# Performance Evaluation of Cognitive Radio Networks with Imperfect Spectrum Sensing and Bursty Primary User Traffic

Authors: Osama Salameh; Herwig Bruneel; Sabine Wittevrongel
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we introduce a four-dimensional continuous-time Markov chain model to evaluate the performance of cognitive radio networks. In such networks, secondary (unlicensed) users may opportunistically use the frequency channels not currently occupied by primary (licensed) users in order to increase the utilization of the wireless spectrum. Secondary users perform channel sensing before as well as during transmission in order not to interfere with primary users. The proposed model assumes that primary users arrive according to a bursty arrival process and moreover takes the possible occurrence of sensing errors (false alarms and misdetections) into account. Several performance measures including the collision rate between primary and secondary users, the blocking probabilities of primary or secondary users, and the mean delay of secondary users are derived and illustrated through numerical examples. The results show that the system performance strongly depends on the degree of burstiness in the arrival process of primary users. It is also observed that the quality of service of the primary network can be seriously compromised due to misdetection by secondary users.


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## Body

## 1. Introduction

Cognitive radio networks (CRNs) are an efficient paradigm to address the scarcity of the wireless spectrum [1, 2]. Under this paradigm, two types of networks operate simultaneously. The first type is a primary network that owns the spectrum and coordinates the access of primary users (PUs) to individual channels. The second type is a secondary network where secondary users (SUs) are allowed to make use of the channels not occupied by PUs. The sole purpose of the secondary network is to increase the utilization of the underutilized spectrum. We assume that both networks operate independently of each other and that the SUs have no knowledge of the access patterns of the PUs. Whenever a SU needs to transmit, it must perform channel sensing to ensure that a channel is idle. Also, a SU must perform sensing while transmitting in order to avoid collision with a possible arrival of a PU. A comparison of different sensing techniques is available in [3, 4].In practice, two kinds of SU sensing errors are common: false alarm and misdetection of a PU. In the event of a false alarm, a SU incorrectly detects the presence of a PU on a channel, which results in a lost transmission opportunity. In case of a misdetection of a PU, a SU fails to detect the actual presence of a PU, which results in a collision between the PU and the SU and the loss of both users. Both kinds of SU sensing errors can occur before and during a SU transmission. As a consequence and following [5], we distinguish two types of false alarm and two classes of misdetection. Type-I false alarm occurs when a sensing SU is unable to detect an idle channel, and Type-II false alarm occurs when a transmitting SU evacuates the channel without PU arrival. Class-A misdetection occurs when a sensing SU incorrectly decides that a channel is idle; when in reality, it is occupied by a PU, and class-B misdetection occurs when a transmitting SU fails to detect the arrival of a PU on the channel. In this paper, both kinds of sensing errors are considered.Most of the results of the research on CRNs obtained so far are based on the assumption that the PU network is time-slotted, and the SU network is perfectly synchronized with the primary network as, e.g., in [6–11]. While the synchronization between SU and PU networks can be difficult to implement as it may require the SU to know the details of the PU network like the exact slot structure, some of the PU networks are not time-slotted. In this paper, we therefore investigate the performance of CRNs where both PU and SU networks are not time-slotted. The problems that emerge from such assumptions are viewed as a new research direction [12]. Furthermore, the majority of works reported in the literature assumes that PUs arrive according to a Poisson process. This model may not be accurate for PUs in a practical environment [13, 14], which in turn affects the accuracy of the obtained performance measures. Incorporating more realistic models in the context of cognitive radio is reported as a research challenge [15], to be considered for the future work [12], where the interrupted Poisson process (IPP) is explicitly mentioned as an example. To fill this gap, we study in this paper the performance of a CRN with asynchronous interaction between SUs and PUs, and our aim is to gain insight into the CRN performance with more realistic (bursty) traffic models of PUs. In particular, we model the arrival process of PUs as an IPP. This model has been reported suitable for web browsing [16] or data traffic [17].Our main contribution in the present paper is the analysis of an analytic model of a CRN under bursty PU traffic and including SU sensing errors. In particular, we introduce a four-dimensional continuous-time Markov chain (CTMC) model to evaluate the CRN performance under bursty PU arrivals and SU sensing errors. To the best of our knowledge, such a model is not available in the literature. For further background on the theory of stochastic processes and CTMCs, we refer to [18–20].The rest of this paper is organized as follows. In Section2, the related work and our contribution are discussed. In Section 3, the system model is presented, and in Section 4, the related CTMC model is derived. We formalize the performance measures in Section 5, and we discuss some numerical results in Section 6. In Section 7, the conclusion is given.


