# Throughput Analysis of IEEE 802.11 WLANs with Inter-Network Interference
## 1 Introduction  
Recently, due to the popularity of mobile Wi-Fi routers and Wi-Fi-based Bring Your Own Device (BYOD), the Wireless Local Area Network (WLAN) environment, which has a one-to-one relationship between an access point (AP) and a device, is increasing. For example, on business trips, the usage of a portable Wi-Fi router or enabling a tethering function on a cell phone (sharing the Internet via Wi-Fi) is widespread. Moreover, multiple players normally use wireless (i.e., Wi-Fi-based) remote controllers for a competition (e.g., Nintendo switches). Another example is related to the Olympic Games in 2020. To prepare for that, Japan plans to offer a vast number of wireless routers for foreign visitors. Each router has wireless broadband interfaces (i.e., 5G, LTE, mmWave) to the Internet while sharing the Internet connection via Wi-Fi. In such an environment, many WLANs are coexisting in a small area. In each network, the AP-device distance is so close that the networks are within the carrier-sensing range of their neighbor. Therefore, the throughput of specific WLAN reduces significantly because of the interaction of carrier-sensing. Constructing mathematical models is an effective way of clarifying that phenomenon. There have been many analytical models for WLANs' performance [1; 2; 3; 4; 5; 6; 7; 8]. The most famous one is presented by [PERSON] [1]. In [PERSON]'s model, expressing the Backoff-Timer (BT) decrement makes it possible to reveal the node operation in each time slot with the saturated condition. In [2], the authors have applied the queuing theory to [PERSON]'s model to express the node behavior in the non-saturation state. The works in [3; 4; 5; 6] include several extended versions of the model. The [PERSON] model is, however, not applicable to express the network characteristic when considering the inter-network interference. That is because that model assumes all nodes of WLAN are in the same carrier-sensing range. When the number of networks in the same carrier-sensing range is large, it is difficult to express the carrier-sensing duration due to the simultaneous transmission. A potential approach for expressing that simultaneity is applying the airtime concept as in [9; 10; 11; 12; 13; 14]. The airtime concept is useful for revealing the frame collisions due to hidden nodes and the carrier-sensing duration at each node [9; 10; 11; 12]. Moreover, it is possible to express transmission probability, carrier-sensing duration and frame-existence probability concerning each node in simple forms. In [13], the authors show the network performance when saturation and non-saturation nodes coexist. Moreover, the analysis in [14] can be applied in various network topologies, including multi-hop ones. However, the previous works typically consider the main reason for throughput decrement is the frame collisions without the consideration of simultaneous transmission [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12]. Moreover, most of the existing models appraise the interference (i.e., including carrier-sensing and frame collision) but within one network. Hence, they do not apply to the investigated networks. Several previous airtime models can express the carrier-sensing duration. However, they omit the simultaneous transmission time because the effect caused by the simultaneous transmission in such scenarios is negligible. We aim to derive analytical models that precisely express each network carrier-sensing duration by subtracting the simultaneous transmission time. This paper proposes analytical expressions of WLANs' throughputs using the airtime concept with the consideration of inter-network interference. Each network throughput can be obtained successfully by expressing transmission duration, carrier-sensing duration, idle duration, and frame-existence probabilities concerning the impacts of neighboring networks. We have applied the expressions for three network scenarios including two string topologies and a grid one. We have also conducted extensive evaluations of the investigated scenario.