The paper extends our earlier work on the sum of a complex Gaussian and the product of two complex Gaussians (a scalar and a vector) by incorporating multiple targets. We derive the joint probability density function (pdf) of the resulting vector, after which we derive the pdf, cumulative distribution function (cdf), and moments of the square norm of the vector (signal energy). A novel integral function is introduced to make the analysis possible. The case of high signal-to-noise ratio (SNR) and a single observation has been treated as a special case. The obtained theoretical results are then used to develop maximum likelihood and moment matching estimators, for estimating the number of targets. Numerical results validate the obtained results and characterize the performance of the proposed estimators.

The joint probability density function (pdf) of a complex vector, which is the product of a complex Gaussian scalar and a complex Gaussian vector, added to another complex Gaussian vector, is derived. The pdf, cumulative distribution function, moments, and moment generating function of the squared norm and norm of this vector are then derived. The derived result is then used to characterize the performance of energy detector (square law detector) for radar system.

In order to reduce the latency of data delivery, one of techniques is to cache the popular contents at the base stations (BSs) i.e. edge caching. However, the technique of caching at edge can only reduce the backhaul delay, other techniques such as BS densification will also need to be considered to reduce the fronthaul delay. In this work, we study the trade-offs between BS densification and cache size under delay constraint at a typical user (UE). For this, we use the downlink SINR coverage probability and throughput obtained based on stochastic geometrical analysis. The network deployment of BS and cache storage is introduced as a minimization problem of the product of the BS intensity and cache size which we refer to the product of “cache intensity” under probabilistic delay constraint. We examine the cases when (i) either BS intensity or the cache size is held fixed, and (ii) when both BS intensity and the cache size are vary. For the case when both BS intensity and the cache size are variable, the problem become nonconvex and we convert into a geometric programing which we solve it analytically.

The conflicting problems of growing mobile service demand and underutilization of dedicated spectrum has given rise to a paradigm where mobile network operators (MNOs) share their infrastructure among themselves in order to lower their operational costs, while at the same time increase the usage of their existing network resources. We model and analyze such an infrastructure sharing system considering a single buyer MNO and multiple seller MNOs. Assuming that the locations of the BSs can be modeled as a homogeneous Poisson point process, we find the downlink signal-to-interference-plus-noise ratio (SINR) coverage probability for a user served by the buyer MNO in an infrastructure sharing environment. We analyze the trade-off between increasing the transmit power of a base station (BS) and the intensity of BSs owned by the buyer MNO required to achieve a given quality-of-service (QoS) in terms of the SINR coverage probability. Also, for a seller MNO, we analyze the power consumption of the network per unit area (i.e., areal power consumption) which is shown to be a piecewise continuous function of BS intensity, composed of a linear and a convex function. Accordingly, the BS intensity of the seller MNO can be optimized to minimize the areal power consumption while achieving a minimum QoS for the buyer MNO. We then use these results to formulate a single-buyer multiple-seller BS infrastructure market. The buyer MNO is concerned with finding which seller MNO to purchase from and what fraction of BSs to purchase. On the sellers' side, the problem of pricing and determining the fraction of infrastructure to be sold is formulated as a Cournot oligopoly market. We prove that the iterative update of each seller's best response always converges to the Nash Equilibrium.