## 2. Related Works and Contribution

In this section, we review some of the studies that assume asynchronous interaction between SUs and PUs and use a CTMC modeling approach. For a summary of our comparison, we refer to Table1. In [21], a CRN comprising one PU and N SUs was studied, where the whole spectrum is treated as one channel exclusively allocated to one user at a time. A steady-state analysis is conducted. In [22], Markov models were developed to study a multichannel CRN. Several performance measures are derived including the mean delay of the SUs, the variance of the SU delay, and the interruption probability of SUs by PU arrival. Both works assume that SUs have perfect sensing abilities, and the presence of SUs does not degrade the performance of the primary network, which is not always true in practice. In [23], the authors analyzed a system where both PUs and SUs access the spectrum through a PU base station (PU-BS). SUs have sensing abilities, and an arriving SU accesses the PU-BS with a certain probability β based on the current system load. A transmitting SU may be dropped with probability α upon the arrival of a PU when all the channels are busy. This paper also considers a collision-free environment between PUs and SUs, but the performance of the PUs may get affected by the presence of SUs when all the channels are busy.Table 1
Comparison among various CTMC-based models.
Reference no.Sensing timeArriving SU errorsTransmitting SU errorsBursty PU arrival (IPP)[21, 23]✗✓✗✗[22]✓✗✗✗[24–27]✗✓✗✗[5, 28]✗✓✓✗[29]✓✓✓✗Proposed model✓✓✓√An analysis of a CRN with imperfect sensing is conducted in [24–27]. The PU termination probability due to collision with SUs is obtained in [24]. This analysis is conducted for the case where the number of channels N=3. A limitation of this work is the assumption that a transmitting SU will always evacuate the channel upon PU arrival, i.e., no misdetection of an arriving PU is considered. Other limitations are that the SU sensing time is considered negligible, and false alarms of transmitting SUs are not taken into consideration. In [25, 26], different extensions of the study of [24] were reported, where new performance measures were investigated. The restriction on the number of channels is removed in [25, 26], but all other limitations of [24] mentioned above still hold. In [27], the authors investigated the case where the SUs are able to sense all the channels in the system to determine the access probability. The blocking probability of PUs and the probability of a SU being dropped are derived. It is important to note that sensing errors of transmitting SUs are not taken into consideration in all the studies [24–27].Sensing errors of sensing SUs and transmitting SUs are considered in [5, 28]. In [28], the case when a colliding PU and SU are dropped from the system was analyzed. In [5], the analysis was extended to other cases when only one of the colliding users is dropped if a collision occurs. The limitations of these studies include the absence of SU sensing time and the absence of SU handoff time, where the latter is the time needed for a transmitting SU to switch to another channel upon the arrival of a PU. The effect of interruption of a transmitting SU by an arriving PU is only considered when no idle channels are available. Furthermore, these studies assume that collisions are not influenced by the number of transmitting SUs in the system. It is worth mentioning that all the studies reviewed in this section are based on an appropriate CTMC, and the arrival of PUs is modeled as a Poisson process. Our previous study [29] is free of all the limitations mentioned above, except that it also models the PU arrival by means of a Poisson process.Our present paper is an extension of [29] in the sense that PUs arrive according to an IPP. Our objective is to analyze CRNs with imperfect SU sensing and bursty PU traffic using a CTMC model in terms of a number of performance measures. We evaluate SU-related performance measures including the delay of SUs, blocking probability of SUs, and throughput rate of SUs. Also, we evaluate measures related to the PU performance, such as the blocking probability of PUs and the collision rate between PUs and SUs affecting both networks. Closed-form expressions for the calculation of all these measures are derived in terms of the steady-state distribution of the involved CTMC.


## 3. System under Study and Modeling Assumptions

We consider a CRN where primary and secondary user networks operate in parallel, as illustrated in Figure1. In this section, we describe the operation of both networks and the related modeling assumptions. The system parameters are summarized in Table 2.Figure 1
Cognitive radio network environment with primary and secondary user networks.Table 2
Cognitive radio system parameters.
NNumber of frequency channelsλ1∗PU arrival rate in active state1/γ1Mean inactive period of IPP for PU arrivals1/γ2Mean active period of IPP for PU arrivalsλ1Average PU arrival rateλ2SU arrival rateμ1PU transmission rateμ2SU transmission rateKMaximum number of sensing SUsσChannel sensing ratepm1Class-A misdetection probabilitypm2Class-B misdetection probabilitypf1Type-I false alarm probabilityδf2Type-II false alarm rate


