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

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.

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 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

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.

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.

Records and Oscillating Random Walk

1976 ◽  
Vol 25 (1-4) ◽  
pp. 55-64 ◽  
Author(s):  
M. L . Aggarwal
Keyword(s):  
Random Walk ◽  
Simple Random Walk ◽  
Oscillating Random Walk

The distribution of the place of occurrence of the m-th record and of the maximum have been obtained in the case of an oRcillating random walk. Similar distributions have already been obtained for a simple random walk (both symmetric and unsymmetric) by F eller (1968) .

scholarly journals Oscillating Brownian motion

1978 ◽  
Vol 15 (02) ◽  
pp. 300-310 ◽  
Author(s):  
Julian Keilson ◽  
Jon A. Wellner
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
First Passage Time ◽  
Passage Time ◽  
Brownian Motion Process ◽  
First Passage ◽  
Special Cases ◽  
Motion Process ◽  
Transition Densities ◽  
Oscillating Random Walk

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

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