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On the effect of false alarm rate on the performance of cognitive radio networks
EURASIP Journal on Wireless Communications and Networking volume 2015, Article number: 244 (2015)
Abstract
Spectrum sensing plays a significant role in enabling utilization of spectrum holes by unlicensed secondary users (SUs) in cognitive radio networks (CRNs). Most of the related work concerning spectrum sensing has focused on sensing carried out by incoming secondary users (SUs) aiming at locating spectrum opportunities. However, in order to appropriately protect returning licensed primary users (PUs), SUs should continuously perform spectrum sensing during their ongoing transmissions. An important issue associated with the continuous sensing is the false alarm rate (FAR), which is defined as the average number of false alarms per unit of time and can be modeled by a Poisson process with Poisson parameter λ _{FAR}. In this paper, we address this issue and develop a continuous time Markov chain (CTMC)based analytical model to evaluate the effect of the false alarm rate on the performance of CRNs. A major feature of the proposed analytical framework is that it takes into account the effects of sensing errors by both incoming SUs looking for free channels to transmit on and the already transmitting SUs expecting the presence of returning PUs. The analytical model also examines the interference tolerance among PUs and SUs as well as the impact of SUs residual selfinterference. The performance results show that high λ _{FAR} can severely degrade PUs performance and reduce the overall system resource utilization. However, with increasing PU interference tolerance, PUs performance improves as well. SU residual interference was found to decrease the detection probability resulting in a low PU performance. Extensive simulations validate the analytical model, demonstrating excellent agreement with the theoretical results.
Introduction
With today’s inefficient utilization of the scarce radio spectrum, cognitive radio (CR) [1–3] is becoming an important tool for solving the problem of spectrum underutilization. As a result, there has been considerable research effort focusing on CR techniques that enable using radio spectrum efficiently. In CR networks (CRNs), unlicensed secondary users (SUs) employ spectrum sensing [4–6] to discover spectrum holes during the absence of licensed primary users (PUs) before attempting network access. Energy detection [7–9] is the simplest method for detecting the presence of PUs. It is based on calculating the energy of the received samples which is compared to a threshold. If the threshold is exceeded, it is decided that a signal or signals are present. If the sensed channel is free, SUs may be allowed to transmit on that channel.
No matter which detection scheme is used for protecting returning PUs, it will lead to the occurrence of false detection of returning PUs, i.e., we will erroneously assume that a PU has returned, when in fact the PU’s channel is free. A simple way to characterize the occurrence of false alarms for already transmitting SUs is to use the average number of false alarms per a time unit, similar to the rate parameter of a Poisson process. We call this parameter the false alarm rate parameter λ _{FAR} [10, 11]. It is to be mentioned that in multichannel systems with handoff capability (as studied here), an SU moving from its current operating channel (for example, due to false detection of a returning PU) will attempt to locate another free channel to continue its ongoing data transmission.
To protect reappearing PUs, SUs perform spectrum sensing on a continuous basis along with data transmission. Figure 1 presents the concept of simultaneous spectrum sensing and data transmission. When a new SU arrives at a channel for the first time, it senses the whole PU channel with a bandwidth W for T _{1} seconds. After discovering that the channel is free, the SU starts data transmission over a bandwidth of W−W _{ GB }−W _{2}. While the SU is transmitting data over the lower part of the PU channel, it continuously monitors the PU channel using the upper part with bandwidth W _{2} to detect the presence of a returning PU. A guard band (GB) subchannel is used to prevent the leakage from SU’s selfinterference signal. The SU makes sensing decision every T _{2} seconds. As a PU reappears (if correctly detected by the SU), the SU ceases its data transmission and leaves the channel.
Spectrum sensing errors impose several challenges into the design of CRNs and specially in models that employ simultaneous spectrum sensing and data transmissions. For example, the frequent occurrence of false alarm events is highly undesirable since it makes it challenging for SUs to fully utilize the spectrum opportunities and severely degrades their quality of service (QoS). The results reported in [12] have revealed the importance of investigating the effect of unnecessary spectrum handoff due to false alarms during spectrum sensing. Motivated by these issues, we develop a comprehensive CTMC analytical framework which models all related sensing factors that have not been fully accounted for in previous works. Our model can incorporate the effect of FAR as well as handling of the effect of the residual selfinterference that is left over after SIC process. To our knowledge, the effect of the FAR on the operation of CRNs has not been investigated in the literature. The contribution of this paper can be summarized as follows:

It motivates and develops a CTMCbased analytical framework that precisely evaluates the performance of CRNs. Unlike existing approaches, the proposed model thoroughly investigates the effect of FARs on the performance of CRNs. The model also takes into consideration the SUs residual selfinterference as well as interference tolerance among PUs and SUs.

It models the occurrence of FAR events as a Poisson process with parameter λ _{FAR} with a theoretical justification based on a shrinking Bernoulli process [13].

It proposes a new performance evaluation measure, the SU selftermination probability. The proposed metric can precisely measure the percentage of SU calls that are terminated because of the FAR occurrence. The new metric also allows for measuring the SUs’ ability of utilizing spectrum opportunities.

Extensive simulations to validate the theoretical results.
We believe that the work presented in this paper contributes towards a better understanding and provides a new insight into the operation of CRNs and can be used to develop more accurate and realistic CRNs performance analysis models. The rest of the paper is organized as follows. Section 2 presents the related work. The system model is presented in Section 3. Section 4 contains a description of the continuous spectrum sensing and data transmission. Section 5 presents the CTMCbased analytical framework. In Section 6, we discuss performance evaluation metrics for the CRN. Section 7 summarizes results and provides comparison of simulation and theoretical results. Finally, Section 8 provides the conclusions and remarks on future work.
