Outage Probability Analysis of Cognitive Radio Networks Under Self-Coexistence Constraint

Transcription

1 1 Outage Probability Analysis of Cognitive Radio Networks Under Self-Coexistence Constraint Syed Ali Raza Zaidi, Des. C. McLernon and Mounir Ghogho EEE,University of Leeds, LS2 9JT, Leeds,U.K. Abstract Cognitive radio networks (CRNs are envisioned to eradicate the artificial scarcity caused by today s stringent spectrum allocation policy. In this article, we develop a statistical framework to model the outage probability at any arbitrary primary/licensed user, while operating in the presence of a collocated secondary network/crn. A system model based on stochastic geometry (utilizing the theory of a Poisson point process is introduced to model the transmission/reception/detection uncertainty due to the random locations and topology of both primary and secondary users. The primary beacon enabled interweave spectrum sharing model is utilized for evaluating outage and interference at a typical primary receiver. It is shown that the self-coexistence constraint ignored in past studies plays a vital role in the determination of outage and interference. A statistical model for interference is developed to incorporate the self-coexistence constraint in terms of medium access probability (MAP. Our study also indicates that under the availability of multiple channels/sub-channels, outage probability also depends on the probability of picking the same channel. Optimal selection of MAP and channel selection probability is briefly discussed. Our analytical and simulation results further consolidate the argument that past studies which do not cater for the selfcoexistence constraint, over-estimate the interference. I. INTRODUCTION COGNITIVE radios (CRs, as the name implies, are intelligent, environment aware, agile and adaptive radios which are blessed with optimal decision making capabilities. Such intelligent radio networks are envisioned to eliminate the spectrum scarcity problem, which is essentially a consequence of inefficient spectrum utilization. Regulatory bodies such as the FCC and Ofcom have already reported that spectrum scarcity is artificial and can be avoided by intelligent and opportunistic radio resource utilization. In this regard, regulatory bodies are considering to relax the stringent spectrum allocation policy and introduce a dynamic spectrum access (DSA mechanism. In the recent past, there has been a lot of effort dedicated by both academia and industry to study the DSA mechanisms for intelligent/cognitive adaptive transceivers. Several academia-industry alliances like CogNeA, PHYDYAS and ARAGON are already investigating the wide range of possibilities for making CRs commercially viable. IEEE has also formed the workgroup to develop an air interface for DSA in the TV frequency band. CRs, often refered to as secondary terminals, are based on the principle of opportunistic exploitation of spectrum vacancies across space and time. However, the operational constraint on such radio devices is to ensure that they do not cause any harmful interference to the primary/licensed or legacy user. Nevertheless, even with the current state-of-the art DSA algorithms [1], it is not possible to completely avoid interference with the primary user. This dilemma has warranted the studies in the domain of statistical characterization of the interference encountered by a primary receiver in the presence of secondary nodes. Past studies [2], [3], [4] have developed the theoretical framework for modeling the interference at the primary receiver considering the presence of multiple secondary transmitters. However, to the best of our knowledge none of these studies consider the fundamental self-coexistence [5] constraint. Generally the term self-coexistence is employed to refer to the peaceful coexistence of CRs within a CRN. Hence, these studies over-estimate the interference encountered by a typical primary receiver. In this article, we extend past studies to develop a statistical framework for outage probability and interference at a typical primary receiver, not only catering for the self-coexistence constraint but also for the channel selection. In other words, when multiple primary and secondary users exist in the network, secondary nodes only pick the same channel/sub-channel as the primary with a certain probability. We demonstrate that the simplest possible way of catering for self-coexistence is to enforce the medium access control (MAC in the secondary network. Slotted ALOHA is perhaps the simplest and easiest to implement decentralized MAC scheme. In this paper, we show that even under such a simplistic MAC mechanism and with a high transmission probability, interference encountered by any arbitrary primary node is significantly less than what has been observed by the past studies. This stems from the fact that the number of interferers are significantly reduced under the self-coexistence constraint and hence the outage probability. It is also shown that the number of interferers can be further reduced by uniform random channel selection when the choice of multiple channels is available for transmission. II. PREVIOUS WORK AND OUR CONTRIBUTION In [4] Ghasemi et al. developed an interference model for cognitive ad hoc networks. He approximated the probability density function (PDF of interference by a shifted log-normal distribution and showed that it closely matched the exact distribution. In [2] Vu et al. developed an interference model for cognitive radio networks in the presence of a primary based beaconing scheme. Authors in [2] have derived average aggregate interference and outage. Scaling laws for the interference and outage were derived by Mitran [6], using an information

