The Annals of Applied Probability

AIMD algorithms and exponential functionals

Fabrice Guillemin, Philippe Robert, and Bert Zwart

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The behavior of a connection transmitting packets into a network according to a general additive-increase multiplicative-decrease (AIMD) algorithm is investigated. It is assumed that loss of packets occurs in clumps. When a packet is lost, a certain number of subsequent packets are also lost (correlated losses). The stationary behavior of this algorithm is analyzed when the rate of occurrence of clumps becomes arbitrarily small. From a probabilistic point of view, it is shown that exponential functionals associated to compound Poisson processes play a key role. A formula for the fractional moments and some density functions are derived. Analytically, to get the explicit expression of the distributions involved, the natural framework of this study turns out to be the $q$-calculus. Different loss models are then compared using concave ordering. Quite surprisingly, it is shown that, for a fixed loss rate, the correlated loss model has a higher throughput than an uncorrelated loss model.

Article information

Ann. Appl. Probab., Volume 14, Number 1 (2004), 90-117.

First available in Project Euclid: 3 February 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 90B18: Communication networks [See also 68M10, 94A05]
Secondary: 68M12: Network protocols

Communication protocols exponential functionals compound Poisson processes autoregressive processes $q$-hypergeometric functions


Guillemin, Fabrice; Robert, Philippe; Zwart, Bert. AIMD algorithms and exponential functionals. Ann. Appl. Probab. 14 (2004), no. 1, 90--117. doi:10.1214/aoap/1075828048.

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