On Scaling Limits of Power Law Shot-noise Fields

This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a $\alpha$-stable limit, intensity of the underlying point process goes to infinity. It is also shown that the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limte the $\alpha$- stable white noise field. Analogous results are also obtained for the extremal shot-noise field which converges to a Fr\'{e}chet white noise field. Finally, these results are applied to the analysis of wireless networks.Comment: 17 pages, Typos are correcte

Non-stationary self-similar Gaussian processes as scaling limits of power law shot noise processes and generalizations of fractional Brownian motion

We study shot noise processes with Poisson arrivals and non-stationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: 1) the conditional variance functions of the noises have a power law and 2) the conditional noise distributions are piecewise. In both cases, the limit processes are self-similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self-similar Gaussian processes with non-stationary increments, via the time-domain integral representation, which are natural generalizations of fractional Brownian motions.Published versio

Capacity of Cellular Wireless Network

Earlier definitions of capacity for wireless networks, e.g., transport or transmission capacity, for which exact theoretical results are known, are well suited for ad hoc networks but are not directly applicable for cellular wireless networks, where large-scale basestation (BS) coordination is not possible, and retransmissions/ARQ under the SINR model is a universal feature. In this paper, cellular wireless networks, where both BS locations and mobile user (MU) locations are distributed as independent Poisson point processes are considered, and each MU connects to its nearest BS. With ARQ, under the SINR model, the effective downlink rate of packet transmission is the reciprocal of the expected delay (number of retransmissions needed till success), which we use as our network capacity definition after scaling it with the BS density. Exact characterization of this natural capacity metric for cellular wireless networks is derived. The capacity is shown to first increase polynomially with the BS density in the low BS density regime and then scale inverse exponentially with the increasing BS density. Two distinct upper bounds are derived that are relevant for the low and the high BS density regimes. A single power control strategy is shown to achieve the upper bounds in both the regimes. This result is fundamentally different from the well known capacity results for ad hoc networks, such as transport and transmission capacity that scale as the square root of the (high) BS density. Our results show that the strong temporal correlations of SINRs with PPP distributed BS locations is limiting, and the realizable capacity in cellular wireless networks in high-BS density regime is much smaller than previously thought. A byproduct of our analysis shows that the capacity of the ALOHA strategy with retransmissions is zero.Comment: A shorter version to appear in WiOpt 201

Performance analysis of mobile users in Poisson wireless networks

Stochastic geometry is a widely accepted mathematical tool used to analyze cellular networks, where the location of base stations are modeled by spatial point processes. It is used to derive closed-form or semi-closed-form expressions for the SINR or for the functions of the SINR which determine various network performance metrics such as coverage probability, "edge" capacity, 90% quantile rate, spectral efficiency, and connectivity without resorting to complicated simulation methods. Predominantly, it is used in deriving marginal distributions of SINR by considering a typical user assumed to be located anywhere on the plane. Models beyond the typical user approach have been proposed with the aim of analyzing QoS metrics of a population of users, and not just a single user. Most of which include considering networks at certain times by representing instances or snapshots of active users as realizations of spatial (usually Poisson) processes or users occurring at random locations that last for some random duration. Analyzing the performance of a typical mobile user on the move or that of a population of such mobile users is complicated since it requires studying not just the marginal but the spatial stochastic fields associated with wireless networks. In this thesis, we model and analyze the fields associated with wireless networks where the locations of base stations are distributed according to a homogeneous Poisson point process. We focus on characterizing the level crossings, extremes, and variability of the Shannon rate fields in noise limited (SNR based) environment by establishing a connection to queueing theory. In interference limited (SIR based) environments, we rely on the theory of Gaussian random fields which arise as natural limits of standardized interference under densification. Using this, we characterize the spatial correlations, and variability of the Shannon rate fields in the limiting regime. We leverage the spatial characterization of the fields to study the temporal variations and various Quality of Service (QoS) metrics seen by the users on the move. The quantification of such metrics as a function of a small number of network parameters, e.g., the density of base stations, path loss, should allow network operators to appropriately tune the density of the base stations to meet the demands of mobile users for a required performance level. In noise limited environments, we study the performance of mobile users in dense networks by incorporating the cost of handovers along with the temporal variability in the Shannon rate. We study the tradeoff between the cost of handover and the Shannon rate by proposing a new class of association policies. Associating with a base station that is farthest in the known users' direction of motion leads to fewer handovers but may lead to a decrease in the rate. Thus, we attribute a local association region to the mobile user to restrict the greediness in the association, which also models the constraint on the available information about the locations of the base stations. We propose a class of greedy association policies and once again leverage stochastic geometry to characterize the performance of such policies. We then optimize the shape and size of the association region by establishing a connection to the theory of Markov processes and compare the performance of this policy to traditional association policiesElectrical and Computer Engineerin