### 3.1. Primary User Network

In the primary network, PUs shareN frequency bands (channels). Arriving PUs are ready to transmit data, and their access to the channels is coordinated by a primary base station such that the primary network is collision-free, i.e., no collisions between PUs can occur. In particular, an arriving PU is randomly assigned an available channel, i.e., a channel not already occupied by another PU. If there are no available channels, the arriving PU is blocked. Note that the primary network has no knowledge about the SU activity, and therefore, all channels occupied by SUs are considered as available channels, and PUs can be assigned to them (the transmitting SU will need to evacuate the channel in such a case). All channels have the same bandwidth and are not time-slotted. Moreover, the frequency channels are error-free, i.e., if a PU starts transmission, it is considered successful unless there is a collision with a SU during transmission due to a misdetection (see below). The PU transmission duration (or channel holding time) is assumed to be exponentially distributed with rate μ1, i.e., with mean value 1/μ1.The arrival of PUs is modeled by an interrupted Poisson process (IPP). This is a two-state Markovian process with an active and an inactive state, as shown in Figure2. In the active state, PUs arrive in accordance with a Poisson process with rate λ1∗, whereas no PUs arrive in the inactive state. The active and inactive periods are both exponentially distributed. We let γ1 denote the rate from the inactive to the active state and γ2 the rate from the active to the inactive state. Then, the steady-state probability that the IPP is in the active state is given by(1)Pactive=γ1γ1+γ2,and the average PU arrival rate equals λ1=λ1∗Pactive. The mean active and inactive periods are given by 1/γ2 and 1/γ1, respectively. For a given value of Pactive, i.e., a given ratio γ1/γ2, smaller values of γ1 and γ2 correspond to longer active and inactive periods and hence a more bursty PU arrival process.Figure 2
Interrupted Poisson process model.


### 3.2. Secondary User Network

SUs are allowed to make use of licensed channels that are temporarily not occupied by PUs. The secondary network is an infrastructureless ad hoc wireless network. There is no central entity like a base station to coordinate the channel access by SUs and avoid interference with PUs. On the contrary, each SU is equipped with all the cognitive capabilities to sense the spectrum and to determine if a channel is occupied by a PU or idle based on its own measurements. SUs are also aware of the channels occupied by other SUs by the exchange of messages among SUs, as also indicated in, e.g., [5, 28, 30], so no collisions between SUs can occur. The SU network is not time-slotted. SUs not only sense the spectrum upon arrival in order to find an idle channel for transmission but also during transmission to avoid collision with arriving PUs. With respect to the latter, we assume that SUs operate in full duplex mode, i.e., a SU can sense and transmit simultaneously, as also considered in [30–33]. This allows an efficient evacuation of the channel by the SU upon arrival of a PU on the channel.We assume in our model that SUs arrive according to a Poisson process with rateλ2. Several admission control policies of SUs are used in CRNs. These policies aim at ensuring quality of service for both SUs and PUs [34] and are implemented by limiting the access of users to the system [35]. We consider the case where an arriving SU starts sensing to find an idle channel if the number of sensing SUs in the system is below a given number K and gets blocked, otherwise. We refer to the maximum number K of sensing SUs as the sensing room size. Once admitted to the sensing room, a SU randomly selects a channel from the set of idle channels or those occupied by PUs and senses the channel. The channel sensing time is assumed to be exponentially distributed with rate σ for mathematical tractability; such assumption of nonfixed sensing time has been used before in, e.g., [36]. With respect to the outcome of the sensing of a channel by a SU, the following options are possible:(i)
The SU correctly detects PU presence on the channel. The SU then picks another channel and starts to sense again.(ii)
The SU perceives the channel as occupied by a PU while it actually is idle, so the SU fires a type-I false alarm. We denote the type-I false alarm probability bypf1, i.e., pf1 is the probability that a SU who has finished sensing an idle channel erroneously thinks a PU is occupying the channel. Also, in this case, the SU picks another channel and starts to sense again.(iii)
The SU correctly detects an idle channel and starts transmission on that channel.(iv)
The SU perceives the channel as idle while it in fact is occupied by a PU, which corresponds to a class-A misdetection. We letpm1 denote the class-A misdetection probability. In this case, there is a collision between the SU and the PU, and both users are dropped from the system.A SU transmits on a single frequency channel, and we assume that the SU transmission duration is exponentially distributed with rateμ2. While transmitting, the SU continuously senses its channel in order to detect the possible arrival of a PU. The following events may then occur instantaneously during the sensing:(i)
A PU arrives and the transmitting SU correctly detects its arrival. The SU then evacuates the channel and starts sensing to find an idle channel provided the sensing room is not full or gets lost, otherwise.(ii)
The transmitting SU incorrectly records PU arrival and thus fires a type-II false alarm. We denote the rate of firing a type-II false alarm byδf2. Likewise, in this case, the SU stops its transmission and either starts sensing for an idle channel in case of a non-full sensing room or gets lost, otherwise.(iii)
The transmitting SU misdetects PU arrival on the channel. The SU and the PU will then collide and both be dropped from the system. This corresponds to a class-B misdetection, and the probability of such misdetection is denoted bypm2.Note that, as long as none of the above 3 events occurs during the sensing, the SU will simply continue its transmission until it is completed. Also, note that the occurrence of type-II false alarms is modeled by the rateδf2, whereas the other error parameters pf1, pm1, and pm2 are probabilities. This is because a SU can fire a type-II false alarm at any time instant during transmission, whereas the other sensing errors are tied to specific instants. Specifically, the parameters pm1 and pf1 are related to the instant when a SU finishes sensing a channel, while pm2 is related to the instant of arrival of a PU to a channel.


