Multicast Capacity Scaling for Cognitive Networks: General Extended Primary Network

Transcription

1 Multicast Capacity Scaling for Cognitive Networks: General Extended Priary Network Cheng Wang, Xiang-Yang Li, Shaojie Tang, Changjun Jiang Departent of Coputer Science, Tongji University, Shanghai, China Key Laboratory of Ebedded Syste and Service Coputing, Ministry of Education, Shanghai, China Tsinghua National Laboratory for Inforation Science and Technology, Tsinghua University, Beijing, China Departent of Coputer Science, Illinois Institute of Technology, Chicago, IL, 6066 Abstract We study the capacity scaling laws for the cognitive network that consists of the priary hybrid network (PhN) and secondary ad hoc network (SaN). PhN is further coprised of an ad hoc network and a base station based (BS-based) network. SaN and PhN are overlapping in the sae deployent region, operate on the sae spectru, but are independent with each other in ters of counication requireents. The priary users (PUs), i.e., the ad hoc nodes in PhN, have the priority to access the spectru. The secondary users (SUs), i.e., the ad hoc nodes in SaN, are equipped with cognitive radios, and have the functionalities to sense the idle spectru and obtain the necessary inforation of priary nodes in PhN. We assue that PhN adopts one out of three classical types of strategies, i.e., pure ad hoc strategy, BS-based strategy, andhybrid strategy. We ai to directly derive ulticast capacity for SaN to unify the unicast and broadcast capacity under two basic principles: () The throughput for PhN cannot be underined in order sense due to the presence of SaN. () The protocol adopted by PhN does not alter in the interest of SaN, anyway. Depending on which type of strategy is adopted in PhN, we design the optial-throughput strategy for SaN. We show that there exists a threshold of the density of SUs according to the density of PUs beyond which it can be proven that: () when PhN adopts the pure ad hoc strategy or hybrid strategy, SaN can achieve the ulticast capacity of the sae order as it is stand-alone; () when PhN adopts the BS-based strategy, SaN can asyptotically achieve the ulticast capacity of the sae order as if PhN were absent, if soe conditions of the relations aong the nuber of SUs, PUs, the destinations of each ulticast sessions in SaN, and the base stations in PhN hold. Index Ters Multicast capacity, Scaling laws, Cognitive networks, Extended networks, Dense networks, Percolation theory I. INTRODUCTION Nowadays, wireless networks are regulated by fixed spectru assignent policy. A large portion of the assigned spectru is used sporadically and geographical variations in the utilization of assigned spectru ranges with a high variance in tie [], []. The liited available spectru co-exists with the inefficiency in the spectru usage. To cope with this proble, dynaic spectru access with cognitive radio has recently been investigated, which is a novel paradig, called cognitive network, that iproves the spectru utilization by allowing secondary users to exploit the existing wireless spectru opportunistically without having a negative ipact on priary users, i.e., licensed users. In this paper, we focus on scaling laws of ulticast capacity for cognitive networks. We construct the cognitive network as a superposition of two independent networks, called priary network and secondary network, that operate at the sae tie, space and frequency. The secondary users are assued to be equipped with cognitive radios and have the functionalities to sense the idle spectru and obtain the necessary inforation of priary users [] [3]. We set the priary network as a hybrid network, denoted by PhN, consisting of base stations (BSs) and ad hoc nodes (priary users, PUs) [4], [5]. We assue the secondary network as an ad hoc network, denoted by SaN. To atch the reality of spectru consuption better, we assue that the network odel has a Pyraid structure. That is, the nuber of PUs, which are licensed to access to the spectru at any tie, is relatively less than the nuber of secondary users (SUs), which can opportunistically access to the spectru. Our odel has three novel points relative to ost existing works: () Since ulticast capacity can be regarded as the general result of unicast and broadcast capacity [6], we directly study the ulticast capacity for cognitive networks in order to enhance the generality of this study. () Since pure ad hoc networks and BS-based networks (static cellular networks) can be regarded as the special case of hybrid networks in ters of the nuber of BSs [4], [5], [7] [], we consider the case that the priary network is a hybrid network, which further increases the generality of our odel. (3) We use the Gaussian channel odel [], [3] that can capture better the nature of wireless channel than other classic interference odels, such as protocol odel and physical odel [4]. We ai to derive the ulticast capacity for SaN under two basic principles: () The throughput for PhN cannot be decreased in order sense due to the presence of SaN. () The protocol adopted by PhN will not alter anyway because of the presence of SaN. These two basic principles are coincident with the abstract of practical techniques of cognitive networks. We first derive the upper bounds on ulticast capacity for a stand-alone network isoorphical to SaN, called stand-alone SaN. Obviously, we can use such upper bounds as that for SaN whatever strategy is adopted by PhN, because under Gaussian channel odel PhN and SaN always have negative influence (interference) on each other under the noncooperative counication schee as long as they share the sae spectru /0/$ IEEE 6