Related work
In recent years, several studies has been proposed to detect returning PUs in CRNs. In [14], the authors investigate the issues of how to maximize the overall discovery of opportunities in the licensed channels and how to minimize the delay in locating an idle channel in order to minimize interference on returning PUs. Similarly, the authors of [15] presented a dynamic spectrum access mechanism in a network where SUs do not have perfect knowledge of PUs’ communication behavior. The interference issue has also been studied. However, this study only considered perfect spectrum sensing and a network with only one PU. In [16], the authors consider a preemptive priority approach for the channel access where SUs must vacate their channels whenever the corresponding PUs appear. The work presented in [17] formulates a joint spectrum sensing and access problem as an evolutionary game by considering the mutual influence between spectrum sensing and access. Although the interference problem has been addressed in these works, the problem of the FAR has not been investigated.
In [18], continuous time Markovian process (CTMP) is used to model PU traffic in opportunistic spectrum access (OSA) systems. However, for analyzing SU’s behavior, discrete time queuing was used. In contrary to our work, the underlying assumption made therein is that sensing and data transmission cannot be carried out simultaneously and therefore the SU has to periodically suspend its data transmission in order to perform spectrum sensing. The problems with this technique are the overheads associated with the scheduling and synchronization of the suspension periods among SUs as well as the frequent interruption in the SU’s data transmission. Additionally, the SU can only detect a reappearing PU during the suspension period, even if the PU reappeared before the suspension period. This work also differs from our study because it only supports CRNs with one channel and the assumption that the spectrum sensing is perfect.
Simultaneous spectrum sensing and data transmission approach have been studied in [19–22]. The issue of selfinterference due to transmitting and receiving in the same band has been studied in [23, 24]. In spite of considering the problem of unnecessary false alarms, the authors of [12, 25] did not investigate their effect on performance metrics such as blocking and termination probabilities. Furthermore, the authors of [26] analyzed different types of unreliable sensing for both incoming and ongoing SUs and their impact on the performance of CRNs without addressing the FAR.
Most of the existing CTMC models [27–29] do not cover all the aspects of the spectrum sensing and CRN operation and some important factors were not fully addressed. In our previous work [30], we analyzed the performance of CRNs using a CTMC framework that supports multichannel, spectrum handoff, fullstate dependent transition rates, and the ability to handle spectrum sensing errors. In this paper, we extend the analysis in [30] to capture the effect of the FAR and to handle the residual selfinterference within the SU transceivers.
System model
We consider a CRN with N number of channels in which SUs are allowed to opportunistically utilize licensed spectrum bands with the constraint that the QoS of PUs remains at an acceptable level. There are two approaches for enabling PUs and SUs to coexist and share radio resources in CRNs: spectrum sharing (SS) and opportunistic spectrum access (OSA) [31]. In the SS model, SUs are allowed to transmit simultaneously with PUs on the same band. On the other hand, the OSA approach, which is more suitable for the model presented in this paper, allows SUs to access the licensed channels opportunistically when PUs are not present.
Primary user model
We assume that the primary channel occupancies are time varying alternating between idle and busy periods, and thus SUs must perform spectrum sensing continuously to detect the presence of returning PUs. PU connections arrive at the network according to a Poisson process at a rate of λ _{1}. The PU service rate which is assumed to be exponentially distributed is μ _{1}. We also assume that PUs can obtain primary channel occupancy information, for example, by accessing a core network that makes signaling or querying of the PUs’ base station [32], and thus it is further assumed that PUs do not collide with each other [28].
We assume that both PUs and SUs have some interference tolerance T _{TOL} of how many seconds of interference they will tolerate before withdrawing from the system. If the PU interference tolerance time T _{TOL} is 0, no SU transmission is allowed [33]. We assume equal interference tolerance for both systems leading to both colliding users withdrawing from the system simultaneously. A similar assumption has been considered previously in [28].
Secondary user model
We assume that SU connections arrive at the network according to a Poisson process with λ _{2}. The SU service rate is assumed to be exponentially distributed with μ _{2}. During the absence of PUs, SUs can opportunistically access the free channels if they are not occupied by other SUs. We also assume that SUs are capable of broadcasting control messages on a common control channel (CCC) [34] to show their existence to neighboring SUs in the proximity. Therefore, SUs do not attempt accessing channels occupied by other SUs. Upon detection of the presence of a returning PU, a SU leaves its current channel and starts the spectrum handover process in order to find a new free channel. If the channel search process ends without finding a free channel, the SU terminates its call and leaves the network.
As illustrated in Fig. 1 and similar to the distributed (coordination function) interframe space (DIFS) operation in IEEE 802.11, the SU has to keep sensing the PU channel from the beginning of its transmission, since the PU can arrive at any time instant of a slot. This process forms a continuous sequence of sensing slots with length equals T _{2}. For example, if the PU appears in the middle of a time slot, then the first slot will not get full PU energy, leading to a smaller detection probability than the later full slots. Since the first T _{2} may be wasted, we assume that the first partial slot sensing never leads to detection, i.e., the detection probability is close to zero. Hence, we should detect the PU arrival during T _{TOL}−T _{2} seconds which corresponds to \(\widehat {T_{\text {TOL}}}=\left \lfloor \frac {T_{\text {TOL}}T_{2}}{T_{2}} \right \rfloor \) slots. Although partial slot sensing can enable the SU to perform sensing immediately after the arrival of the PU and hence have a prompt reaction to protect PUs, for the sake of simplifying the analysis, we consider only the full slot sensing by assuming that the detection process will begin from the first full time slot following the SU arrival.
