## 1 Introduction  
With the development of Internet of Things (IoT) technology, low-power wide-area networks (LPWAN) technologies are able to fill the gap between short-range wireless multi-hop ad hoc networks and cellular networks [1]. Wireless sensor network technologies [2; 3] have become an important research area. Currently, LPWANs are suitable for multiple IoT applications, such as smart city [4], smart energy [5; 6], environmental monitoring [7] and other fields. One of the main technologies of LPWAN is long -range wide-area network (LoRaWAN) [8], which is an open standard framework that allows long-range communication between thousands of low-power-consumption battery-powered devices and a continuously online gateway with a cellular structure. Research into LoRaWAN communications can be mainly divided into two categories: uplink communication and downlink communication. Related research of uplink communication mainly includes the network access reliability of equipment, the performance optimization of frequency hopping transmission, the quality of service of the LoRa network, the physical performance of the LoRa signal, etc. In [9], the reliability of the terminal equipment network transmission mechanism and the network communication delay in LoRaWAN uplink communication are discussed. In [10; 11; 12], the dual key activation update method is used to ensure the security of physical devices in the LoRaWAN network. The LoRaWAN uplink communication research helps to solve the reliability problems of stable network access for terminal equipment and data collection. However, the reply and message confirmation of the network server are usually ignored in such types of research. As for downlink communication, the transmission of business information such as control instructions and interactive data is usually involved, which makes the downlink communication an important part of the LoRaWAN network. It is of great significance to develop the optimal LoRaWAN network downlink routing control strategy to construct the high-performance and high-reliability LoRaWAN network. In the downlink communication of the LoRaWAN network, transmission delay and packet loss rate are important parameters to evaluate the downlink communication service quality. In [13; 14], it is shown that the downlink communication performance is usually limited by the arrival rate, the retransmission times, and the data rate of uplink data. It is proposed to employ downlink channel monitoring, the cyclic broadcast, and the multicast method to improve the downlink communication performance. However, most of these studies try to improve the communication performance based on the optimization of the communication polling mechanism. The routing control is hardly studied. It is known that in LoRaWAN communication, data control and forwarding are closely related. With the increase of the downlink load rate, communication congestion is very likely to be induced if the proper routing control is missing. In recent years, software defined networks (SDN) technology has evolved into a flexible strategy for ensuring heterogeneous networks support a wide range of application requirements, as shown in Table 1.  
In the process of data downlink transmission, when the network is overloaded or congested, by applying the proper routing control strategy, the resource allocation mechanism in SDN offers the possibility to ensure highly efficient network communication. It is known that the bandwidth occupancy rate of the downlink communication link is a system with periodic dynamic changes [22; 23]. It is needed to construct the time series regression analysis for the data processing. In this paper, the auto-regressive integrated moving average (ARIMA) time series algorithm is developed to model and analyze the LoRaWAN downlink bandwidth occupancy.