Delay performance of cognitive radio networks for pointtopoint and pointtomultipoint communications
 Hung Tran^{1}Email author,
 Trung Q Duong^{1} and
 HansJürgen Zepernick^{1}
https://doi.org/10.1186/1687149920129
© Tran et al.; licensee Springer. 2012
Received: 26 August 2011
Accepted: 10 November 2011
Published: 10 November 2011
Abstract
In this article, we analyze the packet transmission time in spectrum sharing systems where a secondary user (SU) simultaneously accesses the spectrum licensed to primary users (PUs). In particular, under the assumption of an independent identical distributed Rayleigh block fading channel, we investigate the effect of the peak interference power constraint imposed by multiple PUs on the packet transmission time of the SU. Utilizing the concept of timeout, exact closedform expressions of outage probability and average packet transmission time of the SU are derived. In addition, employing the characteristics of the M/G/1 queuing model, the impact of the number of PUs and their peak interference power constraint on the stable transmission condition and the average waiting time of packets at the SU are examined. Moreover, we then extend the analysis for pointtopoint to pointtomultipoint communications allowing for multiple SUs and derive the related closedform expressions for outage probability and successful transmission probability for the best channel condition. Numerical results are provided to corroborate our theoretical results and to illustrate applications of the derived closedform expressions for performance evaluation of cognitive radio networks.
Keywords
1. Introduction
Radio spectrum is one of the most precious and limited resources in wireless communications. It has become scarce due to the rapid growth of a variety of mobile devices and the emerging of many new mobile services. However, recent measurement campaigns conducted by the Federal Communications Commission in the United States have revealed that vast portions of the allocated spectrum are heavily underutilized [1]. Clearly, the scarcity of the spectrum is due to its inefficient usage rather than a shortage of spectrum resources. As a consequence, the spectrum utilization problem has become more crucial and has stimulated new research such as extensive work on cognitive radio networks (CRN) [2]. In CRNs, there are two types of users who are referred to as primary user (PU) and secondary user (SU). The PU licenses the spectrum while a SU may access the spectrum owned by the PU provided that it does not compromise the quality of service (QoS) delivered to the PU. Therefore, a major challenge with the design of CRNs is to maintain the desirable QoS at the PU while offering a sufficiently high transmission rate to the SUs.
Recently, the spectrum sharing approach is considered as a promising solution to utilize the licensed radio frequency. Particularly, the SU and the PU can transmit simultaneously as long as the interference caused by the SU to the PU is lower than a predefined threshold. In [3], considering different fading channels, the ergodic capacity of the spectrum sharing system is investigated for either peak interference power constraint or average received interference power at the primary receiver (PURx). This work has revealed that if the link from the secondary transmitter (SUTx) to the PURx resides in a deep fade, the power of the SUTx can be increased to improve the link to the SURx without compromising the peak interference power constraint. Later, the fundamental capacity limits with imperfect channel knowledge have been studied in [4, 5]. In [6], the authors have considered a new sophisticated approach for spectrum sharing systems where the impact of channel knowledge on the performance of a secondary user has been studied. The results show that the channel knowledge of the primary transmitter (PUTx)→PURx link is important to mitigate the interference from the SUTx→PURx link while the channel knowledge of the SUTx→PURx link has little impact on the SU capacity. In [7], different notions of capacity are investigated for the Rayleigh fading channel subject to both the peak and average interference power constraints. Especially, the ergodic capacity and outage capacity which are considered suitable for delayinsensitive and delaysensitive applications are studied. In [8–10], the novel concept of effective capacity has been introduced to investigate the QoS requirements such as delay constraint in wireless communication systems. In particular, the effective capacity is defined as the maximum constant arrival rate that can be provided by the channel while the delay constraint of the spectrum sharing system is satisfied [9]. The results in [9] have also shown that for a given peak and average interference power constraint at the PURx, the maximal effective capacity is achieved under the optimal power control policy. In relation to the delay constraint in the spectrum sharing system, in [11, 12], we have used another approach, which is based on the packet transmission time to investigate the performance of CRN. These results have revealed the impact of the peak interference power constraint on the delay of packets for different types of fading channels. However, we analyzed the spectrum sharing system with peak interference power constraint only for a single PU.
In this article, we therefore extend our previous work [11] to consider the more realistic case of a CRN under the peak interference power constraint in the presence of multiple PUs. Specifically, we examine the delay performance for two scenarios, pointtopoint and pointtomultipoint communications. In the latter scenario, we extend the investigation from multiple PUs to also allow for multiple SUs at the receiving end. We assume that each packet of the SUTx has a delay constraint. In order to not cause harmful interference to any surrounding PURx, the SUTx needs to adapt its transmit power and commence transmission before the packet delay threshold is reached. Given this setting, in the pointtopoint scenario, we derive the probability density function (PDF) and cumulative density function (CDF) for the packet transmission time, outage probability and average transmission time of packets at the SUTx. Furthermore, assuming that packet arrivals at the SUTx follow a Poisson process, the queueing model for pointtopoint scenario can be described as an M/G/1 system in which packet interarrival times are exponentially distributed, service time is a general distribution and traffic is processed by a single server. In the pointtomultipoint scenario, also known as multicast, a secondary base station (SBS) transmits a common packet to all SURx while keeping the peak interference power to the surrounding PURx below a given threshold. By applying the obtained PDF and CDF for the pointtopoint scenario, a closedform expression for the outage probability that the SBS cannot transmit the common packet successfully to a number of SURx are obtained. Moreover, a closedform expression for the probability that the SBS can transmit the common packet successfully to all SURx, i.e. the best channel condition, is also achieved.
