Multilayer Random Sequential Adsorption

In this work, we present a variant of the multilayer random sequential adsorption (RSA) process that is inspired by orthogonal resource sharing in wireless communication networks. In the one-dimensional (1D) version of this variant, the deposition of overlapping rods is allowed only if they are assigned two different colors, where colors are symbolic of orthogonal resources, such as frequency bands, in communication networks. Owing to a strong spatial coupling among the deposited rods of different colors, finding an exact solution for the density of deposited rods of a given color as a function of time seems intractable. Hence, we propose two useful approximations to obtain the time-varying density of rods of a given color. The first approximation is based on the recursive use of the known monolayer RSA result for the indirect estimation of the density of rods for the multilayer version. The second approximation, which is more accurate but computationally intensive, involves accurate characterization of the time evolution of the gap density function. This gap density function is subsequently used to estimate the density of rods of a given color. We also consider the two-dimensional (2D) version of this problem, where we estimate the time-varying density of deposited circles of a given color as a function of time by extending the first approximation approach developed for the 1D case. The accuracy of all the results is validated through extensive Monte Carlo simulations.