Analysis of cognitive radio networks with imperfect sensing Isameldin Suliman, Janne Lehtomäki and Timo Bräysy Centre for Wireless Communications CWC University of Oulu Oulu, Finland Kenta Umebayashi Tokyo University of Agriculture and Technology TUAT Tokyo, Japan Abstract Recently, cognitive radio access has received much attention. Spectrum sensing methods are often used for finding free channels to be used by cognitive radios secondary users. State diagram based approach can be used for analyzing the effects of imperfect spectrum sensing with false alarms and misdetections. The state diagram consists of two-tuples like 1,2 meaning one primary user and two secondary users present. We note that state dependent transition rates are very important for accurate modeling. This is because for example in state, all channels occupied by primary users collisions happen with increased probability. Our contribution is as follows. Explicit expressions for state dependent transition rates are presented for the case with three channels. However, the approach can be used also for more channels. Primary termination probability is used for evaluating the level of interference to primary users caused by secondary users. Secondary success probability is used to find out how often does a secondary call start and terminate successfully. and analysis results agree very well. I. INTROUCTION Cognitive radio [1] has been considered as key mechanism in addressing the problem of spectrum scarcity in wireless communication systems [2]. Main system assumption in cognitive radio use is that there is a primary operator who owns or has licensed the frequency band, and an opportunistic system who attempts to use this band on a non-interfering basis, i.e., as a secondary operator. Secondary users i.e. users who have associated with the opportunistic system are users who seeks free frequency channels for their own communication purposes. Fig. 1 shows example of primary user occupancy in N channels and also the free resource that could be utilized for secondary user transmissions. Primary users i.e. users who are associated with the primary operator however, have strict priority over secondary users. This requires accurate detection of the presence of the primary user by the secondary users. ifferent types of spectrum sensing mechanisms have been used for detecting presence of primary users in wireless networks. These mechanisms include energy [] and cyclostationary [4] based methods. For example, in energy detection see, e.g., [5], the detector measures the energy / power of the received signal during some time period and in some frequency channel. Then the measured value of energy / power is compared to a threshold. If the threshold is exceeded, the detector decides that a primary signal was present. Perfect detection of presence of primary users cannot be obtained by Fig. 1. Channel occupancy by primary users in N channels. using spectrum sensing methods. There will be false alarms and misdetections. The interactions between primary and secondary users can be studied by using continuous-time Markov chains CTMC as in [6]. Therein, perfect sensing was assumed. The effects of imperfect sensing can also be analyzed using Markov chains as recently performed in [7]. Therein, a multichannel system was assumed and the system state was reduced to two numbers: the number of channels occupied by primary users and the number of channels occupied by secondary users. State diagram based approach was also used in [8] with perfect radio resource detection for modelling dynamic spectrum access. According to our best knowledge, in the existing literature about Markov based analysis of cognitive radios, the state dependence of events such as collisions with primary users or finding free channels for secondary users is not fully taken into account. In practice, the transition rates do not depends only on the arrival/service rate but also on the system state. For example, if there are several primary users present, then the probability of collision is increased and also the probability of finding a free channel is decreased. In this paper, we apply Markov modeling approach similar to [7], [8]. Our goal is to perform performance analysis of a multichannel cognitive radio system with primary and secondary users and with imperfect sensing and random channel search order. The performance metrics used are the probability that a secondary user call is normally terminated and the primary termination probability. The analysis is carried out 978-1-4244-521-4/9/ $26. 29 IEEE 1616
by using the state diagram based approach. The prominent feature of our state diagram is that it has state-dependent transition rates at the nodes. These transition rates are found by going through all the possible search sequences and by taking into account the channel state. We believe that the state dependency is very important for modeling cognitive radio systems with higher accuracy. Explicit state diagram and results are presented for case with three channels. However, the approach can be used also for more channels. II. SYSTEM MOEL False alarm probability P F refers to the probability that a free channel is classified as being occupied. The misdetection probability P M =1 P is the probability that an occupied channel is classified as vacant. The detection probability P is the probability that a occupied channel is correctly classified as occupied. False alarms reduce spectrum utilization of secondary users while misdetections cause interference to the primary users. We assume that there are N channels available. These channels are shared