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This paper investigates two fundamental characteristics of a wireless multihop network: its minimum node degree and its k-connectivity. Both topology attributes depend on the spatial distribution of the nodes and their transmission range. Using typical modeling assumptions -a random uniform distribution of the nodes and a simple link model -we derive an analytical expression that enables the determination of the required range r0 that creates, for a given node density ρ, an almost surely k-connected network. Equivalently, if the maximum r0 of the nodes is given, we can find out how many nodes are needed to cover a certain area with a k-connected network. We also investigate these questions by various simulations and thereby verify our analytical expressions. Finally, the impact of mobility is discussed.The results of this paper are of practical value for researchers in this area, e.g., if they set the parameters in a network-level simulation of a mobile ad hoc network or if they design a wireless sensor network.

Abstract-The random waypoint model is a commonly used mobility model for simulations of wireless communication networks. In this paper, we present analytical derivations of some fundamental stochastic properties of this model with respect to: (a) the length and duration of a movement epoch, (b) the chosen direction angle at the beginning of a movement epoch, and (c) the cell change rate of the random waypoint mobility model when used within the context of cellular networks. Our results and methods can be used to compare the random waypoint model with other mobility models. The results on the movement epoch duration as well as on the cell change rate enable us to make a statement about the "degree of mobility" of a certain simulation scenario. The direction distribution explains in an analytical manner the effect that nodes tend to move back to the middle of the system area.

Abstract-The random waypoint model is a commonly used mobility model for simulations of wireless communication networks. In this paper, we present analytical derivations of some fundamental stochastic properties of this model with respect to: (a) the length and duration of a movement epoch, (b) the chosen direction angle at the beginning of a movement epoch, and (c) the cell change rate of the random waypoint mobility model when used within the context of cellular networks. Our results and methods can be used to compare the random waypoint model with other mobility models. The results on the movement epoch duration as well as on the cell change rate enable us to make a statement about the "degree of mobility" of a certain simulation scenario. The direction distribution explains in an analytical manner the effect that nodes tend to move back to the middle of the system area.

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This paper presents a framework for the calculation of stochastic connectivity properties of wireless multihop networks. Assuming that n nodes, each node with transmission range r 0 , are distributed according to some spatial probability density function, we study the level of connectivity of the resulting network topology from three viewpoints. First, we analyze the number of neighbors of a given node. Second, we study the probability that there is a communication path between two given nodes. Third, we investigate the probability that the entire network is connected, i.e. each node can communicate with every other node via a multihop path. For the last-mentioned issue, we compute a tight approximation for the critical (r 0 , n) pairs that are required to keep the network connected with a probability close to one. In fact, the problem is solved for the general case of a k-connected network, accounting for the robustness against node failures. These issues are studied for uniformly distributed nodes (with and without 'border effects'), Gaussian distributed nodes, and nodes that move according to the commonly used random waypoint mobility model. The results are of practical value for the design and simulation of wireless sensor and mobile ad hoc networks.

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