We consider a wireless network comprising two types of nodes, namely, sinks and relays . The sink nodes are connected to a wireline infrastructure, while the relay nodes are used to extend the region covered by providing multi-hop paths to the sink nodes. Restricting to the one-dimensional setting, our objective is to characterize the fraction of covered region as a function of sink and relay node densities. We first compare and contrast our infrastructure-based model with the traditional setting where every node is a sink, and hence a location is covered if it simply lies within the range of some node. Then, drawing an analogy between the connected components of the network and the busy periods of an $M/D/\infty $ queue (and using renewal theoretic arguments) we derive a closed-form expression for the average vacancy (complement of coverage). We also compute an upper bound for vacancy by introducing the notion of left-coverage (i.e., coverage by a node on the left); a lower bound is derived by coupling our model with an independent-disk model , where the sinks’ coverage regions are independent and identically distributed. Through an extensive theoretical and numerical study, we investigate the problem of minimizing network deployment cost subject to a constraint on the average vacancy. We also conduct simulations to understand the properties of a general notion of coverage, obtained by introducing hop-counts into the definition. Parameterized approximations for the hop-constrained cluster lengths (around a sink) are proposed, whose efficacy is evaluated numerically. In particular, there exists a range of parameter values where our cluster-length approximation is good. Finally, hop-constrained cost optimization is conducted to demonstrate the efficacy of the infrastructure-based design.

中文翻译：

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具有基础结构节点和中继扩展的一维无线网络的覆盖范围

我们考虑的无线网络包括两种类型的节点，即 下沉 和 接力 。宿节点连接到有线基础设施，而中继节点用于通过提供到宿节点的多跳路径来扩展覆盖的区域。限于一维设置，我们的目标是根据接收器和中继节点密度来表征覆盖区域的比例。我们首先比较和对比基于基础架构使用传统设置的模型，其中每个节点都是一个接收器，因此，如果仅在某个节点的范围内，则可以覆盖该位置。然后，在网络的连接组件与网络繁忙时段之间进行类比。 $ M / D / \ infty $ 排队（并使用更新的理论论据），我们得出平均空缺（覆盖率的补充）的封闭形式的表达式。我们还通过引入以下概念来计算空缺的上限左覆盖（即，左侧节点的覆盖范围）；通过将我们的模型与独立磁盘模型 ，汇的覆盖区域相互独立且分布均匀。通过广泛的理论和数值研究，我们研究了在平均空缺情况下将网络部署成本降至最低的问题。我们还进行了模拟，以了解通过将跳数引入定义中而获得的一般覆盖概念的特性。提出了跃点约束簇长度（在汇附近）的参数化近似值，其有效性通过数值评估。特别是，在我们的簇长近似值良好的情况下，存在一系列参数值。最后，进行跳数受限的成本优化，以证明基于基础架构的设计的有效性。