Limit Theorems for Branching Stochastic Processes of Special Form

Sevast'yanov, Boris Alexandrovich

doi:10.1137/1102022

Cited by 49 publications

(30 citation statements)

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“…They were studied extensively in the literature; the classical references are Refs. [6][7][8]22,23,27,31,36,38,44] . The process considered in this article also appears in various models in queueing theory (see Refs.…”

Section: Introductionmentioning

confidence: 99%

Heavy-Tailed Branching Process with Immigration

2013

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In this article, we analyze a branching process with immigration defined recursively by random variables independent of B t . We assume that one of generic variables A and B has a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solution X t .We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.

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Section: Introductionmentioning

confidence: 99%

Heavy-Tailed Branching Process with Immigration

2013

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“…Branching processes find their origins in the work of Bienaymé [11] and Galton and Watson [12], the first asymptotic result in the theory of branching processes being obtained by Kolmogorov [13]. The first reference on branching with migration is [14]. Multi-type branching processes in a varying environment without migration have been studied in [15].…”

Section: Introductionmentioning

confidence: 99%

Expected waiting time in symmetric polling systems with correlated walking times

Altman

Fiems

2007

Queueing Syst

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Polling systems have been extensively studied, and have found many applications. They have often been used for studying wired local area networks such as token passing rings and wireless local area networks such as bluetooth. In this contribution we relax one of the main restrictions on the statistical assumptions under which polling systems have been analyzed. Namely, we allow correlation between walking times. We consider (i) the gated regime where a gate closes whenever the server arrives at a queue. It then serves at that queue all customers who were present when the gate closes. (ii) The exhaustive regime in which the server remains at a queue till it empties.Our analysis is based on stochastic recursive equations related to branching processes with migration with a random environment. In addition to our derivation of expected waiting times for polling systems with correlated walking times, we set the foundations for computing second order statistics of the general multi-dimensional stochastic recursions.

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“…For the generating functions F(t, x) Z V{Pkl(t) 1, k2(t) 2 pk3(t) 311'w2w32 3 the system of equations (2) d (F,, F2, V3), F,(0, x) x, k 1, 2, 3, was obtained in [1]. In the case under consideration, the last two equations in (2) are solved separately.…”

Section: Introductionmentioning

confidence: 99%

“…[2]. All limit distributions for the case when final type particles are present in a process with two types of particles have been obtained in [3]. In Section of the present note a new form of the limit distribution for processes with transitions T1 T2 T3 is obtained.…”

Section: Introductionmentioning

confidence: 99%

“…In Section of the present note a new form of the limit distribution for processes with transitions T1 T2 T3 is obtained. In Section 2 it is shown that, in some instances, the multidimensional limit distributions for processes with transitions T2 T1 T3 are products of the one-dimensional distributions obtained in [3]. A study of processes with n 3 types of particles permits it to establish the form of the limit distributions which may occur in the case of processes of the form T1 T2 T, for arbitrary n. Xj lx Xk Since fl(x) is linear in xl, ali bl 0.…”

Section: Introductionmentioning

confidence: 99%

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