between primary and secondary users, with primary users having priority over the secondary users. Calls of the primary users arrive with rate and secondary calls arrive with rate λ 2. The corresponding service rates are μ 1 and. We have made some simplifications similar to those used in [7] so as to enable finding theoretical solutions. The assumptions are: 1 When primary user arrives to a channel occupied by secondary users, the secondary user will always notice the primary user [7]. Note that this results into a very short collision with the primary user. After this, secondary user starts to search for a new channel. uring this phase, the secondary user will perform detection on the remaining channels with random order until it finds a free channel or all channels are determined to be busy. Free channel is decided to be occupied with false alarm probability P F and occupied channel is determined to free with miss detection probability P M = 1 P, where P is the detection probability 2 All state transitions are instantaneous, i.e., the time it takes to search for a free channel is assumed to be negligible. Note that in practice, the acquisition time can be quite large, depending on the search method used [9]. A secondary user knows the channels occupied by other secondary users and it will not use them. The necessary information can be distributed over, for example, some signaling channel. 4 A primary user knows the channels occupied by other primary users so that there will be no collisions between primary users [7] 5 In case of collisions between primary user and secondary user, both colliding users withdraw from the channel [7]. Note that the collision when primary comes to secondary channel is assumed to be short and does not cause the primary to leave the channel. P M : Misdetection => Collision P S F F P F P P F : False alarm Fig.. Movement from state 1,1 to state 1,. 6 The search order for new free channels is random similar to random search in [9]. The search stops after an idle channel is found or all channels are found to be occupied. In practice, it might be allowed to search all the channels several times until giving up. III. ANALYSIS We use a two-dimensional Markov chain to model the system. The system states are given by two-tuples i, j where: i is number of channels used for primary users calls and j is number of channels used for secondary users calls. For example 1,2 refers to state with one primary user and two secondary users. Let N be the number of channels available in the system. The total number of channel occupied by primary and secondary users does not exceed N. Therefore, we have the following restrictions: i N, j N, i + j N. Let Q i,j denotes the steady state probability that the system is in state i, j, which can be interpreted as the proportion of time that the system spends in state i, j. A. State transition diagram Fig. 2 shows the state transition diagram when there are three channels. The state-dependent transition rates have been derived by going through all the possible sequences of channels and detection events. Some simple examples are given next. Let us consider transition from state 1,1 to state 1,. The detection events and channel search orders that lead to this transition are illustrated in Fig.. There are two possibilities for this transition. In the first possibility, the existing secondary user call ends with rate so that the number of secondary users is reduced by one. In the second possibility, a new primary user comes to the existing secondary user channel forcing it to search for free channels. If the existing secondary user ends up with colliding the existing primary user then both secondary 1617
, 1 + 2 1 P F, 2 1, 2 μ 1 + λ 2 P M λ21 P 2 F λ21 PF μ2 2μ2 2 P F 2+ 1 P 2 F λ21 P F 1 + P 2 2μ2 + λ1pm 2 1+ 1 2 1 P F 1+P λ21 P F, 1 1, 1 2, 1 μ 1 + λ 2P M 2 1 + P F 2μ 1 + λ 2 1 P 2 μ2 λ1 P 2 F λ1 P P 2 2+P + P F +2P F P λ 2 1 PF μ2 + P M 1 + 4 PF λ1 2 P F P 2 k= P k λ21 P F 1 P 2 μ2 + λ1, 1, 2,, μ 1 + λ 2P M 1+PF + PF 2 2μ 1 + λ 2P M 2 + P + P F +2P F P μ 1 + λ 2 1 P Fig. 2. State transition probabilities and primary calls will be terminated thus leaving only the new primary thus resulting to the state 1,. Let us discuss this in more detail and start from the state 1,1. Because primary knows about other primaries, the new primary has two possible channels to use, one which is occupied by secondary user and one which is totally free. Therefore, the new primary user comes to secondary user channel with probability primary is not concerned about secondary users. Then the secondary user goes straight to the existing primary user channel with probability and collision happens with probability P M. Alternative is that the secondary user goes first to the free channel with probability and has false alarm and then goes to the primary user channel and has misdetection. All these terms result into rate of + P M 4 1 + P F Movement from state 2,1 to state,. The new primary user always forces the secondary user to leave the channel and to search new free channels. The secondary user correctly detects that the two other channels are occupied by primary user with probability P 2. Thus the transition rate is P 2. B. Balance equations The balance equations can be written by considering the transition rates using the rule that input must equal output for 1618