## 3.1. Primary User Network

In the primary network, PUs shareN frequency bands (channels). Arriving PUs are ready to transmit data, and their access to the channels is coordinated by a primary base station such that the primary network is collision-free, i.e., no collisions between PUs can occur. In particular, an arriving PU is randomly assigned an available channel, i.e., a channel not already occupied by another PU. If there are no available channels, the arriving PU is blocked. Note that the primary network has no knowledge about the SU activity, and therefore, all channels occupied by SUs are considered as available channels, and PUs can be assigned to them (the transmitting SU will need to evacuate the channel in such a case). All channels have the same bandwidth and are not time-slotted. Moreover, the frequency channels are error-free, i.e., if a PU starts transmission, it is considered successful unless there is a collision with a SU during transmission due to a misdetection (see below). The PU transmission duration (or channel holding time) is assumed to be exponentially distributed with rate μ1, i.e., with mean value 1/μ1.The arrival of PUs is modeled by an interrupted Poisson process (IPP). This is a two-state Markovian process with an active and an inactive state, as shown in Figure2. In the active state, PUs arrive in accordance with a Poisson process with rate λ1∗, whereas no PUs arrive in the inactive state. The active and inactive periods are both exponentially distributed. We let γ1 denote the rate from the inactive to the active state and γ2 the rate from the active to the inactive state. Then, the steady-state probability that the IPP is in the active state is given by(1)Pactive=γ1γ1+γ2,and the average PU arrival rate equals λ1=λ1∗Pactive. The mean active and inactive periods are given by 1/γ2 and 1/γ1, respectively. For a given value of Pactive, i.e., a given ratio γ1/γ2, smaller values of γ1 and γ2 correspond to longer active and inactive periods and hence a more bursty PU arrival process.Figure 2
Interrupted Poisson process model.


## 3.2. Secondary User Network

SUs are allowed to make use of licensed channels that are temporarily not occupied by PUs. The secondary network is an infrastructureless ad hoc wireless network. There is no central entity like a base station to coordinate the channel access by SUs and avoid interference with PUs. On the contrary, each SU is equipped with all the cognitive capabilities to sense the spectrum and to determine if a channel is occupied by a PU or idle based on its own measurements. SUs are also aware of the channels occupied by other SUs by the exchange of messages among SUs, as also indicated in, e.g., [5, 28, 30], so no collisions between SUs can occur. The SU network is not time-slotted. SUs not only sense the spectrum upon arrival in order to find an idle channel for transmission but also during transmission to avoid collision with arriving PUs. With respect to the latter, we assume that SUs operate in full duplex mode, i.e., a SU can sense and transmit simultaneously, as also considered in [30–33]. This allows an efficient evacuation of the channel by the SU upon arrival of a PU on the channel.We assume in our model that SUs arrive according to a Poisson process with rateλ2. Several admission control policies of SUs are used in CRNs. These policies aim at ensuring quality of service for both SUs and PUs [34] and are implemented by limiting the access of users to the system [35]. We consider the case where an arriving SU starts sensing to find an idle channel if the number of sensing SUs in the system is below a given number K and gets blocked, otherwise. We refer to the maximum number K of sensing SUs as the sensing room size. Once admitted to the sensing room, a SU randomly selects a channel from the set of idle channels or those occupied by PUs and senses the channel. The channel sensing time is assumed to be exponentially distributed with rate σ for mathematical tractability; such assumption of nonfixed sensing time has been used before in, e.g., [36]. With respect to the outcome of the sensing of a channel by a SU, the following options are possible:(i)
The SU correctly detects PU presence on the channel. The SU then picks another channel and starts to sense again.(ii)
The SU perceives the channel as occupied by a PU while it actually is idle, so the SU fires a type-I false alarm. We denote the type-I false alarm probability bypf1, i.e., pf1 is the probability that a SU who has finished sensing an idle channel erroneously thinks a PU is occupying the channel. Also, in this case, the SU picks another channel and starts to sense again.(iii)
The SU correctly detects an idle channel and starts transmission on that channel.(iv)
The SU perceives the channel as idle while it in fact is occupied by a PU, which corresponds to a class-A misdetection. We letpm1 denote the class-A misdetection probability. In this case, there is a collision between the SU and the PU, and both users are dropped from the system.A SU transmits on a single frequency channel, and we assume that the SU transmission duration is exponentially distributed with rateμ2. While transmitting, the SU continuously senses its channel in order to detect the possible arrival of a PU. The following events may then occur instantaneously during the sensing:(i)
A PU arrives and the transmitting SU correctly detects its arrival. The SU then evacuates the channel and starts sensing to find an idle channel provided the sensing room is not full or gets lost, otherwise.(ii)
The transmitting SU incorrectly records PU arrival and thus fires a type-II false alarm. We denote the rate of firing a type-II false alarm byδf2. Likewise, in this case, the SU stops its transmission and either starts sensing for an idle channel in case of a non-full sensing room or gets lost, otherwise.(iii)
The transmitting SU misdetects PU arrival on the channel. The SU and the PU will then collide and both be dropped from the system. This corresponds to a class-B misdetection, and the probability of such misdetection is denoted bypm2.Note that, as long as none of the above 3 events occurs during the sensing, the SU will simply continue its transmission until it is completed. Also, note that the occurrence of type-II false alarms is modeled by the rateδf2, whereas the other error parameters pf1, pm1, and pm2 are probabilities. This is because a SU can fire a type-II false alarm at any time instant during transmission, whereas the other sensing errors are tied to specific instants. Specifically, the parameters pm1 and pf1 are related to the instant when a SU finishes sensing a channel, while pm2 is related to the instant of arrival of a PU to a channel.