2 at the sae tie. To copute such upper bounds, we exploit the hoogeneity property and randoness property of network topology, [5]. Our ain work is to design ulticast strategies for SaN under two principles entioned above, by which the ulticast throughput, i.e., the lower bounds on ulticast capacity, for SaN can be achieved of the optial order atching the upper bounds. We design two types of ulticast strategies for SaN. In one type of strategy, we devise the hierarchical ulticast routing based on the highway syste consisting of the first-class highways (FHs) and second-class highways (SHs), and we use a hierarchical TDMA schee to schedule those highways. In the other type, we build the routing based on the highway syste only coprised of second-class highways (SHs), in order to avoid the bottleneck on the accessing path into highways for soe cases []. Cobining with the two types of strategies, we obtain the achievable ulticast throughput as the lower bounds of ulticast capacity for SaN. How to protect the capacity for PhN fro decreasing in order sense is the precondition in the designing of any strategy for SaN. Our solution is to set a protection area (PA) for every node in PhN. As one iportant characteristics different fro other related work such as [6] [8], we allow a PA to be dynaic according to the state of the corresponding priary node. Benefitting fro the dynaics of PAs, secondary users (SUs) can access opportunistically into the spectru fro both tie and space doains. While, static protection areas used in [6] [8] ake soe SUs be never served. In our solution, an intuitive view is that: At a certain tie, the receiver in PhN can possibly receive data at a rate of the sae order as in the scenario where PhN onopolizes the spectru, as long as all transitters in SaN are out of a large enough PA of this receiver; siilarly, a receiver in SaN can also possibly receive data at the sae rate (in order sense) as that for the standalone SaN, as long as this receiver is out of all PAs of the active transitters in PhN. Two ain technical challenges of the specific designing of ulticast strategies for SaN are listed as follows. How large of the PAs are optial with respect to the capacities for both SaN and PhN? As discussed above, the larger PAs are better for protecting the throughput for PhN. Meanwhile, too large PAs will result in decreasing the throughput for SaN. In other words, there is a tradeoff between the throughputs for PhN and SaN in ters of the size of PAs. Furtherore, it is easy to understand that the designing of ulticast strategies for SaN depends on the specific strategy adopted by PhN. As a hybrid network, PhN could generally adopt three broad categories of ulticast strategies, according to [4], [5], [9]. The first one is the classical BS-based strategy under which counications between any users are relayed by soe specific BSs. The second one is the pure ad hoc strategy, i.e., the ultihop schee without any BS-supported. The third one is the hybrid strategy, i.e., the ultihop schee with BSsupported. According to these three strategies adopted by PhN, we define the appropriate PAs for each PU and BS, and call the A-Type PA and B-Type PA, respectively. Specifically, under pure ad hoc strategy, the B-Type PAs are never active; under BS-based strategy, the B-Type PAs are always active; and under hybrid strategy, both A-Type PAs and B-Type PAs ight be active in a certain tie. How to build the highways, including FHs and SHs? Different fro the traditional highways in [], [3], the construction of highways in SaN is ore coplicated because it is involved with the blocking of soe active PAs. For FHs, we design a detouring schee under which every FH detours the PAs, and we can prove that the produced FHs have a large enough density and large enough capacity to support the relay of data in SaN. For SHs, we design a hierarchical TDMA scheduling schee by which enough SHs can be scheduled in a constant scheduling period, and all SUs have chance to be served via accessing to the SHs, except when PhN adopts BS-based strategy. As the final result, cobining the upper bounds and lower bounds, we show that: () When PhN adopts the pure ad hoc strategy or hybrid strategy, the per-session ulticast capacity for SaN is of order Θ( d ) when d = O( (log ) ), and 3 is of order Θ( ) when d = Ω( ), where is the log total nuber of SUs and d is the nuber of destination nodes of each ulticast session in SaN. () When PhN adopts the BS-based strategy, an infinitesial fraction of SUs cannot be served. The per-session ulticast capacity for SaN is asyptotically of the sae order as in Case (). The rest of the paper is organized as follows. In Section II, we introduce the syste odel. In Section III, we present our ain results. We copute the upper bounds of ulticast capacity for SaN in Section IV. In Section V, we derive the achievable ulticast throughput as the lower bounds on ulticast capacity for SaN by designing the specific ulticast strategies. In Section VI, we review the related work. In Section VII, we conclude this paper. II. SYSTEM MODEL A. Network Topology The network odel has a two-layer structure over a square region A(n) = [0, n]. The first layer is the priary hybrid network (PhN) consisting of Θ(n) priary users (PUs, priary ordinary nodes) and b(n) base stations (BSs). In PhN, PUs are placed according to a Poisson point process of unit intensity over the region A(n); the region A(n) is partitioned into b(n) square subregions of side length n/b(n); one base station (BS) is located at the center of each subregion. We assue that BSs are connected via the high-bandwidth wired links that are certainly not the bottleneck throughout the routing. The second layer is the secondary ad hoc network (SaN) consisting of Θ() secondary users (SUs, secondary ad hoc nodes). In SaN, SUs are distributed according to a Poisson point process of intensity n over the region A(n). We randoly choose n s (or s ) nodes fro all PUs (or SUs) as the sources of the ulticast sessions in PhN (or SaN), and for each PU v p (or SU v s ), pick uniforly at rando n d PUs (or d SUs) as the destinations. Fro Chebychev s inequality, 63