each state [1]. Additionally, N i= j= N Q i,j =1 1 For example, for the simple case of the state, the balance equation results into Q, = Q [ ] 1, μ1 + λ2pm 1+PF + PF 2 + Q,1 + λ 2 1 PF 2 For the case of state 1,2 we have the following balance equation Probability 5.15.1.5 Q 1,2 λ 1 P +2 + λ 1 P M + μ 1 + λ 2 P M =Q, + Q λ21 P F 1+P 1,1 + Q λ11+21 P F,2 The resulting set of simultaneous linear equations balance equations for every state and the normalization constraint can be easily solved using, for example, MATLAB. C. Primary user termination probability We use the term primary termination probability to refer to the probability that a primary user call, which has not been blocked in start, is terminated due to collisions with secondary users because of misdetections. The probability that a primary user call is terminated due to collisions with secondary users can be found by going through the state diagram states. The result is P PT = N i=1 N i j= + N 1 N i i=1 j=1 Q i,j T i,j i 1,j iμ 1 Q i,j T i,j i,j 1 j 1 QN, 4 where Q i,j is the state probability of state i,j and T i,j i 1,j is the transition from state i,j toi 1,j. IV. NUMERICAL AN SIMULATION RESULTS First, we performed simulation to verify the solution to the state equations. The simulations were performed with MAT- LAB using an event-based approach and Poisson arrival processes. The simulation setup used the assumptions mentioned in Section II, i.e., the acquisition time was negligible. However, during acquisition the channels were randomly searched using the specified false alarm and detection probabilities, i.e., we did not simulate the Fig. 2 directly thus providing verification for the derived state transition probabilities. The theoretical results are compared with simulation in Fig. 4 for the case N =. It can be seen that the theory and simulation agree very well. The state numbers are explained in Table I. Fig. 5 shows the primary termination probability theory and simulation. Of course, when P =1, the primary termination probability is zero. From the results we can see that if 5 % 1 2 4 5 6 7 8 9 1 State index Fig. 4. State probabilities, simulation versus theory, P F =.15, P =.71, =7, λ 2 =.5, μ 1 = =4, N =channels. Primary termination probability 5.15.1.5.6.65.7.75.8.85.9.95 1 P Fig. 5. Primary termination probabilities, simulation versus theory, P F =.15, =7, λ 2 =.5, μ 1 = =4, N =channels. is the maximum allowed termination probability caused by secondary users interference, then the detection probability P must be around.95 or greater. Fig. 6 shows the probability that a secondary call is started and terminated normally. It can be seen that as expected the success probability goes down when increases. This means that the high primary arrival rate means that the channels are more often occupied by primary users reducing opportunities for secondary users to access the network. 1619
Probability that a secondary call is succesfull 1.9.8.7.6.4..1 =2 =4 =8 1 2 4 5 6 7 8 9 1 Fig. 6. Probability that a secondary user call is normally terminated, P F =.15, λ 2 =.5, μ 1 =4, N =channels. TABLE I MAPPING BETWEEN i,j AN STATE INEX i,j state index, 1 1, 2 2,, 4,1 5 1,1 6 2,1 7,2 8 1,2 9, 1 REFERENCES [1] J. Mitola III, Cognitive radio: An integrated agent architecture for software defined radio, Ph.. dissertation, Royal Institute of Technology, Sweden, 2. [2] S. Haykin, Cognitive radio: Brain-empowered wireless communications, IEEE J. Select. Areas Commun., vol. 2, pp. 21 22, Feb. 25. [] H. Urkowitz, Energy detection of unknown deterministic signals, Proc. IEEE, vol. 55, no. 4, pp. 52 51, Apr. 1967. [4] W. A. Gardner, Signal interception: a unifying theoretical framework for feature detection, IEEE Trans. Commun., vol. 6, no. 8, pp. 897 96, Aug. 1988. [5] J. Lehtomäki, Analysis of energy based signal detection, Ph.. dissertation, Acta Univ Oul Technica C 229. Faculty of Technology, University of Oulu, Finland, ec. 25. [Online]. Available: http://herkules.oulu.fi/isbn9514279255/ [6] B. Wang, Z. Ji, M. Abdulrehem, R. Liu, and T. Clancy, Primaryprioritized markov approach for dynamic spectrum allocation, IEEE Trans. Wireless Commun., vol. 8, pp. 1854 1865, Apr. 29. [7] S. Tang and B. L. Mark, Modeling and analysis of opportunistic spectrum sharing with unreliable spectrum sensing, IEEE Trans. Wireless Commun., vol. 8, pp. 194 194, 29. [8] Y. Xing, R. Chandramouli, S. Mangold, and S. S. N., ynamic spectrum access in open spectrum wireless networks, IEEE J. Select. Areas Commun., vol. 24, no., pp. 626 67, 26. [9] L. Luo and S. Roy, Analysis of search schemes in cognitive radio, in SECON, 27, pp. 647 654. [1] L. Kleinrock, Queueing Systems, Volume I:. John Wiley and Sons, Mar. 1975. V. CONCLUSIONS AN FUTURE WORK We have presented analysis of cognitive radio networks with imperfect sensing. Our approach employed a multidimensional Markov chain state diagram to obtain exact theoretical probabilities. We used state dependent transition rates provide more accurate analysis. As performance metrics we used the probability that a primary call is terminated by secondary users due to misdetections and the probability that a secondary call is successful. The main results of the paper are that the probability of collision to primary users increases with the probability of miss-detection and that the probability of successful secondary communications decreases with the primary traffic arrival rate. In the future work, it would be useful to investigate the case where secondaries will not always notice the arrival of the primary users and also the case where only the secondary will leave when collisions occur. Additionally, the state equations should be generalized to larger number of channels and the effects of timing offset between primary user arrival and secondary user sensing period could be studied. The simulation setup could be extended to take into account non-negligible acquisition times. 162