Continuous spectrum sensing model
One possibility for implementing continuous sensing is to leave the upper part of the PU channel empty (i.e., free from SU transmissions) [35]. As shown in Fig. 1, we split the PU channel into three subchannels: (A) SU communication channel, (B) a sufficient vacant guard band to reduce the effect of SU’s selfinterference, and (C) SU sensing channel. When the PU is active, it uses the whole bandwidth (A+B+C) for its communication. The secondary user uses subchannel (A) for its communication. A reappearing PU can be detected by sensing, during ongoing SU transmission from the subchannel (C). It is obvious that a problem here is the selfinterference due to the leakage of the SU’s transmitted signal back to its sensing device. However, the emergence of a large variety of selfinterference cancellation techniques [36–38] in the literature enabled efficient reduction in selfinterference and therefore allowing radios to operate in fullduplex mode. For example, the authors of [39] present a method for canceling a passband selfinterference signal using adaptive filtering in the digital domain. Therefore, in addition to the vacant guard band and bandpass filtering, selfinterference cancellation has also been assumed to remove most of the residual selfinterference. In this model, q∈{1,2} denotes an index with the interpretation that q=1 if the spectrum sensing is carried out by incoming SUs and q=2 if the spectrum sensing is performed by ongoing SUs.
Energy detectorbased spectrum sensing
Without loss of generality, we consider that initial and ongoing spectrum sensing are done using an energy detector [9] with an integrate and dump operation mode as described in [10, 11]. The analysis techniques presented in this paper are generic and not limited to any particular detector, provided that the used detector can be mapped to false alarm probabilities, probability of detections, and false alarm rates. Let y _{ q }(t) denote the SU received signal process. We express the incoming SU received signal process in the form
and the ongoing SU received signal process can be formulated as
In Eqs. (1) and (2), \(s_{\!_{\text {PU}}}(t)\) is the PU transmitted signal, \(s_{\!_{\text {SU}}}(t)\) represents the leakage from the SU transmitted signal, n(t) is the additive white Gaussian noise (AWGN), \(h_{\!_{\text {PU}}}\) is the PU channel gain while \(h_{\!_{\text {SU}}}\) represents the SU leakage signal gain, and t is the time. In the above equations, H _{0} is the null hypothesis meaning that PU is not present in the sensed band, and H _{1} represents the alternative hypothesis referring to the presence of the PU signal.
The received signal is filtered by a bandpass filter to remove the outofband and selfinterference noise. The filtered signal is then squared by the squaring device and applied to the integrator. The integrator output Y _{ q } (also denotes the decision variable of the energy detector) is sampled every T _{ q } seconds. Then, the integrator is reset before integrating the next sample over the next T _{ q } seconds. Finally, Y _{ q } is compared with the decision threshold to decide about the presence of the PU. Let W _{ q } denote the sensed bandwidth. Let γ _{ q } denote the signal to noise ratio SNR and η _{ q } denote the energy detection threshold. According to [9]
where \(\chi _{2u_{q}}^{2}\) is a chisquare distribution with 2u _{ q } degrees of freedom (i.e., the timebandwidth product u _{ q }=W _{ q } T _{ q }) and \(\chi _{2u_{q}}^{2}(2\gamma _{q})\) is a noncentral chisquare distribution with 2u _{ q } degrees of freedom and a noncentrality parameter (2γ _{ q }). It has been shown in [40] that the probability of detection P _{ Dq } and the false alarm probability P _{FAq } can be given as follows:
where Q _{ m(.,.)} is the generalized mth Marcum Qfunction [41].
Referring to Fig. 1, where W represents the PU channel bandwidth. To obtain the received signal energy, let P _{PU} denote the PU transmitted signal power. Let also N _{0} denote the onesided power spectrum density (PSD). Let T _{1} denote the incoming SU initial sensing time. Assuming that the PU signal power is uniformly distributed over the PU channel, then the incoming SU signal energy within the initial sensed area can be obtained as
and the initial sensing SNR is then obtained with
Similarly, we can obtain the signal energy within the ongoing SU sensed area with a sensing time duration T _{2} as
Although we use guard band, bandpass filtering, and selfinterference cancellation to eliminate the effect of SU leakage signal, we assume that the SU selfinterference has not been fully removed. To handle the effect of any remaining selfinterference, we follow the results presented in [42] to model the residual interference. Let assume that the SU operate with a singleantenna fullduplex transceiver. Let \(\alpha _{_{\text {SU}}}\) denote the SU’s residual interference distortion factor. By using Eq. (2), the effective ongoing sensing SNR can be expressed as [42]
In continuous spectrum sensing with fullduplex communication, the consideration of selfinterference is particulary important since the selfinterference can affect the sensing outcome and degrade SUs performance. Although the SU selfinterference signal can have nonzero mean, it has been been assumed in the majority of related works to have a zero mean. For instance in [43], the authors mentioned that in practical fullduplex systems, the selfinterference cannot be completely canceled, such that the signals received at each node is a combination of the signal transmitted by the other source, the residual selfinterference (RSI), and the noise. They also assume that the RSI can be typically modeled as zeromean additive white Gaussian noise (AWGN). The work reported in [42] assumed that the Gaussian distortion and noise follows central chisquare distribution in the absence of PU signals but potentially including RSI and noncentral chisquare distribution when PU signal is present.
Selfinterference mitigation in fullduplex MIMO relays has been investigated in [23] where the authors focused on minimizing the residual loop interference so that it can be regarded as additional relay input noise. They assumed that all signal from the relay output to the relay input (including loop interference (LI) signal) and noise vectors have zero mean. Furthermore, the authors of [44–46] assumed that the SU selfinterfering signal before carrying out selfinterference suppression (SIS) to be a zeromean random signal with selfinterference channel coefficient equal one. In [47], the residual selftransmitted signal is modelled with circular symmetric complex Gaussian variables. Following the common practice in existing models, the use of the assumption that SU’s leakage signal can be zero mean and follows central chisquare distributions is justified and can be hold in order to take into account the RSI signal and perform the analysis.