2 2 theoretic formulation. Hong et al. in [3] utilized an Alphastable distribution for modeling interference in cognitive radio networks. Contributions and Organization As previously discussed, to the best of our knowledge none of the studies on interference modeling for CRs include the self-coexistence constraint in the secondary network. In other words, previous studies over-estimate the interference encountered by the primary receiver and the outages incurred at the primary. So in this paper: (1 We consider a network model (Section III where the spatial distribution of primary and secondary nodes is characterized by the Poisson point process. We formalize the stochastic geometry based network model and the intuition behind such a model. Moreover, it is formally shown that in a multiple primary and multiple secondary nodes scenario, modeling of interference at a typical primary receiver is similar to the modeling of interference at a single primary receiver in the presence of multiple Poisson distributed secondary transmitters; (2 Based on the network and spectrum sharing model introduced in Section III, we develop the statistical framework for evaluating the outage probability of a typical primary receiver under the self-coexistence constraint, while utilizing uniform channel selection; (3 We then briefly discuss the optimal selection of MAP and channel selection probability (Section V. Based on this analysis, we observe the trade-off between the usefulness of the secondary network and the outage incurred at the primary receiver; (4 Lastly, we demonstrate the accuracy of our approach using Monte Carlo simulations and analytical results (Section VI. We also show that the interference at a typical primary receiver follows a fat tailed distribution. We demonstrate that under the self-coexistence constraint this fat tailed behavior can be significantly reduced. Moreover, the outage probability observed under the self-coexistence criteria is significantly less than the reported results in past studies. III. STOCHASTIC GEOMETRY BASED NETWORK MODEL A. Primary and Secondary Network Geometry In this paper, we consider infinite spatially collocated primary and secondary ad hoc networks (see Fig. 1. The spatial distribution of the nodes is captured using two collocated homogeneous Poisson point processes (HPPPs [7] i.e., Φ p and Φ s with intensities λ p and λ s respectively. In other words, the location of the nodes of the primary network constitute a HPPP Φ p, where λ p is the average number of the primary nodes per unit area. Similar statement can be made for the HPPP formed by the secondary nodes Φ s with intensity λ s. Note that the HPPP is a very well established statistical model for the spatial distribution of the nodes in ad hoc wireless networks [8]. Many studies in the past [4], [2] have utilized the HPPP for modeling the spatial distribution of the secondary network. Considering Φ i (B (i ɛ{s, p} 1 as a counting process defined over a bounded Borel set B, the number of nodes (i.e., points of Φ i in B have a Poisson distribution with a finite mean λ i v d (B for some constant λ i. v d (B is the Lebesgue measure defined on the measurable space [R d, B d ]. In other words, v d (B is the volume of a d dimensional bounded Borel set B. If B is a d dimensional sphere v d (B = b d r d, where r is the radius of the sphere and b d is the volume of the unit sphere in R d, then b d = π d /Γ(1+d/2with Γ(a = x a 1 exp( xdx. Note that by the Superposition Theorem [8], the overall network formed by both the primary and the secondary nodes also follows a HPPP. Furthermore, such a network model by construction satisfies the required regularity conditions of the HPPP [7], i.e., the HPPP Φ i is locally finite and simple. At a given instant, any arbitrary primary/secondary node can act either as a transmitter or a receiver. Hence the points x n of the HPPP Φ i formed by the primary/secondary networks can be marked by attaching the appropriate signifier. Such a point process is refered to as a Marked Poisson point process (MPPP. The marks are assumed to take the values in the space of marks M. It is beyond the scope of this paper to provide detailed discussion on the theory of the marked point processes. The interested reader is refered to [7] and [8] for the theoretical background. In order to simplify the theory, rather than considering the MPPP, we formulate this classification in an alternative but equivalent manner. Utilizing the Superposition theorem, the HPPP Φ i can be constructed by the superposition of two independent HPPPs Φ tx i and Φ rx i with intensity λ tx i and λ rx i respectively, such that λ i = λ tx i + λ rx i, with: λ tx i = λ i ρ and λ rx i = (1 ρλ i. (1 where ρ 1. Since the interference analysis is fundamentally concerned with the distribution of the transmitters, we can invoke the above-mentioned property to model the spatial distribution of the transmitters in the secondary network (HPPP Φ s with intensity λ tx i. B. Channel Model The large scale path-loss between any arbitrary transmitter y t ɛ Φ tx i and receiver y r ɛ Φ rx i is given by l( y r y t, where, l(. is a distance dependent path-loss function. Generally, the large scale path-loss is modeled by considering the power law function, i.e.,l(r = CR α R 1, where C is the frequency dependent constant, R is the distance between the transmitter and the receiver and α > 2 is the terrain or environment dependent path-loss exponent. Unfortunately, this type of power law path-loss function suffers from a singularity near zero. In other words, for distances of the order of R < 1, transmit power is amplified. This is, however, contrary to the physically observed phenomenon. Luckily, this problem does not arise while modeling interference in cognitive radio. The main reason behind this stems from the widely accepted regulatory principle, i.e., none of the cognitive radios inside the primary exclusion region of the primary can transmit during 1 We utilize the notation of i instead of s or p to avoid repetition, when both primary and secondary networks can be treated under the same framework.

