arXiv subjects
Publications and source records attributed to Rafał Kapelko.
This paper studies the fundamental problem of data persistency for a general family of redundancy schemes in distributed storage systems, called replicated erasure codes. Namely, we analyze two strategies of replicated erasure codes distribution: random and symmetric. For both strategies we derive closed analytical and asymptotic formulas for expected data persistency despite nodes failure.
This paper deals with random sensors initially randomly deployed on the line according to general random process and on the plane according to two independent general random processes. The mobile robot with carrying capacity $k$ placed at the origin point is to move the sensors to achieve the general scheduling requirement such as coverage, connectivity and thus to satisfy the desired communication property in the network. We study tradeoffs between the energy consumption in robot's movement, the numbers of sensors $n$, the sensor range $r$, the interference distance $s$, and the robot capacity $k$ until completion of the coverage simultaneously with interference scheduling task. In this work, we obtain upper bounds for the energy consumption in robot's movement and obtain the sharp decrease in the total movement cost of the robot so as to provide the coverage simultaneously with interference requirement.
This paper investigates the problem of the minimilization of energy consumption in reallocation of wireless mobile sensors network (WMSN) to assure good communication without interference. Fix $d\in\mathbb{N}\setminus\{0\}.$ Assume $n$ sensors are initially randomly placed in the hyperoctant $[0,\infty)^d$ according to $d$ identical and independent Poisson processes each with arrival rate $λ>0.$ Let $0< s \le v$ be given real numbers. We are allowed to move the sensors, so that every two consecutive sensors are placed at distance greater than or equal to $s$ and less than or equal to $v.$ Fix $a\ge 1.$ Assume that $i-$th sensor is displaced a distance equal to $m(i).$ The cost measure for the displacement of the team of sensors is the sum $\sum_{i=1}^{n}d_i^a$ ($a-$total movement). In this work, we discover and explain a sharp decline and a sharp increase (a threshold phenomena) in the expected minimal $a-$total movement around the interference-connectivity distances $s,v$ equal to $\frac{1}λ.$
The present paper contains additional asymptotic result over an earlier investigation of Kapelko and Kranakis. Consider $n$ mobile sensors placed independently at random with the uniform distribution on the unit interval $[0,1]$. Fix $a$ an odd natural number. Let $X_i$ be the the $i-$th closest sensor to $0$ on the interval $[0,1].$ Then the following identity holds $$\sum_{i=1}^n\mathbf{E}\left[\left|X_i-\left(\frac{i}{n}-\frac{1}{2n}\right)\right|^a\right]=\frac{Γ\left(\frac{a}{2}+1\right)}{2^{\frac{a}{2}}(1+a)}\frac{1}{n^{\frac{a}{2}-1}}+O\left(\frac{1}{n^{\frac{a-1}{2}}}\right),$$ when $a$ is an odd natural number, where $Γ(z)$ is the Gamma function.
Consider the distance between two i.i.d. and independent Poisson processes with arrival rate $λ>0$ and respective arrival times $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ on a line. We give a closed analytical formula for the %expected distance to the power $a$ $\E{|X_{k+r}-Y_k|^a}, $ for any integer $k\ge 1, r\ge 0$ and $a\ge 1.$ The expected difference of the arrival times to the power $a$ between two i.i.d. and independent Poisson processes we represent as the combination of the Pochhammer polynomials. Especially, for $r=0$ and any positive integer $a,$ the following identity is valid $$ \E{|X_k-Y_k|^a}=\frac{a!}{λ^a}\frac{Γ\left(\frac{a}{2}+k\right)}{Γ(k)Γ\left(\frac{a}{2}+1\right)}, $$ where $Γ(z)$ is Gamma function.
Consider $n$ mobile sensors placed independently at random with the uniform distribution on a barrier represented as the unit line segment $[0,1]$. The sensors have identical sensing radius, say $r$. When a sensor is displaced on the line a distance equal to $d$ it consumes energy (in movement) which is proportional to some (fixed) power $a > 0$ of the distance $d$ traveled. The energy consumption of a system of $n$ sensors thus displaced is defined as the sum of the energy consumptions for the displacement of the individual sensors. We focus on the problem of energy efficient displacement of the sensors so that in their final placement the sensor system ensures coverage of the barrier and the energy consumed for the displacement of the sensors to these final positions is minimized in expectation. In particular, we analyze the problem of displacing the sensors from their initial positions so as to attain coverage of the unit interval and derive trade-offs for this displacement as a function of the sensor range. We obtain several tight bounds in this setting thus generalizing several of the results of [10] to any power $a >0$.