This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings Dynamic Spectrum Access with Imperfect Sensing in Open Spectrum Wireless Networks David Tung Chong Wong, Anh Tuan Hoang, Ying-Chang Liang and Francois Po Shin Chin Institute for Infocomm Research Agency for Science, Technology and Research (A*STAR) 21 Heng Mui Keng Terrace, Singapore 119613 {wongtc, athoang, ycliang, chinfrancois}@i2r.a-star.edu.sg Abstract - Analytical formulations of dynamic spectrum access (DSA) with perfect sensing (PS) and imperfect sensing (IS) for two radio systems are presented The DSA with PS model is solved explicitly using a two-dimensional Markov chain, respectively, while the DSA with IS model is solved numerically using a two-dimensional Markov chain Grades of service (GoSs) like system airtime and blocking probabilities are considered The performance of these GoS measures and the effect of the probability of false alarm and probability of misdetection for DSA with IS are evaluated Numerical results illustrate that the effect of DSA with IS as compared with DSA with PS Keywords – Dynamic Spectrum Access, Perfect Sensing, Imperfect Sensing INTRODUCTION Dynamic Spectrum Access (DSA) needs to allocate the available bandwidth in an intelligent manner [1] The usual ways of assigning different fixed bandwidths to different system are not getting the full benefits of having dynamically shared bandwidths for different systems only as and when they need them Dynamic spectrum access can help to minimize unused spectral bands or white spaces [1] Two ways of sharing radio spectrum between dissimilar radio systems has been considered in [1] Both queueing and without queueing schemes are considered These schemes assume perfect sensing The problem of opportunistic dynamic spectrum access in an ad hoc network is considered in [2] The foundation of our paper is based on [1] and not [2] In this paper, we classify the scheme without queueing in [1] as DSA with perfect sensing (PS) for two radio systems In this two radio systems, the frequency channels in the spectrum are dynamically shared by the two radio systems The number of basic frequency channels occupied by one radio system is assumed to be a multiple of that by the other system The arrival processes and departure processes of the two radio systems are all assumed to be Poissonian The problem considered here is at the spectrum level and not at the frequency channel level We also extend this case at the spectrum level to the case where there is imperfect sensing (IS) of the frequency channels of the spectrum are considered Imperfect sensing consists of false alarm and misdetection The former reflects the case where it is decided that there is no signal in the frequency spectrum when there is none and it is decided that there is signal in the frequency spectrum when the signal is there The latter reflects the case where it is decided that there is signal when there is none and it is decided that there is no signal when the signal is there Deciding that a signal is there when there is none causes false alarm, while deciding that a signal is not there when there is a signal causes misdetection Sensing of the usage of the frequency channels can be done using spectrum sensing mechanisms In [1], the DSA scheme with IS is formulated using a twodimensional Markov chain and the steady state probabilities are solved using a minimum-mean-squared error (MMSE) solution In our paper, the steady state probability of the Markov chain is explicitly expressed as a closed-form expression The DSA scheme with IP for a special case is formulated using a two-dimensional Markov chain and the flow balance equations are explicitly formulated which can be solved using linear algebraic equations solution The performance of grade of service (GoS) measures like radio system airtime and blocking probability and the effect of probability of false alarm and probability of misdetection in the frequency spectrum channels are evaluated