6533b7dafe1ef96bd126d8cb

RESEARCH PRODUCT

Scaling behavior of an airplane-boarding model

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An airplane-boarding model, introduced earlier by Frette and Hemmer [Phys. Rev. E 85, 011130 (2012)], is studied with the aim of determining precisely its asymptotic power-law scaling behavior for a large number of passengers $N$. Based on Monte Carlo simulation data for very large system sizes up to $N={2}^{16}=65\phantom{\rule{0.16em}{0ex}}536$, we have analyzed numerically the scaling behavior of the mean boarding time $\ensuremath{\langle}{t}_{b}\ensuremath{\rangle}$ and other related quantities. In analogy with critical phenomena, we have used appropriate scaling Ans\"atze, which include the leading term as some power of $N$ (e.g., $\ensuremath{\propto}$${N}^{\ensuremath{\alpha}}$ for $\ensuremath{\langle}{t}_{b}\ensuremath{\rangle}$), as well as power-law corrections to scaling. Our results clearly show that $\ensuremath{\alpha}=1/2$ holds with a very high numerical accuracy ($\ensuremath{\alpha}=0.5001\ifmmode\pm\else\textpm\fi{}0.0001$). This value deviates essentially from $\ensuremath{\alpha}\ensuremath{\simeq}0.69$, obtained earlier by Frette and Hemmer from data within the range $2\ensuremath{\le}N\ensuremath{\le}16$. Our results confirm the convergence of the effective exponent ${\ensuremath{\alpha}}_{\mathrm{eff}}(N)$ to $1/2$ at large $N$ as observed by Bernstein. Our analysis explains this effect. Namely, the effective exponent ${\ensuremath{\alpha}}_{\mathrm{eff}}(N)$ varies from values about $0.7$ for small system sizes to the true asymptotic value $1/2$ at $N\ensuremath{\rightarrow}\ensuremath{\infty}$ almost linearly in ${N}^{\ensuremath{-}1/3}$ for large $N$. This means that the variation is caused by corrections to scaling, the leading correction-to-scaling exponent being $\ensuremath{\theta}\ensuremath{\approx}1/3$. We have estimated also other exponents: $\ensuremath{\nu}=1/2$ for the mean number of passengers taking seats simultaneously in one time step, $\ensuremath{\beta}=1$ for the second moment of ${t}_{b}$, and $\ensuremath{\gamma}\ensuremath{\approx}1/3$ for its variance.