Parking on the integers

Models of parking in which cars are placed randomly and then move according to a deterministic rule have been studied since the work of Konheim and Weiss in the 1960s. Recently, Damron, Gravner, Junge, Lyu, and Sivakoff introduced a model in which cars are both placed and move at random. Independently at each point of a Cayley graph $G$, we place a car with probability $p$, and otherwise an empty parking space. Each car independently executes a random walk until it finds an empty space in which to park. In this paper we introduce three new techniques for studying the model, namely the space-based parking model, and the strategies for parking and for car removal. These allow us to study the original model by coupling it with models where parking behaviour is easier to control. Applying our methods to the one-dimensional parking problem in $\mathbb{Z}$, we improve on previous work, showing that for $p<1/2$ the expected journey length of a car is finite, and for $p=1/2$ the expected journey length by time $t$ grows like $t^{3/4}$ up to a polylogarithmic factor.