It should be noted that when we use a dedicated part of the bandwidth (subchannel C) for continuous sensing, the effect of the residual interference becomes much lower than when we use the full bandwidth for simultaneous sensing and transmission. Each incoming SU correctly detects channel occupancy with probability P _{ D1}, and falsely classifies a free channel as occupied with P _{FA1}. Similarly, each SU with ongoing calls detects the arrival of a PU with probability P _{ D2} and falsely classifies a free channel as occupied with P _{FA2}. The corresponding misdetection probabilities for incoming and outgoing SUs are P _{ M1}=1−P _{ D1} and P _{ M2}=1−P _{ D2}, respectively. The detection probability P _{ D2} refers to the probability of detecting incoming PU during the first \(\widehat {T_{\text {TOL}}}\) full slots of its arrival, instead of the perslot detection probability. If the perslot detection probability is denoted as z then \(P_{D2}=1(1z)^{\widehat {T_{\text {TOL}}}}\). This does not affect the FAR process since one perslot false alarm event is enough to initiate the spectrum handoff and channel searching process. Modeling of partial slot sensing is left for future work.
Poisson process approximation
We model the occurrence of the false alarm at each sensing decision with the Bernoulli process. The energy detector makes only one sensing decision in each slot which results into a binary variable (0 or 1). Since the sensing decisions with only white Gaussian noise present are independent, the resulting binary output of the sensing clearly follows the Bernoulli process (i.e., independent and identically distributed process generating 1 and 0 s), and the Bernoulli parameter corresponds to the probability of FAR occurrence (binary output 1) in each spectrum sensing decision.
At each spectrum sensing decision epoch T _{2}, a false alarm occurs with probability P _{FA2} and does not occur with probability 1−P _{FA2}, independently of the decision outcome of the last sensing period. The λ _{FAR} parameter is the product of the decision rate and the false alarm probability [10, 11, 48]. Therefore, λ _{FAR} is given by P _{FA2}/T _{2}. Let us assume that the sensing interval T _{2} is short and therefore we assume that the decision rate given by 1/T _{2} is large, and that the false alarm P _{FA2} is small as otherwise there would be too many false alarms for successful SU operation. Then, the arrival process of false alarms can be approximated by a Poisson process as a limit of a shrinking Bernoulli process [13] with parameter λ _{FAR}.
Continuous time Markov chain model (CTMC)
We consider a twodimensional continuous time Markov chain (CTMC) to describe the CRN system. At any time, the system state is determined by (i,j) where i represents the number of channels occupied by PUs and j represents the number channels occupied by SUs with restriction that 0≤i≤N, 0≤j≤N, 0≤i+j≤N. Let i μ _{1} and j μ _{2} denote the the service completion time for PUs and SUs, respectively. The transition rate from state (i,j) to state (h,l) is given by \(T_{(\textit {h,l})}^{(\textit {i,j})}\). Note that the parameter λ _{FAR} affects all state transitions from states with the number of SUs j>0. As the number of channels increases, the number of states of the CTMC grows exponentially. Since the transition rates depend on the system states, the large number of states combined with the channel searching process under imperfect sensing conditions would make it not trivial to compute the state transition rates. Because it is impractical to present a state transition diagram for a CRN with an arbitrary number of channels, we present an illustrative example in Fig. 2 which shows the allowable state transitions from and to state (1,1) in a Markov chain with three channels.
As an example, consider a CRN with three channels denoted by C1, C2, and C3. Let us assume that channel C1 is occupied by an PU, channel C2 is occupied by a SU, and the last channel C3 is free. The Markov chain is in state (1,1). We now explain a series of events that trigger the system to move from state (1,1) to state (0,0). State (0,0) indicates that all channels are free. On the occurrence of λ _{FAR}, the SU leaves C2 and starts the channel searching process. There are two channel selection possibilities for the SU for continuing its data transmission. The SU can first select C1 with probability 1/2 and then misdetect the presence of the PU on C1 with probability P _{ M1}. The second possibility is to select C3 with probability 1/2, then falsely classify the free channel as occupied by the PU with a false alarm probability P _{FA1}, and finally misdetect the presence of the PU on C1 with probability P _{ M1}. Both selections lead the SU to collide with the PU, and eventually both of them leave the network. Combining all these events, the transition rate from state (1,1) to state (0,0) can be obtained by \(\frac {{{\lambda _{\text {FAR}}}{P_{M1}}}}{2}(1 + {P_{\text {FA1}}})\).
Another example is the transition from state (1,1) to state (2,0). This happens with the arrival of a PU with rate λ _{1} and with probability 1/2 to channel C2 which is occupied by the SU. The SU correctly detects the presence of the PU with detection probability P _{ D2} and vacates the channel. After leaving the channel, the SU has two possibilities with probability 1/2 for each. The SU first falsely classifies the free channel C3 as being occupied by a PU with false alarm probability P _{FA1} and then detects the presence of the PU in channel C1 with probability P _{ D1}, ending up leaving the network. The other possibility is that the SU correctly detects the existence of the PU in channel C1 with probability P _{ D1} and then erroneously classifies the free channel C3 as occupied by a PU with false alarm probability P _{FA1}. The resulting transition rate is \(\frac {\lambda _{1}{P_{D2}}{P_{D1}}{P_{FA1}}}{2}\). Proceeding in a similar manner, the transition rates to and from the remaining states can be obtained.
Generalization of the CTMC
The goal of this section is to extend the results presented previously to describe a CRN with an arbitrary number of channels N. When N is large, constructing a state transition diagram and finding a solution to the corresponding balance equations is complicated and time consuming. Similar to [30], we use a recursive method to calculate the state transition rates to and from all different states of the CTMC representing the CRN network. The state transition rates are used to get all the possible balance equations. Recalling that the number of PUs is denoted by i and the number of SUs is denoted by j. Let us also assume that the number of free channels is denoted by k which is given by k=N−i−j.