3 (a) PU Layer (b) BS Layer (c) SU Layer Fig.. PhN consists of PU layer and BS layer; SaN has only one layer, i.e., SU layer. (a) The black sall square is the source of a given ulticast session. The bigger shaded squares are the A-Type PAs. (b) The sall black hexagons are the BSs that are placed in the center positions of the subregions of area n/b(n). The shaded squares around BSs are the B-Type PAs. (c) Dashed lines denote the Euclidean spanning tree of a given ulticast session. we can assue that the nubers of PUs and SUs are n and, respectively, as in [], [0], which does not change our results in order sense. The following are our basic assuptions. Assuption : PhN operates as if SaN were absent. That is, PhN does not alter its protocol due to SaN anyway. Assuption : Secondary nodes can sense the locations of priary nodes and know the protocols adopted by PhN. Assuption 3: PhN and SaN are overlapped into a layered network with a Pyraid structure. Specifically, n = o( log ). According to Assuption 3, SaN is indeed dense scaling [6], [4], [5], [], [], while PhN is an extended network [4], [5], [], [3] [5]. More discussions about two types of scaling networks can be found in Section II-B of our technical report [6]. B. Counication Model We assue that all priary users (PUs) and BSs transit with the constant wireless transission power P, [4], [5]. Let V(τ) denote the set of transitters in tie slot τ. Then, during any tie slot τ, a node v i V(τ) can counicate with another node v j via a direct link, over a channel with bandwidth B, ofrate ( ) P l(v R(v i,v j ; τ) =B log + i,v j), N 0+ P l(v k,v v k V(τ)/v j) i where the constant N 0 > 0 is the abient noise. The wireless propagation channel typically includes path loss with distance, shadowing and fading effects. As in ost related works, such as [], [3], we assue that the channel gain depends only on the distance between a transitter and receiver, and ignore shadowing and fading. Following the setting in [] (defined in Section 3 of []), we give PhN and SaN different channel power gain according to their different scaling characteristics. Since PhN is extended scaling, the channel power gain l(v i,v j ) is given by l(v i,v j ) = in{,d α ij }; Since SaN is dense scaling, the channel power gain l(v i,v j ) is given by l(v i,v j ) = d α ij. Here, dij = d(vi,vj) = vivj is the Euclidean distance between two nodes v i and v j, α > denotes the power attenuation exponent []. C. Capacity Definition Let V = {v,v,,v n} denote the set of all nodes in the network and let the subset S V denote the set of source nodes of ulticast. Let the nuber of ulticast sessions be S = n s. For each source v S,i S, we uniforly choose n d nodes at rando fro other nodes to construct D S,i = {v S,i,v S,i,,v S,ind } as the set of destinations, where obviously n d n. We call U S,i = {v S,i} D S,i the spanning set of ulticast session M S,i. Denote Λ S,nd = (λ S,,λ S,,,λ S,ns ) as a rate vector of the ulticast data rate of all ulticast sessions. We say a rate vector asyptotically feasible if it is (, )-feasible [8]. Based on a ulticast rate vector, we define the per-session ulticast throughput (PMT) as Λ P S,n d (n) = n s v S,i S(,) λ S,i. Furtherore, the PMT Λ P S,n d (n) = n s λs,i v S,i is S(,) asyptotically achievable if Λ S,nd =(λ S,,λ S,,,λ S,ns ) is asyptotically feasible. Definition (Asyptotic Multicast Capacity): The asyptotic per-session ulticast capacity (Asyp-PMC) of a class of rando networks is of order Θ(g(n)) if there are deterinistic constants 0 <c<d<+ such that li n + Pr(ΛP S,n d (n) =c g(n) isasyp-achievable) =, li inf n + Pr(ΛP S,n d (n) =d g(n) isasyp-achievable) <. The traditional definition of per-session ulticast capacity (PMC) [3] can be regarded as a special case of asyptotic per-session ulticast capacity when S(, ) = S. III. MAIN RESULTS We present the upper bounds and lower bounds on ulticast capacity for the secondary ad hoc network (SaN), respectively. Finally, cobining the lower bounds and upper bounds, we 64

4 obtain the ulticast capacity. We assue that s =Θ(), where is the total nuber of secondary nodes and s is the nuber of ulticast sessions in SaN. Such assuption is ade in ost works related to capacity scaling laws. To siplify the description, let the expression φ(n) : [φ 0 (n),φ (n)] represent that φ(n) =Ω(φ 0 (n)) and φ(n) =O(φ (n)). A. Upper Bounds on Multicast Capacity We first derive the upper bounds on ulticast capacity for SaN as if the priary hybrid network (PhN) were absent. Straightforwardly, such results are also the upper bounds for SaN when PhN works. Theore : The PMC for SaN is at ost of order O( d ) when d :[, (log ) ] Λ P = O( d log )when d :[(log ), log ] O( ) when d :[ log,] Here, d denotes the nuber of destination nodes of each ulticast session in SaN. Clearly, the result in Theore always holds whatever strategy is adopted in PhN. B. Lower Bounds on Multicast Capacity We derive the lower bounds on ulticast capacity by designing the strategies for SaN corresponding to three classical types of strategies adopted in PhN [4], [5]. ) When PhN Adopts Pure Ad Hoc Strategy: In this case, all secondary users (SUs) involved in the ulticast sessions can be served. We have Theore : The achievable PMT for SaN is of order Ω( d ) when d :[, (log ) ] 3 Ω( ) when Λ P = d (log ) 3 d :[ (log ), 3 (log ) ] Ω( )when d :[ d log (log ), log ] () Ω( ) when d :[ log,] ) When PhN Adopts BS-based Strategy: In this case, soe SUs are covered by the B-Type PAs that are always active, then they cannot be served. Under our strategy for SaN, we can ensure that there are at least ρ s () ulticast sessions of SaN with at least ρ d () d destinations can be served, where ρ s (), ρ d (), as. Theore 3: The Asyp-achievable PMT for SaN is of order Λ P, where Λ P is defined in (). 3) When PhN Adopts Hybrid Strategy: In this case, we set SaN to be idle when the down-links and up-links involved with the BSs are scheduled in PhN, and we schedule SaN in the other phases. Under such strategy, we get the achievable ulticast throughput of the sae order as in Theore. C. Multicast Capacity for SaN Cobining the upper bounds in Theore and the lower bounds in Section III-B, we can obtain Theore 4: When PhN adopts the pure ad hoc strategy or the hybrid strategy, the PMC for SaN is of order { Θ( d )when d :[, C P = (log ) 3 ] Θ( ) when d :[ log,] () When PhN adopts the BS-based strategy, Asyp-PMC for SaN are also of order C P. Note that there exists a gap between the upper bounds and lower bounds for the case when d :[/(log ) 3,/log ]. We leave out closing this gap for our future work. IV. UPPER BOUNDS ON MULTICAST CAPACITY FOR SAN In this section, we propose the upper bounds on ulticast capacity for SaN when PhN is absent. In Section V, we will prove that such bounds can be achieved indeed. Therefore, we focus on the stand-alone SaN, where the nodes are distributed into the square region [0, n] according to a Poisson distribution of the density of n with n = o( log ). Now, we