Markov chain of distances between parked cars

We describe the distribution of distances between parked cars as a solution of certain Markov process and show that its solution is obtained with the help of a distributional fixed point equation. Under certain conditions the process is solved explicitly. The resulting probability density is compared with the actual parking data measured in the city. We focus on the spacing distribution between cars parked parallel to the curb somewhere in the city center. We will assume that the street is long enough to enable a parallel parking of many cars. Moreover we assume that there are no driveways or side streets in the segment of interest and that the street is free of any kind of marked parking lots or park meters. So the drivers are free to park the car anywhere provided they find an empty space to do it. Finally we assume that many cars a cruising for parking. So there are not free parking lots and a car can park only when another car leaves the street. The standard way to describe random parking is the continuous version of the random sequential adsorption model known also as the "random car parking problem" see [1], [2] for review. In this model it is assumed that all cars have the same length l0 and park on randomly chosen places. Once parked the cars do not leave the street. This model leads to predictions that can be easily verified. First of all it gives a relation between the mean bumper to bumper distance D and the car length: D ∼ 0.337 l0. Further the probability density Q(D) of the car distances D behaves like [3],[4], [5], [6] Q(D) ≈ −ln(D) (1) To test this results real parking data were collected recently in the center of London [7]. The average distance between the parked cars was 152 cm which fit perfectly with the relation D ∼ 0.337 l0 for l0 = 450 cm. The gap density (1) was however fully incompatible with the observed facts. The relation (1) gives Q(D) → ∞ for D → 0 . The data from London showed however that in reality Q(D) → 0 as D → 0. Rawal and Rogers [7] used therefore an amended version of the model with a car re-positioning process (describing the car manoeuvering) to fit the data. Later Abul-Magd [8] pointed out that the car spacings observed in London can be well described by the Gaussian Unitary Ensemble of random matrices [9],[10] if the parked cars are regarded as particles of a one dimensional interacting gas. In both cases the authors assumed that Q(D) ∼ D for small D. However the origin of the particular car re-positioning or car interaction remained unclear. Our aim here is to show that the car parking can be understood as a simple Markov process. The main difference to the previous approach is that we enable the parked cars to leave the street vacating the space for a new car to parks there. The distance distribution is obtained as a steady solution of the repeated car parking and car leaving process. To derive the equations we describe a car parking on a roundabout junction (we know that it is not allowed to park there, but it simplifies the consideration). The idea is simple: Assume that all cars have the same length l0 and park on a roundabout junction with a circumference L. The length L is chosen in interval 3l0 < L < 4l0) so that maximally 3 cars can park there. These cars define three spacings D1, D2, D3 with D1 +D2 +D3 + 3l0 = L (2) The parking process goes as follows: one randomly chosen car leaves the street and the two adjoining lots merge into a single one. In the second step a new car parks into the empty space and splits it into two smaller lots. Such fragmentation