On the Characteristics of Spectrum-Agile Communication Networks

Transcription

1 1 On the Charateristis of Spetrum-Agile Communiation Networks Xin Liu Wei Wang Department of Computer Siene University of California, Davis, CA Abstrat Preliminary studies as well as general observations indiate the presene of a signifiant amount of white spae in radio spetrum, varying on frequeny, time, and geographi loations. Thus, it is likely that spetrum aess, instead of true spetrum sarity, is the limiting fator of potential growth of wireless servies. Enabled by regulatory hanges and radio tehnologies advanes, opportunisti usage of the white spae has the potential to signifiantly mitigate the spetrum sarity. In this paper, we study the harateristis of opportunisti spetrum availability and its exploration. We present two new metris to apture unique harateristis in networks with spetrum agility. We also study the spatial and temporal properties of opportunisti spetrum availability. Last, the existing resoure sharing problem formulation is modified to inorporate the effet of opportunisti spetrum availability. I. INTRODUCTION Spetrum is among the most heavily regulated and expensive nature resoure around the world. However, although almost all spetrum suitable for wireless ommuniations have been alloated, preliminary studies and general observations indiate that muh of the radio spetrum is not in use for a signifiant amount of time, and at large numbers of loations. For instane, experiments onduted by Shared Spetrum Company indiates 62% perent of white spae (unused spae) below 3GHz band even in the most rowded area near downtown Washington, DC, where both governmental and ommerial spetrum usage are intensive [10]. In the experiment, a band is ounted as white spae if it is wider than 1MHz and remains unoupied for 10 minutes or longer. Furthermore, spetrum usages vary dramatially in time, geographi loations, and frequeny. A lot of preious spetrum (below 5GHz), perfet for wireless ommuniations that worth billions of dollars, sit there silently. The large portion of white spae indiates that opportunisti or dynami spetrum usage may signifiantly mitigate the spetrum sarity. In this paper, we fous on the opportunisti spetrum utilization by users other than the primary liensed ones on a non-interfering or leasing basis. Suh usage is being enabled by regulatory poliy initiatives and radio tehnology advanes. First, both the Federal Communiations Commission (FCC) and the federal government have made important initiatives towards more flexible and dynami spetrum usage, e.g., [4], [3], [2], [5], [1]. Furthermore, opportunisti spetrum sharing is enabled by software-defined radio or ognitive radio tehnologies, where suh tehnology advanes provide the apability for a spetrum-agile radio devie to sense and operate on a wide range of frequenies using appropriate ommuniation mehanisms, and thus enable dynami and more intense spetrum reuse in spae, time, and frequeny dimensions. We fous on the study of the seondary users who observe hannels available dynamially and explore them opportunistially. We refer seondary users as spetrum users who are not owners of the spetrum and operate based on agreements/etiquettes imposed by the primary users of the spetrum. Beause spetrum availability of a seondary user is determined by primary users, it is alled opportunisti spetrum availability. We study the harateristis of opportunisti spetrum availability and its potential benefits. (Note that the seondary users may have their own liensed/alloated bandwidth where they are primary users, whih is not the onern of this paper.) The objetive of the paper is to better understand systems with opportunisti spetrum availability. We have the following ontributions. We introdue two new metris, effetive nonopportunisti bandwidth and spae-bandwidth produt, to apture the inherent properties of white spae and the benefits of opportunisti spetrum utilization. Spetrum availability observed by a seondary user is time-varying and loation-dependent. We analyze suh spatial and temporal properties. We modify the existing resoure sharing problem formulation to inorporate opportunisti spetrum availabilities. The paper is organized as follows: we first disuss the system model in Setion II. We then study two new metris in Setion III. The temporal and spatial orrelation of hannel availability for seondary users are analyzed in Setion IV. Last, we propose modifiations to the existing resoure alloation problem formulation so that opportunisti spetrum availability is aptured. (This part is omitted in the extended abstrat due to spae limitation.) We onlude the paper in Setion V. II. SYSTEM MODEL We onsider two types of users. Primary users are the rightful owners and have strit priority on spetrum aess. Seondary users are spetrum-agile devies that an sense the environment and adapt to appropriate frequeny, power, and transmission shemes. They an opportunistially aess

