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.

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.

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

scholarly journals An Exploration of the Triplet Periodicity in Nucleotide Sequences with a Mature Self-Adaptive Spectral Rotation Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Bo Chen ◽  
Ping Ji
Keyword(s):  
Random Walk ◽  
Complex Plane ◽  
Query Sequence ◽  
Nucleotide Sequences ◽  
Sequence Pattern ◽  
Coding Regions ◽  
Persistent Pattern ◽  
Triplet Periodicity ◽  
Self Adaptive

Previously, for predicting coding regions in nucleotide sequences, a self-adaptive spectral rotation (SASR) method has been developed, based on a universal statistical feature of the coding regions, named triplet periodicity (TP). It outputs a random walk, that is, TP walk, in the complex plane for the query sequence. Each step in the walk is corresponding to a position in the sequence and generated from a long-term statistic of the TP in the sequence. The coding regions (TP intensive) are then visually discriminated from the noncoding ones (without TP), in the TP walk. In this paper, the behaviors of the walks for random nucleotide sequences are further investigated qualitatively. A slightly leftward trend (a negative noise) in such walks is observed, which is not reported in the previous SASR literatures. An improved SASR, named the mature SASR, is proposed, in order to eliminate the noise and correct the TP walks. Furthermore, a potential sequence pattern opposite to the TP persistent pattern, that is, the TP antipersistent pattern, is explored. The applications of the algorithms on simulated datasets show their capabilities in detecting such a potential sequence pattern.

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