposition of many earlier results.

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