Category: Vehicular networks

The transdimensional approach

In vehicular networks, transceivers are inherently confined to a subset of the two-dimensional Euclidean space. This subset is the street system where cars are allowed to move. Accordingly, stochastic geometry models for vehicular networks usually consist of two components: A set of streets and a set of point processes, one for each street, representing the vehicles. The most popular model is the Poisson line process (PLP) for the streets, combined with one-dimensional PPPs of vehicles on each line (street).

This PLP-PPP model does not have T-junctions, only intersections. Accordingly, all vehicles are of order 2 or 4, per the taxonomy introduced here. The order is determined by the number of directions in which a vehicle can move.

The PLP-PPP is a Cox process due to the independent one-dim. PPPs, and the underlying street system determining the intensity measure (the line process) is also based on the PPP. Consequently, the PLP-PPP inherits a certain level of tractability from the PPP, in the sense that exact expressions can be derived for some quantities of interest. In particular, the SIR distribution (complementary cumulative distribution function, ccdf) at the typical vehicle for a transmitter at a fixed distance can be derived without difficulty. However, the expression requires the evaluation of two nested improper integrals. While such a result certainly has its value, it does not give direct insight how the resulting SIR distribution depends on the network parameters. Also, other metrics that depend on the SIR often require further integration, most importantly the SIR meta distribution, which is calculated from the higher moments of the conditional SIR ccdf (given the point process).

This raises the question whether it is possible to find a closed-form result that is much more quickly evaluated and provides a tight approximation. Simply replacing the PLP-PPP by a two-dimensional PPP produces poor results, especially in the high-reliability regime (where the SIR ccdf is near 1). Similarly, considering only the one street that the typical vehicle lies on (i.e., using only a one-dimensional PPP) ignores all the interference from the vehicles on the other streets, which strongly affects the tail of the distribution.

How about a combination of the two – a superposition of a one-dimensional PPP for the typical vehicle’s street and a two-dimensional PPP for the other vehicles? In this approach, PPPs of two different dimensions are combined to a transdimensional PPP (TPPP). It accurately characterizes the interference from nearby vehicles, which are likely to lie on the same street as the typical vehicle, and captures the remaining interference without the complexity of the PLP. The three key advantages of this approach are:

  • The TPPP leads to closed-form results for the SIR ccdf that are asymptotically exact, both in the lower and upper tails (near 0 and near infinity).
  • The results are highly accurate over the entire range of the SIR ccdf, and they are obtained about 100,000 times faster than the exact results. Hence, if fast evaluation is key and a time limit of, say, one μs is specified, the transdimensional approach yields more accurate results than the exact expression. Put differently, the exact expression only leads to higher accuracy if ample computation time is available.
  • The simplicity of the TPPP extends to all moments of the conditional success probability, which greatly simplifies the calculation of the SIR meta distribution.

The TPPP approach is also applicable to other street systems, including the Poisson stick model (where streets are modeled as line segments of random length) and the Poisson lilypond model, which forms T-junctions (where vehicles are of order 3). For the stick model with independent lengths, the exact expression of the nearest-neighbor distance distribution involves six nested integrals, hence a transdimensional is certainly warranted. More details can be found here.

A case for T junctions

It has been established (for example, here) that the standard two-dimensional homogeneous PPP is not an adequate model for vehicular networks, since vehicles are mostly confined to streets. The Poisson line Cox process (PLCP) has naturally emerged as the model of choice. In this process, one-dimensional PPPs are placed on a street system formed by a Poisson line process. This model is somewhat tractable and thus has gained some traction in the community. With probability 1 each line (or street) intersects with each other line, so intersections are formed, and the communication performance at the typical intersection vehicles can be studied. This is important since vehicles at intersections are more accident-prone than other vehicles.

How about T junctions? Clearly, the PLCP has no T junctions a.s. But while not quite as frequent as (four-way) intersections, they are an important building block of the street systems in every city, and it is reasonable to assume that they inherit some of the dangers of intersections. However, the performance of vehicles at T junctions have barely been modeled and analyzed. The reason is perhaps not that it is not worthy of study but the lack of a natural model. Let’s say we wanted to construct a Cox model of vehicles that is supported on a street system that has no intersections but only T junctions, with the T junctions themselves forming a stationary point process (in the same way the intersections in the PLCP form a stationary point process). What is the simplest (most natural, most tractable) model?

One model we came up with is inspired by the so-called lilypond model. From each point of a PPP, a line segment grows in a random orientation in both directions. All segments grow at the same speed until one of their endpoints hit another segment. Once all growth has stopped, the lilypond street model is obtained. Here is a realization:

Figure 1. Realization of Lilypond street model, starting with a PPP of density 0.1.

Then PPPs of vehicles can be placed on each line segment to form a Lilypond line segment Cox process. Some results for vehicular networks based on this model are available here. The model has the advantage that it has only a single parameter – the density of the underlying PPP of the center points of each line segment. On the other hand, the distribution of the length of the line segments can only be bounded, and the construction naturally creates a dependence between the lengths of nearby segments, which limits the tractability. For instance, in a region with many initial Poisson points, segments will be short on average, while in a region with sparse Poisson points, segments will be long. Also, the construction implies that simulating this process takes significantly more time than simulating a PLCP.

Given the shortcoming of the model, it seems quite probable that there are other, simpler and (even) more natural models for street systems with T junctions. Let’s try and find them!