scholarly journals Convergence of excursion point processes and its applications to functional limit theorems of Markov processes on a half-line

Bernoulli ◽  
2008 ◽  
Vol 14 (4) ◽  
pp. 963-987 ◽  
Author(s):  
Kouji Yano
Keyword(s):  
Markov Processes ◽  
Limit Theorems ◽  
Point Processes ◽  
Half Line ◽  
Functional Limit ◽  
Functional Limit Theorems

A Markovian analysis of additive-increase multiplicative-decrease algorithms

2002 ◽  
Vol 34 (01) ◽  
pp. 85-111 ◽  
Author(s):  
Vincent Dumas ◽  
Fabrice Guillemin ◽  
Philippe Robert
Keyword(s):  
Communication Networks ◽  
Markov Processes ◽  
Limit Theorems ◽  
Invariant Measures ◽  
Point Of View ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Transient Behaviour ◽  
Additive Increase ◽  
Probabilistic Point

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

A Markovian analysis of additive-increase multiplicative-decrease algorithms

2002 ◽  
Vol 34 (1) ◽  
pp. 85-111 ◽  
Author(s):  
Vincent Dumas ◽  
Fabrice Guillemin ◽  
Philippe Robert
Keyword(s):  
Communication Networks ◽  
Markov Processes ◽  
Limit Theorems ◽  
Invariant Measures ◽  
Point Of View ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Transient Behaviour ◽  
Additive Increase ◽  
Probabilistic Point

The additive-increase multiplicative-decrease (AIMD) schemes designed to control congestion in communication networks are investigated from a probabilistic point of view. Functional limit theorems for a general class of Markov processes that describe these algorithms are obtained. The asymptotic behaviour of the corresponding invariant measures is described in terms of the limiting Markov processes. For some special important cases, including TCP congestion avoidance, an important autoregressive property is proved. As a consequence, the explicit expression of the related invariant probabilities is derived. The transient behaviour of these algorithms is also analysed.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (03) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

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