begin to prove Theore. Firstly, we have Lea : The PMC for SaN is at ost of order O(ax{/ d, log /}). Proof: Please refer to Appendix B-A of [6]. Lea : The PMC for SaN is at ost of order { O( d log ) when d = O( log ) O( ) when d =Ω( log ) Proof: Please refer to Appendix B-B of [6]. Cobining Lea with Lea, we get Theore. V. LOWER BOUNDS ON MULTICAST CAPACITY FOR SAN Generally, the lower bounds on the capacity can be obtained by designing the specific ulticast strategies. To be convenient to describe the strategies, we first recall a notion. Definition (Schee Lattice, [7]): Divide a square deployent region of side length a into a lattice consisting of square cells of side length l, we call the lattice schee lattice and denote it as L(a, l,θ), where θ [0, π 4 ] is the iniu angle between the sides of deployent region and those of produced cells. A. Overview of Multicast Strategy Denote the strategy for PhN as consisting of routing schee r and transission scheduling t. Assue that every subphase of t, i.e., tj (for j =,,,ς) operates under an independent TDMA schee. Denote the scheduling periods of those TDMA schee as Kj, where K j 3. We will design the ulticast strategy for SaN according to the specific strategy adopted by PhN. We first construct the specific protection areas (PAs) for each priary user (PU) and each BS in PhN, and call the A-Type PA and B-Type PA, respectively. Please see the illustration in Fig. (a) and Fig. (b). Then, at slot τ(τ j,j), j ς and τ j Kj, which represents the τ j th tie slot in a scheduling period in phase j, we set the status of PAs of the nodes scheduled in τ(τ j,j) as active. Thus, in any tie slot τ, the region A(n) =[0, n] is 65

5 partitioned into two regions: the occupied region O(τ), which is the region covered by all active PAs in tie slot τ, and the vacant region V(τ), which is the copleent of region O(τ). Accordingly, we denote the set of the SUs covered by the occupied region O(τ) or surrounded by the active PAs in tie slot τ, asthesetp(τ). Finally, for a given ulticast session M S,i with the source node v S,i, when v S,i τ(τ j,j) P(τ), the ulticast session M S,i will be ignored; otherwise, by using the algorith in [3], we construct the Euclidean spanning tree (EST), denoted by EST( U S,i), based on the set US,i, where U S,i is the spanning set and U S,i = U S,i τ(τ j,j) P(τ). Then, siilar to the ulticast routing designed in [3], our routing for SaN is guided by the spanning tree EST( U S,i). The counication of each link in EST( U S,i) is routed via the highway syste siilar to that in [4], if applicable. However, intuitively, the routing paths ight be broken by the active PAs in soe (or even all) tie slots. Thus, how to deal with such intractability? Is it possible that the optial throughput for SaN can be achieved? Here, the called optial order of throughput is the upper bounds of ulticast capacity for the stand-alone SaN. Then, given a specific protocol in PhN, we have three questions in designing ulticast strategies for SaN. Question : How to construct and schedule the first-class highways (FHs) and second-class highways (SHs) such that, in any tie slot when SaN is scheduled, no link along the highways is across the active PAs? Question : How large is the density of the highway syste in SaN, including the FHs and SHs, if exists? Question 3: How to ensure our ulticast strategy to serve the SUs (or ulticast sessions) as uch as possible? Obviously, the status of PAs and the ethod of constructing the highway syste in SaN are deterined by the strategy adopted by PhN. Thus, all of three questions should be answered depending on the protocol of PhN. According to the existing works [4], [5], when the TDMA transission scheduling schee is adopted in PhN, the strategy for hybrid network can be classified into three types, i.e., pure ad hoc strategy, BS-based strategy and hybrid strategy. Next, we introduce concisely these strategies, and answer the three questions above according to the specific protocol of PhN. B. When PhN Adopts Pure Ad Hoc Strategy In PhN, under the pure ad hoc strategy, since no base station is used, all B-Type PAs are always inactive. For an A-Type PA, its status (active or inactive) is deterined by the routing and transission scheduling adopted by PhN. To achieve the optial order of throughput for PhN, we assue that the ulticast strategy in [4] is adopted in PhN. The strategy is divided into two phases. Denote the routing schee in the first phase in PhN as r and the transission scheduling in PhN as t.inthis phase, the strategy is designed based on the schee lattice L( n, c, π 4 ), where c>0 is a constant defined in [4]. The routing is constructed based on the first-class highways (FHs) consisting of the short links of constant length; and those short links are scheduled by a TDMA schee. Assue that the constant scheduling period is K (K =3in [4]). Denote the routing schee in the second phase in PhN as r and the transission scheduling as t. In this phase, the strategy is designed based on the schee lattice L( n, σ log n ε n, 0), where σ is a constant defined in Lea 4 of [4] and ε n is an adjusting constant to ensure the value of n/(σ log n ε n ) to be an integer; the routing is constructed based on the second-class highways (SHs) consisting of the links of length of order Θ( log n); and those links are also scheduled by a TDMA schee of constant period K (K = 4 in [4]). An iportant ethod is called the parallel transission scheduling under which Θ(log n) links initiating fro each active cell are siultaneously scheduled. As in PhN, the highway syste in SaN also consists of two levels highways: first-class highways (FHs) and second-class highways (SHs). We first introduce the fro the situation where PhN is not considered, and then extend the to the real situation in which the priority of PhN is inviolable. ) Highways for SaN absent of PhN: When the PhN is ignored, the highway syste can be constructed by the siilar ethod in [4]. The first-class highways (FHs) are indeed the highways constructed in []. The second-class highways (SHs) are built without using percolation theory []. Existence and Density of FHs: The FHs are constructed and scheduled based on the schee lattice L( n, c n, π 4 ) as illustrated in Fig.. Since the distribution of