It is important to notice that the state transition rates presented in this paper are different from those defined in [30] for several reasons: (1) the inclusion of the effect of the FAR in the CTMC. (2) In our previous work [30] we assumed that ongoing SUs perfectly detect the arrival of PUs and that there are no false alarms during ongoing data transmission. However, in this paper, we assume that SUs with ongoing connections do not perfectly detect the arrival of PUs and also that the false alarm probability during ongoing calls is not negligible. With this assumption, ongoing SUs detect PU arrivals with P _{ D2} where P _{ D2} is an arbitrary value between 0 and 1 and that the false alarm probability during ongoing calls equal P _{FA2} which is also an arbitrary value. Note that this is a more realistic assumption for practical CRNs and represents a significant improvement over our previous work [30] that leads to obtaining accurate state transition rates and state probabilities. (3) Because of the false alarms during ongoing sensing, we require a new state transition that defines the transition from state (i,j) to state (i−1,j−1), with i>0 and j>0. In addition to state transitions because of the FAR events, we refer the reader to [30] for details concerning the other different events that trigger state transitions. All possible state transition types are described as follows:

Transition type 1: (i,j)→(i,j+1). This transition defines the increase in the number of SU by one and can be obtained by
$$ \begin{aligned} f(\textit{i,k}) &= \frac{k}{{i + k}}(1  {P_{\text{FA1}}}) \\&\quad+ \frac{k}{{i + k}}{P_{\text{F}\text{A1}}}f(\textit{i,k}  1)\\ &\quad+ \frac{i}{{i + k}}{P_{D1}}f(i  1,k) \end{aligned} $$((9))where function f(.) is used to define the increase in the number of SUs by 1 [30]. P _{ D1} and P _{FA1} have been defined earlier to denote the initial sensing’s detection and false alarm probabilities. They have been used to obtain more accurate state transition rates and state probabilities in comparison to results obtained in [30]. The overall state transition rate for this case is given by λ _{2} f(i,k)

Transition type 2: (i,j)→(i−1,j). This transition defines the decrease in the number of PUs by one. We use the recursive function g(.) [30] to define this transition
$$ \begin{aligned} g(\textit{i,k}) &= \frac{i}{{i + k}}{P_{M1}} + \frac{i}{{i + k}}{P_{D1}}g(i  1,k)\\ &\quad+ \frac{k}{{i + k}}{P_{\text{F}\text{A1}}}g(\textit{i,k}  1), \end{aligned} $$((10))where P _{ M1}, P _{ D1}, and P _{ F A1} denote the initial sensing’s misdetection, detection, and false alarm probabilities, respectively. The overall state transition rate for this case can be obtained by i μ _{1}+λ _{2} g(i,k) [30].

Transition type 3: (i,j)→(i+1,j). This transition is given by
$$\begin{array}{*{20}l} T_{(i + 1,j)}^{(i,j)} = {\lambda_{1}}\left({\frac{{N  i  j}}{{N  i}}} \right)+ \qquad\frac{{j{\lambda_{1}}{P_{{D2}}}}}{{N  i}}f(i,N  i  j) \end{array} $$((11))to reflect the increase in the number of PUs by one.

Transition type 4: (i,j)→(i,j−1). The state transition rate for decreasing the number of SUs by one is given by
$${} \begin{aligned} T_{(i,j  1)}^{(i,j)} &= j{\mu_{2}} + {\lambda_{1}}{P_{{M2}}}\frac{j}{{N  i}} \\ &\quad+ j{\lambda_{\text{FAR}}}\left({1  f(i,N  i  j)  g(i,N  i  j)} \right)\\&\quad+ {\lambda_{1}}{P_{{D2}}}\frac{j}{{N  i}}g(i,N  i  j) \end{aligned} $$((12)) 
Transition type 5: (i,j)→(i+1,j−1). The state transition rate for this case is given by
$$ \begin{aligned} T_{(i + 1,j  1)}^{(i,j)} &= \left({{\lambda}_{1}}{{P}_{{D2}}}\frac{j}{N  i} \right)\\ & \quad{\times} \left({1  f(i,N  i  j)  g(i,N  i  j)} \right) \end{aligned} $$((13)) 
Transition type 6: (i,j)→(i−1,j−1). The number of PUs is decreased by one and the number of SUs is decreased by one. This transition occurs if after the occurrence FAR, the SU ends up colliding with a PU. We get the transition rate as
$$ T_{(i  1,j  1)}^{(\textit{i,j})} = j{\lambda_{\textit{F}{AR}}}\left[ {g(i,N  i  j)} \right] $$((14))
Construction of state transition rate matrix and computation of the steady state probability vector
Let Q denote the state transition rate matrix (also known as infinitesimal generator) of the CTMC. Let π denote the steady state probability vector with π _{ (i,j) } denoting the probability that the system is in the steady state (i,j). When the system is in the steady or equilibrium state, the normalization condition is given by \(\sum \limits _{i = 0}^{N} {\sum \limits _{j = 0}^{N} {{\boldsymbol {\pi _{(\textit {i,j})}} = 1} }}\) [49] with the condition that 0≤i≤N, 0≤j≤N, and 0≤i+j≤N. Let D equals the total number of states in CTMC. We map the elements of the steady state probability vector π _{ (i,j) } from state to index by assigning a unique integer index to identify each state. Therefore, the steady state probability vector can be represented as π=(π _{1},π _{2},...,π _{ D }) and the normalization condition is given by \(\sum \limits _{d} {{\boldsymbol {\pi }_{d}}} = 1\). The steady state probabilities of the CTMC can be found by applying the following procedure:

Step 1: Solve the recursive Eqs. (10–15) to obtain the state transition rates.

Step 2: Drive the balance equations using the rule that incoming transition rates to each state must equal outgoing transition rates from that state [50].