3 3 the primary s communication session. In the case of ad hoc networks, the primary s exclusion region is defined by the disk of radius r e, centered at the primary transmitter. Hence the exclusion disk defines the interference footprint created by any other concurrent transmission. Alternatively, it is also possible to center the exclusion disk on the primary receiver rather than the primary transmitter. The detailed discussion on the primary exclusion region (also refered to as the primary guard zone can be found in [9]. It is worth highlighting that the primary s exclusion region is an important design parameter, which is indeed dictated by the maximum tolerable interference threshold, secondary node density, fading, environment dependent path-loss and the medium access control mechanism. Since the focus of this paper is inclined towards the study of interference as seen by a typical primary user from the collocated secondary network, the path-loss function is modified to accommodate the exclusion region as,l(r = CR α 1 r e R. Besides the path-loss attenuation. We consider that the channel effects due to multipath fading between any arbitrary transmitter y t ɛ Φ tx i and receiver y r ɛ Φ rx i can be modeled using a random variable G with the PDF f G (., cumulative distribution function (CDF F G (. and mean µ. We also consider that G is independent and identically distributed (i.i.d. both in the spatial and temporal domain (i.e., across different links in space and different symbols on the same link. Hence the overall impact of the communication channel is modeled using a random variable H = Gl(r. C. Spectrum Sensing Model for the Secondary Network The outage probability of a typical primary receiver is governed by the spectrum sharing mechanism. As mentioned earlier, past studies have explored several potential mechanisms for the secondary networks to dynamically access the spectrum. However, in this paper we restrict our discussion to the spectrum interweave mechanism. [1] and the references therein provide detailed discussion on the spectrum access mechanisms. As proposed by the FCC, each secondary user is obliged to detect the primary s presence in the licensed frequency band and hence refrain from transmitting during the primary s communication session. Many studies have investigated several potential algorithms to detect the presence of the primary. It is beyond the scope of this paper to elaborate the discussion on this topic. Interested readers may refer to [1] for an overview of the primary detection algorithms and spectrum sharing models [1]. In this paper, we consider a beacon/control channel based spectrum sensing model. The primary transmitter explicitly sends a control signal such as grant and inhibit when it leaves or enters the transmission mode. Such a scheme is also known as out of band sensing [4]. Numerous studies on the interference modeling [2], [4] have utilized this model. In this paper, we assume that beacon channel is interference free. In practice this is assured by the ad hoc network employing handshaking(cts/rts type mechanisms. In such a network all primary users which receive CTS from a primary in their contention domain, refrain to send their own beacon. Hence before initiating its communication a primary receiver can send inhibit beacon without suffering significant interference. IV. OUTAGE PROBABILITY ANALYSIS Consider a typical primary receiver x ɛ Φ rx p ; the interference encountered at this receiver determines its outage probability, P out = Pr{I x > I th } (2 where, I x is the aggregate interference power measured at x and I th is the maximum tolerable interference under which the primary can communicate while maintaining the required quality of service (QoS. The upper and lower bounds on the tolerable interference of the primary is defined by the regulatory bodies such as FCC. Essentially, FCC has proposed the use of the metric similar to I th refered to as Interference temperature. In practice, it is useful to consider a regulatory constraint of the form, P out Θ. Such a constraint is also useful in studying how network transmission capacity (TC scales with different parameters of the secondary access mechanisms. In order to establish the outage probability P out of the typical primary receiver, as indicated by (2, we need to evaluate the complementary CDF (CCDF of the interference (shot noise process at x I x = yɛφ I G xy l( x y P s (3 where, G xy is the channel gain between the primary receiver x and the secondary transmitter y, l(. accounts for the large scale path-loss and P s is the transmit power of the secondary node. We