arXiv subjects

Michael A. Zazanis

Publications and source records attributed to Michael A. Zazanis.

6 recordsLinked to original sources

arXiv2026

Gated Infinite Server Queues in Light Traffic

We consider $M/G/\infty$ queues with gated service and obtain results on the distribution of the stage length and the number of customers served in a stage when the system is stationary. The stage length density is expressed as an infinite series of terms, involving the solution of an infinite system of linear equations. The convergence of a sequence of solutions arising from truncations of the infinite system is established in the light traffic case. Analogous results are established for a similar $GI/M/\infty$ gated system.

math.PR↗

arXiv2026

Generalized Coverage Processes with Infinitely Divisible Finite Dimensional Distributions

In this paper we define a class of coverage processes with infinitely divisible finite dimensional distributions and a particular type of correlation structure that can be thought of as generalizations of the classical Ornstein--Uhlenbeck process and which include coverage processes such as the $M/GI/\infty$ process. We show how such processes arise naturally as limits of superpositions of independent ON/OFF Markov processes with different parameters by formulating an appropriate limit theorem. Various examples of processes of this type are given.

math.PR↗

arXiv2023

Ruin Theory Problems in Simple SDE Models with Large Deviation Asymptotics

We examine hitting probability problems for Ornstein-Uhlenbeck (OU) processes and Geometric Brownian motions (GBM) with respect to exponential boundaries related to problems arising in risk theory and asset and liability models in pension funds. In Section 2 we consider the OU process described by the Stochastic Differential Equation (SDE) $dX_t = μX_t dt + σdW_t$ with $X_0=x_0$ evolving between a lower and an upper deterministic exponential boundary. Both the finite horizon ``ruin probability'' problem and the corresponding infinite horizon problem is examined in the low noise case, using the Wentzell-Freidlin approach in order to obtain logarithmic asymptotics for the probability of hitting either the lower or the upper boundary. The resulting variational problems are studied in detail. The exponential rate characterizing the ruin probability and the ``path to ruin'' are obtained by their solution. Logarithmic asymptotics for the meeting probability in a pair of OU processes with different positive drift coefficients, driven by independent Brownian motions is also obtained using Wentzell-Freidlin techniques. The optimal paths followed by the two processes and the meeting time $T$ are determined by solving a variational problem with transversality conditions. In Section 3 a corresponding problem involving a Geometric Brownian motion is considered. Since in this case, an exact, closed form solution is also available and we take advantage of this situation in order to explore numerically the quality of the Large Deviations results obtained using the Wentzell-Freidlin approach.

math.PR↗

arXiv2021

Age of information without service preemption

When designing a message transmission system, from the point of view of making sure that the information transmitted is as fresh as possible, two rules of thumb seem reasonable: use small buffers and adopt a last-in-first-out policy. In this paper, we measure freshness of information using the "age of information" performance measure. Considering it as a stochastic process operating in a stationary regime, we compute not just the first moment but the whole marginal distribution of the age of information (something important in applications) for two well-performing systems. In neither case do we allow for preemption of the message being processed because this may be difficult to implement in practice. We assume that the arrival process is Poisson and that the messages have independent sizes (service times) with common distribution. We use Palm and Markov-renewal theory to derive explicit results for Laplace transforms. In particular, this approach can be used to analyze more complex last-in-first-out systems with larger buffer sizes.

cs.PF↗

arXiv2021

Age of information distribution under dynamic service preemption

Age of Information (AoI) has emerged as an important quality-of-service measure for applications that prioritize delivery of the freshest information, e.g., virtual or augmented reality over mobile devices and wireless sensor networks used in the control of cyber-physical systems. We derive the Laplace transform of the stationary AoI for the M/GI/1/2 system with a "dynamic" service preemption and pushout policy depending on the existing service time of the in-service message. Thus, our system generalizes both the static M/GI/1/2 queue-pushout system without service preemption and the M/GI/1/1 bufferless system with service preemption - two systems considered to provide very good AoI performance. Based on our analysis, for a service-time distribution that is a mixture of deterministic and exponential, we numerically show that the dynamic policy has lower mean AoI than that of these two static policies and also that of the well studied M/GI/1/1 blocking system.

cs.PF↗

arXiv2016

Polynomial approximations to continuous functions and stochastic compositions

This paper presents a stochastic approach to theorems concerning the behavior of iterations of the Bernstein operator $B_n$ taking a continuous function $f \in C[0,1]$ to a degree-$n$ polynomial when the number of iterations $k$ tends to infinity and $n$ is kept fixed or when $n$ tends to infinity as well. In the first instance, the underlying stochastic process is the so-called Wright-Fisher model, whereas, in the second instance, the underlying stochastic process is the Wright-Fisher diffusion. Both processes are probably the most basic ones in mathematical genetics. By using Markov chain theory and stochastic compositions, we explain probabilistically a theorem due to Kelisky and Rivlin, and by using stochastic calculus we compute a formula for the application of $B_n$ a number of times $k=k(n)$ to a polynomial $f$ when $k(n)/n$ tends to a constant.

math.PR↗