Signal-to-noise-plus-interference ratio (SINR) outage probability is among one of the key performance metrics of a wireless network. In this paper, we propose a semi-analytical method based on the saddle point approximation (SPA) technique to calculate the SINR outage of a wireless system whose SINR can be modeled in the form (Σi=1MXi/(1 + Σi=1NYi)) where Xi denotes the useful signal power and Yidenotes the power of the interference signal. Both M and N can also be random variables. The proposed approach is based on the saddle point approximation to cumulative distribution function as given by Wood-Booth-Butler formula. The approach is applicable whenever the cumulant generating function of the received signal and interference exists, and it allows us to tackle distributions with large skewness and kurtosis with higher accuracy. In this paper, we exploit a four parameter normal-inverse Gaussian (NIG) distribution as a base distribution. Given that the skewness and kurtosis satisfy a specific condition, NIG-based SPA works reliably. When this condition is violated, we recommend SPA based on normal or symmetric NIG distribution, both special cases of NIG distribution, at the expense of reduced accuracy. For the purpose of demonstration, we apply SPA for the SINR outage evaluation of a typical user experiencing a downlink coordinated multi-point transmission from the base stations that are modeled by homogeneous Poisson point process. Numerical results are presented to illustrate the accuracy of the proposed set of approximations.

One of the many problems faced by current cellular network technology is the underutilization of the dedicated licensed spectrum of network operators. An emerging paradigm to solve this issue is to allow multiple operators to share some parts of each other's spectrum. Previous works on spectrum sharing have failed to integrate the theoretical insights provided by recent developments in stochastic geometrical approaches to cellular network analysis with the objectives of network resource allocation problems. In this paper, we study the non-orthogonal spectrum assignment with the goal of maximizing the social welfare of the network, defined as the expected weighted sum rate of the operators. We adopt the many-to-one stable matching game framework to tackle this problem. Moreover, using the stochastic geometrical approach, we show that its solution can be both stable as well as socially optimal. To obtain the maxima of social welfare, the computation of the game theoretical solution using the generic Markov Chain Monte Carlo method is proposed. We also investigate the role of power allocation schemes using Q-learning, and we numerically show that the effect of resource allocation scheme is much more significant than the effect of power allocation for the social welfare of the system.

In this paper, we study a problem where multiple operators (OPs) need to share a common pool of spectrum with each other. Our objective is to maximize the social welfare, defined as the overall weighted sum rate of the OPs. The problem is decomposed into two parts: the first part is to allocate RBs to OPs, which we do so by extending the framework of many-to-one matching game with externalities. The second part is to allocate power of small cell base stations (SBSs) belonging to each OP, which is accomplished using reinforcement learning. Assuming that the SBSs associated with each OPs are spatially distributed according to Poisson point process (PPP), we show that pairwise stable matchings achieve local maximas of the social welfare function. We propose two algorithms to search for the stable matchings. Simulation results show that these algorithms are well behaved in terms of convergence and efficiency of the solutions.

In this paper, we apply the concepts of network multiple-input-multiple-output (MIMO) to small cell networks. To do so, the issue of imperfect channel state information (CSI) at the transmitter is considered when frequency-division duplexing is used, for which the feedback channel is limited. We first introduce a regret based learning approach to optimize the transmit beamforming parameters for the cases when the feedback channel is temporarily unavailable during deep fades. We then propose a coalition formation game model to cluster the small cell base stations so that they can perform cluster-wise joint beamforming. We take the \tit{recursive core} as the solution concept of the coalition formation game. To obtain the recursive core, we first consider a typical merge-split algorithm. However, we show that this algorithm can be unstable. Alternatively, we adopt the merge-only algorithm which guarantees the formation stability and show that its outcome belongs to the recursive core. Finally, we analyze the average number and the average size of coalitions that can form during such a coalition formation process. Numerical simulations are given to illustrate the behavior of the coalition formation among small cell base stations.

This paper considers the problem of downlink power allocation in an orthogonal frequency-division multiple access (OFDMA) cellular network with macrocells underlaid with femtocells. The femto-access points (FAPs) and the macro-base stations (MBSs) in the network are assumed to compete with each other to maximize their capacity under power constraints. This competition is captured in the framework of a Stackelberg game with the MBSs as the leaders and the FAPs as the followers. The leaders are assumed to have foresight enough to consider the responses of the followers while formulating their own strategies. The Stackelberg equilibrium is introduced as the solution of the Stackelberg game, and it is shown to exist under some mild assumptions. The game is expressed as a mathematical program with equilibrium constraints (MPEC), and the best response for a one leader-multiple follower game is derived. The best response is also obtained when a quality-of-service constraint is placed on the leader. Orthogonal power allocation between leader and followers is obtained as a special case of this solution under high interference. These results are used to build algorithms to iteratively calculate the Stackelberg equilibrium, and a sufficient condition is given for its convergence. The performance of the system at a Stackelberg equilibrium is found to be much better than that at a Nash equilibrium.