The rest of the article is organized as follows. In Section 2, the system model and assumptions for the pointtopoint and pointtomultipoint scenarios are introduced. In Section 3, analytical formulations for the pointtopoint scenario such as the PDF and CDF of the packet transmission time, the outage probability, and the moment of packet transmission time is derived. On this basis, queueing theoretical conclusions are drawn. In Section 4, we present the delay performance for the pointtomultipoint scenario. Section 5 provides numerical results and discussions. Finally, conclusions are presented in Section 6.
2. System model
In the sequel, we introduce the pointtopoint and pointtomultipoint scenarios in the context of a spectrum sharing system where the SU operates in the area of multiple PUs. As for the radio links between the different entities, we assume identical and independent distributed (i.i.d.) Rayleigh block fading channels with unitmean in the presence of additive white Gaussian noise (AWGN). The additive noises at both SURx and PURx constitute independent circular symmetric complex Gaussian random variables with zeromean and variance N_{0}, denoted as $\mathcal{C}\mathcal{N}$ (0,N_{0}). As the SU and the PUs may transmit simultaneously, the interference caused by the SU to the PUs should not exceed a certain threshold.
2.1. Pointtopoint scenario
2.1.1. Peak interference power constraint
2.1.2. Delay constraint
2.1.3. Queuing model for pointtopoint communications
The packets arriving at the SU are stored in a buffer and served in firstin firstout (FIFO) order. Assuming that the packet arrival follows a Poisson process with arrival rate λ, the considered pointtopoint scenario may be modeled as an M/G/1 queueing system [17–19] with service time given as general distribution and the system being equipped with a single server [20].
where $\rho =\lambda E\left[T\right]$ is referred to as channel utilization and E[T^{ i }], i = 1,2 denotes the first and second moment of packet transmission time, respectively. Furthermore, the following result from queueing theory can be applied for the stability of transmission of the SU.
2.2. Pointtomultipoint scenario
2.2.1. Peak interference power constraint
2.2.2. Delay constraint
Clearly, if the SBS receives ACKs from all SURx before t_{out}, it can be considered as the best channel condition. On the other hand, the SBS may not transmit the common packet successfully to all SURx due to the fading environment.
3. Performance analysis for pointtopoint communications
In this section, we derive closedform expressions for the PDF and CDF of packet transmission time as well as outage probability. Based on these results, we not only quantify the first and second moment of packet transmission time but also investigate the queueing theoretical characteristics of the considered spectrum sharing system.
3.1. PDF of packet transmission time
In this scenario, the SUTx wants to transmit with maximum transmission rate in order to reduce dropped packets due to timeout. On the other hand, the SUTx not only needs to adjust its transmission power in response to changes of the transmission environment but also guarantee the QoS of any PURx around.
It is easy to see that the packet transmission time, T, now turns out to be a function of multiple random variables, i.e. h_{1}, g_{ m }, m = 1, 2,..., M. Therefore, in order to investigate the delay performance, we need to derive the PDF of T in the sequel.
where $G=\frac{{Q}_{\mathsf{\text{pk}}}}{{N}_{0}}1$ is introduced for brevity. It is noted that (20) exactly leads to the PDF of [[11], Eq.(10)] for the peak interference power constraint of a single PURx by setting M = 1.
In the subsequent sections, the important result in (20) will be used to investigate the outage probability, the average transmission time and the average waiting time of packets.
3.2. Outage probability
while ${f}_{{T}_{suc}}\left(x\right)=0$ for x ≥ t_{out}. In the following, the PDF ${f}_{{T}_{suc}}\left(x\right)$ given in (27) will be used to derive the moment of packet transmission time.
3.3. Moment of packet transmission time
Let us recall that a transmitted packet can be received successfully or not due to the fading channel. Therefore, examining average transmission time shall consider both packet transmission time without and with timeout.
where P_{ out }is given by (22) and $\mathbb{E}\left[{T}_{suc}^{i}\right]$, i = 1, 2 can be calculated by (29) and (32), respectively
3.4. Queuing theoretical characteristics
Secondly, the transmission of an SU is stable if and only if the average arrival rate is less than the average transmission rate. Thus, we can make a statement about the stable transmission condition as follows:
The inequality (36) is derived by substituting (34) for i = 1 into (7).
where $E\left[W\right]$ is given by (35).
4. Performance analysis for pointtomultipoint communications
In this section, we consider pointtomultipoint communications, in which both SU and PU links undergo Rayleigh fading. We first derive the exact closedform expression for the outage probability of the secondary system, and then we consider the probability for the special case that the SBS can transmit the common packet successfully to all SURx in its coverage range.