2 2 unused spetrum vaated by idle primaries. Primary users are onventional legay users whose hardware and protools should not be required to retrofit seondary user aess needs. Therefore, hannel availability of seondary users is inherently determined by the ativities and properties of primary users. We fous on the harateristis of suh opportunisti hannel availability. We assume that when a primary user is using the band, seondary users are not allowed to o-exist in viinity. We assume that information on hannel availabilities is given to seondary users. Although it is a great hallenge to obtain suh information, it is not the fous of the paper and there are various shemes proposed. We use footprint to abstrat the spae oupany of a primary user in a hannel. Seondary users outside the footprint of primary users are allowed to transmit on the hannel given the maximum transmission power onstraint. This is motivated by the idea that eah ommuniation onsumes spae, e.g., [7] and many others. It is also justified by the oexistene study of TV stations and WLAN devies. It has been shown that one or more WLAN devies an operate outside a ertain range from the TV station without violating the D/U (desired/undesired signal strength) requirement of legitimate TV reeivers that are within the servie ontour of the TV station [8], [9]. Similar analysis an be applied to other types of primary users. Therefore, the footprint of a primary user is modelled by a disk of radius r. The hannel used by an ative primary user annot be utilized by seondary users in its footprint. On the hand, if it is outside the footprints of all primary users of a hannel, the seondary user an operate on the hannel. Intuitively, footprint an be onsidered as a model for the interferene/ommuniation range of a user. Transmission power and its effet are inorporated in the size of a footprint. For instane, in the ase of TV/WLAN sharing, the (maximum) transmission power of a WLAN devie affets both the footprint of itself and that of a TV station [8], [9]. Footprints of different users may or may not overlap depending on appliation senarios. For instane, two TV stations using the same hannel annot have overlapping servie ontours. Otherwise, TV reeivers in the overlapping area may not reeive desired TV signal quality. On the other hand, in many appliation senarios, espeially mobility and ubiquitous overage are required, overlapping is a neessity. For instane, the overage areas of different base stations have to overlap to support user mobility and good overage. Furthermore, multiple WLANs an have overlapping footprints and share a hannel using CSMA/CA. On the other hand, they an also avoid overlapping by hoosing different hannels providing the number of available hannels is large enough. We use the following model for hannel availability that is observed by seondary users at a snapshot. We first abstrat the networks of seondary users into a graph, where vertexes represent seondary users. If two wireless users are within the ommuniation/interferene range of eah other, then the two nodes are onneted by an edge. If two vertexes are onneted by an edge in the graph, we assume that these two nodes annot use the same spetrum simultaneously. This abstration is widely used in the literature, suh as [11]. In addition, we assoiate with eah vertex a set, whih represents the available Fig. 1. (A, B) I (hn C) (A, B) III (hn C) II (hn B) 5 (A, C) (A) 3 6 (A, B, C) (B, C) IV (hn A) A snapshot of hannel availability at seondary users. hannels of the user. As mentioned earlier, if a seondary user is outside the footprint of all (ative) primary users of a hannel, then the hannel is available at the seondary user. We show an example in Figure 1. The six vertexes 1-6 represent six different seondary users. There are three frequeny bands, namely A, B, and C, whih are ommuniation hannels that are opportunistially available to the seondary users. We assume that all hannels have the same bandwidth, whih an be generalized easily. In addition, four (ative) primary users I-IV are present, using bands C, B, C, A, respetively. The footprint of a primary user is represented by the dashed irle entered at it. For instane, Node 5 is within the footprints of primary users II and III, who use hannels B and C, respetively. Therefore, hannels B and C are not available at Node 5 while hannel A is. Beause of loation differenes, eah node may have aess to a different set of hannels. In our figure, the available hannels are (A,B) at vertex 1, (A,C) at vertex 2, et. The resoure alloation problem is how they should share these hannels. Note that Figure 1 shows a snapshot of the network. At different time instane, due to the traffi variation of primary users, hannel availabilities at seondary users vary. The paper fouses on harateristis of spetrum agile networks. In other words, we study the inherent properties instead of the design/ontrol of the system. To elaborate, primary users are legay users and we annot ontrol their loations, traffi patterns, ommuniations, and other parameters. Furthermore, seondary users, e.g., WLAN devies, have their own properties suh as maximum transmission power, loation, and density. We onsider these properties given as well. III. METRICS FOR OPPORTUNISTIC SPECTRUM UTILIZATION The basi onept of opportunisti spetrum utilization is to utilize spetrum unutilized at time, frequeny, and spae. Beause suh spetrum availability is dynami and different from traditional spetrum availability defined by ommandand-ontrol manner, we first introdue two new metris for its haraterization.

