Simulation of a Robust Communication Protocol for Sensor Data Acquisition

Panholzer, Georg; Veichtlbauer, Armin; Dorfinger, Peter; Schrittesser, Ulrich

doi:10.1109/icwmc.2010.85

Cited by 2 publications

(1 citation statement)

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“…Finally, we would like to point out that Alois Panholzer has independently obtained related results [15], notably a detailed description of limiting distributions.…”

Section: Resultsmentioning

confidence: 88%

Counting Defective Parking Functions

Cameron¹,

Johannsen²,

Prellberg

et al. 2008

Electron. J. Combin.

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Suppose that m drivers each choose a preferred parking space in a linear car park with n spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with a larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if k drivers fail to park, we have a defective parking function of defect k. Let cp(n, m, k) be the number of such functions.In this paper, we establish a recurrence relation for the numbers cp(n, m, k), and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of cp(n, m, k). In particular, for the case m = n, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of n, is the Rayleigh distribution. On the other hand, in the case m = ω(n), the probability that all spaces are occupied tends asymptotically to one.

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“…Finally, we would like to point out that Alois Panholzer has independently obtained related results [15], notably a detailed description of limiting distributions.…”

Section: Resultsmentioning

confidence: 88%