Step 3: Use the balance equations to build the infinitesimal generator matrix Q. All elements not on the main diagonal of Q represents state transition from one state to another. The elements on the main diagonal of Q make the sum of the elements in the respective row equal zero [51].

Step 4: Apply the normalization condition \(\sum \limits _{d} {{\boldsymbol {\pi }_{d}}} = 1\)

Step 5: Solve the system of linear equations π Q=0 to obtain the CTMC’s steady state probabilities.
Each element in the steady state probability vector π represents the percentage of time that the system spends in that state.
Since the number of states of the CTMC grows exponentially with the number of the channels in the network, it would be impossible to derive the CTMC transition rates by hand for large number of states. In this sense, the utilized recursive approach solves one part of this problem. However, the number of states is still exponential, which lead to higher memory and processing time requirements when the number of channels increases since the fullstate transition matrix is used to obtain exact results. With very large number of channels, approximation solutions with reduced number of channel states would be beneficial. In the literature, some approximation methods have been presented for CTMCs with a large number of states [51, 52]. The results presented in this paper have been obtained using the exact full CTMC. However, when the number of channels is very large which brings some inefficiency, we can apply approximate solutions of large CTMCs to overcome this problem [51, 52].
Performance evaluation measures
In order to measure the performance of the CRN, we define several performance evaluation measures: secondary forced termination probability (S U _{ FTP }), primary forced termination probability (P U _{ FTP }), and secondary self termination probability (S U _{ STP }). Those performance metrics are calculated by using the state transition rates and the steady state probabilities π _{(i,j)} and state transition rates derived in the previous section. The reader is referred to [30] for more details on the definition and derivation of other performance metrics such as secondary successful probability (S U _{ SP }), primary blocking probability (P U _{ BP }), secondary blocking probability (S U _{ BP }), as well as system resource utilization.
Secondary forced termination probability (S U _{ FTP })
The secondary forced termination probability, denoted by S U _{ FTP }, is the probability of terminating SU calls because of SU’s failure to find a new free channel after moving from its current channel. The S U _{ FTP } is calculated and defined by Eq. (15). It reflects the ratio of terminated SUs’ call to total SU call arrivals λ _{2}.
Primary forced termination probability (P U _{ FTP })
The primary forced termination probability, denoted by P U _{ FTP } and given by Eq. (16), is calculated as the ratio of terminated PU calls because of collisions with SUs to the total primary call arrivals λ _{1}.
Secondary self termination probability (S U _{ STP })
Here, we introduce a new performance metric that measures the secondary selftermination probability. The motivation of proposing this new metric starts with the fact that in CRNs, the occurrence of false alarms is of critical importance since significant amount of SUs’ calls could be terminated because of FARs. The new metric helps in accurately determining the percentage of SUs’ connection terminated due to SUs’ own errors. By identifying this metric, it allows for measuring the SUs’ ability of utilizing spectrum opportunities and helps in designing CRNs by setting correct spectrum sensing parameters. The metric determines the ratio of terminated SU calls because of FAR occurrence to the total secondary call arrivals λ _{2}. As mentioned earlier, upon λ _{FAR} arrival, an SU has to terminate its own active call if it finishes the channel searching process without finding a new idle channel. The S U _{ STP } can be calculated as:
where \({{\overline {T}}_{(i,j  1)}^{(\textit {i,j})}}\) represents the portion of the transition rate from state (i,j) to state (i,j−1) that occur because of the λ _{FAR}.
Simulation and numerical results
In this section, we report results obtained both through theoretical analysis and simulations. We conduct simulations with MATLAB using an eventbased approach and Poisson arrival processes. The parameters in simulations and theory are chosen as follows: We set the primary licensed bandwidth as 20 MHz, the initial sensing bandwidth is 20 MHz meaning that a SU carries out spectrum detection over the whole spectrum. However, the continuous sensing bandwidth is chosen to be 2 MHz. The initial sensing time =20 μs. We set the primary and secondary service rates as μ _{1}=μ _{2}=4. PU signal power is –91 dBm. The PU power has been set to a low level since SUs should be able to detect even weak PUs signals. The noise level is –160 dBm/Hz. To include the effect of the residual interference signal, we set the interference distortion factor to 0.1. We present plots for different performance metrics. It can be observed from all plots that the analytical results are in excellent agreement with the simulation results, which demonstrates the accuracy and validity of the CTMC analytical model.
ROC curves
The receiver operating characteristic (ROC) for both initial and continuous spectrum sensing is shown in Fig. 3. For continuous spectrum sensing, it can be clearly seen that better detection performance is achieved when large values of continuous sensing durations are used. However, for initial sensing, small initial sensing time T _{1} is enough for good detection level. This improved initial detection performance can be attributed to the fact that incoming SUs perform spectrum sensing over the whole PU bandwidth and hence their time bandwidth product is improved.
The impact of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the detection and false alarm probabilities is demonstrated by the ROC curves shown in Fig. 3. The SNR value for the initial sensing is 19 dB. However, the SNR values for the continuous sensing vary depending on the spectrum sensing time duration T _{2} that affects the time bandwidth product. We assume that the PU signal power is uniformly distributed over the PU channel. It can be seen that the residual interference affects the ROC curves, as the ROC performance drops significantly with increases in \(\alpha _{_{\text {SU}}}\) values. We can also see from the figure that the curve for the initial sensing with spectrum sensing time duration T _{1}=10 μs is identical with the curve for continuous sensing with spectrum sensing time duration T _{2}=100 μs and \(\alpha _{_{\text {SU}}}=0\). This is due to the fact that their time bandwidth products are the same and equal to 200. Figure 3 also shows the effect of the residual interference distortion factor \(\alpha _{_{\text {SU}}}\) on the false alarm and detection probabilities.