assume that the secondary nodes transmit with a constant power. It is, however, possible to consider the situation where the secondary nodes transmit with a power P s with PDF f PS (. and CDF F ps (. under the same framework. At this juncture, it is worth registering that not all the secondary transmitters will contribute towards the aggregate interference, i.e., only those secondary transmitters which at minimum satisfy all of the following conditions to qualify as potential contributors. Condition 1: Only the secondary transmitters outside the exclusion region of the typical primary receiver can potentially transmit and cause harmful interference (the effect of the exclusion region is incorporated into the path-loss model. Condition 2: Condition 1, is not sufficient to characterize the secondary transmitter as a potential contributor, rather only those secondary transmitters that lie outside the exclusion region and cannot detect the inhibit beacon from the primary will continue transmitting and cause interference to the primary receiver. Past studies such as [4], [2] have developed the theoretical framework for modeling aggregate interference at the primary receiver considering a single primary transmitter with the above two constraints. Nevertheless, these studies ignore the fundamental self-coexistence constraint of cognitive radios (CRs in the secondary network. In particular, even if every secondary transmitter always has data to transmit to some secondary receiver, not all of them can transmit at the same time. This constraint stems from the fact that, if all the secondary transmitters transmit data all the time none of them will be successful in their transmission. In practice, this is assured by the MAC mechanism employed at the secondary transmitter.

4 4 An additional channel selection constraint further reduces the number of interferers and hence the outage probability. Hence beside condition (1 and (2 two additional constraints which a secondary transmitter should satisfy to qualify as a potential interferer are: Condition 3: (Channel selection constraint Considering, an ad hoc network where multiple channels are available, any arbitrary secondary transmitter contributes towards the interference at the primary receiver in its vicinity, if it satisfies both condition (1 and (2 and randomly picks the same channel as the primary to perform its communication with a secondary receiver. Condition 4: (Self-coexistence constraint Beside condition 3, since the number of sub-channels is limited, multiple secondary transmitters can pick the same sub-channel (on which the primary is also communicating. They only transmit with probability in a fixed time-slot and defer the transmission with probability 1. The time slots are assumed to be synchronized. In essence, for multiple primary and secondary nodes, the medium access control mechanism lends itself as multi-band slotted ALOHA. In order to model the conditions 2-4, the HPPP of the secondary transmitters needs to be thinned. Using Slivnyak s theorem [7] then (3 becomes: I o = I x = yɛφ I G y l( y P s. (4 where, o denotes origin. Hence the problem is translated into a scenario where the network is centered at the primary receiver and the interference is encountered from all the secondary transmitters satisfying the aforementioned conditions located in the area R d \b(, r e (where b(x, y is ball of radius y centered at point x. Before applying the conditions 2-4 by performing the thinning operation, it is useful to translate the point process into the polar cordinates, hence using the Mapping theorem [7] the intensity of the HPPP Φ tx s is given as λ tx s (r = λ tx s dr d 1 b d. (5 Here, r is the distance between the secondary transmitter and the primary s receiver. In order to incorporate the detection process, the HPPP Φ tx s is further decomposed into two HPPPs, Φ det s Φ tx s, s.t. λ det s (r λ tx s (r Φ md s Φ tx s, s.t. λ md s (r λ tx s (r (6 Φ tx s = Φ md s Φ det s, and λ tx s (r = λ det s (r + λ md s (r where the Φ md s representing the transmitters which misdetected and is constructed using dependent thinning of Φ tx s. Assuming that primarys do not interfere with themselves and multiple control channels are available to transmit the beacon (in other words, the beacon channel does not suffer from interference so any secondary transmitter detects the beacon with probability 1 if the signal-to-noise ratio (SNR of the beacon channel from the primary receiver to secondary transmitter is greater than some threshold SNR. Mathematically, 1 md (γ(r = { 1 γ(r < γ th γ(r γ th (7 where 1 md (γ(r is the indicator random