Numerical results illustrate that the effect of DSA with IS as compared with DSA with PS Section describes the system models and the parameters used throughout this paper The analytical models are presented in Section The analytical models include DSA with PS and DSA with IS schemes Section presents the numerical results from analyses Some concluding remarks are found in Section SYSTEM MODEL We consider a frequency spectrum with physical capacity C frequency channels For easy reference we define the basic unit of capacity as a frequency channel A radio system may use a frequency bandwidth of one frequency channel, while another radio system may use a frequency bandwidth of a multiple number of frequency channels The following system parameters used throughout the paper Parameters C: total frequency channels in the spectrum Ck: nominal number of frequency channels for radio system k, k=1,2 nk: number of frequency bandwidths used by radio system k, k=1,2, in progress rk: number of basic channels required by each radio system k bandwidth Bk: blocking probability for radio system k, k=1,2 Pfak: probability of false alarm for radio system k, k=1,2 1525-3511/08/$25.00 ©2008 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings ij Pmd : probability of radio system i misdetect radio system j, i,j=1,2 λk: arrival rate of radio system k, k=1,2 µ k−1 : mean access duration of radio system k, k=1,2 11 22 Pmd and Pmd are assumed to be zero respectively, and A DSA with Perfect Sensing (PS) Two-dimensional finite state Markov chains are used to solve for DSA with PS for two radio systems as shown in Fig 1(a) Instead of using MMSE solution for the Markov chains, we express the steady state probability for the Markov chains as a closed-form expression The GoS measures like airtime and blocking probability can be derived from this steady state probability expression (1 − 2,0 µ1 (1 − Pfa ) 2λ1 µ1 µ1 λ2 0,0 0,1 µ2 (a) µ1 (1 − Pfa )3λ1 0,0 3,1 µ2 12 Pmd λ1 21 Pfa ) Pmd λ2 2,1 ( ) µ2 (1 − Pfa ) 1,0 1,0 3λ1 21 md 3µ1 (1 − Pfa1 )λ1 2,0 2λ1 (P ) λ 3,0 3µ1 λ1 12 Pmd 2λ1 21 Pmd λ2 3µ1 µ1 1,1 µ2 12 Pmd 3λ1 (1 − Pfa )3 λ2 µ1 0,1 µ2 (b) Let {P(n1,n2}} be the steady state probabilities of the Markov chain for two radio systems Solving the Markov chain, we get P( n1 , n ) C C C λi ∏ − 1 − ni + 1 i =1 ri ri ri ni ! µ i C − n2 λ2 P(n1, n2 ) C / r1 (C −n1r1 ) / r2 r B2 = ∑ ∑ C / r ( C −m r ) / r , 1 2C n1 = n2 = ∑ 1 (4) − λ m P m m ( , ) ∑ 2 2 m =0 r m = C if (C − n1r1 − n2r2 ) < r2 excluding state 0, , r B DSA with Imperfect Sensing (IS) Fig Dynamic Spectrum Access having C=3 (a) with Perfect Sensing, and (b) with Imperfect Sensing = by C if (C − n1r1 − n2r2 ) < r1 excluding state ,0 , r 1 ANALYTICAL MODELS 3,0 given C − n1 λ1P (n1 , n2 ) C / r1 (C − n1r1 ) / r2 r1 B1 = ∑ ∑ , C / r (C − m r ) / r 1 C n1 = n2 = (3) ∑ ∑ − m1 λ1P(m1, m2 ) m =0 m2 =0 r1 ni λi C / r1 (C −n1r1) / r2 C C C ∑ ∑ ∏ − 1 − n i + 1 n1 = i =1 ri ri n2 = ri ni ! µ i ,(1) ni where x denotes the greatest integer smaller than or equal to x The airtime of radio system k is given by C / r1 (C −n1r1 ) / r2 n k Airtime k = ∑ P (n1 , n ) , (2) ∑ C / rk n1 = n2 =0 As the PASTA property does not apply here [1], the blocking probability seen by an arriving radio system or arrival are, Next, let us consider two radio systems with IS for radio system We can model the frequency channel occupancy as a two-dimensional Markov chain as shown in Fig 1(b) with parameters in Section This Markov chain can be modeled and solved using the techniques in [3] Let n=(n1,n2) denote the state of the system with the number of frequency bandwidths used by radio system k (nk) in the two radio systems, and let r=(r1,r2) denote the number of basic frequency channels (rk) required for each frequency channel of radio system k Let λk(n) denote the arrival rate and µk(n) the departure rate in the spectrum system With C denoting the total number of nominal channels, the state space of the system, denoted by S, is given by