AIMD algorithms and exponential functionals

Nonparametric estimation of jump rates for a specific class of piecewise deterministic Markov processes

“…and the law of the random variable Y δ k is an exponential translated by Z δ k−1 and of parameter λ/δc. Then (20,20) i.i.d. All the conditions of Assumption (S) are satisfied, except point a.…”

“…The TCP (transmission control protocol) (see Dumas et al [17], Guillemin et al [20] for instance) is one of the main data transmission protocol in Internet. The maximum number of packets that can be sent at time t k in a round is a random variable X t k .…”

Long-time behavior and Darwinian optimality for an asymmetric size-structured branching process

“…This result generalizes part the results in Hall Wake 1990). As we will see in its proof, using (Guillemin et al 2004), we can also simplify this expression in some special cases. However, we will not use this explicit expression to calculate the moments of γ .…”

“…where P(Θ = θ) = θ , P(Θ = 1 − θ) = 1 − θ and E is exponentially distributed with parameter 1/α. This equation was studied in particular in (Guillemin et al 2004, Section 3). By using (Guillemin et al 2004, Proposition 5) with the following notation, coming from their setting: β ∈ (0, 1), X = ln(Θ)/ ln(β) and let E 0 ∼ E(1), we obtain…”

“…To retrieve a fully symmetric model, simply take α 0 = α 1 (physiological symmetry) and θ 0 = θ 1 = 1/2 (morphological symmetry). This model belongs to the class of growth-fragmentation dynamics that has attracted a lot of attention in the litterature, see for instance (Michel 2006a, b;Olivier 2017;Bertoin and Watson 2018;Campillo et al 2017;Doumic et al 2015;Guillemin et al 2004;Cloez 2017; and references therein, among many others. However, it is more general than classical size-structured growth-fragmentation models as, to our best knowledge, it is the first model to take physiological and morphological asymmetry into account.…”