3 3 A. Effetive Non-opportunisti Bandwidth Consider the following question: suppose that there are 62% of white spae under 3GHz. If we an fully utilize suh white spae, is it equivalent to gaining an additional spetrum band of = 1.86GHz? The answer is that it depends. To address this question, we introdue the notion of effetive non-opportunisti bandwidth (ENOB). It is defined as the equivalent non-opportunisti bandwidth required to ahieve the same performane as in the ase of opportunisti spetrum availability. A non-opportunisti hannel is referred to a hannel that is always available to all (seondary) users in onsideration, whih is how spetrum is alloated in the traditional ommand-and-ontrol manner. This metri is designed to study the impat of opportunisti spetrum availability. We elaborate the idea with a naive example. Consider a simple network with only two nodes. They annot use the same hannel simultaneously due to interferene. Consider a hannel with bandwidth W that is opportunistially available at these two nodes. The hannel is available at eah node with probability p independently. Suppose that a user obtains one unit of throughput per unit of spetrum. Then the total throughput gained by the two users is: W (p p(1 p) 1 + (1 p) 2 0) = W p(2 p). The first term represents the ase where the hannel is available to both nodes and only one of them an use it due to interferene. The seond term is for the ase where only one of the users observes the spetrum availability and uses it. The last term is the ase where the spetrum is available to neither of the two users. To ahieve the same total throughput, B e = W p(2 p) unit of non-opportunisti bandwidth is required. To elaborate, when a spetrum of B e is available to both users in the traditional way (i.e., the spetrum is always available to both users), the throughput is W p(2 p) beause only one of the users an use it at any given time. Thus, we laim that B e = W p(2 p) as ENOB in this simple example. Suppose W = 3GHz and p = 62%. In this example, we see the impat of 62% white spae under 3G is equivalent to W p(2 p) = 2.76GHz of spetrum in the traditional way. The orrelation on spetrum availabilities at seondary users also plays an important role. For instane, if two seondary users are very lose and observe the same spetrum opportunity, then B e = W p instead of W p(2 p) as in the independent ase in this example. Note that spetrum is not being reated by seondary users. Instead, they simply explore the spetrum holes generated by primary users. Inherent harateristis of primary users, suh as ommuniation range, transmission power, traffi pattern, node density, and topology, determine the spetrum opportunities of seondary users. The purpose of ENOB is to quantify the potential of suh spetrum opportunities for a given set of seondary users. The intuition is similar to that of the effetive bandwidth used to apture the statisti multiplexing gain. However, what is being apture here is the degree of spatial reuse and statistial multiplexing between primary and seondary users. In general, beause of the harateristis of primaries mentioned Fig N A Chain Topology above, users observe different hannel availabilities and yield utilization gain, whih is quantified by ENOB. 1) ENOB of a Chain Topology: We use a hain topology to further illustrate the idea of ENOB. We onsider a hain topology of length N. We assume that only the two nearest nodes interfere with eah other and thus annot use the same spetrum simultaneously, as shown in Figure 2. We alulate its ENOB as follows. Consider a spetrum band of width W. Let p 0 be the probability that a node observes the hannel available. Let p be the probability that node A observes the hannel available given that its neighbor B observes the hannel available. Note that p is indeed a funtion of distane between A and B, as disussed in Setion IV-A. Here we assume all nodes on the hain are evenly spaed and thus omit the distane in the notation. Let q be the probability that A observes the hannel available given that B does not, where q = p 0 1 p 1 p 0. As shown in the Appendix (in the full paper), the ENOB of the hain topology is