S:={n:r⋅n≤C} When the spectrum system is in state n and a radio system k arrival arrives, an admission policy determines whether or not the call is admitted into the radio system Here, the admission policy is a DSA scheme with preemption for radio system We can specify the admission policy by mapping f:=(f1,f2) for new arrivals in the non-collision states, fC:=(fC1, fC2) for new arrivals in the collision states, where fk and fCk: S→{0,1}, and fk(n), and fCk(n) each takes on the value or if radio system k arrival is rejected or admitted, respectively, when the system state is n They are defined by the following equations: 1, if r ⋅ n + rk ≤ C (5) f k (n ) = 0, otherwise 1, if r ⋅ n + rk > C , r1 n1 ≤ C , r2 n ≤ C f Ck (n) = (6) 0, otherwise ,(7) for which S (f ) + S (f C ) = S Let P(n) denote the equilibrium probability that the system is in state n The global balance equations for the Markov process under the policies f and fC are This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings ∑ [λ (n) f k k =1 k C / r1 AirtimeCk = ∑ (n ) + λCk (n ) f Ck (n ) + µk (n )]P (n ) = ∑ λk (n − e k ) f k (n − e k )P (n − e k ) k =1 + ∑ µk (n + e k )P (n + e k ) k =1 where ek is a K-dimensional vector of all zeros except for a one in the kth place, C / r −n C − P fa1 1 − n1 λ1 , if f1 (n) = r λ1 (n) = , (9) 0, otherwise ) C − P fa C / r1− n1 − n λ , if f (n) = r , (10) 0, otherwise µ k (n ) = n k µ k , < r ⋅ n ≤ C , (11) ( ) and 12 C Pmd − n1 λ1 , if f C1 (n) = r 0, otherwise λC1 (n) = (12) ( ) 21 n1 C − n λ , if f C (n) = − P fa C / r1−n1 Pmd r λ C (n ) = 0, otherwise ( ) ∑ The blocking probability of radio system 1, B1, is given by + ∑ λCk (n − e k ) f Ck (n − e k )P (n − e k ) λ (n ) = nk P (n1 , n ) , (15) n1 = n2 = (C − n1r1 ) / r2 +1 C / rk (8) k =1 ( C / r2 (13) Equation (5) allows frequency bandwidths of the two radio systems to be admitted to C frequency channels without collisions and false alarm, while equation (6) allows allows frequency bandwidths of the two radio systems to be admitted to C frequency channels with collisions and misdetection Equation (11) allows these radio systems to be serviced with the total channel occupancy to be less than or equal to C frequency channels Equation (12) allows radio system frequency channels to be admitted under misdetection Equation (12) allows radio system frequency channels to be admitted under misdetection Equation (8) can be solved using LU decomposition [4] together with the condition for the total probability of all states to obtain P(n) LU decomposition is a common numerical technique for solving linear algebraic equations This gives exact solution However, efficient approximate computational algorithm [5] could be used for a large system state space Note that equation (8) without the λCk(n) terms represents the case of DSA with PS The good airtime for radio system k is given by C / r1 (C − n1r1 ) / r2 nk P (n1 , n ) , (14) AirtimeGk = ∑ ∑ n1 =0 n2 =0 C / rk while collision airtime for radio system k is given by (1 − P12 )3λ P(0,1) + (1 − P12 ) 2λ P (1,1) + (1 − P12 )λ P( 2,1) 1 md md md (16) + + + P λ P P λ P P λ P ( , ) ( , ) ( , ) fa1 fa1 fa1 , B1 = C / r1 C / r2 C ∑ ∑ − m1 λ1P (m1, m2 ) m1 =0 m2 = r1 while, the blocking probability of radio system 2, B2, is given by [ ] [ ] ] − (1 − P )3 λ P(0,0) + − (1 − P )2 P 21 λ P(1,0) fa fa md 21 21 + − (1 − Pfa )( Pmd ) λ2 P(2,0) + − ( Pmd ) λ2 P (3,0) (17) B2 = C / r1 C / r2 C m λ P m m − ( , ) ∑ ∑ 2 2 m1 = m2 = r2 [ ] [ NUMERICAL RESULTS The numerical results have been obtained by means of numerical analysis in Section We consider two radio systems with PS and IS The parameters used in the numerical examples are as follows: C=3, P fa1 = P fa = 0.05 and 12 21 Pmd = Pmd = 0.05 unless otherwise stated Figs and show the airtime and blocking probability, respectively, for DSA with PS and DSA with IS schemes with C=3 Radio system (RS) for the good state in DSA with IS has an airtime that is slightly higher than that in DSA with PS, while RS for the