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Heavy traffic limit theorems in fluctuation theory

Published online by Cambridge University Press:  01 July 2016

A. J. Stam
A. J. Stam*
Affiliation:
Rijksuniversiteit Groningen
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Abstract

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.

Keywords

CONVERGENCE TO OSCILLATING RANDOM WALKSLADDER VARIABLESMAXIMUMQUEUEING

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978, pp. 852 - 866
Copyright
Copyright © Applied Probability Trust 1978 

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