Harish Pillai | Searcharxiv

arXiv2024

Optimal moments on redundancies in job cloning

We consider the problem of job assignment where a master server aims to compute some tasks and is provided a few child servers to compute under a uniform straggling pattern where each server is equally likely to straggle. We distribute tasks to the servers so that the master is able to receive most of the tasks even if a significant number of child servers fail to communicate. We first show that all \textit{balanced} assignment schemes have the same expectation on the number of distinct tasks received and then study the variance. We show constructions using a generalization of ``Balanced Incomplete Block Design''\cite{doi:10.1111/j.1469-1809.1939.tb02219.x,sprott1955} minimizes the variance, and constructions based on repetition coding schemes attain the largest variance.

cs.DC↗

arXiv2012

Analysis of Stable Periodic Orbits in the 1-D Linear Piecewise-smooth Discontinuous Map

In this paper we consider a one dimensional liner piecewise-smooth discontinuous map. It is well known that stable periodic orbits exist in this type of map for a specific parameter region. It is also known that the corresponding bifurcation phenomena termed as "period adding bifurcation" exhibits a special structure for such case. In last couple of years, various authors have analyzed this structure using border collision bifurcation (BCB) curves and given characterization for various parameter regions. In this paper we have analyzed a specific parameter space using a different method. We show that this method enables us to pose some interesting problems like: is there any relationship between the period of an orbit and the cardinality of orbits? We prove that such a relationship exists and exactly ϕ(n) orbits exist for a period n where ϕ is the Euler's number. We propose an algorithm which gives the location of fixed points of all such ϕ(n) distinct orbits and gives their range of existence with respect to parameters. Further, we show that using a novel algorithm the computations to find their range of existence can be reduced.

math.DS↗