good states in DSA with IS has lower airtime than that of DSA with PS as shown in Fig The airtime wasted due to collisions for RS and RS are low as shown in Fig Figs and show the airtime and blocking probability, respectively, for DSA with PS and DSA with IS schemes with C=3 and varying the probability of false alarm at λ1 / µ1 = λ / µ = 0.2 In general, the airtimes for RSs and for DSA with IS decrease under heavy probability of false alarm At a probability of one for the false alarm, both RSs and two for DSA with IS will always not transmit Thus their airtimes drop to zero The airtimes for DSA with PS are shown as horizontal lines just for illustration, though they are strictly speaking not functions of the probability of false alarm On the other hand, the airtime for RS in DSA with IS under probability of false alarm between 0.05 and 0.35 can be slightly larger than that for DSA with PS The reason for this is that both RSs and RS will miss the chance to transmit and RS occupies three frequency channels as compared to only one frequency channel for RS Thus RS for DSA with IS will have higher chances of transmissions at the expense of RS 2, resulting in better airtime time for RS within this range of probability of false alarm This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings CONCLUDING REMARKS Analytical formulations of dynamic spectrum access with perfect sensing and imperfect sensing are presented in Section Markov chains are used to model these schemes Explicit closed-form steady state probability expressions are presented for the DSA with perfect sensing, and global balance equations are formulated for the DSA with imperfect sensing Numerical results demonstrated the effect of DSA with imperfect sensing over DSA with perfect sensing in terms of airtime and blocking probability under the influence of probability of false alarm and probability of misdetection Future work includes joint optimization of spectrum sensing and dynamic spectrum access REFERENCES [1] Y Xing, R Chandramouli, S Mangold and S Shankar N., “Dynamic Spectrum Access in Open Spectrum Wireless Networks” IEEE Journal on Selected Areas in Communications, vol 24, no 3, pp 626-637, March 2006 [2] Q Zhao, L Tong and A Swami, “Decentralized Cognitive MAC for Dynamic Spectrum Access,” IEEE SynSpan 2005, pp 224-232, 2005 [3] K.W Ross, “Multiservice Loss Models for Broadband Telecommunication Networks,” Springer, 1995 [4] W.H Press, S.A Teukolsky, W.T Vettering and B.P Flannery, “Numerical Recipes in C,” Cambridge University Press, 2nd Edition, 2002 [5] S.C Borst and D Mitra, “Virtual Partitioning for Robust Resource Sharing: Computational Techniques for Heterogeneous Traffic,” IEEE Journal on Selected Areas in Communications, vol 16, no 5, pp 668678, June 1998 0.5 RS 1- PS 0.45 RS 1- IS Good St at e 0.4 RS 1- IS Colli St at es RS - PS Airtime (share) 0.35 0.3 RS - IS Good St at es 0.25 RS - IS Colli St at es 0.2 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 Offered traffic per radio system (Erlang) Fig Airtime with C = RS 1- PS 0.9 RS 1- IS 0.8 Blocking Probability From Fig 5, the blocking probabilities for RSs and for DSA with IS increase toward one under heavy probability of false alarm However, the blocking probability for RS under probability of false alarm between 0.05 and 0.35 can be slightly lower than that for DSA with PS The reasons for these observations are the same in the previous paragraph Similar trends can be seen for the airtime and blocking probability, respectively, for DSA with PS and DSA with IS schemes with C=3 and varying the probability of false alarm at λ1 / µ1 = λ / µ = 0.8 in Figs and Figs and show the airtime and blocking probability, respectively, for DSA with PS and DSA with IS schemes with C=3 and varying the probability of misdetection at λ1 / µ1 = λ / µ = 0.2 In general, the airtimes for RSs and decrease at a more gentle rate as the probability of misdetection increases From Fig 9, the blocking probabilities for RSs