variable, i.e. the detection process can be expressed as a Bernoulli trial with the probability of misdetection Pr{γ(r < γ th } and probability of detection 1 Pr{γ(r < γ th }. The SNR γ(r = P bl(rg N o, where N o is the noise power at the secondary transmitter and P b is the beacon transmission power. Note that the point process formed by such dependent thinning is non-homogeneous. The intensity of Φ md s is given as λ md s (r = λ tx s (r1 md (γ(r. In order to satisfy the channel selection constraint, Φ md s is further thinned using independent thinning, i.e. if probability of picking a channel from a pool of available channels is p ch. Φ md,ch s is constructed by p ch thinning, where each point of Φ md s is retained with probability p ch, and hence the intensity of the process becomes, λ md,ch s (r = λ md s (rp ch. Finally applying another independent thinning to accommodate the self-coexistence constraint we obtain the point process of interferers Φ I, with intensity λ I (r = λ md,ch s (r. Notice, selection of p ch and is an important design parameter and indeed the selection is based on the trade-off between the interference encountered that can be tolerated by the primary and usefulness of the secondary network. We will come back to this issue in later discussion. We adopt a very well established method of calculating the moment generating function (MGF and inverting it back to obtain the PDF of the interference f I (.. M I (s = E(exp(sI o = E(exp(s yɛφ I G y l( y P s, = E E G (exp(sg y l( y P s. (8 yɛφ I Using the definition of the Generating functional [8] for the Poisson point process, i.e.,g(f(x = exp ( R (1 f(xλ(dx, then (8 can be written d as: {ˆ }] M I (s = exp [ E G (1 exp(sp s Gl(rλ I (rdr. R d \b(,r e (9 It is not possible to obtain the closed-form expression for such an MGF. Hence using the method of cumulants introduced in [4], the n th cumulant is obtained as κ n = dn ln(m I (s/ds n. Due to the limited space we will not provide detailed derivation of κ n here, the interested reader is refered to [11]. Without any loss of generality it is assumed that P b,p s and C are all unity. The final expression obtained for κ n is, [ κ n = p ch dλ tx s b d EG low(n, γ thre d αn d re αn d + Eup G ( d α, γ ] thre d γ d α n th (1 where, γ th = γ th N o, EG low(n, a = a gn f G (gdg and E up G (n, a = a gn f G (gdg. Notice that the results of [4] become a special case and can be obtained if both the conditions 3-4 are relaxed, i.e.,,p ch,r e is 1 and d = 2. The PDF of the aggregate interference can be obtained from the cumulants using the Method of Moments [4]. Authors in [4] have employed a similar approach in the absence of conditions 3 and 4 to model the interference by the lognormal or shifted log-normal distribution. Note that it is well

5 5 established in the literature that the Gaussian approximation is not valid for interference due to its skewed and fat-tailed behavior [4]. The main problem with these distributions is that although they closely fit in the body, they do not fit accurately in the tail for all parameters. Moreover, the expressions for matching such moments (see [4] are quite complex. An alternative choice can be the log-logisitic distribution. However, it does not result in a good fit both in body and tail. It is possible to accurately fit the distribution using simulation and maximum likelihood estimate or L-moments type statistics. After conducting several experiments and goodness of fit testing, we found that the Gamma distribution is the best fit for the interference distribution. An added advantage with the Gamma distribution is that moment matching expressions lend themselves into a very simple form. Hence interference I Gamma(k, θ with k being the shape parameter and θ is the scale parameter with k = κ2 1 and θ = κ 2. (11 κ 2 κ 1 The outage probability of a typical primary receiver is then obtained as, ( P out = Γic κ 2 ( 1 κ, I th κ 1 κ 2 κ 2 /Γ 2 1 κ 2, (12 where Γ ic (a, b = x a 1 e x dx, is the upper incomplete b Gamma function. V. OPTIMAL CHANNEL SELECTION AND SELF-COEXISTENCE Due to space limitation, in this section we briefly discuss how the optimal values for the channel selection probability and transmission probability for the secondary transmitter can be determined. A. Channel Selection Probability Although, in this study we assumed that the secondary transmitter uniformly selects the channel from the pool of operational channels, in practice channel selection probability also depends on the historical information. In principle, channels will be picked uniformly amongst those channels for which the secondary transmitter has not received