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Tail asymptotics for cumulative processes sampled at heavy-tailed random times with applications to queueing models in Markovian environments
This paper considers the tail asymptotics for a cumulative process sampled at a heavy-tailed random time . The main contribution of
this paper is to establish several sufficient conditions for the asymptotic
equality as , where and is a certain
positive constant. The main results of this paper can be used to obtain the
subexponential asymptotics for various queueing models in Markovian
environments. As an example, using the main results, we derive subexponential
asymptotic formulas for the loss probability of a single-server finite-buffer
queue with an on/off arrival process in a Markovian environment
Universal behavior of QCD amplitudes at high energy from general tools of statistical physics
We show that high energy scattering is a statistical process essentially
similar to reaction-diffusion in a system made of a finite number of particles.
The Balitsky-JIMWLK equations correspond to the time evolution law for the
particle density. The squared strong coupling constant plays the role of the
minimum particle density. Discreteness is related to the finite number of
partons one may observe in a given event and has a sizeable effect on physical
observables. Using general tools developed recently in statistical physics, we
derive the universal terms in the rapidity dependence of the saturation scale
and the scaling form of the amplitude, which come as the leading terms in a
large rapidity and small coupling expansion.Comment: 14 pages, 2 figures; v2: Secs. 2 and 3 substantially rewritten and
Sec. 4 expanded in order to make more explicit the connection with
statistical physics. Acknowledgment and reference added. Results and
conclusions unchanged. To appear in Phys. Lett.
Customer sojourn time in GI/G/1 feedback queue in the presence of heavy tails
We consider a single-server GI/GI/1 queueing system with feedback. We assume
the service times distribution to be (intermediate) regularly varying. We find
the tail asymptotics for a customer's sojourn time in two regimes: the customer
arrives in an empty system, and the customer arrives in the system in the
stationary regime. In particular, in the case of Poisson input we use the
branching processes structure and provide more precise formulae. As auxiliary
results, we find the tail asymptotics for the busy period distribution in a
single-server queue with an intermediate varying service times distribution and
establish the principle-of-a-single-big-jump equivalences that characterise the
asymptotics.Comment: 34 pages, 4 figures, to appear in Journal of Statistical Physic
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