Delay performance of cognitive radio networks for point-to-point and point-to-multipoint communications
© Tran et al.; licensee Springer. 2012
Received: 26 August 2011
Accepted: 10 November 2011
Published: 10 November 2011
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 closed-form 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 point-to-point to point-to-multipoint communications allowing for multiple SUs and derive the related closed-form 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 closed-form expressions for performance evaluation of cognitive radio networks.
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 under-utilized . 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) . 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 , 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 (PU-Rx). This work has revealed that if the link from the secondary transmitter (SU-Tx) to the PU-Rx resides in a deep fade, the power of the SU-Tx can be increased to improve the link to the SU-Rx without compromising the peak interference power constraint. Later, the fundamental capacity limits with imperfect channel knowledge have been studied in [4, 5]. In , 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 (PU-Tx)→PU-Rx link is important to mitigate the interference from the SU-Tx→PU-Rx link while the channel knowledge of the SU-Tx→PU-Rx link has little impact on the SU capacity. In , 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 delay-insensitive and delay-sensitive 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 . The results in  have also shown that for a given peak and average interference power constraint at the PU-Rx, 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  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, point-to-point and point-to-multipoint 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 SU-Tx has a delay constraint. In order to not cause harmful interference to any surrounding PU-Rx, the SU-Tx needs to adapt its transmit power and commence transmission before the packet delay threshold is reached. Given this setting, in the point-to-point 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 SU-Tx. Furthermore, assuming that packet arrivals at the SU-Tx follow a Poisson process, the queueing model for point-to-point scenario can be described as an M/G/1 system in which packet inter-arrival times are exponentially distributed, service time is a general distribution and traffic is processed by a single server. In the point-to-multipoint scenario, also known as multicast, a secondary base station (SBS) transmits a common packet to all SU-Rx while keeping the peak interference power to the surrounding PU-Rx below a given threshold. By applying the obtained PDF and CDF for the point-to-point scenario, a closed-form expression for the outage probability that the SBS cannot transmit the common packet successfully to a number of SU-Rx are obtained. Moreover, a closed-form expression for the probability that the SBS can transmit the common packet successfully to all SU-Rx, 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 point-to-point and point-to-multipoint scenarios are introduced. In Section 3, analytical formulations for the point-to-point 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 point-to-multipoint scenario. Section 5 provides numerical results and discussions. Finally, conclusions are presented in Section 6.
2. System model
In the sequel, we introduce the point-to-point and point-to-multipoint 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 unit-mean in the presence of additive white Gaussian noise (AWGN). The additive noises at both SU-Rx and PU-Rx constitute independent circular symmetric complex Gaussian random variables with zero-mean and variance N0, denoted as (0,N0). 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. Point-to-point scenario
2.1.1. Peak interference power constraint
2.1.2. Delay constraint
2.1.3. Queuing model for point-to-point communications
The packets arriving at the SU are stored in a buffer and served in first-in first-out (FIFO) order. Assuming that the packet arrival follows a Poisson process with arrival rate λ, the considered point-to-point 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 .
where 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. Point-to-multipoint scenario
2.2.1. Peak interference power constraint
2.2.2. Delay constraint
Clearly, if the SBS receives ACKs from all SU-Rx before tout, it can be considered as the best channel condition. On the other hand, the SBS may not transmit the common packet successfully to all SU-Rx due to the fading environment.
3. Performance analysis for point-to-point communications
In this section, we derive closed-form 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 SU-Tx wants to transmit with maximum transmission rate in order to reduce dropped packets due to timeout. On the other hand, the SU-Tx not only needs to adjust its transmission power in response to changes of the transmission environment but also guarantee the QoS of any PU-Rx 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. h1, 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.
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 for x ≥ tout. In the following, the PDF 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 , 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 is given by (35).
4. Performance analysis for point-to-multipoint communications
In this section, we consider point-to-multipoint communications, in which both SU and PU links undergo Rayleigh fading. We first derive the exact closed-form 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 SU-Rx in its coverage range.
4.1. Outage probability
In the point-to-multipoint scenario, the SBS transmits common packets to SU-Rx in its coverage. Some SU-Rx 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 SU-Rx cannot receive the common packets successfully, known as outage probability.
Similar to point-to-point communications, the event that the SU-Rx n cannot receive a packet successfully is formulated as T n ≥ tout 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 point-to-multipoint communications, the SBS may transmit common packets successfully to all SU-Rx 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 tout, i.e., .
where (39) can be calculated with the help of (19).
5. Numerical results
We first provide numerical results for point-to-point 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 point-to-multipoint communications. The system parameters are selected following  as follows:
System bandwidth: B = 1 MHz
Packet size: L = 4096 bits (512 bytes)
Timeout: tout = 10 ms
Noise power spectral density: N0 = 1 W/Hz
5.1. Point-to-point 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, Qpk, 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 Qpk, the channel utilization 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 SU-Tx 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 Qpk ≥ 16.5 dB, as the channel utilization is sufficiently low.
5.2. Point-to-multipoint communications
In this article, we have analyzed the delay performance of spectrum sharing systems for point-to-point and point-to-multipoint 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. Closed-form expressions for the outage probability and average transmission time for point-to-point 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 point-to-point communications, we have also derived closed-form expressions for the outage probability and the successful transmission probability for point-to-multipoint 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 point-to-point and point-to-multipoint communications in spectral sharing systems may serve to efficiently examine system performance. For example, it may be used to deduce a trade-off between QoS requirements of the secondary system and interference constraints posed by the primary system.
Part of this study was presented at the IEEE International Symposium on Wireless and Pervasive Computing, Hong Kong, China, February 2011.
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