Intracell Interference Characterization and Cluster Interference for D2D Communication

The homogeneous spatial Poisson point process (SPPP) is widely used for spatial modeling of mobile terminals (MTs). This process is characterized by a homogeneous distribution, complete spatial independence, and constant intensity measure. However, it is intuitive to understand that the locations of MTs are neither homogeneous, due to inhomogeneous terrain, nor independent, due to homophilic relations. Moreover, the intensity is not constant due to mobility. Therefore, assuming an SPPP for spatial modeling is too simplistic, especially for modeling realistic emerging device-centric frameworks such as device-to-device (D2D) communication. In this paper, assuming inhomogeneity, positive spatial correlation, and random intensity measure, we propose a doubly stochastic Poisson process, a generalization of the homogeneous SPPP, to model D2D communication. To this end, we assume a permanental Cox process (PCP) and propose a novel Euler-Characteristic-based approach to approximate the nearest-neighbor distribution function. We also propose a threshold and spatial distances from an excursion set of a chi-square random field as interference control parameters for different cluster sizes. The spatial distance of the clusters is incorporated into a Laplace functional of a PCP to analyze the average coverage probability of a cellular user. A closed-form approximation of the spatial summary statistics is in good agreement with empirical results, and its comparison with an SPPP authenticates the correlation modeling of D2D nodes.

[1]  B. Ripley.  The Second-Order Analysis of Stationary Point Processes , 1976 .

[2]  Robert J. Adler,et al.  Excursions above a fixed level by n-dimensional random fields , 1976, Journal of Applied Probability.

[3]  François Baccelli,et al.  Stochastic Geometry and Wireless Networks, Volume 1: Theory , 2009, Found. Trends Netw..

[4]  P. McCullagh,et al.  The permanental process , 2006 .

[5]  Jeffrey G. Andrews,et al.  Statistical Modeling and Probabilistic Analysis of Cellular Networks With Determinantal Point Processes , 2014, IEEE Transactions on Communications.

[6]  Karl J. Friston,et al.  Assessing the significance of focal activations using their spatial extent , 1994, Human brain mapping.

[7]  A. M. Hasofer,et al.  Level Crossings for Random Fields , 1976 .

[8]  Carl E. Rasmussen,et al.  Gaussian processes for machine learning , 2005, Adaptive computation and machine learning.

[9]  J. Cao.  The size of the connected components of excursion sets of χ2, t and F fields , 1999, Advances in Applied Probability.

[10]  Angel R. Martinez,et al.  Computational Statistics Handbook with MATLAB , 2001 .

[11]  Muhammad Ali Imran,et al.  Distance Based Cooperation Region for D2D Pair , 2015, 2015 IEEE 81st Vehicular Technology Conference (VTC Spring).

[12]  Sheldon M. Ross,et al.  Introduction to Probability and Statistics for Engineers and Scientists , 1987 .

[13]  Jesper Møller,et al.  The permanent process , 2005 .

[14]  Y. Peres,et al.  Determinantal Processes and Independence , 2005, math/0503110.

[15]  J. Møller,et al.  Determinantal point process models and statistical inference , 2012, 1205.4818.

[16]  Florence Puech,et al.  Generalizing Ripley's K function to inhomogeneous populations , 2009 .

[17]  Jeffrey G. Andrews,et al.  Fitting determinantal point processes to macro base station deployments , 2014, 2014 IEEE Global Communications Conference.

[18]  Robert W. Heath,et al.  Modeling heterogeneous network interference , 2012, 2012 Information Theory and Applications Workshop.

[19]  T. Shirai,et al.  Random point fields associated with certain Fredholm determinants I: fermion, Poisson and boson point processes , 2003 .

[20]  Jeffrey G. Andrews,et al.  Modeling and Analysis of K-Tier Downlink Heterogeneous Cellular Networks , 2011, IEEE Journal on Selected Areas in Communications.

[21]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[22]  Evgeny Spodarev,et al.  Stochastic geometry, spatial statistics and random fields : asymptotic methods , 2013 .

[23]  K. Worsley,et al.  Local Maxima and the Expected Euler Characteristic of Excursion Sets of χ 2, F and t Fields , 1994, Advances in Applied Probability.

[24]  Wuyang Zhou,et al.  The Ginibre Point Process as a Model for Wireless Networks With Repulsion , 2014, IEEE Transactions on Wireless Communications.

[25]  J. Møller,et al.  Log Gaussian Cox Processes , 1998 .

[26]  Muhammad Ali Imran,et al.  Coverage Gain and Device-to-Device User Density: Stochastic Geometry Modeling and Analysis , 2015, IEEE Communications Letters.

[27]  R. Adler.  The Geometry of Random Fields , 2009 .

[28]  Andrea Bergmann,et al.  Statistical Parametric Mapping The Analysis Of Functional Brain Images , 2016 .

[29]  Yu-Sheng Chen,et al.  Stochastic geometry based models for modeling cellular networks in urban areas , 2012, Wireless Networks.

[30]  J. Møller,et al.  Handbook of Spatial Statistics , 2008 .

[31]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[32]  O. Macchi.  The coincidence approach to stochastic point processes , 1975, Advances in Applied Probability.

[33]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[34]  Arthur W. Toga,et al.  Brain mapping : the disorders , 2000 .

[35]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[36]  G. Shedler,et al.  Simulation of Nonhomogeneous Poisson Processes by Thinning , 1979 .