Effect of the interference tolerance \(\widehat {T_{\text {TOL}}}\)
We start the analysis by investigating the impact of the PU interference tolerance on the performance of the CRN. We tested the cases of \(\widehat {T_{\text {TOL}}}=0, 1, 2, 3, 4, 5\) slots. By recalling the fact that PU detection starts from the first full \(\widehat {T_{\text {TOL}}}\) slots, if \(\widehat {T_{\text {TOL}}}=0\), PUs tolerate only very little interference from an SU, and strict interference constraints should be satisfied. This means that the SU should leave the channel immediately upon the arrival of a PU. However, in practice, this is not possible since the SU user needs some sensing time to detect the presence of the PU. In Fig. 4, we compare the PU successful probability P U _{ SP } for different values of T _{TOL} and against the PU arrival rate λ _{1}. We can observe from the figure the improvement in PU performance as we relax the interference constraint by increasing \(\widehat {T_{\text {TOL}}}\). We can also observe that the P U _{ SP } drops significantly when \(\widehat {T_{\text {TOL}}}=0\). However, when \(\widehat {T_{\text {TOL}}}\) is sufficiently large, PU performance starts to improve, indicating that SUs are getting enough sensing time to complete the process of detecting incoming PUs. This implies that the \(\widehat {T_{\textit {TOL}}}\) has to be chosen in a way that meets the PUs interference constraint, and at the same time also maximizes the CRN performance. Figure 4 also presents the impact of the primary arrival rate λ _{1} on the CRN’s performance in term of P U _{ SP }. It can be easily seen that there are peak values in P U _{ SP } curves. As λ _{1} increases, the P U _{ SP } improves. However, as λ _{1} become large, most of the channels will be occupied by PUs, thus increasing the possibility of collisions with SUs who move away from their channels because of the λ _{FAR}.
Figure 5 confirms what we asserted above regarding the impact of \(\widehat {T_{\text {TOL}}}\) on the PU performance. While varying λ _{1} from 1–10, the figure compares primary forced termination probability P U _{ FTP } for each T _{TOL} value. It can be observed that P U _{ FTP } monotonically decreases with increasing λ _{1}. It is also evident that the P U _{ FTP } decreases with the increase of the \(\widehat {T_{\text {TOL}}}\). The reason behind this trend can be attributed to the fact that employing small \(\widehat {T_{\text {TOL}}}\) reduces the SUs capability of correctly detecting incoming PUs. In such a case, SUs collisions with incoming PUs increases forcing PUs to leave the network and hence increasing the P U _{ FTP }.
In Fig. 6, we plot the secondary self termination probability S U _{ STP } against λ _{1} for different values of \(\widehat {T_{\textit {TOL}}}\). The figure shows that increasing \(\widehat {T_{\textit {TOL}}}\) can result in a significant SU performance degradation. For example, when \(\widehat {T_{\textit {TOL}}}=0\), SUs collide with incoming PUs, making life easier for other secondary users since they could find free channels more easily. On the one hand, when λ _{1} is small, PUs will have a smaller network resource share, leaving more opportunities for SUs. For example, SUs who leave their channels due to λ _{FAR} could find free channel and therefore reduce the S U _{ STP }. On the other hand, when λ _{1} is large, most of the channels will be occupied by PUs, leaving smaller network resources for SUs opportunistic access. In this case, the effect of S U _{ STP } is more noticeable since with FAR occurrence (which trigger SUs to leave their channels), SUs either correctly detect the presence of PUs or collide with them. In both causes this leads to an increase in the S U _{ STP }. Intuitively, PUs are better protected by employing large \(\widehat {T_{\text {TOL}}}\) as it provides SUs with enough sensing time to complete the detection process.
Effect of the false alarm rate
Figure 7 shows the effect of λ _{FAR} on the primary forced termination probability P U _{ FTP } for different number of channels N. It can be seen from Fig. 7 that P U _{ FTP } curves for different N have unique minimums at different λ _{FARs}. The minimum points represent the optimum sensing parameters for SUs that would strongly protect PUs against forced connection termination. According to Fig. 7, the P U _{ FTP } decreases with the increase in λ _{FAR} until it reaches the minimum point after which it starts to monotonically increase. On the one hand, too low λ _{FAR} reduces the PU’s performance. The degradation in PUs performance is due to the fact that small values of λ _{FAR} leads to small P _{ D2}, and therefore existing SUs frequently collide with incoming PUs and thereby increase P U _{ FTP }. On the other hand, with high values of λ _{FAR} (and consequently high P _{ D2}), it would be easy for SUs to detect incoming PUs and therefore they can avoid collision with them. However, too high λ _{FAR} is not good since SUs initiate unnecessary spectrum handoffs, leading to sharp increases in P U _{ FTP }. As shown in the figure, the gap between P U _{ FTP }’s curves shrinks as N increases. This reflects the fact that when N is sufficiently large, most of the channels are occupied by PUs because λ _{1}>λ _{2}, and the combined effect of λ _{FAR}, P _{ M2} and P _{ M1} forces some PUs to terminate their calls. With further increases in N, the effect of λ _{FAR} on P U _{ FTP } is flat, as there would be enough radio resources to meet SUs demands. After SUs avoid incoming PUs and initiate a handoff and channel switching process, SUs will most likely find new free channels and hence will not harm existing PUs as there are more channels to accommodate them.
Figure 8 presents the effect of λ _{FAR} on the secondary self termination probability S U _{ STP } for different number of channels N. It can be seen that there are peak values in the S U _{ STP } curves. The following explanation is related to this behavior: on the one hand, when λ _{FAR} is relatively low, S U _{ STP } increases with the increase in λ _{FAR}, until it reaches the peak value and then starts to decline. The degradation in SU performance can be explained by the fact that as λ _{FAR} increases, a growing number of SUs leave their current channels. If they cannot find new free channels elsewhere they terminate their calls, and thereby increase the S U _{ STP }. On the other hand, when λ _{FAR} is relatively large, the detection probability P _{ D2} improves, which enables SUs to detect incoming PUs and move away from their channels. Hence, the proportion of SUs in the system is reduced, leading to a decrease in the S U _{ STP }. It came as no surprise that increasing N greatly influences the S U _{ STP }. When N is small, it would not be easier for SUs who initiate spectrum handoff to find new free channels to move to and the SUs have to leave the network. This results in a sharp increase in S U _{ STP }.