an inhibit beacon. This can be accommodated in the definition of p ch by considering it as a conditional probability, i.e., if M is the total number of channels, considering that N is the number of channels for which the beacon is received; the secondary transmitter picks a channel randomly from the remaining(m N channels. Notice that N depends on the connectivity properties and hence follows the Weibull distribution. Nevertheless p ch is conditionally uniform. B. Self-coexistence/Medium Access Probability The medium access probability which enforces the selfcoexistence constraint not only depends on the usefulness of the secondary networks but also depends on the tolerable interference of the primary receiver I th and its outage probability. When the primary s tolerable interference is of main concern, Markov s inequality or the Chernoff bound can be employed to determine the optimal value of p corresponding to the I th, e.g. P out κ1 I th given that P out Θ, we obtain ΘI th = κ 1, substituting the expression for κ 1 and by arithmetic manipulation the value of p can be determined. However, contradictory requirements arise from the usefulness of the secondary network and when this is of importance principles from ad hoc networks can be adopted and p can be determined as in [12]. VI. RESULTS AND DISCUSSION In this section, we highlight several insights obtained by performing Monte Carlo Simulations (which also validates the theoretical framework. Fig. 2 shows the PDF of the interference at a typical primary receiver. Results are obtained by executing a hundred thousand iterations for Exponential (corresponding to Rayleigh fading and Log-normal distributions. These histograms are plotted against the fitted Gamma distribution, obtained by evaluating the cumulants and using the Methods of Moments. Expressions for the upper and lower moments required to compute the cumulants are straightforward to calculate and thus not included here (due to limited space. Fig. 2 illustrates that both experimental and theoretical results closely match. In order, to test goodness of fit, a Chi2 test was also performed. Both simulation and experimental results shown here are obtained by considering following numeric values of the parameters: inital intensity of secondary transmitters λ tx s =.1, path-loss exponent α = 4, radius of exclusion region r e = 1. Normalized detection SNR threshold γ th = γ th N o = 4 db and dimension of network space d = 2. Choice of these parameters is arbitrary but we have used same parameters as in [4] in order to ensure that the results are comparable. The empirical CDF was obtained from a histogram and is plotted against theoretical results in Fig. 3. Note that the simulation results slightly deviate from the theoretical results for a high threshold because the order of outage probability is 1 4 or 1 5 while the number of iterations is 1 5. This can be improved by increasing number of iterations. Note that the PDF of the interference is skewed (mean > median > mode and has a fat-tailed behavior (Log-normal or Gamma distribution exhibit the fat tailed behavior in general. This fat tailed behavior implies that with non-negligible probability there will be some nearby secondary transmitters which will cause harmful interference to the primary receiver, hence leading to the outage of the primary. This fat tailed behavior is indeed highly undesirable. Several possible mechanisms can be adapted to get rid off this behavior. In general any mechanism which can reduce the uncertainty about the primary s presence can contribute towards reduction of the interference encountered at the primary. This indeed implies that agility of the cognitive radio needs to be improved. Amongst many alternatives, one simple way is to introduce the cooperation amongst secondary nodes. Such cooperation can reduce the uncertainty in detection encountered due to spatial or temporal variations of the channel between the secondary transmitter and primary receiver. Fig. 4 shows that under the coexistence and channel selection constraints, outage probability is significantly reduced and so is the fat-tailed behavior. Hence this validates that in practice many studies over-estimate the outages encountered by