SUs follows a Poisson with ean of c (derived by the intensity n ties the area of the cell c n ), the cell in L( n, c n, π 4 ) has the sae open probability as the cell of the lattice in Fig. of [], i.e., p e c.leth = c. According to Theore 5 of [], by choosing a large enough c, there are Ω(h) paths crossing the network area fro left to right, w.h.p., and these can be grouped into disjoint sets of δ log h paths, with each group crossing a rectangle slab of size n (κ log h ε h ) n c, for all κ>0, δ sall enough, and a vanishingly sall ε h so that the side length of each rectangle is an integer. Therefore, we can divide such rectangle into δ log h slices of size n π(n, ), where π(n, ) =Θ( n ). Denote the h jth slice in the ith slab as s h (i, j), where i κ log h ε h and j δ log h. Based on this, we allocate the relay burden of nodes in s h (i, j) to a specific first-class highway (FH), denoted by ħ h (i, j) representing the jth horizontal FH in the ith slab. Siilarly, for the vertical case, we can explain the corresponding s v (i, j) and ħ v (i, j), and define the apping between the. Existence and Density of SHs: The SHs are constructed and scheduled based on the schee lattice L( n, σ log n ε, 0). For the dense scaling network odel, the parallel transission scheduling does not work [4]. Then, in SaN, having no parallel SHs like in PhN [5], there exists only one second-class highway (SH) in each colun (or row). Denote each colun as s v(i), where i n σ log n ε. Based on this, we allocate the relay burden of nodes in s v(i) to a specific second-class highway (SH), denoted by ħ v(i) representing the 66

6 l n Fig.. The cells are of side length c = c. The slab is of side length l =(κ log h ε h ) c. The shaded regions are the A-Type PAs. The sall square nodes at the center of A-Type PAs represent the priary users, and the sall circle nodes represent the secondary users. Those cells that contain at least one secondary users and are not shaded are called non-protected open. SH contained in the ith colun. Siilarly, for the vertical case, we can explain the corresponding s h (i) and ħ h (i), and define the apping between the. Reark that we can use a TDMA with the constant period K, to schedule the SHs. We will provide the detailed analysis in Section V-B4. ) Highways for SaN Present of PhN: Consequently, we construct the highway syste for SaN based on percolation theory [], ensuring that no highways in SaN crosses the active PAs in any tie slot. Existence and Construction of FHs: The FHs in SaN will be scheduled in the first phase in PhN, i.e., r or t.in this phase, surrounding each PU, we build its A-Type PA as a cluster of nine cells of side length c n, as illustrated in Fig. In any tie slot τ, an A-Type PA is active or inactive depends on whether the central priary user (PU) is scheduled (including both transitting and receiving) or not. Recall that the transission scheduling of FHs in PhN is a TDMA schee with constant period K. Then, in any tie slot of the scheduling for FHs in PhN, the nuber of scheduled cells will be n of the nuber of all cells, i.e., K c. That is, in soe tie slots, the nuber of scheduled PUs is of order Θ(n). Hence, it has no ipact on our results in ters of order when the dynaic of the status of A-Type PAs in the first phase is ignored, i.e., alla-type PAs are always regarded as active in the first phase. Next, we build the FHs in SaN that co-exists with PhN. We first odify the definition of open cells []. A cell in L( n, c n, π 4 ) is called non-protected open if it is nonepty, i.e., it contains at least one SU, and does not belong to any A-Type PAs. Please see the illustration in Fig.. Then, we have the following lea. Lea 3: When n = o(), acellinl( n, c n, π 4 ) is non-protected open with probability p s p as n. Proof: According to the definition of non-protected open, we have p s = ( e c ) e 9c n, where e 9c n is the probability that a cell in L( n, c n, π 4 ) is not covered by any PAs. Cobining with the condition that = ω(n), we PhN t t K K SaN First Phase Second Phase K -TDMA in First Phase K = K -TDMA in Second Phase K = Fig. 3. Illustration of Scheduling Schee. We describe the case that K = 3 and K = 4. The scheduling for SaN is divided into two phases corresponding to two phases in PhN. In the first (or second) phase, SaN schedules in sequence K K (or K K ) cells in the schee lattice L( n, c n/, π 4 ) (or L( n, σ log n/ ε, 0)) during one period of 3K (or K4 = K K ) slots; that is, each cell will be scheduled continuously 3 (or K ) slots. Reark that, during the continuous K slot for each cell in L( n, σ log n/ ε, 0), the cell is really scheduled only when it is not covered by the active PAs. obtain that p s p as n, which copletes the proof. By Lea 3, we can prove the existence of FH in SaN, and obtain the sae density of FHs in SaN as that in PhN. Thus, we can use the sae notations of FHs in the situation absent of PhN, which will be used in Algorith. Scheduling of FHs in SaN: Let the transitting power of SUs in the first phase be P (c n )α, where P (0,P 0 ] is a constant. Recall that the constant P 0, defined in Section II-B, is the axiu transitting power in SaN. Obviously, P (c n )α (0,P 0 ]. Because the FHs in SaN detour all PAs, the capacity of FHs in PhN can be protected fro increasing in ters of order, which is proved in Theore 5. As illustrated in Fig.3, in the first phase in SaN, the scheduling unit is also the cluster of K K cells. Unlike in PhN, each cell in a scheduling unit is scheduled continually 3 slots. By this ethod, it holds that there is at least one slot out of these three slots during which the nearest distance between the transitter in PhN and the receiver in SaN is of a constant order. Scheduling of SHs in SaN: As the new schee lattice, denoted by L( n, σ log n ε, 0), is used, we define the new A-Type PA that is a cluster of nine cells in L( n, σ log n ε, 0) centered at a priary user (PU). Please see Fig. 4(a). Let the transitting power of SUs in the second phase be P (σ log n ε ) α, where P (0,P 0 ] is a constant. Obviously, it holds that P (σ log n ε ) α (0,P 0 ]. Do the SHs, constructed for SaN absent of PhN, still work now? The following Lea 4 will answer this question. 67