where B e (N) = W U(N) = 2(1 p 0)q 1 p 2 ( 1 p 2 +p 0 1 p 2 Note that if p = 1, then N+1 2 U(N) (N + 1)/2 N2 1 p2 N 2 1 p 2 p 2 ) 1 p N (1 p 0 )q 1 p 2 1 {N is even} B e (N) = W p 0. Figure 3 illustrates ENOB for different values of p, where p 0 is 0.1 and 0.7, respetively. In the figures, the x-axis is the value of p, where p [p 0, 1]. The y-axis is the ENOB normalized over W p 0, where is W p 0 is the ENOB when p = 1. It is lear that the larger the value of p, i.e., the higher the orrelation between neighboring nodes, the lower the value of ENOB. In addition, the normalized ENOB is higher for smaller values of p 0. The trend is similar for different values of N. The value of ENOB depends on many fators, suh as the probability of the spetrum availability, the orrelation among different users, and the topology of the network. In general, the higher the orrelation of spetrum availability among users, the lower the value of ENOB. The intuition is that a system with high orrelation behaves similarly to a system with the traditional spetrum availability, and thus has low statisti multiplexing gain. On the other hand, the more omplex the

4 4 Eqv. Bw. Fig p 0 = p = N=2 N=9 1.1 N=2 N= p Normalized equivalent bandwidth of a hain topology. topology, the higher the value of ENOB. For instane, onsider a lique of size N. Assume that eah node observes hannel available with probability p independently. Then the effetive bandwidth of the hannel is B e = W ( 1 (1 p) ) N. For very small p, B e NW p, whih shows a gain of roughly N-fold. The intuition is that the multiplexing gain is higher for a dense network, espeially when p is small. In summary, the metri, ENOB, is introdued to apture the impat of the opportunisti spetrum availability. In general, ENBO is diffiult to alulate beause the network topologies are more ompliated and the availability of hannels at different nodes are orrelated in a omplex manner. Thus, we may need to resort to numerial evaluations. B. Spae-bandwidth Utilization Consider a wireless devie. It is lear that a wireless devie onsumes a ertain amount of spetrum for its ommuniation, whih is an irreproduible natural resoure. Equivalently important is that a ertain amount of spae is also oupied by the ommuniation [7]. Wireless devies, espeially primary users and seondary users, are different in ommuniation power and therefore oupy different amount of spae for ommuniations. For instane, the servie range of a TV station is about 100km. In omparison, the transmission range of an unliensed wireless LAN (WLAN) devie is about 100m. Therefore, 6MHz of bandwidth (the bandwidth of an analog TV hannel) oupied by a TV station has a different ommuniation result than the same bandwidth of a WLAN devie. Thus, to quantify suh differenes in utilization, we use a simple metri, spae-bandwidth produt, whih is defined as the produt of the spae and bandwidth utilization of a user in ommuniation. This definition is analogous to the transport apaity defined in [7] to study apaity saling laws in large wireless networks. Another analogy is the man-mile metri used by airline industry. We note that open spetrum enables spetrum reuse by seondary users in both spae and frequeny dimensions. Thus, this metri is important to understand the effet of opportunisti spetrum utilization. In this setion, we use footprint to abstrat the spae utilization of both primary and seondary users. We model footprint of a wireless devie as a disk of radius r. The radii of primary and seondary users an be dramatially different. For instane, it has been proposed that unliensed users an transmit in the unused TV band [6]. Another potential is for unliensed wireless devie to share radar bands. In both appliations, a primary user has a dramatially larger transmission power and thus larger footprint than that of a seondary user. The spae-bandwidth produt has a lear physial meaning in broadast appliations. For instane, a TV station with overage area of 100 square miles has the same spaebandwidth produt as 10 smaller TV station, eah with 10 square miles of overage. Assume reeivers are uniformly distributed in the area. Then they serve the same number