## 4. Continuous-Time Markov Chain Model

Given the above modeling assumptions, the performance of the considered system can be investigated by means of a four-dimensional continuous-time Markov chain (CTMC) model. In particular, we define system statex as the quadruplet x=x1,x2,x3,x4, where x1 denotes the number of PUs in the system, x2 and x3 are the numbers of transmitting and sensing SUs, respectively, and x4 indicates whether the PU arrival process is in the active state (x4=1) or in the inactive state (x4=0). The finite state space S of the CTMC contains all states x1,x2,x3,x4 satisfying the conditions(2)x1+x2≤N,x3≤K,andx4=0,1.In the rest of this section, we focus on the derivation of the infinitesimal generatorQ of the CTMC. On the one hand, it is a well-known property that the row sums of this matrix are equal to zero such that the diagonal element qx,x of Q (x∈S) is expressed as(3)qx,x=−∑y∈S,y≠xqx,y.On the other hand, the nondiagonal elementqx,y (x≠y) of the matrix Q is the total transition rate from state x to another state y. To obtain qx,y (x≠y), we therefore need to list all events that cause a change from system state x=x1,x2,x3,x4 to another state y and to determine the corresponding transition rates.A first event is related to PU arrival, which occurs with rateλ1∗ if the IPP process of PUs is in the active state, i.e., if x4=1. Such PU arrival will cause a change in the system state if there are still available channels for PUs, i.e., if x1<N, and one of the N−x1 available channels will then randomly be assigned to the arriving PU. In case the assigned channel is an idle channel, the number of PUs in the system simply increases by one and we get a transition to state y=x1+1,x2,x3,x4. In case the assigned channel is already in use by a transmitting SU and the PU arrival is correctly detected by the transmitting SU (i.e., there is no class-B misdetection), the SU will moreover evacuate the channel and start to sense again if x3<K or leave the system if x3=K. Finally, in case of an assigned channel already occupied by a SU together with a class-B misdetection, a collision occurs between the arriving PU and the transmitting SU, and both users are dropped from the system such that the system state changes to y=x1,x2−1,x3,x4. These observations lead to the following transition equations related to PU arrival:(4)qx,y=λ1∗N−x1−x2N−x1,ify=x1+1,x2,x3,x4,x1<N,x4=1,(5)qx,y=λ1∗x2N−x11−pm2,ify=x1+1,x2−1,x3+1,x4,x1<N,x4=1,x3<K,(6)qx,y=λ1∗x2N−x11−pm2,ify=x1+1,x2−1,x3,x4,x1<N,x4=1,x3=K.Note that the case of a class-B misdetection is not included in equations (4)–(6); since also other events may cause a transition to state y=x1,x2−1,x3,x4, the expression for the total transition rate to this state is postponed until later.A second event similarly corresponds to SU arrival, which occurs with rateλ2 and causes a state change if the number of sensing SUs already in the system is less than K. This leads to(7)qx,y=λ2,ify=x1,x2,x3+1,x4,x3<K.Next, each of thex1 transmitting PUs can complete transmission with rate μ1 such that(8)qx,y=x1μ1,ify=x1−1,x2,x3,x4.Similarly, a SU transmission completion occurs with ratex2μ2, which changes the system state to state y=x1,x2−1,x3,x4. Moreover, a type-II false alarm occurs with rate x2δf2, and such a false alarm makes the corresponding transmitting SU stop its transmission and return to the sensing state if x3<K or leave the system if x3=K. Taking into account that a change to state y=x1,x2−1,x3,x4 may also be caused by PU arrival together with a class-B misdetection, the next set of transition equations we get is as follows:(9)qx,y=x2μ2,ify=x1,x2−1,x3,x4,x4=0,x3<K,(10)qx,y=x2μ2+δf2,ify=x1,x2−1,x3,x4,x4=0,x3=K,(11)qx,y=λ1∗x2N−x1pm2+x2μ2,ify=x1,x2−1,x3,x4,x1<N,x4=1,x3<K,(12)qx,y=λ1∗x2N−x1pm2+x2μ2+δf2,ify=x1,x2−1,x3,x4,x1<N,x4=1,x3=K,(13)qx,y=x2δf2,ify=x1,x2−1,x3+1,x4,x3<K.Another event corresponds to a sensing SU finishing the sensing of a channel, which occurs with ratex3σ, provided that not all channels are already occupied by transmitting SUs, i.e., if x2<N. Indeed, since SUs are aware of the channels in use by other SUs, SUs will only sense the N−x2 channels not already occupied by other SUs. In case the sensed channel is idle and this is correctly detected by the SU, i.e., there was no type-I false alarm, the SU starts transmission. In case the sensed channel is in use by a PU and this is not correctly detected by the SU, i.e., there was a class-A misdetection, the SU starts transmission, and the collided PU and SU are both dropped from the system. The corresponding transition equations are(14)qx,y=x3σN−x2−x1N−x21−pf1,ify=x1,x2+1,x3−1,x4,x2<N,(15)qx,y=x3σx1N−x2pm1,ify=x1−1,x2,x3−1,x4,x2<N.A final set of events that cause a system state change corresponds to state changes in the PU arrival process, which occur with rateγ1 from the inactive to the active state and rate γ2 vice versa. We thus have(16)qx,y=γ1,ify=x1,x2,x3,x4+1,x4=0,qx,y=γ2,ify=x1,x2,x3,x4−1,x4=1.For all other combinations of statex and y (x≠y), the transition rate qx,y equals 0:(17)qx,y=0,otherwise.We now denote byπx1,x2,x3,x4 the steady-state probability that the system is in state x=x1,x2,x3,x4∈S. The row vector π with elements πx1,x2,x3,x4 is then calculated from the set of equilibrium conditions for the CTMC:(18)πQ=0,in combination with the normalization condition π1=1, where 1 is a column vector of ones of appropriate size.