7 00 PU 0 PU PA (a) SHs Meets A-Type PA BS (b) SHs Meets B-Type PA Fig. 4. SHs built based on L( n, σ log n ε, 0). (a) When PhN adopts pure ad hoc strategy, the SHs in SaN need not detour the PAs, but wait for their inactive status. (b) When PhN adopts BS-based strategy, since the PAs are always active, SHs in SaN have to detour all B-Type PAs along the SHs adjacent to the PAs. The bold polylines denote the detouring paths. Lea 4: K τ P(τ(τ =, )) =, where P(τ(τ, )) represents the set of SUs covered or surrounded by the active PAs in the tie slot τ(τ, ), i.e., theτ th scheduling slot of the second phase in PhN. Proof: We will use the apagoge to prove this result. Assue that there were an SU, say v, such that the SU v K τ P(τ(τ =, )), in other words, K τ P(τ(τ =, )) =. This eans that during the second phase in PhN, the distance between any pairs of PUs scheduled in different tie slots is of order O( (σ log n ε )), i.e., the length of the diagonal of the A-Type PA. On the other hand, since the schee lattice L( n, σ log n ε n, 0) is adopted by PhN in the second phase, there ust be two PUs that are scheduled in different tie slots between which the distance is of order Ω( K (σ log n ε n )). Furtherore, since = ω(n), n it holds that log n = o( log ), then (σ log n/ ε )=o( K (σ log n ε n )). There is a contradiction about the distance between specific pairs of PUs. Hence, we get that K τ P(τ(τ =, )) =. This lea eans that for any SU, there is at least one slot out of the scheduling period of the second phase in PhN, i.e., K tie slots, in which the SU can be possibly scheduled. Recall that K is the constant period of TDMA schee used for SHs in SaN absent of PhN. Hence, we can use a TDMA schee with the period of K 4 = K K to schedule the SHs at least once. Since K (0, + ), we can obtain the sae capacity, in order sense, of the SHs in SaN regardless of the presence of PhN. 3) Multicast Strategy for SaN: For a given ulticast session M S,i with the source node v S,i and the spanning set U S,i,we first construct the Euclidean spanning tree EST(U S,i ) by the ethod in [3]. Then, we can build the ulticast routing tree based on the highway syste and spanning tree EST(U S,k ). More specifically, for each counication pair in EST(U S,k ), i.e., an edge, the packets will access to the specific FH via the specific SH. The strategy for SaN is divided into two phases that are synchronous to the two phases in PhN. Please see the illustration in Fig.3. The detailed ulticast routing schee is presented in Algorith. To clarify the description, we first recall the forulation of the highway syste. s h (x, y): The yth horizontal slice in the xth horizontal slab in the schee lattice L( n, c n, π 4 ). s h (z): The zth row in L( n, σ log n ε, 0). ħ h (x, y): The horizontal FH bearing the relay load initiated fro the nodes in the slice s h (x, y). ħ h (z): The horizontal SH bearing the relay load initiated fro the nodes in the row s h (z). The forulations for the vertical case are siilarly defined. 4) Analysis of Multicast Throughput for SaN: First, we should guarantee the priority of PhN in ters of throughput. Then, we propose the following theore. Theore 5: Using Algorith to construct the ulticast routing for SaN, denoted by r s, and using the transission schee described in Fig. 3, denoted by t s, to schedule SaN, we can protect the capacity of highways in PhN, including FHs and SHs, fro decreasing in order sense due to SaN. Proof: Due to the liited space and the siilarity to the proofs of Leas 8 and 0 in [8], we oit the proof. Please refer it to Appendix B-C (Proof of Theore 5) in [6]. Because SaN does not add the load of any highway in PhN, by Theore 5, we obtain that the presence of SaN has no ipact on the order of throughput for PhN, when the strategy for SaN is designed as in Theore 5. Having ensured the priority of PhN, we now study the throughput for SaN. First, we answer Question 3 above. Theore 6: Under the ulticast routing r s and transission schee t s, all ulticast sessions in SaN can be served. Proof: In the second phase, the area of the A-Type PA is of order o(log n), while the area of the cell in the schee lattice of the second phase in PhN, is of order Θ(log n). Hence, for any secondary user (SU) in SaN, there exists a tie slot τ(τ, ) during which this SU is not covered by any active PA. That is, τ(τ P(τ(τ,), )) =, where τ K, which copletes the proof. Now, we start to analyze the ulticast throughput for SaN under r s and t s We first study the capacity of the FHs and SHs of SaN. Theore 7: Under the transission scheduling t s, the capacity of FHs and SHs in SaN can be achieved of order Ω(). Proof: The first phase in SaN has a TDMA scheduling period of 3K. There are at least K slots in which the su interference produced by PhN at any receiver in SaN, denoted by I ps is of a constant order, while the interference derived by SaN at this receiver, denoted by I ss, is also sued up to a constant order. This eans that PhN will not underine the capacity of the FHs in SaN in order sense. Then, siilar to the result of single dense networks in [4], the capacity of FHs in SaN can be achieved of order Ω(). The second phase in SaN has a TDMA scheduling period of K 4 = K K. There are at least K slots in which the corresponding cells in L( n, σ log n ε, 0) are really scheduled. During such slots, we can get that I ps = O() and I ss =Θ(), which eans PhN will not underine the 68