of users. In other words, the spae-bandwidth produt is proportional to the number of users served. In the ase of peer-to-peer ommuniation, spae-bandwidth produt is an indiation of overage. It is also an indiret impliation of the tradeoff between the ost of an AP (and its installation) and the data rate for eah user. A spetrum-agile system exploits holes in both spatial and frequeny dimensions. However, there ould be spetrum intervals that annot be (easily) utilized. For instane, if a seondary user requires a spetrum band of 5MHz, then an isolated free TV band of 6MHz may not be fully utilized. We note that OFMD tehnology may be able to utilize sattered spetra. However, stringent requirements on band filters may limit its pratial implementation. Spae-bandwidth produt quantify suh holes in utilization in both spetrum and spae dimensions. We next illustrate its usage in a speial ase that fouses on the spatial dimension. 1) An Illustration Example: Consider one spetrum band that both primary and seondary users an fully utilize. Assume primary and seondary users have dramatially different footprint sizes. Suh differene gives us another type of spatial gain. An analogy is to put big stones in a jar. After the jar is filled with big stones, one an still fill a large amount of sand into the jar by taking advantage of the spae among big stones. In this analogy, it is unfair to ompare a big stone with a grind of sand unless the spae/volume is taken into aount. Similarly, we use spae-bandwidth produt to illustrate the differene in footprint size. We first onsider the ase where footprints are not allowed to overlap. We assume that primary users are stationary. They deide their loations in the following manner. Consider a large area with no users. The first primary user randomly selets its loation. A following primary user also hooses its loation randomly, but the hoie is limited to the area where its footprint does not overlap with those of the previous users. In the end, if no additional primary users an be assigned, we onsider the system full and an alulate the maximum spae utilization of the primary users. The proess mimis the loation determination of primaries, say TV stations. The loation of a TV station is in general determined by population density and geographi areas. For instane, it is more likely

5 5 to loate a TV station in a big ity than in a deserted area just to fully utilize the spae. Subsequent TV stations have to respet the footprints of prior alloations. After the primary users pik their loations, seondary users subsequently loate themselves without overlapping footprints with previous users, both primary and seondary. An example of the senario is that an AP avoids hannels used by primary users and other WLANs operating in viinity when it selets a hannel. We onsider an area large enough so that boundary effets are ignored to simplify analysis. We onsider the full apaity ase. In other words, primary users randomly pak in as muh as possible followed by seondary users. It is interesting to see that the footprints of primary users an only oupy about µ portion of the spae. Numerial evaluation indiates that µ If the radii of seondary users are muh smaller than that of a primary user, seondary users an oupy µ portion of the unoupied spae of primaries. In other words, seondary users an oupy µ (1 µ) 0.25 portion of the total spae. This is a total spae-bandwidth utilization of 70%. Consider a heterogenous network with TV stations, WLAN devies, pio sensors, then the overall utilization is 0.83 instead of 0.45 with only primary users. We have the following observations. First, this type of paking results in no ost at primary users. In other words, with or without seondary users, primary users ahieve the same spatial utilization. Furthermore, the seondary users paked in spae holes are able to use the same hannel regardless whether primary users are ative or not. Seondly, any one type of users an only ahieve a utilization of µ, even if it is pio sensors that oupy the entire spae. In other words, the gain omes from the heterogeneity in spae onsumption of different types of devies. Note that the sequene