Airplane Boarding, Disk Scheduling and Space-Time Geometry

Analysis of Airplane Boarding Times

“…The analysis carried out in Bachmat et al (2005Bachmat et al ( , 2006 suggests that, as the number of passengers n becomes large, t ≈ T √ n. More precisely, for any ε > 0, with probability approaching 1 as n −→ ∞, 1 − ε < t T √ n < 1 + ε.…”

“…We will recall the basic facts about this case from Bachmat et al (2005Bachmat et al ( , 2006 and state a few more which will serve us later. When there is no boarding policy, the joint row/queue position distribution is uniform, namely, p(q, r) = 1.…”

“…An analytical model which takes airplane design parameters into account was recently introduces in Bachmat et al (2005Bachmat et al ( , 2006. The authors express the expected boarding time T , as the number of passengers is large, in terms of the solution to a variational problem.…”

“…The variational problem turns out to have a beautiful geometric interpretation in terms of spacetime (Lorentzian) geometry. A major issue that is not addressed in Bachmat et al (2005Bachmat et al ( , 2006 is the validation of the model against detailed simulations of the boarding process. This is particularly important since the model makes many simplifying assumptions.…”

“…In Section 3 we present, following Bachmat et al (2005Bachmat et al ( , 2006, a constrained variational problem whose solution provides the expected critical path and boarding time in the limit as the number of passengers tends to infinity. The input to the variational problem is a probability measure p, determined by the airline boarding policy and a congestion parameter k which is related to the interior design of the airplane.…”