Bandwidth allocation is important for a mesh network to meet quality of service (QoS) requirements of the users. This bandwidth allocation has to ensure that the maximum throughput is achieved. However, this problem becomes more challenging when the information about a network (e.g., channel quality and hence link capacity) is not accurately known, especially, in a real system whose parameters are uncertain. Therefore, in this paper, we apply the robust optimization technique to develop a model that can tolerate the uncertainty of the system parameters. First, we formulate the bandwidth allocation problem as a constrained optimization problem and its robust counterpart is obtained. Then, we suggest an approach to choose the suitable system parameters so that with a certain high probability, the feasibility of the solution from optimization formulation is ensured for all meaningful realizations of the system parameters. Moreover, we introduce the reduced conservativeness approach for the bandwidth allocation. In this approach, the tradeoff between throughput and robustness to the uncertainty of a system parameter is explored. Specifically, the decrease in throughput will be minimized given the target robustness.

This paper considers the problem of access control in the uplink transmission of an OFDMA femtocell network. An underlying noncooperative power game has been devised, based on which a coalition game is formulated by taking a suitable value function. Only two complementary coalitions are allowed to exist in order to reflect the set of transmitters connected to either the macro base station or the femto access point. The transmitters in the same coalition cooperate by operating on non-interfering subchannels, while those in the complementary coalition are assumed to operate so as to cause maximum jamming. The value of a coalition is obtained as the max-min of utility sum of each individual in the given coalition. In the process, we also examine the optimal jamming strategy of the complementary coalition. Finally, we argue that the obtained value function cannot be super-additive. Since the super-additivity property is required for some of the solutions of cooperative game theory, we resort to the Shapley value solution, for which the super-additivity need not hold, to allocate the payoff to each user in a given coalition. Assuming that the transmitters want to be in the coalition that maximizes their payoff, we form a Markov model to obtain the stable coalition structure. We take these stable coalition structures as the required solution of our access control problem.

In this paper, the \emph{strategic} coexistence between macro and femtocell tiers is studied using tools from evolutionary game theory and reinforcement learning. In the first case, femto base stations (FBSs) exchange information through a central controller, and adapt their strategies based on their instantaneous payoffs and average payoffs of the femtocell population. A fictitious play formulation is also examined where FBSs maximize their payoffs given the empirical frequency of other femtocells' actions. In the second case, when information exchange among femtocells is no longer possible, each femtocell gradually learns by interacting with its local environment through trials-and-errors, and adapt its strategies. Variant of the evolutionary game approach (referred to as replication by imitation) is also investigated where femtocells probabilistically review their strategies and imitate other femtocells in the network. Finally, the overall performance of the network in terms of spectral efficiency and convergence is shown to be adamantly driven by the type of information available at femtocells.

This paper considers the downlink power allocation problem in a cellular network where a bi-level hierarchy exists. The network is comprised of the macrocells underlaid with femtocells. The objective of each station in the network is to maximize its capacity under power constraints. The problem is formulated as a Stackelberg game with the macrocell base stations as the leaders and the femtocell access points as the followers. The leaders are assumed to have enough information and foresight to consider the response of the followers while formulating their strategies. To characterize such interaction between leaders and followers, Stackelberg equilibrium is introduced; and it is shown to exist under the assumption of continuity of best response function of the leader sub-game. %For the case of Nash games, the relationship between the upper and lower sub-game equilibrium is explored.