The Annals of Probability

On a Class of Stochastic Recursive Sequences Arising in Queueing Theory

Francois Baccelli and Zhen Liu

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This paper is concerned with a class of stochastic recursive sequences that arise in various branches of queueing theory. First, we make use of Kingman's subadditive ergodic theorem to determine the stability region of this type of sequence, or equivalently, the condition under which they converge weakly to a finite limit. Under this stability condition, we also show that these sequences admit a unique finite stationary regime and that regardless of the initial condition, the transient sequence couples in finite time with this uniquely defined stationary regime. When this stability condition is not satisfied, we show that the sequence converges a.s. to $\infty$ and that certain increments of the process form another type of stochastic recursive sequence that always admit at least one stationary regime. Finally, we give sufficient conditions for this increment sequence to couple with this stationary regime.

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Ann. Probab., Volume 20, Number 1 (1992), 350-374.

First available in Project Euclid: 19 April 2007

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Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 60F20: Zero-one laws 60G10: Stationary processes 60G17: Sample path properties 60G55: Point processes 60K25: Queueing theory [See also 68M20, 90B22] 68Q75 68Q90 68R10: Graph theory (including graph drawing) [See also 05Cxx, 90B10, 90B35, 90C35] 93D05: Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.) 93E03: Stochastic systems, general 93E15: Stochastic stability

Stochastic recursive sequences queueing theory ergodic theory stationary processes coupling


Baccelli, Francois; Liu, Zhen. On a Class of Stochastic Recursive Sequences Arising in Queueing Theory. Ann. Probab. 20 (1992), no. 1, 350--374. doi:10.1214/aop/1176989931.

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