# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 1 Introduction
Inspired by this work, many researchers have engaged in the study of the capacity of mobile networks [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40]. Of all the mobility models, the i.i.d. model is the simplest and the most commonly used one. [PERSON] _et al._[31] analyzed the delay-throughput tradeoff of i.i.d. mobility networks. They prove that the throughput capacity is \(\Omega\left(\dfrac{D}{n}\right)\), where \(n\) is the number of nodes and \(D\) is the transmission delay. The mobile network in [31] is substantially a fast mobility model. In this model, the data transmissions and the node mobility are assumed to be on the same time scale. Hence there exists only one-hop transmission in a time slot. In [32], [PERSON] and [PERSON] considered the case that the node mobility is much slower than the data transmissions. In their network model, in single time slot, packets can be delivered by multi-hop transmissions. They show that the delay-throughput tradeoff is \(\Theta\left(\sqrt{\dfrac{D}{n}}\text{log }n\right)\) in their i.i.d. slow mobility network model. Other researchers have also contributed a lot to the studies on the capacity of i.i.d. mobility networks. The random walk model is also an important mobility model. [PERSON] _et al._[33] first introduced the random walk model. Later on, the throughput capacities of per source-destination pair in fast mobility networks [34] and slow mobility networks were investigated [35]. Besides i.i.d. mobility model and random walk model, other mobility models such as Brownian model and hybrid random walk model are also well studies [36; 37]. However, most of the literature above focuses on the two-dimensional mobility model or one-dimensional mobility model. In fact, as wireless technologies are rapidly developing, wireless networks are extending from two-dimensional to three-dimensional space. Moreover, in many scenarios, the network can be better modeled by a three-dimensional space instead of a two-dimensional space. For instance, in modern battlefields, a large amount of wireless networks are emerging which consist of lots of military units such as fleets, troops and aircrafts. Hence, we can foresee that studies on three-dimensional networks will be of great significance. In previous studies, only a few researchers studied the capacity problems in three-dimensional networks [41; 42; 43]. The three-dimensional arbitrary network model and three-dimensional random network model are, for the first time, proposed in [41]. The authors calculated the transport capacity and throughput capacity of the two types of models, respectively. [PERSON] _et al_. [42] proposed a three-dimensional network model where the distributions of nodes are inhomogeneous. They obtained the asymptotic expressions on the throughput capacity of their model. [PERSON] _et al_. [43] considered the three-dimensional networks with UWB technologies. They also obtained the throughput capacity. In this paper, we consider heterogeneous networks with \(n\) normal nodes and \(m\) helping nodes. In our network model, the helping nodes are more powerful than the normal nodes and just work in routing and relaying. They are neither the source nodes nor the destination nodes. Moreover, we adopt the delay constraint \(D\) in our routing strategy. Concretely, if the source can send the packets to its destination within a time slot \(D\), the packets will be transmitted in multi-hop fashion, which is also called _ad hoc_ transmission mode. If the transmission delay exceeds \(D\) time slots, the packets will be delivered to the destination with the help of helping nodes, which is called helping transmission mode. We adopt the i.i.d. mobility model in this paper. Moreover, we consider two time scales: fast mobility and slow mobility. Besides, we also consider two cases, where all the nodes are distributed in two-dimensional space and where all of them are distributed in three-dimensional space. We first give an intuitive analysis under virtual channel systems. Then the asymptotic expressions are obtained by strict proof.