Lonely passengers and the “detachment process”
Journal Article
Overview
abstract
We introduce the detachment process, a novel, time-inhomogeneous Markov process inspired by I. P. Tóth’s problem [4, 13] concerning the number of “lonely passengers” (those without companions) when n passengers are seated independently and uniformly in k initially empty buses. Tóth showed that this number is stochastically non-decreasing in k for fixed n. We complement [4, 13] by treating the number of buses k as a time parameter. Specifically, for a fixed number of passengers n, the state of our Markov process at time k≥1 is exactly Tóth’s configuration (n,k). (We extend the process’s definition for all t∈[1,∞).) These processes can be coupled for all n≥1, and this larger coupled process is what we dub the detachment process. Our investigation focuses on properties related to detachment, clumping, the number of lonely passengers and of non-empty buses. The central notion is detachment, which occurs at time k if each passenger occupies a distinct bus; the process is in a state of detachment at k. At a detachment time k the process transitions from a non-detached state at k−1 to a detached state at k. Four critical time-scales are identified — linear, quadratic, and log-corrected linear or quadratic in the number of passengers, n. For example, we provide a scaling limit result for the {0,1}-valued process X(n) on (0,∞), defined by Xt(n):=�(state of detachment at timetn2), in the sense of finite-dimensional distributions. On the other hand, it is shown that if, for n≫1,y>0, one defines the time scale k(n)=k(n,y):=n2[2(2logn−2loglogn+y)]−1, and e(n,k) is the expected number of times the process is in a state of detachment up to time k, then limn→∞e(n,k(n,y))=c(y):=e−y∕8. We also investigate (relative) clumping and explore why modeling the number of passengers with a Poisson distribution simplifies the analysis of Tóth’s original model. In the latter derivation, a comparison theorem for binomial distributions, originally obtained by J. Najnudel [9], is presented with a novel proof. A comparison with the literature on occupancy problems is given in Subsection 1.6.