## 5. Performance Measures

Based on the vector of steady-state probabilitiesπ, several performance measures of the cognitive radio system can be computed. In this section, we give expressions for the collision rate between PUs and SUs, the blocking probabilities of PUs and SUs, the throughputs of PUs and SUs, and the mean delay of SUs.First, the collision rateα between PUs and SUs is the rate at which PUs and SUs run into a collision due to a class-A or class-B misdetection and are dropped from the system. In view of the transition equations (11), (12), and (15), the collision rate α is calculated as follows:(19)α=∑x2=0N−1∑x1=0N−x2∑x3=0K∑x4=01πx1,x2,x3,x4x3σx1N−x2pm1+∑x1=0N−1∑x2=0N−x1∑x3=0Kπx1,x2,x3,1λ1∗x2N−x1pm2.Note that, in the first term, the summation runs over all states in the state spaceS with x2<N since a class-A misdetection can only occur in such states. Similarly, the summation in the second term runs over all states with x1<N and x4=1 since a class-B misdetection is only possible in such states.Secondly, the blocking probabilityβPU of PUs is the probability that an arriving PU finds all N channels already occupied by other PUs and therefore gets blocked. Taking into account the IPP arrival process of PUs, the blocking probability βPU is given by(20)βPU=1Pactive∑x3=0KπN,0,x3,1.The blocking probabilityβSU of SUs corresponds to the probability that an arriving SU finds already K sensing SUs in the system and therefore equals(21)βSU=∑x1=0N∑x2=0N−x1∑x4=01πx1,x2,K,x4.Next, the throughputηPU of PUs is the rate at which PUs successfully complete transmission and is hence given by(22)ηPU=∑x1=0N∑x2=0N−x1∑x3=0K∑x4=01πx1,x2,x3,x4x1μ1.Similarly, the throughputηSU of SUs is calculated as(23)ηSU=∑x1=0N∑x2=0N−x1∑x3=0K∑x4=01πx1,x2,x3,x4x2μ2.Finally, from Little’s law [19], the mean delay EdSU of SUs, i.e., the total time SUs spends in the cognitive radio system between the instant of effectively entering the system and their departure instant from the system, can be obtained as(24)EdSU=EnSU,tr+EnSU,seλ21−βSU,where λ21−βSU is the effective arrival rate of SUs in the system and EnSU,tr and EnSU,se are the mean numbers of transmitting and sensing SUs in the system, respectively:(25)EnSU,tr=∑x1=0N∑x2=0N−x1∑x3=0K∑x4=01x2πx1,x2,x3,x4,EnSU,se=∑x1=0N∑x2=0N−x1∑x3=0K∑x4=01x3πx1,x2,x3,x4.


