arXiv subjects

Gianmarco Bet

Publications and source records attributed to Gianmarco Bet.

24 records · Page 2Linked to original sources

arXiv2020

Effect of energy degeneracy on the transition time for a series of metastable states: application to Probabilistic Cellular Automata

We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model is peculiar because of the presence of three metastable states, two of which are degenerate with respect to their energy. We identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability. Further, we also identify the metastable states according to the potential-theoretic approach to metastability, and this allows us to give precise asymptotics for the expected transition time from any such metastable state to the stable state.

math.PR↗

arXiv2020

Weighted Dyck paths for nonstationary queues

We consider a model for a queue in which only a fixed number $N$ of customers can join. Each customer joins the queue independently at an exponentially distributed time. Assuming further that the service times are independent and follow an exponential distribution, this system can be described as a two-dimensional Markov process on a finite triangular region $\mathfrak S$ of the square lattice. We interpret the resulting random walk on $\mathfrak S$ as a Dyck path that is weighted according to some state-dependent transition probabilities that are constant along one axis, but are rather general otherwise. We untangle the resulting intricate combinatorial structure by introducing appropriate generating functions that exploit the recursive structure of the model. This allows us to derive a fully explicit expression for the probability density function of the number of customers served in any busy period (equivalently, of the length of any excursion of the Dyck path above the diagonal) as a weighted sum with alternating sign over a certain subclass of Dyck paths, whose study is of independent interest.

math.PR↗

arXiv2018

An alternative approach to heavy-traffic limits for finite-pool queues

We consider a model for transitory queues in which only a finite number of customers can join. The queue thus operates over a finite time horizon. In this system, also known as the $Δ_{(i)}/G/1$ queue, the customers decide independently when to join the queue by sampling their arrival time from a common distribution. We prove that, when the queue satisfies a certain heavy-traffic condition and under the additional assumption that the second moment of the service time is finite, the rescaled queue length process converges to a reflected Brownian motion with parabolic drift. Our result holds for general arrival times, thus improving on an earlier result which assumes exponential arrival times.

math.PR↗

arXiv2017

Big jobs arrive early: From critical queues to random graphs

We consider a queue to which only a finite pool of $n$ customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement $S$ arrives to the queue after an exponentially distributed time with mean $S^{-α}$ for some $α\in[0,1]$; so larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: $α= 0$ gives the so-called $Δ_{(i)}/G/1$ queue and $α= 1$ is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size $n$ grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.

math.PR↗

arXiv2016

Finite-pool queues with heavy-tailed services

We consider the $Δ_{(i)}/G/1$ queue, in which a a total of $n$ customers independently demand service after an exponential time. We focus on the case of heavy-tailed service times, and assume that the tail of the service time distribution decays like $x^{-α}$, with $α\in (1,2)$. We consider the asymptotic regime in which the population size grows to infinity and establish that the scaled queue length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the headstart that is needed to create a long period of activity. This result should be contrasted with the case of light-tailed service times, which was shown to have a similar scaling limit, but then with a Brownian motion instead of an α-stable process.

math.PR↗

arXiv2015

Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

We consider a single server queue that serves a finite population of $n$ customers that will enter the queue (require service) only once, also known as the $Δ_{(i)}/G/1$ queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets $n$ and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.

math.PR↗