# Stochastic Network Calculus-Aided Delay Analysis of Wireless-Power Line Mixed Networks
## 1 Introduction  
Due to the rapidly availability and huge geographical area coverage of power lines, power line communication (PLC) has been widely utilized to meet the ever-increasing user demand for speedy access to data [1]. For example, PLC has been applied for smart grid communication from transmission and distribution networks to consumer-end home area networks, for enabling the local area network and for home automation using indoor wiring infrastructure [2]. However, different from conventional wired communication, PLC has its own set of challenges, since the power lines were not initially designed for communication purposes. In particular, the channel fading, noise power and attenuation in PLC randomly fluctuate as the location, time and frequency vary. Thus, it remains a great challenge to integrate PLC into modern communication system and to model PLC channels. Additive noise in PLC systems is more complex than that in wired communications since it is a mixture of background noise and impulsive noise, which can be described by the Nakagami-m distribution and middleton class A distribution [3]. In addition to these two noise models, Bernoulli-Gaussian distributions are widely utilized to model both types of noises in the literature. Similar to conventional wired and wireless communications, the PLC channel is also affected by multiplicative noise due to the adverse influence of multipath and mismatches in the impedances at the joints. These effects can be well imitated by Rician and Rayleigh fading models [4]. Furthermore, the log-normal distribution has been proven to be an excellent model for PLC communications by the measurement campaign conducted in US urban and sub-urban areas over medium voltage power lines across frequencies of 1.8 to 30 MHz [5]. The PLC channel model has been thoroughly investigated, and the performance of PLC system has been well analyzed over recent decades. The attention and the focus ofresearchers has become the wireless-power line mixed communications. It has been proven that network performance can be sufficiently improved when we use a wireless-power line diversity system by employing diverse combinations of the received wireless and PLC signals [6]. The authors of [7] investigated a PLC-assisted wireless relay network wherein power lines initialize and synchronize wireless amplify-and-forward relays and broadcast data between these relays. Similarly, a wireless-PLC relay network was considered in [8] where the wireless channel is occupied by the master node while the PLC channel is occupied by slave nodes. The standards set by INSTEON Technologies combine wireless and PLC technologies together by utilizing the wireless channel to relay control commands [9]. Furthermore, the physical layer performance can be strengthened with emerging technologies of modern wireless networks, such as beamforming and 5G. The performance of PLC networks and PLC-wireless networks has been well-investigated. In [10], the bit error rate (BER) performance of OFDM PLC considering the impulsive noise was demonstrated, and the results showed that the OFDM-based PLC systems were subject to much higher error rates in the presence of impulsive noise. Furthermore, the BER performance for a medium-voltage MIMO-OFDM PLC system was studied in [11]. In [12], the trade-off between the algorithm complexity and symbol error rate (SER) was studied in the case of interfacing between wired and wireless communications. Most of the performance analysis works focused on modeling the channel model and noise model for PLC systems and concentrated on BER and SER metrics [13; 14]. Though several wireless-PLC mixed networks have been investigated, few works have dealt with the delay analysis problem thus far. To address this problem, we not only need to evaluate the delay for wireless transmission and PLC links but also have to consider that the output of the wireless link is the input of the PLC link.