## 6. Numerical Results

In this section, we investigate the performance of the considered cognitive radio system by means of several numerical examples. In all examples, the number of frequency channels isN=20. We define the PU load as ρPU=λ1/μ1N and the SU load as ρSU=λ2/μ2N. We consider the case where ρSU=0.5. The transmission durations of the PUs and the SUs both have mean values 1/μ1=1/μ2=10ms. We consider a mean SU sensing time 1/σ=10ms, and the maximum number of sensing SUs equals K=50.In the first set of examples (Figures3–7), we investigate the effect of the IPP parameters λ1∗, γ1, and γ2 representing the PU traffic on the performance measures. We assume that γ1=γ2 which makes Pactive=Pinactive=0.5. This assumption is close to the approximation of real PU traffic [37]. In this set of examples, we consider the case where SUs have perfect sensing abilities, i.e., where pf1=pm1=pm2=0 and δf2=0/s. In Figures 3–7, we also compare the performance to the case where PUs are assumed to arrive according to a Poisson process with the same average PU arrival rate λ1. The results for the case of Poisson arrivals are obtained based on our earlier model in [29].Figure 3
Blocking probabilityβPU of PUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pf1=pm1=pm2=0, δf2=0/s, and different values of γ1=γ2.Figure 4
Mean delayEdSU of SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pf1=pm1=pm2=0, δf2=0/s, and different values of γ1=γ2.Figure 5
ThroughputηPU of PUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pf1=pm1=pm2=0, δf2=0/s, and different values of γ1=γ2.Figure 6
ThroughputηSU of SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pf1=pm1=pm2=0, δf2=0/s, and different values of γ1=γ2.Figure 7
Blocking probabilityβSU of SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pf1=pm1=pm2=0, δf2=0/s, and different values of γ1=γ2.Figure3 shows the blocking probability βPU of PUs versus the average PU arrival rate λ1, for different values of γ1=γ2. As expected, for small λ1, the blocking probability βPU is close to 0 regardless of the value of γ1=γ2. For higher values of λ1, we observe that the PU blocking probability is higher for lower values of γ1=γ2. This is an expected result since smaller values of γ1=γ2 correspond to longer active and inactive periods and a more bursty PU arrival process. Note that this higher blocking probability for lower γ1=γ2 also means that, for a given value of λ1, i.e., for a given PU load, the effective arrival rate λ11−βPU of PUs entering the system decreases as γ1=γ2 decreases. From Little’s law and in view of the constant PU transmission duration considered in Figure 3, it then immediately follows that, for a given value of λ1, the mean number of PUs in the system decreases as γ1=γ2 decreases. Finally, Figure 3 shows that the PU blocking probability is lower for the case of a Poisson arrival process than for IPP arrivals. This is intuitively clear since the Poisson process does not take any time correlation in the PU arrival process into account, while under the IPP model, burstiness is not neglected and PUs tend to arrive to the system in a more clustered way.Figure4 shows the mean delay of SUs versus the average PU arrival rate λ1, again for different values of γ1=γ2. When the PU load ρPU is small, the effect of γ1=γ2 on the mean SU delay is rather small. In this case, for a fixed PU load, the number of PU packet arrivals during an active period is higher for small γ1=γ2, and the more bursty PU arrival process also causes a slight increase in the number of sensing SUs and the mean SU delay. For higher values of λ1, more PU packet arrivals are blocked for small values of γ1=γ2 (see also Figure 3), and thus, more white spaces are available for SUs. Consequently, the mean SU delay decreases as γ1=γ2 decreases.In Figure5, the throughput ηPU of PUs is plotted versus λ1. An increase in the PU arrival rate results in a smaller increase in the throughput ηPU for smaller values of γ1=γ2. This is again due to the increased blocking probability of PUs for smaller γ1=γ2 as shown above (see Figure 3).In Figure6, the throughput ηSU of SUs is given versus λ1, for different values of γ1=γ2. Obviously, for increasing λ1, the throughput of SUs decreases regardless of the value of γ1=γ2. When λ1 is small, it can be seen that the SU throughput is smaller for low γ1=γ2. This is again explained by the fact that, for small values of γ1=γ2, PU arrivals during active periods occupy more channels, and less white spaces are available for SUs. For a high PU load, the number of PU packets admitted to the system is higher in case of high γ1=γ2 due to a decreased blocking probability of PUs. So, for high λ1, the effect of γ1=γ2 on the throughput is reversed, i.e., the throughput of SUs increases for decreasing γ1=γ2.Figure7 shows the SU blocking probability βSU versus λ1, for different γ1=γ2. We observe that the SU blocking probability increases as the average PU arrival rate λ1 increases. For small values of λ1, the blocking probability βSU is higher when γ1=γ2 is small. For high λ1 (e.g., λ1=1600/s), the order of the curves is reversed. These observations are in full agreement with those of Figure 6. Also, note that, intuitively, the SU blocking probability is expected to decrease as the maximum number K of sensing SUs increases. A detailed study of the impact of K on the CRN performance under Poisson PU arrivals and perfect spectrum sensing has been performed in [22].In all the figures above, it can be seen that, for the highest considered value ofγ1=γ2=1000/s (when the system switches between the active and inactive states more quickly and the PU arrival process is thus less bursty), the performance measures of PUs and SUs are closest to the case of Poisson arrivals, whereas already indicated burstiness or time correlation is completely absent.In our next examples, we explore the effect of SU sensing errors on the performance of PUs and SUs when PU packet arrivals are represented by an IPP. We first focus on the impact of the class-A and class-B misdetection probabilitiespm1 and pm2 in Figures 8–10. In these figures, we consider an IPP with mean active and inactive periods of 1/γ1=1/γ2=10ms, i.e., γ1=γ2=100/s. We set pf1=0 and δf2=0/s, which means there are no false alarms.Figure 8
