The Buffon needle problem for Lévy distributed spacings and renewal theory

What is the probability that a needle dropped at random on a set of points scattered on a line segment does not fall on any of them? We compute the exact scaling expression of this hole probability when the spacings between the points are independent identically distributed random variables with a power-law distribution of index less than unity, implying that the average spacing diverges. The theoretical framework for such a setting is renewal theory, to which the present study brings a new contribution. The question posed here is also related to the study of some correlation functions of simple models of statistical physics.

Availability and maintenance modeling of a batch service queuing system

PurposeThe purpose of the paper is to analyze reliability characteristics of batch service queuing system with a single server model that envisages Poisson input process and exponential service times under first come, first served (FCFS) queue discipline.Design/methodology/approachWith the help of renewal theory and stochastic processes, a model has been designed to discuss the reliability and its characteristics.FindingsThe instantaneous and steady-state availability along with the maintenance model of the systems subject to generalized M/Mb/1 queuing model is derived, and a few particular cases for availability are obtained as well. For supporting the developed model, a case study on electrical distribution system (EDS) has been illustrated, which also includes a comparison for the system subject to M/Mb/1 queuing model and the system without any queue (delay).Originality/valueIt is a quite realistic model that may aid to remove congestion in the system while repairing.

Optimizations of generational garbage collections based on continuous damage process

In the computer science community, garbage collection is a dynamic storage management technology to ensure the reliability of computer systems. In this paper, we consider two garbage collection policies to meet the goal of time consumption for a generational garbage collector when increase in objects might be unclear at discrete times for the high frequency of computer processes. That is, (a) tenuring collection is triggered at the [Formula: see text]th minor collection preventively or at a threshold amount [Formula: see text] of surviving objects correctively, and (b) major collection is made at discrete times [Formula: see text] for a given [Formula: see text] or at the [Formula: see text]th collection including minor and tenuring collections. Using the damage process and renewal theory, the expected cost rates are obtained, and their optimal policies for tenuring and major collection are discussed analytically and computed numerically.

Reconceptualising Atrial Fibrillation Using Renewal Theory: A Novel Approach to the Assessment of Atrial Fibrillation Dynamics

Despite a century of research, the mechanisms of AF remain unresolved. A universal motif within AF research has been unstable re-entry, but this remains poorly characterised, with competing key conceptual paradigms of multiple wavelets and more driving rotors. Understanding the mechanisms of AF is clinically relevant, especially with regard to treatment and ablation of the more persistent forms of AF. Here, the authors outline the surprising but reproducible finding that unstable re-entrant circuits are born and destroyed at quasi-stationary rates, a finding based on a branch of mathematics known as renewal theory. Renewal theory may be a way to potentially unify the multiple wavelet and rotor theories. The renewal rate constants are potentially attractive because they are temporally stable parameters of a defined probability distribution (the exponential distribution) and can be estimated with precision and accuracy due to the principles of renewal theory. In this perspective review, this new representational architecture for AF is explained and placed into context, and the clinical and mechanistic implications are discussed.

Optimizations of discrete garbage collections in computer reliability systems

In computer science community, garbage collection is a dynamic storage management technology to ensure the reliability of computer systems. In this paper, we consider two discrete garbage collection policies to meet the goal of time consumption for a generational garbage collector. That is, garbage collections occur at a nonhomogeneous Poisson process, (a) tenuring collection is triggered at the Nth minor collection preventively or at a threshold amount [Formula: see text] of surviving objects correctively, whichever takes place first, and (b), tenuring collection is triggered at the first collection when the amount of surviving objects has exceeded a threshold level [Formula: see text] and major collection is triggered at discrete times kT for a given T. Using the damage process and renewal theory, the expected cost rates are obtained, and their optimal policies for tenuring and major collections are discussed analytically and computed numerically.