4.1. Outage probability
In the pointtomultipoint scenario, the SBS transmits common packets to SURx in its coverage. Some SURx may not receive the common packets successfully due to fading environment. In order to analyze the performance of this scenario, we will calculate the probability that k out of the total of N SURx cannot receive the common packets successfully, known as outage probability.
Similar to pointtopoint communications, the event that the SURx_{ n }cannot receive a packet successfully is formulated as T_{ n }≥ t_{out} where T_{ n }is an i.i.d. random variable distributed following the CDF given by
where (38) is obtained by using the binomial theorem and the help of (19).
4.2. Best channel condition
For pointtomultipoint communications, the SBS may transmit common packets successfully to all SURx if the channel condition is ideal. This is known as the best channel condition which can be expressed as the longest transmission time for one common packet to be less than t_{out}, i.e., $\left\{\underset{n}{\text{max}}\left\{{T}_{n}\right\}<{t}_{\mathsf{\text{out}}}\right\}$.
where (39) can be calculated with the help of (19).
5. Numerical results
We first provide numerical results for pointtopoint communications. In particular, we study the impact of the peak interference power constraint and the number of PUs on the outage probability, average transmission time and queuing theoretical characteristics of the secondary system. We then discuss results about the outage probability and the probability that the SBS can transmit the common packet successfully under the best channel condition for pointtomultipoint communications. The system parameters are selected following [16] as follows:

System bandwidth: B = 1 MHz

Packet size: L = 4096 bits (512 bytes)

Timeout: t_{out} = 10 ms

Noise power spectral density: N_{0} = 1 W/Hz
5.1. Pointtopoint communications
In the sequel, we focus on the impact of the peak interference power constraint and the number of PUs on the performance of an SU.
5.1.1. Outage probability
5.1.2. Average transmission time
5.1.3. Queuing theoretical results
Figure 8 illustrates that the average waiting time increases as the number of PUs and arrival rate increase. Apparently, these results are in line with the behavior observed for the outage probability and average transmission time above and may be explained as follows. At a fixed value of Q_{ pk }, an increasing number of PUs leads to an increase of average transmission time due to the reasons explained above and hence an increase of average waiting time. Similarly, when the arrival rate increases, the number of packets to be stored in the buffer increases as well and await transmission. On the other hand, as the transmission rate is restricted due to the peak interference power constraint, the packets have to stay longer in the buffer before they are transmitted.
Figure 9 provides insights into the stable transmission condition as a function of the peak interference power, Q_{pk}, with the number of PUs given as M = 1,3,5 and arrival rates being λ = 10,50 packets/s. The results show that for a given value of the number PUs, M, and fixed value of the peak interference power Q_{pk}, the channel utilization $\rho =\lambda E\left[T\right]$ for arrival rate λ = 10 packets/s outperforms the result for λ = 50 packets/s. In other words, the significant lower channel utilization for λ = 10 packets/s compared to λ = 50 packets/s provides a more stable transmission with respect to the service rate μ in terms of the stable condition formulated in (36). Clearly, the service rate μ of an SUTx is restricted for a fixed value of the peak interference power Q_{ pk }while a higher arrival rate causes more packets to be processed by the buffer expecting timely transmission. Accordingly, the ratio of arrival rate to service rate, relating to the stable transmission condition λ/μ < 1, has to be carefully considered in order to not exceed the capacity of the secondary system. It can also be observed from the figure that the stable transmission condition can be easily satisfied in the high regime of the peak interference power, say Q_{pk} ≥ 16.5 dB, as the channel utilization is sufficiently low.
5.2. Pointtomultipoint communications
6. Conclusions
In this article, we have analyzed the delay performance of spectrum sharing systems for pointtopoint and pointtomultipoint communications. In particular, we have assumed that each packet has a delay threshold, transmission channels undergo Rayleigh fading, SUs posses perfect CSIs and ACKs are transmitted without error and delay. Closedform expressions for the outage probability and average transmission time for pointtopoint communications are obtained. In addition, we have utilized the M/G/1 queuing model to analyze the queueing characteristics of such systems including the average transmission time, the packet waiting time and the stable transmission condition of an SU. Based on the analytical framework established for pointtopoint communications, we have also derived closedform expressions for the outage probability and the successful transmission probability for pointtomultipoint communications under best channel conditions. Numerical results for representative scenarios have been provided to quantify the impact of an increase of the number of SUs and PUs on system performance. In particular, it has been shown that an increasing number of SUs or PUs significantly increases packet delay if the peak interference power is constraint by the PUs to be low while small performance degradation is observed if the PUs tolerate sufficiently large peak interference power. Accordingly, the developed analytical framework for pointtopoint and pointtomultipoint communications in spectral sharing systems may serve to efficiently examine system performance. For example, it may be used to deduce a tradeoff between QoS requirements of the secondary system and interference constraints posed by the primary system.
Declarations
Acknowledgements
Part of this study was presented at the IEEE International Symposium on Wireless and Pervasive Computing, Hong Kong, China, February 2011.
Authors’ Affiliations
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