Stochastic Geometry and Wireless Networks, Part II: Applications

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Hence, the network can be adaptively and optimally evolved by responding to the network dynamics. The proposed strategy is used to maximize long-term utility, which is achieved by considering both current network conditions and future network dynamics. We define the utility of an action to include network throughput gain and the cost of transmission power. We show that the resulting network of the proposed strategy eventually converges to stationary networks, which maintain the states of the nodes.

Moreover, we propose to determine initial transmission ranges and initial network topology that can expedite the convergence of the proposed algorithm.

Our simulation results confirm that the proposed strategy builds a network which adaptively changes its topology in the presence of network dynamics. Moreover, the proposed strategy outperforms existing strategies in terms of system goodput and successful connectivity ratio. The connected world which began with representative services such as connected cars, networked unmanned aerial vehicles UAVs and the Internet of things IoT , results in network with inherent dynamics. The network entities of such services generally have high mobility, which causes frequent changes in member nodes associated with these networks and unstable channel conditions with high link failure rates.

Hence, it is essential to form robust networks against such dynamics by adaptively reformulating inter-connections among network entities. However, solving this problem based on conventional centralized solutions requires too high computational complexity such that it cannot be practically considered. Rather, it can be solved by decentralized and spontaneous network formation strategies that can proactively respond to the network dynamics by modifying the network topology based on the decision-making processes of each network entity.

However, it is not straightforward to design decentralized strategies that enable each network entity to make its own and optimal decisions, because the network entities are intimately coupled. Specifically, the network entities can be tightly inter-connected, so that the impact of small changes from a network entity may propagate over a large number of entities. Therefore, each network entity should consider the corresponding responses associated with its decisions to make optimal decisions.

Stochastic Geometry and Wireless Networks : Francois Baccelli :

This may require significantly high computational complexity or may not be feasible in practice. Therefore, it is essential for the design of decentralized strategies to decouple the inter-connections among network entities. In this paper, we show that the inter-connections among network entities can be decoupled by deploying network coding, which is referred to as network decoupling.

For a network coding enabled wireless ad-hoc network which is widely considered as a network model of a connected world a packet passes through many intermediate entities. Thus, it can be mixed with other packets multiple times. This leads to packet anonymity , where all packets in the network eventually have identical information including their terminal nodes. Packet anonymity allows an entity to consider the other entities as its environment so that complicated inter-connections among network entities can be decoupled, and only the connection directly associated with the entity is considered as a one-hop connection.

This leads to network decoupling so that the interactions between network entities can be interpreted as a node-environment interaction at each entity. Motivated by the node-environment interaction, we use an MDP to find a decentralized strategy, which is referred to as a policy, for network formation.

We consider wireless entities to be autonomous decision-making agents and the state of an agent is defined as the number of effective nodes. Here, effective nodes are the entities that have successfully received packets from the agent. The probability density function of the states is modeled by the Poisson point process PPP , which is widely used to characterize the behavior of mobile nodes. The action of an agent is defined as the amount of increasing or decreasing transmission range, which is the outcome of the policy for the current state of the agent.

The policy is optimal if it enables the agent to maximize long-term utility.

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As a node increases its transmission range, the number of hops required to reach the terminal node decreases, without loss of generality, leading to an improvement in network throughput. However, extending the transmission range increases power consumption and causes more inter-node interference. This is explicitly captured by the utility function, which represents both network throughput improvement and the additional corresponding transmission power. Therefore, the optimal policy enables each entity to successively determine the optimal changes in transmission range at each state, such that the entities can strike a balance between network throughput gain and power consumption.

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Finally, the consequences of the distributed decisions from each entity eventually determine the network topology. Note that the proposed strategy allows the resulting network topology to evolutionarily adapt against network dynamics. This is because the state is defined by the effective nodes, which are directly dependent on link failure rates i. For example, a larger transmission range may be required in a channel with higher link failure rates for the target number of effective nodes. Similarly, an agent can increase its transmission range to sustain connectivity in the case of sparse node density.

The proposed strategy is also robust against frequent changes in member nodes of the considered network, which is widely observed in mobile networks. This is because the behavior of existing nodes is not affected by individual network members, instead it is only affected by the number of effective nodes included in its own transmission range.

Unlike conventional optimal solutions that focus on maximizing immediate utility, the proposed optimal policy determined by the MDP can provide a long-term strategy, which determines actions by explicitly considering future dynamics in the network. Specifically, the actions taken by the optimal policy can maximize the long-term utilities, which are expressed as the sum of discounted utilities over time. The discount factor can be determined by considering the consistency of network conditions. Therefore, the actions determined by the proposed policy can consider both current and future network dynamics.


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The proposed system consists of two phases: initialization and adaptation. In the initialization phase, the optimal policy for each intermediate node is found and the state can be initialized. As will be shown in this paper, optimal actions can lead the network formation result to certain topologies, referred to as stationary networks. Hence, we design the initial network to be close to the stationary network. In the adaptation phase, each node adaptively and optimally changes its transmission range based on the optimal policy for the current state induced by network dynamics.

We show that network coding can lead to packet anonymity where both the information and terminal of all packets in network asymptotically become identical,. We show that the packet anonymity of network coding allows inter-connections among network nodes to be decoupled into node-environment interactions at each node, which is referred to as network decoupling,. We formulate the problem of network topology formation in an MDP framework and provide a decentralized solution to the network formation strategy,.

The proposed strategy improves network robustness by adaptively rebuilding its topology in the presence of network dynamics which includes unstable channel conditions with high link failure rates, and high mobility of network nodes that causes frequent changes in member nodes associated with the considered network,. The proposed strategy is a foresighted strategy that chooses the action maximizes a long-term utility by considering future network dynamics,.

The proposed strategy can determine the optimal transmission range that balances network throughput improvement and transmission power consumption,. The resulting network of the proposed strategy converges to stationary networks, and. We propose how to initialize a network such that the speed of convergence to the stationary network can be improved. Rather, our focus is on robust network formation based on network coding, which can proactively reform network topology against network dynamics in a decentralized manner. The rest of the paper is organized as follows.

Since network coding was first introduced in [ 1 ] , it has shown excellent ability to improve throughput, robustness and complexity. The beginning of network coding was for throughput gain in a multicast scenario. Since packet loss is very prevalent in wireless network, typical erasure coding schemes have been widely employed by inserting a degree of redundancy at the source node. Using network coding, redundancy can be inherently included at the intermediate nodes because wireless networks have broadcast links that reach multiple end nodes. Another advantage of network coding is that there is a lower complexity requirement for network formation compared to a conventional store-and-forward approach.

In a conventional store-and-forward approach, it is difficult to find the optimal routing path that can achieve the capacity upper bound. Network coding, however, can transform complex network formation problems into low-complexity distributed problems. For example, a distributed solution that satisfies optimality condition to minimum delay and minimum energy consumption is proposed in [ 18 ].


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Hence, suboptimal but practical solutions are often studied [ 21 , 22 ]. Using a game theoretical approach, each node in the network determines its transmission power and the use of network coding operations. Network formation strategies for dynamic network conditions in conventional routing schemes have been widely studied in the context of a self-organizing network. Protocols for self-organization of wireless sensor networks where there exists a large number of static nodes with energy constraints are described in [ 23 ].

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Stochastic geometric analysis of massive MIMO networks

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