is also important larger users need to be alloated first to ahieve better utilization. In a spetrumagile ommuniation networks, it is likely that seondary users are smaller than primaries. Therefore, spetrum-agile networks an ahieve better utilization of spae, whih an be aptured by the spae-bandwidth produt. If the dynamis in spetrum utilization of primary users is also taken into aount, then the gain is even higher. Assume eah primary user is ative with probability p 0 on average. Consider the ase where there are enough seondary users that demand spetrum. Then the spae-bandwidth utilization of a primary-user-only system is only p 0 µ on average while one tier of seondary users an improve the utilization to p 0 µ + µ(1 p 0 µ). In the above, we onsider the ase where the overlap of footprints is not allowed. On the other hand, if suh overlap is allowed, then the utilization depends on the density of primary and seondary users. Let λ p and λ s, and R p and R s be the density and radius of primary and seondary users, respetively. The utilization of primary users is 1 exp( λ p πrp). 2 One tier of seondary users improves the utilization by exp( λ p πrp) 2 (1 exp( λ s πrs)). 2 In summary, the main purpose of spae-bandwidth produt is to quantify the gain of opportunisti spetrum utilization, in both the spae and frequeny dimensions. The spaebandwidth produt is an indiator of resoure onsumed and thus an be diretly linked eonomi values of spetra. We note that there are limitations in the definition of spaebandwidth produt. For instane, the definition does not diretly apply to spreading spetrum ommuniation systems. Another limitation is that the produt is not a diret indiation of throughput or transport apaity in peer-to-peer ommuniation systems. IV. SPATIAL AND TEMPORAL PROPERTIES The hannel availability observed by a seondary user is time-varying and loation-dependent. In this setion, we quantify suh spatial and temporal properties. A. Spatial Correlation It is intuitively lear that when two seondary users are lose to eah other, they are more likely to observe the same hannel availability. When they are far away, the observations are more likely to be independent. We quantify this spatial orrelation next. Consider a hannel. Primary users of the hannel loate in the area following a Poisson distribution with density λ. This is a typial assumption used to study random wireless networks, e.g., in [7]. The footprint of eah primary user is a disk with radius R p. In this model, overlapping of footprints is allowed. The loation of a seondary user is independent of that of primaries. Then a seondary user, A, observes the hannel available when there is no ative primary users within radius R p. Denote P (A) as the probability that A observes the spetrum availability. We have P (A) = e λπr2 p. Consider another seondary user, B, at distane d. We are interested in the probability that B also observes the availability given that A observes spetrum availability, denoted as P (B A). We have { exp( λ(πr 2 P (B A) = p a 0 )), 0 d 2R p P (B), d 2R p, ( ) d a 0 = 2Rp 2 os 1 d Rp 2R 2 (d/2) 2. p When d 2R p, we have P (B A) = P (B); i.e., two nodes are so far away that they observe independent hannel availabilities. We use a relative radio between P (B A) and P (B) as an indiation of the spatial orrelation. If B observes availability independent of A, then the ratio is 1. The larger the value, the higher the spatial orrelation. Figure 4 illustrate this orrelation. In the figure, the x-axis is the normalized distane, i.e., d/r p. The y-axis is the radio between P (B A) and P (B) to show the impat of A s availability at B. In the figure, R p = 1, and the value of λ is indiated on the orresponding urve. It is lear from the figure that the loser the two seondary nodes, the more likely that they observe the same hannel availability; i.e., the higher the spatial orrelation. For larger values of λ, the higher the normalized orrelation. In fat, when the value of λ is large, both P (B A) and P (B)

