Diffusion limit of a modified Erlang-B system with sensing time of secondary users

Annals of Operations Research https://doi.org/10.1007/s10479-022-05153-w ORIGINAL RESEARCH Diffusion limit of a modified Erlang-B system with sensing time of secondary users Kazuma Abe1 · Tuan Phung-Duc2 Accepted: 19 December 2022 © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022 Abstract We analyze a queueing model for cognitive wireless networks using the asymptotic-diffusion method (Moiseev et al. (2020); Nazarov et al. (2020)). Cognitive wireless is a technology that resolves radio spectrum shortages by allowing secondary users (SUs, unlicensed users) to occupy channels initially assigned to primary users (PUs, licensed users). SUs need to sense the channel availability upon arrival. After sensing, an SU can transmit if there is an idle channel; otherwise, the SU must continue sensing. We consider the situation where SUs may be interrupted by the arrivals of PUs when all channels are occupied. We derive a diffusion limit for the queueing model when the SUs’ mean sensing time tends to infinity. The diffusion limit leads to an approximate probability distribution of the number of sensing SUs. Finally, we derive a necessary stability condition which turns out to be consistent with the sufficient condition obtained in previous research. Keyword Cognitive wireless networks, Diffusion limit, Asymptotic behavior, Retrial queue Mathematics Subject Classification 60K25 · 60K05 · 90B22 1 Introduction Recent progress in information technology has led to an explosive increase in Internet traffic. In addition to the development of the Internet of Things (IoT), COVID-19 drives online communication needs, making well networked environments essential. As a result, there is a concern about the shortage of wireless resources, such as radio bandwidth allocated to unlicensed users. B Tuan Phung-Duc Kazuma Abe 1 Graduate School of Science and Technology, University of Tsukuba, 1-1-1, Tennoudai, Tsukuba, Ibaraki, Japan 2 Institute of Systems and Information Engineering, University of Tsukuba, 1-1-1, Tennoudai, Tsukuba, Ibaraki, Japan 123 Annals of Operations Research Cognitive wireless, which is used to develop 5G or Beyond (B5G) technologies, is expected to alleviate the bandwidth scarcity problem. Cognitive wireless networking separates users into two classes: primary users (PUs) and secondary users (SUs). PUs are licensed and have a dedicated bandwidth, while SUs can only opportunistically use that bandwidth. There are three main types of cognitive technology: overlay access, underlay access and interweave access (Nasser et al., 2021) . In underlay access, SUs are allowed to transmit concurrently with PUs over the same channels. However, their traffic must not exceed a certain threshold so as to keep the interference on PUs below a fair value. In overlay access, SUs may, simultaneously with PUs, occupy the same channel until the capacity of the channel is maximized. In this case, the SU sends its data by relaying the PUs. This type of technology thus needs the cooperation of each user and may result in an invasion of PUs’ privacy. In interweave access, SUs are allowed to transmit at maximum power only when PUs are not present. This paradigm is also known as the classical cognitive radio (CR) (Nasser et al., 2021) . In this paper, we focus on interweave access. SUs must sense the availability of the channels before using the frequency bands. If an SU finds an idle channel after sensing, it occupies the channel and starts communication; otherwise, the SU must sense again to find an idle channel at a later time. This sensing behavior of SUs resembles retrial queues. The reader can refer to Artalejo and Gómez (2008); Falin and Templeton (1997) for research in retrial queues. In retrial queues, arriving customers are blocked when the servers are already fully occupied. These blocked customers instead enter a virtual waiting room, called the orbit, and seek service again after a random waiting time until they successfully complete the communication (Phung-Duc, 2019) . In the cognitive radio model, every arriving SU is sent to the sensing pool (the orbit in the retrial queue) and senses to find an idle channel. Queueing systems for cognitive radio networks are extensively studied (PalunčIć et al., 2018) . Salameh et al. (2017) considered a model with a stochastic choice of channels and a finite number of simultaneously sensing users (i.e., a finite sensing pool). In Akutsu and Phung-Duc (2019) and Phung-Duc et al. (2021), they assume that the size of the sensing pool is infinite. Diffusion limits for queueing systems were deeply studied in Halfin and Whitt (1981) and Whitt (2004), and those for retrial queues were studied in Moiseev et al. (2020), Nazarov et al. (2019), Nazarov et al. (2020) and Nazarov et al. (2020b). The latter used the characteristic function approach. In this paper, using the asymptotic-diffusion method (Moiseev et al., 2020; Nazarov et al., 2020; Nazarov et al., 2019, 2020b) , we focus on the situation where it takes SUs a long time to sense the availability of channels. In this case, the evaluation of the number of SUs in the orbit is extremely difficult using a conventional method such as level-dependent quasi-birthand-death process (QBD) (Phung-Duc et al., 2010) . Because the number of SUs in the orbit is large, we need to truncate the orbit at an extremely large truncation level, denoted by N ∗ . The complexity of the algorithm is proportional to that for computing N ∗ inverse matrices and thus is extremely large. From the computational point of view, the asymptotic diffusion method is more useful in this case and complements the level-dependent QBD approach. When σ takes closer to 0, the number of SUs in the sensing pool diverges, but a scaled version of this number converges to a deterministic process. Furthermore, we study the second-order asymptotics, in which the scaled and centered number of SUs in the orbit weakly converges to a diffusion process. Finally, the limiting results are used to approximate stationary performance measures. The main result of this study is a necessary condition of stability for the steady-state regime, which turns out to be consistent with the sufficient condition in Phung-Duc et al. (2021). In 123 Annals of Operations Research addition, we show the uniqueness of the stationary solution of the differential equation, which determines the asymptotic number of SUs in the sensing pool in the stationary regime. The paper consists of seven sections. In Sect. 2, we introduce the model description and the mathematical model. Next, we consider the diffusion analysis of cognitive wireless networks in Sect. 3. In Sect. 4, we prove the main results for the stability condition of our model. Using the continuous probability distribution obtained from the diffusion limit, we construct approximations for the discrete distribution in Sect. 5. Section 6 compares the (...truncated)