and decrease as the probability of misdetection increases The blocking probability for RSs and can be better than that of DSA with PS under some range of probability of misdetection The reason for this is that as the probability of misdetection increases towards one, RSs and more likely to jump into the frequency channels Thus the blocking probabilities for them decrease as the probability of misdetection increases Similar trends can be seen for the airtime and blocking probability, respectively, for DSA with PS and DSA with IS schemes with C=3 and varying the probability of misdetection at λ1 / µ1 = λ / µ = 0.8 in Figs 10 and 11 RS - PS 0.7 RS - IS 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 Offered traffic per radio system (Erlang) Fig Blocking probability with C = This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings RS - IS Colli St at es RS - PS 0.14 Airtime (share) RS 1- PS 0.9 RS - IS Good St at e P md 12=P md 21 =0.05 0.16 RS - PS λ /µ1 =λ /µ2 =0.2 0.12 RS - IS Good St at es 0.1 RS - IS Colli St at es 0.08 0.06 RS 1- IS 0.8 Blocking Probability 0.2 0.18 RS - PS 0.7 λ /µ1 =λ /µ2 =0.8 0.6 0.5 0.4 0.3 0.04 0.2 0.02 0.1 0 0.2 0.4 0.6 0.8 Fig Airtime varying probability of false alarm with traffic load of 0.2 and C = 0.2 RS 1- PS 0.9 0.8 RS - IS P md 12=P md 21=0.05 0.4 0.3 RS 1- PS RS 1- IS Good St at e RS 1- IS Colli St at es RS - PS 0.14 Airtime (share) λ /µ1 =λ /µ2 =0.2 0.5 0.8 P fa1 =P fa2 =0.05 0.16 RS - PS 0.6 0.6 λ /µ1 =λ /µ2 =0.2 0.18 RS 1- IS 0.7 0.4 Fig Blocking probability varying probability of false alarm with traffic load of 0.8 and C = Blocking Probability 0.2 Prob of false alarm , Pfa1 (=Pfa2) Prob of false alarm , Pfa1 (=Pfa2) 0.12 RS - IS Good St at es 0.1 RS - IS Colli St at es 0.08 0.06 0.2 0.04 0.1 0.02 0 0.2 0.4 0.6 0.8 0.2 0.4 0.6 Prob of m isdetection, Prob of false alarm, Pfa1 (=Pfa2) Fig Blocking probability varying probability of false alarm with traffic load of 0.2 and C = 0.8 Pmd12 (=Pmd ) 21 Fig Airtime varying probability of misdetection with traffic load of 0.2 and C = 0.4 RS 1- PS RS - PS 0.9 RS - IS Good St at e 0.3 RS - IS Colli St at es RS - PS λ /µ1=λ /µ2=0.8 0.25 RS - IS Good St at es P md 12=P md 21=0.05 0.2 RS - IS Colli St at es 0.15 0.1 0.05 RS 1- IS 0.8 Blocking Probability 0.35 Airtime (share) RS - IS P md 12=P md 21=0.05 RS - PS 0.7 λ /µ1 =λ 2/µ2 =0.2 0.6 RS - IS P fa1 =P fa2 =0.05 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 0.8 Prob of false alarm, Pfa1 (=Pfa2) Fig Airtime varying probability of false alarm with traffic load of 0.8 and C = 0.2 0.4 0.6 0.8 Prob of m isdetection, Pmd12 (=Pmd21) Fig Blocking probability varying probability of misdetection with traffic load of 0.2 and C = This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings 0.4 RS 1- PS RS 1- IS Good St at e 0.35 Airtime (share) 0.3 RS 1- IS Colli St at es RS - PS λ /µ1 =λ 2/µ2 =0.8 0.25 RS - IS Good St at es P fa1 =P fa2 =0.05 0.2 RS - IS Colli St at es 0.15 0.1 0.05 0 0.2 0.4 0.6 0.8 Prob of m isdetection, Pmd12 (=Pmd21) Fig 10 Airtime varying probability of misdetection with traffic load of 0.8 and C = RS 1- PS 0.9 RS 1- IS Blocking Probability 0.8 0.7 λ /µ1 =λ /µ2 =0.8 RS - PS P fa1 =P fa2 =0.05 RS - IS 0.6 0.5 0.4 0.3 0.2 0.1 0 0.2 0.4 0.6 Prob of m isdetection, 0.8 Pmd12 21 (=Pmd ) Fig 11 Blocking probability varying probability of misdetection with traffic load of 0.8 and C = ... system (RS) for the good state in DSA with IS has an airtime that is slightly higher than that in DSA with PS, while RS for the good states in DSA with IS has lower airtime than that of DSA with. .. proceedings RS - IS Colli St at es RS - PS 0.14 Airtime (share) RS 1- PS 0.9 RS - IS Good St at e P md 12=P md 21 =0.05 0.16 RS - PS λ /µ1 =λ /µ2 =0.2 0.12 RS - IS Good St at es 0.1 RS - IS Colli St... respectively, and A DSA with Perfect Sensing (PS) Two-dimensional finite state Markov chains are used to solve for DSA with PS for two radio systems as shown in Fig 1(a) Instead of using MMSE solution for