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Resource allocation games are commonly used to model problems in many domains ranging from security to advertising. One of the most important resource allocation games is the Colonel Blotto game: Two players distribute a fixed budget of resources over multiple battlefields to maximize the aggregate value of battlefields they win, each battlefield being won by the player who allocates more resources to it. Despite its long-standing history and importance, the continuous version of the game---where players can choose any fractional allocation---still lacks a complete Nash equilibrium solution in its general form with asymmetric players' budgets and heterogeneous battlefield values.

In this work, we propose an approximate equilibrium for this general case. We construct a simple strategy (the independently uniform strategy) and prove that it is an epsilon-equilibrium. We give a theoretical bound on the approximation error in terms of the number of battlefields and players’ budgets which identifies precisely the parameters regime for which our strategy is a good approximation. We also investigate an extension to the discrete version (where players can only have integer allocations), for which we proposed an algorithm to compute very efficiently an approximate equilibrium. We perform numerical experiments that guarantee that we can "safely" use this strategy in practice. Our work extends the scope of application of Colonel Blotto games in several practical cases, especially with large games' parameters (e.g. in advertisements, voting, security, etc.,)

The recent evolution of mobile communications and the widespread use of ``smart'' mobile devices have radically changed the behavior and the needs of the mobile user. In the developed world, people that use their mobile devices just to place some calls and to text some SMS can be characterized as endangered species. Ubiquitous access to Internet, videos/songs streaming and upload/download data flows on the fly are some of the modern demands of the mobile user. It would not be an exaggeration to say that modern users have almost the same demands regardless if they are connected through wire to the network from their personal computers or if they have established a wireless connection through their cellular devices.

On the one hand the overall network load increases on the other hand base stations have limited resources due to the fact that are able to operate only to a limited part of the electromagnetic spectrum. Some (temporal) solutions for the aforementioned problem are the expansion of frequency operational bands (mm wave) or the use of more antennas (massive MIMO) or the denser deployment of small cells. In this talk, we are interested in the promising case of denser small cells that will be tighter integrated with the macro cell. Unfortunately to deploy optimally a small cell network is not trivial. A small cell network is usually deployed in an ad-hoc style and not all at once, so a part of the network exists already and cannot be planned. Additionally, there are natural obstacles and physical limitations that do not allow to deploy base stations wherever we want. So, the small cell network's topology is quite different from the traditional (macro cell) well-structure one.

From our point of view, the future of mobile communications will approach the following: A mass of users - will cause a nondescript data traffic - that will be served from an irregular planed and heterogeneous network. This chaotic picture however makes the problem of network modeling and performance analysis extremely challenging.

To this end, the our primary focus is to build up an analytical framework in order to analyze the performance of a randomly placed network, which serves randomly placed users. To achieve this, we based our analysis on two main tools: (a) stochastic geometry, to understand the impact of topological randomness and coverage maps and (b) queueing theory, to model the competition between concurrent flows within the same BS. The second goal is to propose, based on this analysis, some general design guidelines and insights about specific communication scenarios that mainly involve LTE and WiFi networks.