6 6 P(B A)/P(B) λ=1 λ=0.7 the available period of a seondary user is also exponential. Let 1/µ I be the mean of the idle period of a primary user. If footprints of primary users do not overlap, then the available period of a seondary user within a footprint is exponentially distributed with mean 1/µ I. If footprints of primary users an overlap and primary users loate with density λ p, then the available period of a seondary user is exponentially distributed with mean 1/µ a where { 0 w. p. 1 e λ p πr 2 p Fig λ=0.5 λ= Normalized distane Spatial Correlation between seondary users are small; i.e., the lower the probability of a seondary user observing an available hannel. However, the impat of A s availability is relatively large; i.e., the urve is steeper for a larger value of λ. In summary, distane has a signifiant impat on the spatial orrelation on hannel availability between seondary users. We are extending the study to the ase where the footprint of primary users do not overlap and the ase where the footprint of a primary user is irregular. Numerial results indiate that similar trends exist. B. Temporal Correlation The temporal properties of seondary users inlude the distributions of hannel availability and unavailability, and their predition. Clearly, suh temporal harateristis depend on the properties of primary users. Consider a partiular hannel. We note that a seondary user an aess a hannel if it is not within the footprints of any ative primary users of the hannel. Based on the disussion in Setion III-B, we know that there are seondary users that are outside the footprints of any primary users simply by exploring the spae holes. For suh users, this spetrum band is always available. However, there are also seondary users that are within the footprints of one or more primary users. They an aess the hannel only when the orresponding primaries are not ative. We study these users next. Temporal properties of seondaries depend on the ativity pattern of primaries. For instane, if the ativities of primary users are periodi, suh as TV stations, then the hannel availability of a seondary user within footprints is also periodi and is perfetly preditable. Another speial ase we illustrate next is when the arrival proesses of primary users are independent Poisson proesses. Note that we do not assume the servie time distribution or busy period of a primary user. It has been shown that the servie time distribution is well modelled as exponential for voie traffi, but not for data traffi. However, the idle period of primary users is the hannel available period of seondary users. Beause the arrival proess of a primary user an be modelled as Poisson, the idle period of primary users is exponential. Thus, µ a = kµ I w. p. e λpπr2 p (λ pπr 2 p )k k! In the above equation, the first one represents the ase where the user is outside all footprints of primaries and thus always observes the hannel available. The seond one is the ase the user is within the footprint of k primary users. A hallenging issue is to understand the distribution of un-available period for seondary users. Beause the busy period of a primary user is in general not exponential, the distribution analysis is the same as the busy period analysis of an M/G/n queue. V. CONCLUSION In this paper, we haraterize opportunisti spetrum utilization in spetrum-agile ommuniation networks. The harateristis and potentials of spetrum holes in both frequeny and spae are studied. Spatial and temporal properties are quantified. Suh harateristis are onsidered as inherent. The paper ontributes to better understandings of the properties of spetrum agility. VI. APPENDIX A. Calulation of Spatial Correlation We alulate P (B A) given the distane between B and A is d using Figure 5. The event that A observes the hannel available implies that there is no ative primary users in the left irle. Therefore, the overlapping area has no ative primaries and B also observes the hannel available if and only if the rest of the irle around B has no primary users. Let the size of the overlapping area be a 0. We have P (B A) = exp ( λ p (πr 2 p a 0 ) ). We alulate a 0 next. We have ( ) R α = os 1. d/2 The size of the triangle, b 0, is R sin(α) d/2. Therefore, a 0 = πr 2 2α 2π 2b 0 ( ) d = 2Rp 2 os 1 d Rp 2R 2 (d/2) 2. p.