In Fig. 9, for several values of N, we investigate the effect of λ _{FAR} on the primary successful probability P U _{ SP }. The plot shows when λ _{FAR} is low; increasing λ _{FAR} slightly increases P U _{ SP } until it reaches some (flat) peak points. The increase is a reflection of the improved detection P _{ D2}. As can be observed from Fig. 10, increasing λ _{FAR} has an obvious negative impact on the secondary successful probability S U _{ SP }. The reason for the reduction in S U _{ SP } is that when λ _{FAR} is high, SUs increasingly initiate unnecessary spectrum handoff processes. SUs also initiate spectrum handoff processes if they detect the arrival of returning PUs. If SUs cannot find new channels, even though there are some free channel(s), they are forced to terminate their own connections and hence reducing S U _{ SP }. The results shown in Fig. 10 indicate that SU optimal performance is when λ _{FAR} is close to zero. However, in practice, this value means SUs do not perform spectrum sensing and therefore cannot be used since too low λ _{FAR} leads to poor PU performance. Results shown in Fig. 7 confirm that low λ _{FARs} are not the best for PUs performance as they do not satisfy their QoS/interference constraint. Note that in order to improve network performance, there is a critical sensing tradeoff to be made. This result is a good motivation and illustrates the significance for optimizing detection parameters such that the effects of the λ _{FAR} can be kept within limits that would not harm PUs.
Figure 11 shows the influence of λ _{FAR} on the system resource utilization for different number of channels. When λ _{FAR} is small, SUs do not frequently initiate handoff processes and the effect of λ _{FAR} remains flat. However, as λ _{FAR} increases, a growing number of SUs terminate their calls if they cannot find other free channels. This explains the reduction in the system resource utilization. As illustrated in Fig. 11, PUs’ own resource utilization is lower than the overall system resource utilization. However, at high λ _{FAR}, PUs’ and system resource utilization get closer. This indicates that SUs do not complete their service when λ _{FAR} is too high and the network resource is mainly utilized by PUs.
Performance under perfect spectrum sensing
The effect of perfect sensing on system resource utilization is shown in Fig. 12 where we plot a set of curves showing network resource utilization versus the secondary arrival rate λ _{2} for different number of channels N. The dashdotted curves depict the CRN performance when SUs operate without sensing errors. It can be observed that, due to the absence of false alarms, network resources are better utilized. In this case, all unoccupied PUs’ channels can be opportunistically utilized by SUs. This effect is more noticeable when λ _{2} is high. On the other hand, sensing errors can severely reduce the network resource utilization. The impact of the resource underutilization can be reflected in the huge gap between the resource utilization curves. As λ _{2} increases, a growing number of SUs gain network access. However, λ _{FAR} prevents them from using network resources because of the increasing spectrum handoff initiation, which may lead to premature connection termination. With a small N, the system resources will be highly utilized since PUs and SUs will occupy most of the channels. However, when N is large, the radio resources are not fully utilized and therefore decreases the overall system resource utilization.
Conclusions
We studied the effect of the false alarm rates λ _{FARs} on the operation of CRNs. We developed a CTMCbased analytical model to evaluate the performance of CRNs under realistic network operating conditions. The proposed model not only includes sensing errors by incoming SUs but also takes into account the misdetection and false alarm probabilities by ongoing SUs. The modeling approach described here is capable of examining other performance evaluation parameters such as the effect of interference tolerance \(\widehat {T_{\text {TOL}}}\) among PUs and SUs as well as the effect of SU residual selfinterference. We derived formulas for different performance metrics, including primary and secondary forced termination probabilities as well as secondary selftermination probability. Furthermore, we performed extensive simulations to validate the accuracy of the analytical model. Simulation results are in excellent agreement with the analytical results.
Results have shown that λ _{FAR} greatly influences the performance of CRNs by degrading SU performance and reducing network resource utilization. Results have also shown that decreasing the interference tolerance \(\widehat {T_{\text {TOL}}}\) has negative effect on the performance of PUs as it reduces primary successful probability and increases their forced termination probability. A similar effect was also observed with the increase in the SU residual interference distortion factor. Large amount of residual interference deteriorates the detection probability and leads to a reduced PU performance. The incorporation of λ _{FAR} into the CTMC model allows for obtaining exact and accurate state transition probabilities that improves calculation of the performance evaluation measures. The results of the proposed analytical model provide a new insight into the operation of CRNs and can be used to develop practical and more accurate CRN performance evaluation models. In future work, cooperative spectrum sensing can be considered for improving detection performance and/or for mitigating the effects of fading. Further study is also needed to investigate the case where on collisions, only SU calls will be terminated. Additionally, adaptive sensing parameters based on PU channel utilization can also be studied.
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Acknowledgements
The work of Janne Lehtomäki was supported by the Research Council of the University of Oulu. The work of Kenta Umebayashi was supported by the Strategic Information and Communications R&D Promotion Programme (SCOPE).
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Suliman, I.M., Lehtomäki, J. & Umebayashi, K. On the effect of false alarm rate on the performance of cognitive radio networks. J Wireless Com Network 2015, 244 (2015). https://doi.org/10.1186/s1363801504743
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DOI: https://doi.org/10.1186/s1363801504743
Keywords
 Cognitive radio network
 Opportunistic spectrum access
 False alarm rate
 Markov chain
 Spectrum sensing
 Performance analysis