# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 1 Introduction  
Network capacity is one of the key factors of wireless networks. The first study of network capacity is carried out by [PERSON] and [PERSON] in [1]. They investigated the throughput capacity of a wireless network where \(n\) nodes are randomly distributed, and find that the per-node throughput capacity is \(\Theta\left(\sqrt{\frac{W}{n\log n}}\right)\), where \(W\) is the highest transmission rate of each node. Their finding implies that as the size of the network goes to infinity, the per-node throughput will decrease to zero. Following this work, extensive studies have been conducted to achieve a tighter capacity bound [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Some researchers propose to add some base stations to help with long distance transmissions in networks, which are called \"hybrid networks\" [16; 17; 18; 19; 20; 21]. Compared with pure _ad hoc_ networks, the base stations are neither sources nor destinations, they work only as relay nodes. [PERSON] _et al._[16] studied the capacity of networks where the base stations are regularly placed and the distribution of normal nodes is independent and identically distributed (i.i.d.). They obtain the asymptotic expressions on aggregate throughput capacity and maximize the capacity under different channel allocation schemes. [PERSON] _et al._[17] investigated the capacity bound of the networks where base stations and nodes are both randomly deployed. They prove that the per-node throughput is \(\Theta\left(\frac{W}{\log n}\right)\)when the number of nodes and the number of base stations have the same order. [PERSON] _et al._[18] considered the case that the nodes are randomly distributed and the base stations are arbitrarily placed. They proved that the network capacity depends on the number of base stations. [PERSON] _et al._[19] proposed a networkmodel where delay constraint is considered and each node is equipped with directional antennas. They also analyzed how the number of base stations and the beamwidth of directional antennas affect the network capacity. There are also some papers focusing on guaranteeing the QoS of some special scenarios over hybrid wireless networks [22; 23; 24]. The distortion-aware concurrent multipath transfer (CMT-DA) solution, delay stringent coded transmission (ASCOT) and goodput-aware load distribution (GALTON) are proposed in these papers. By evaluating the performance through experiments, the authors prove that the proposed models outperform existing transmission schemes. A large volume of literature proves that adding base stations is an efficient way to improve the network capacity. However, this has some disadvantages. First, to improve the throughput capacity significantly, a large number of base stations should be set up, which is very expensive. Second, it takes a lot of time to establish such a wired network. Third, in some practical cases such as battlefields and rescue tasks, there is little time to set up base stations. Therefore, compared with establishing base stations, deploying some powerful helping nodes is an easy and efficient way to improve the network capacity. Networks which consist of normal nodes and helping nodes are called \"heterogeneous networks\". Though there are plenty of research studies on heterogeneous networks [25; 26; 27; 28; 29], most of them focus on routing protocol design [22; 23] and MAC protocol design [27; 28], instead of network capacity. One study focused on network capacity [29] investigated the capacity of two types of networks: regular heterogeneous networks and random heterogeneous networks. In the literature mentioned above, the nodes are usually assumed to be stable. Some researchers focus on the networks where the nodes are mobile. [PERSON] and [PERSON] for the first time introduced mobility to networks [30]. They proved that the network capacity reaches \(\Theta(W)\) in mobile networks, while the transmission delay will go to infinity in their network model.