Multi-tier Network Performance Analysis using a Shotgun Cellular System
  This paper studies the carrier-to-interference ratio (CIR) and carrier-to-interference-plus-noise ratio (CINR) performance at the mobile station (MS) within a multi-tier network composed of M tiers of wireless networks, with each tier modeled as the homogeneous n-dimensional (n-D, n=1,2, and 3) shotgun cellular system, where the base station (BS) distribution is given by the homogeneous Poisson point process in n-D. The CIR and CINR at the MS in a single tier network are thoroughly analyzed to simplify the analysis of the multi-tier network. For the multi-tier network with given system parameters, the following are the main results of this paper: (1) semi-analytical expressions for the tail probabilities of CIR and CINR; (2) a closed form expression for the tail probability of CIR in the range [1,Infinity); (3) a closed form expression for the tail probability of an approximation to CIR in the entire range [0,Infinity); (4) a lookup table based approach for obtaining the tail probability of CINR, and (5) the study of the effect of shadow fading and BSs with ideal sectorized antennas on the CIR and CINR. Based on these results, it is shown that, in a practical cellular system, the installation of additional wireless networks (microcells, picocells and femtocells) with low power BSs over the already existing macrocell network will always improve the CINR performance at the MS. 
Prasanna Madhusudhanan, Juan G. Restrepo, Youjian (Eugene) Liu,   Timothy X Brown, and Kenneth R. Baker
cs.IT math.IT