8 Algorith Multicast Routing based on FHs and SHs Input: The ulticast session M S,k and EST(U S,k ). Output: A ulticast routing tree T (U S,k ). : for each link u i u j of EST(U S,k ) do : According to the positions of u i and u j, deterine the indexes a i, b i, y i and c j, d j, x j, where u i s h (a i,b i ) s v(y i ); u j s v (c j,d j ) s h (x j). 3: Packets are drained fro u i into the horizontal FH ħ h (a i,b i ) via the vertical SH ħ v(y i ). 4: Packets are carried along the horizontal FH ħ h (a i,b i ). 5: Packets are carried along the vertical FH ħ v (c j,d j ). 6: Packets are delivered fro the vertical FH ħ v (c j,d j ) to u j along the horizontal SH ħ h (x j). 7: end for 8: Considering the resulted routing graph, we erge the sae edges (hops), reove those circles which have no ipact on the connectivity of the counications for EST(U S,k ). Finally, we obtain the final ulticast routing tree T (U S,k ). Algorith Multicast Routing based on Only SHs Input: The ulticast session M S,k and EST(U S,k ). Output: A ulticast routing tree T (U S,k ). : for each link u i u j of EST(U S,k ) do : According to the positions of u i and u j, deterine the indexes x i and y j, where u i s h (x i); u j s v(y j ). 3: Packets are drained fro u i into the horizontal SH ħ h (x i) by a single hop. 4: Packets are carried along the horizontal SH ħ h (x i). 5: Packets are carried along the vertical SH ħ v(y j ). 6: Packets are delivered fro the vertical SH ħ v(y j ) to u j by a single hop. 7: end for 8: Using the siilar procedure in Step 8 of Algorith, we can obtain the final ulticast routing tree T (U S,k ). capacity of the SHs in SaN in order sense. Then, due to the dense scaling, we can derive the capacity of the SHs in SaN is of order Ω(), [4]. Please see the detailed proof in Appendix B-D of [6]. According to Theore 7, we get Theore 8. Theore 8: When d = O( (log ) ), the per-session ulticast throughputs for SaN in the first phase and second phase, can be achieved of Ω( d ) and Ω( d (log ) 3 ), respectively. Proof: Please refer to Appendix B-E of [6]. Like in the single rando dense network [6], when the nuber of destination nodes of each ulticast session is beyond soe threshold, to be specific, d =Ω( (log ) ),the ulticast throughput derived by the ulticast strategy based on the FHs and SHs cooperatively is not optial in order sense. For this case, the ulticast strategy based only on SHs can derive larger throughput. Next, we describe such routing schee in Algorith. Theore 9: By using the ulticast routing based only on SHs,i.e., the ulticast routing constructed by Algorith, and scheduling only for SHs, the per-session ulticast throughput for SaN can be achieved of order Ω(/) Ω( ) d log when d = O( log ) when d =Ω( log ) Proof: According to Theore 7, the capacity for each SH can be achieved of constant order. By a siilar procedure of proof for Theore 8, we can prove that the relay burden of each node in SHs is of order O(in{, d log }). Then, the theore is proved. Cobining Theore 8 and Theore 9, we can get Theore by perforing soe siple algebraic coputations. C. When PhN Adopts BS-based Strategy For this case, PhN adopts the classical BS-based strategy based on the schee lattice L( n, n b, 0), where b = b(n) is the nuber of base stations in PhN. Under this strategy, the sources deliver the data to BSs during the Uplinks phase and BSs deliver the received data to destinations during the Downlinks phase. The counication between any pairs of PUs will be relayed by the BSs. In order to achieve better throughput, the BS-based strategy is adopted in PhN only when b =Ω(n/ log n) [5]. Because BSs are regularly placed, i.e., each BS locates at the center of each cell, all cells can be siultaneously scheduled during both Uplinks phase and Downlinks phase, and sustain a rate of ( b n ) α [5]. In each cell, all downlinks and uplinks are scheduled in sequence. All B- Type PAs will be always active. The SUs contained in such PAs cannot be served. Thus, we study the ulticast capacity for SaN under the general definition of ulticast capacity, i.e., asyptotic ulticast capacity (Definition ). Obviously, the classic definition of capacity [3], [4], can be regarded as a special case of the asyptotic ulticast capacity. Next, we focus on the Downlinks phase in which SaN is scheduled. Whether or not SaN is scheduled in Uplinks phase has no ipact on the ulticast throughput in order sense. ) Highway Syste: SaN still prefers to adopt the ulticast strategy based on the FHs and SHs as in the case that all BSs are always inactive. Also, the B-Type PAs for BSs in the first phase and that in second phase are clusters of nine cells in the schee lattices L( n, c n, π 4 ) and L( n, σ log n ε, 0), respectively. Notice that the effectiveness of such B- Type PA relies closely on the fact that SaN is dense scaling while PhN is extended scaling. Intuitively, as the nuber of BSs is increasing, such FHs or SHs as in the case absent of active BSs ight be daaged by the PAs. Fortunately, the facts that b = O(n) (then b = o( log )) and BSs are regularly placed [5] guarantee SaN the sae density of FHs, in order sense, as in the case absent of BSs. The original SHs in SaN are still reained. When the packets carried along an SH are stopped by a PA, they will detour the PA along the adjacent SHs. Hence, the loads of the SHs around the PAs are probably heavier than that of other 69

9 SHs. Please see the illustration in Fig. 4(b). Throughout the routing, the bottleneck in the second phase should be in those SHs with heavy burden. We will exploit this fact to analyze the ulticast throughput for SaN. ) Served Set of SaN: Now, we deal with the question how any SUs are not served at all. Denote the set of all SUs that are not served as P, and denote the set of all sources in SaN by S. Based on the sets P and S, we propose a definition of the served set of ulticast sessions, which can be divided into two regies depending on d, i.e., the nuber of destination nodes of each ulticast session. Definition 3 (Served Set): A served set, denoted by S, isa subset of S. Define S := S S P, when d = ω(log ); and define S:= {v S,i U S,i P = }, when d = O(log ). We further define a subset of D S,i as D S,i := D S,i D S,i P. Then, we have the following result. Theore 0: As n, it holds that S S ; and for each v S,i S, it holds that unifor w.h.p., D S,i D S,i. Proof: Based on the fact that n = o( log ), and by the tails of Chernoff bound [], we can prove this theore. Please refer to Appendix B-F of [6]. For a ulticast session M S,i, we define a subset of U S,i as U S,i = {v S,i } D S,i, and build the spanning tree EST( U S,i ) to guide the ulticast routing of M S,i. According to Theore 0, the throughput derived by the strategy based on the served set is asyp-achievable. 3) Guarantee of Priority of PhN: In the first phase of SaN, the interference produced by SaN at a receiving PU in PhN, denoted by I sp, is of order O(), due to the setting of PAs. Then, I sp = O(N 0 ), where the constant N 0 > 0 is the abient noise. Thus, the presence of SaN does not change the order of capacity of FHs in PhN. Siilarly, we can prove that SaN does not ipair the capacity of SHs in PhN in order sense. 