Mean delayEdSU of SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, 1/γ1=1/γ2=10ms, pf1=0, δf2=0/s, and different values of pm1=pm2.Figure 9
ThroughputηSU of SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, 1/γ1=1/γ2=10ms, pf1=0, δf2=0/s, and different values of pm1=pm2.Figure 10
Collision rateα between PUs and SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, 1/γ1=1/γ2=10ms, pf1=0, δf2=0/s, and different values of pm1=pm2.Figure8 shows the mean delay EdSU of SUs versus the average PU arrival rate λ1, for different values of pm1=pm2. We observe that, for small values of pm1 and pm2, the mean SU delay substantially increases as λ1 increases. In this case, the number of collisions between PUs and SUs is small (see also Figure 10), and the number of SUs waiting to transmit is large. For higher values of pm1 and pm2, the number of collisions increases; hence, the number of sensing SUs decreases, and an increase of λ1 has a smaller effect on the mean SU delay.In Figure9, the throughput ηSU of SUs is plotted versus λ1, again for different pm1=pm2. For small values of λ1, ηSU is larger for smaller pm1=pm2. For smaller pm1=pm2, the number of collisions is smaller and, as expected, more SUs can access the channels, leading to a higher throughput. However, a more surprising behavior of the throughput ηSU is detected when λ1 is larger. In particular, when pm1=pm2≤0.2, a slightly larger SU throughput is observed for larger pm1=pm2, i.e., ηSU is higher even though there are more collisions, which is counterintuitive. In this case, larger λ1 causes more collisions between PUs and SUs, new white spaces appear that are successfully occupied by SUs, and also ηSU increases. This behavior is found to persist regardless of the value of γ1=γ2 and also when the values of the false alarm parameters are not equal to zero. When pm1=pm2>0.2, the described behavior is found not to exist, and the larger pm1=pm2, the lower ηSU.Figure10 shows the collision rate α between PUs and SUs versus the average PU arrival rate λ1, for different pm1=pm2. From this figure, we notice that, in case PU arrivals are represented by an IPP, the quality of service of the PU network can be seriously compromised due to the misdetection of PUs by SUs even when λ1 is small. For instance, for a PU load ρPU=0.1, i.e., for λ1=200/s, and pm1=pm2=0.1, the collision rate is around 33 collisions per second for the considered IPP as compared to 23 collisions per second for the case of Poisson arrivals.Figure11 shows the collision rate α versus λ1, for pm1=pm2=0.2 and different values of γ1=γ2. We again set pf1=0 and δf2=0/s. It can be seen that, for the considered case of pm1=pm2=0.2, the parameters γ1=γ2 do have an influence on α. For values of pm1=pm2<0.2 and larger values of λ1, the influence of γ1=γ2 on α is found to be more significant and can exceed 40 percent. For values of pm1=pm2>0.2, the influence of γ1=γ2 on α is found to decrease. Finally, it is also found that the false alarm parameters have little influence on the collision rate, as expected.Figure 11
Collision rateα between PUs and SUs versus the average PU arrival rate λ1, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, pm1=pm2=0.2, pf1=0, δf2=0/s, and different values of γ1=γ2.Next, we investigate the effect of the false alarm parameterspf1 and δf2. We now set pm1=pm2=0, and we consider an average PU arrival rate λ1=1000/s.Figure12 shows the throughput ηSU of SUs versus γ1=γ2, for different values of pf1 and δf2. As expected, for increasing values of pf1 and δf2, the throughput ηSU decreases as more SUs stay in the sensing state. Also, it can be seen that, as γ1=γ2 increases, the throughput ηSU first increases and then decreases for higher γ1=γ2. This special effect of γ1=γ2 is also visible in Figure 7 at point λ1=1000/s for the case pf1=0 and δf2=0/s and is due to the considered intermediate value of the PU load. As explained before, for low PU load, ηSU increases as γ1=γ2 increases, whereas for high PU load, the effect is reversed.Figure 12
ThroughputηSU of SUs versus γ1=γ2, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, λ1=1000/s, pm1=pm2=0, and different values of pf1 and δf2.Figure13 shows the blocking probability βSU of SUs versus γ1=γ2, for different values of pf1 and δf2. The behavior of βSU is clearly in full agreement with that of the SU throughput ηSU (see Figure 12). As the values of the false alarm parameters pf1 and δf2 increase, the blocking probability βSU decreases.Figure 13
Blocking probabilityβSU of SUs versus γ1=γ2, for ρSU=0.5, N=20, K=50, 1/σ=1/μ1=1/μ2=10ms, λ1=1000/s, pm1=pm2=0, and different values of pf1 and δf2.


## 7. Conclusion

In this paper, we have presented a CTMC model to evaluate the performance of cognitive radio networks where SUs can experience sensing errors and PUs arrive according to an interrupted Poisson process. The effect of the IPP parameters and the sensing error parameters on the performance measures of both PU and SU networks has been studied extensively. Future work could look into a further generalization of the model to include time correlation in the SU arrival process as well.



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*Source: 4102046-2020-06-02.xml*