6 6 the typical primary receiver. Moreover, as the medium access probability (MAP is reduced, a proportional reduction in outage is observed. Note that even for the high values of MAP =.8, outage is significantly less than the scenario where there is no self-coexistence constraint. Also from Fig. 4 it is obvious that in the presence of multiple channels where each channel is uniformly picked, the outage probability is further reduced. Hence any practical ad hoc network designed to satisfy both self-coexistence and channel-selection constraint can reap the benefit of opportunistic exploitation of spectrum, while possibly causing little or no harmful interference. Due to lack of space, we did not include the results where the interference threshold is fixed, while the normalized detection SNR threshold is varied. In such a scenario it is noticed that even for the high detection threshold, outage probability under the self-coexistence constraint is significantly lesser than the counter case. VII. CONCLUSION In this paper, we have developed a statistical model of the interference encountered by a typical primary receiver due to the secondary transmitters in a collocated secondary network. It is shown that the self-coexistence constraint which is not considered in past studies, plays a vital role in over all characterization of interference at a typical primary receiver. In particular, such a constraint can be accommodated by enforcing simplistic MAC mechanisms such as ALOHA. It is demonstrated that even under high medium access probability, interference is significantly reduced by enforcing such a constraint. Furthermore, simulation and analytical results also show that when multiple channels are available, with the uniform selection of channels, interference can be further reduced. It is also shown that selection of access probability is based on trade-off between the usefulness of the secondary network and tolerable interference by the primary receiver. Lastly, with the help of Monte Carlo simulations and an analytical framework, it is validated that by incorporating the self-coexistence in an interference model is significantly important and past studies which have ignored the selfcoexistence constraint over-estimate the interference caused by the secondary network. REFERENCES [1] A. Goldsmith, S. Jafar, I. Maric, and S. Srinivasa, Breaking spectrum gridlock with cognitive radios: An information theoretic perspective, Proceedings of the IEEE, vol. 97, no. 5, pp , 29. [2] M. Vu, S. Ghassemzadeh, and V. Tarokh, Interference in a cognitive network with beacon, in IEEE Wireless Communications and Networking Conference, 28, pp [3] X. Hong, C. Wang, and J. Thompson, Interference Modeling of Cognitive Radio Networks, in IEEE Vehicular Technology Conference, VTC Spring, 28, pp [4] A. Ghasemi and E. Sousa, Interference aggregation in spectrum-sensing cognitive wireless networks, IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 1, pp , 28. [5] C. Cordeiro, K. Challapali, D. Birru, and S. Shankar, IEEE 82.22: An introduction to the first wireless standard based on cognitive radios, Journal of communications, vol. 1, no. 1, pp , 26. [6] P. Mitran, Interference scaling laws in cognitive networks, in Communications, 24th Biennial Symposium on, 28, pp [7] D. Stoyan, W. Kendall, J. Mecke, and D. Kendall, Stochastic geometry and its applications. Wiley New York, [8] M. Haenggi and R. K. Gant, Interference in Large Wireless Networks. Foundations and Trends in Networking, NOW Publisher, 28, vol. 3, no. 2. [9] M. Vu, N. Devroye, and V. Tarokh, The primary exclusive region in cognitive networks, IEEE Trans. on Wireless Comm, 28. [1] Q. Zhao and B. Sadler, A survey of dynamic spectrum access, IEEE Signal Processing Magazine, vol. 24, no. 3, pp , 27. [11] S. A. R. Zaidi, D. McLernon, and M. Ghogho, Transmission capacity analysis of cognitive radio networks, Technical Report available at: 21. [12] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, An Aloha protocol for multihop mobile wireless networks, IEEE Transactions on Information Theory, vol. 52, no. 2, pp , 26. Figure 1. Y Primary s Exclusion Region r e Secondary Transmitter Secondary Receiver Primary Transmitter Primary Receiver X Outage Probability for α = 4, λ tx s =.1,d = 2 and re = 1. PDF of Interference f I (i PDF of Interference f I (i Normalized Interference x Simulation 3 Fitted Normalized Interference x 1 3 r e Simulation Fitted Exponential Log normal Figure 2. PDF of interference f int (.( Exponential (green and Lognormal fading (blue, = p ch = 1. Figure 3. -see(12. Outage Probability (P out Outage Probability (P out Exponential µ=1 Lognormal σ =6dB Simulation Theoretical Normalized Interference Threshold (I x 1 3 th Outage probability for α = 4, λ tx s =.1 and = p ch = =.8,p ch =1 =.5,p ch =1 =.8,p ch =.2 =.8,p ch =.1 = 1, p ch = Normalized Interference Threshold (I x 1 4 th Figure 4. Outage probability for α = 4, λ =.1,d = 2 and r e = 1 (dashed line for simulation and solid line is for theoretical results-see(12