4) Asyp-Achievable Multicast Throughput for SaN: We first consider the capacities of FHs and SHs in SaN. Lea 5: In the first phase, the su of the interference produced by PhN at a receiver in SaN is of order O(). Proof: The procedure of the proof is siilar to that of Theore 7. Please refer to Appendix B-G of [6]. According to Theore 7, during any tie slot τ in the first phase when a link v i v j is scheduled, the interference on v j produced by SaN itself can be bounded as I ss (v i,v j ; τ )= O(). Cobining with Lea 5, we get that the capacity of FHs in SaN does not decrease due to PhN, and it is still of order Ω(). Using the siilar procedure, we can also prove that the capacity of SHs in SaN is still of order Ω(). Next,we should analyze the loads of FHs and SHs, respectively. The forer are obviously the sae as the loads of FHs in SaN when BSs are absent. For the latter, the detouring schee increases the loads of the SHs adjacent to PAs; but it can be proved that the increent does not change the order of the loads of those SHs non-adjacent to the PAs. Based on the analysis above, we can obtain Theore 3, as one of our ain results. The detailed proof of Theore 3 can be found in Appendix B-H of [6]. D. When PhN Adopts Hybrid Strategy In this case, the strategy for PhN can be divided into four phases: FHs phase, SHs phase, Downlinks phase and Uplinks phase [4], [5], [9]. We can use a siple interission-ethod to achieve the sae order of ulticast throughput for SaN as in the case where no BS is adopted. Specifically, let SaN be idle during Downlinks phase and Uplinks phase of PhN, and schedule SaN in FHs phase and SHs phase of PhN by the siilar schees used in the case when PhN adopts the pure ad hoc strategy. Hence, the results of Theore and Theore 4 for the case when PhN adopts hybrid strategy can be proved. VI. LITERATURE REVIEW The issue of capacity scaling laws for cognitive networks is a relatively new topic. In [3], the priary source-destination and cognitive S-D pairs were odeled as an interference channel with asyetric side inforation. In [8], the counication opportunities were odeled as a two-switch channel. Note that both works [3], [8] had only considered the singleuser case in which a single priary and a single cognitive S-D pairs share the spectru. Recently, a single-hop cognitive network was considered in [9], where ultiple secondary S-D pairs transit in the presence of a single priary S-D pair. They showed that a linear scaling law of the singlehop secondary network is obtained when its operation is constrained to guarantee a particular outage constraint for the priary S-D pair. For ulti-hop and ultiple users case, Jeon et al. [7] first studied the achievable unicast throughput for cognitive networks. In their cognitive odel, the priary network is a rando dense SANET or a dense BS-based network [5], and the secondary network is always a rando dense SANET; two networks operate on the sae space and spectru. Following the odel of [7], Wang et al. [30] studied the ulticast throughput for the priary and secondary networks. In order to ensure the priority of priary users in eanings of throughput, they defined a new etric called throughput decreent ratio (TDR) to easure the ratio of the throughput of PaN in presence of SaN to that of PaN in absence of SaN. Endowing PaN with the right to deterine the threshold of the TDR, they [30] devised the ulticast strategies for SaN. Both the unicast routing in [7] and ulticast routing in [30] are built based on the backbones siilar to the second-class highways in [4], which suggests that the derived throughputs are not optial under the Gaussian Channel odel for ost cases. By introducing percolation-based routing [], [4], Wang et al. [8] iproved the ulticast throughput for the sae cognitive network odel as in [7], [30]; they showed that under soe conditions, there exist the corresponding strategies to ensure both networks to achieve asyptotically the upper bounds of the capacity as they are stand-alone. One of the coon characteristics in [7], [8], [30] is that the priary and secondary networks in all three odels are dense scaling. More iportantly, the coon defect of three work is that all the strategies in [7], [8], [30] shield the tie-doain, which akes the routing path always detour the protection areas (or 70

10 preservation regions), although they are soeties inactive. Under those strategies, there possibly soe secondary users that can never be served. VII. CONCLUSION We study the ulticast capacity for cognitive networks that operate under TDMA schee. The network odel consists of the priary hybrid network (PhN) and the secondary ad hoc network (SaN). PhN and SaN are assued to be extended scaling and dense scaling, respectively, which enhances the reality of our network odel. We devise the dynaic protection area (PA) for each priary node according to the strategy adopted in PhN. Based on the PAs, we design the ulticast strategies for SaN under which the highway syste acts as the ulticast backbone. Under the precondition that SaN should have no negative ipact on the order of the throughput for PhN, our strategy has the following erits. Firstly, by our strategy, the optial throughput for SaN can be (asyptotically) achieved for soe cases. Secondly, under our strategy, unlike ost related works, secondary nodes can access opportunistically into the spectru fro both tie and space doains. Thirdly, under our strategy, all secondary users can be served, except for the case that PhN adopts BS-based strategy. ACKNOWLEDGMENTS The research of authors is partially supported by the National Basic Research Progra of China (973 Progra) under grants No. 00CB3800 and No. 00CB334707, the Progra for Changjiang Scholars and Innovative Research Tea in University, the Shanghai Key Basic Research Project under grant No. 0DJ400300, the Expo Science and Technology Specific Projects of China under grant No. 009BAK43B37, the NSF CNS-0830, the National Natural Science Foundation of China under grant No , the Progra for Zhejiang Provincial Key Innovative Research Tea, and the Progra for Zhejiang Provincial Overseas High-Level Talents. REFERENCES [] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, and S. 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