7 7 Fig. 5. Two nodes with distane d. α B. Calulation of ENOB for the hain Topology R d Let X i be the event that node i observes the hannel available, Y i be the event that node i uses the hannel, and x be that event x does not happen. Let p = P (X a X b ), where p [p 0, 1]. Beause p 0 = P (X a ) = P (X a X b )P (X b ) + P (X a X b )P ( X b ), we have q = P (X a X b ) = p 0 1 p 1 p 0. Let p u i be the probability that node i uses the hannel in onsideration. We find p u i iteratively. We assign hannel from left to right. A node is assigned the hannel if the hannel is available and the neighbor on the left is not assigned the hannel. This sheme an be shown to yield the maximum utilization. We have Furthermore, p u 1 = p 0, (1) p u 2 = P (Y 2 ) = P (Ȳ1)P (X 2 Ȳ1) (2) = (1 p u 1)P (X 2 X 1 ) (3) = (1 p 0 )q. (4) p u i = P (Ȳi 1)P (X i Ȳi 1) For even i, we have = P (Ȳi 1 X i 1 )P (X i Ȳi 1 X i 1 ) +P (Ȳi 1X i 1 )P (X i Ȳi 1X i 1 ) = P ( X i 1 )P (X i X i 1 ) +P (Y i 2 )P (X i 1 X i 2 )P (X i X i 1 ) = (1 p 0 )q + p u i 2p 2 = (1 p 0 )q + (p u i 4p 2 + (1 p 0 )q )p 2 = (1 p 0 )q + (1 p 0 )q p 2 + p u i 4p 4. p u i = (1 p 0 )q (1 + p 2 + p p i 2 i 2 = (1 p 0 )q p 2k k=0 1 p i = (1 p 0 )q 1 p 2. ) For odd i, we have p u i = (1 p 0 )q (1 + p 2 + p p i 3 ) + p u 1p i 1 i 3 = (1 p 0 )q p 2k + p 0 p i 1 k=0 1 p i 1 = (1 p 0 )q 1 p 2 Thus, we have p u i = (1 p 0 )q 1 p i 1 p 2 1 p (1 p 0 )q i 1 1 p 2 + p 0 p i 1. + p 0 p i 1 i is even i is odd Let U(N) be the utilization of a unit spetrum. We have U(N) = = = N i=1 p u i N i=2,even 2(1 p 0 )q 1 p i 1 p 2 + N i=1,odd 1 p N (1 p 0 )q 1 p 2 1 {N is even} N/2 j=1 1 p 2j 2(1 p 0 )q 1 p 2 (N 1)/2 + j=0 1 p N (1 p 0 )q 1 p 2 1 {N is even} = 2(1 p N 2 0)q N2 1 p2 ( 1 p 2 1 p 2 1 p 2 +p 0 1 p 2 N+1 2 p 2 ) p 0 p i 1 p 0 p 2j 1 p N (1 p 0 )q 1 p 2 1 {N is even} On the other hand, if a hannel is always available to the seondary users, then the utilization per unit bandwidth is (N + 1)/2. Thus, the equivalent bandwidth if B e (N) = W U(N) (N + 1)/2. REFERENCES [1] DARPA. Defense advaned researh projets ageny (DARPA) next Generation (XG) ommuniations program. [2] FCC. Establishment of an interferene temperature metri to quantify and manage interferene and to expand available unliensed operation in ertain fixed, mobile and satellite frequeny bands, ET Doket No , FCC [3] FCC. Failitating opportunities for flexible, effiient, and reliable spetrum use employing ognitive radio tehnologies, notie of proposed rule making and order, FCC [4] FCC. In the matter of promoting effiient use of spetrum through elimination of barriers to the development of seondary markets, WT Doket No , FCC Rd [5] FCC. Promoting effiient use of spetrum through elimination of barriers to the development of seondary markets, WT Doket No , report and order and further notie of proposed rulemaking, FCC (seondary markets report and order and FNPRM). [6] FCC. Unliensed operation in the TV broadast bands, ET Doket No ; additional spetrum for unliensed devies below 900 mhz and in the 3 ghz band, ET Doket No , FCC [7] P. Gupta and P. R. Kumar. The apaity of wireless networks. IEEE Transations on Information Theory, 46(2): , 2000.

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