Data Aggregation Scheduling in Wireless Networks with Cognitive Radio Capability

Transcription

1 Data Aggregation Schedling in Wireless Networks with Cognitive Radio Capability Mingyan Yan, Sholing Ji, Meng Han, Yingsh Li, and Zhipeng Cai Department of Compter Science, Georgia State University, ATL, GA 3, USA {myan2, mhan7, yli, School of Electrical and Compter Engineering, Georgia Institte of Technology, ATL, GA 8, USA Abstract Complicated collisions and spectrm ncertainty constrain the sage of Cognitive Radio Networks (CRNs) on heavy transmission and time sensitive applications. On the other hand, data aggregation has been considered as an essential operation in wireless networks. A large amont of effort has been dedicated to the investigation of CRNs and data aggregation in wireless networks. However, the existing literatres rarely concentrate on how to se cognitive radio techniqe to promote the performance of data aggregation in conventional wireless networks. In this paper, we investigate the Minimm Data Aggregation Schedling in wireless networks with Cognitive Radio capability (MLDAS-CR) problem. As the first try, an approximation schedling algorithm based on Integer Linear Programming (ILP) and Linear Programming (LP) is proposed. According to the simlation reslts, this method performances great, however, it is difficlt to theoretically evalate the soltion. Therefore, a heristic schedling algorithm with garanteed latency bond is presented in or frther investigation. The performance of the proposed soltions are evalated throgh extensive simlations. I. INTRODUCTION A report released by Federal Commnications Commission (FCC) [1] shows that a large portion of licensed wireless spectrms are ndertilized while the nmber of wireless sers has explosively increased in the last decade. In order to alleviate the spectrm shortage and ndertilization problem, Cognitive Radio Networks (CRNs) have been proposed. In CRNs, nlicensed sers are eqipped with cognitive radios which are capable of adapting transmitter parameters based on interaction with their operating environment. Unlicensed sers can dynamically access and exploit licensed spectrm holes when the spectrm is noccpied by licensed sers. Therefore, CRNs are called Next Generation Networks [2]. Since nlicensed sers need to avoid collisions with the ongoing transmissions betweeen both licensed sers and other nlicensed sers, the spectrm availability is qite limited for nlicensed sers. Frthermore, de to the npredictable activities of licensed sers, nlicensed sers can only access the licensed spectrm opportnistically. This ncertainty constrains the sage of CRNs on heavy transmission and time sensitive applications. e.g., fast data aggregation in wireless networks. Data aggregation has been considered as an essential operation in wireless networks. It plays an vital role in the smmarized data gathering procedre, sch as tracking critical /14/$31.00 c 2014 IEEE phenomena in continos and periodic monitoring applications. Dring the data aggregation process, raw readings are aggregated and then transferred in the network. Classic aggregation fnctions sch as sm, maximm, minimm, average or cont are widely sed. Since data is aggregated at intermediate nodes dring the transmission process, both data redndancy and the nmber of transmissions are redced. Therefore, data aggregation is an efficient strategy to alleviate energy consmption and medim access contention. A large amont of research on data aggregation can be fond in existing literatres, where some of them focses on energy efficiency (sch as [3]) and some others concern abot time performance ([4]-[10], for example). In this paper, we concentrate on the investigation of data aggregation in wireless networks with cognitive radio capability. Instead of investigating CRNs that data transmissions among nlicensed sers can only rely on the nstable spectrm holes, we stdy time efficient data aggregation in a wireless network, where the sers in the network are eqipped with cognitive radios. As in conventional wireless networks, an nlicensed spectrm is assigned to the network. It is available to the in-network sers all the time. Meanwhile, the cognitive radio enables wireless sers searching and exploiting the spectrm holes. Since a defalt working spectrm is always garanteed, reglar data transmission can still be processed if there is no spectrm hole. Frthermore, when extra idle spectrm exists, some sers can move to that spectrm, so that alleviate contention on the defalt spectrm and speed p the transmission procedre. In this paper, instead of taking noccpied spectrm holes as or only hope for data transmission, we consider them as or assistant and se them to accelerate the data aggregation process. Particlarity, we do not intitively assme that the spectrm holes are only from the licensed bands as discssed in existing literatres. Literatres on coexistence of heterogeneos wireless systems can be fond, sch as the coexistence of ZigBee and Wifi stdied in [11] and [12]. We cold have the faith that with the developed technology in wireless networks, more and more heterogeneos networks can co-exist with each other. Therefore, the investigation of this paper is meaningfl for facilitating data aggregation process by taking advantage of nsed spectrm resorces in other networks. A large amont of effort has been dedicated to the inves-

2 2 tigation of CRNs and data aggregation in wireless networks. Several athors did realize the perspective of introdcing cognitive radio capability to improve the performance of wireless networks. For example, [13]-[16] stdied isses of wireless networks with cognitive radio capability. In all the above for articles, sers can work on a stable nlicensed spectrm or access licensed spectrm opportnistically. The athors of [13] focsed on the network performance when CSMA is employed. [14] stdied the similar isse as [13] with an additional retrial phenomenon. When and how long to perform spectrm sensing in cognitive radio enabled smart grid is investigated in [15]. The performance of wireless mesh networks with cognitive radio capability is analyzed in [16]. However, the existing literatres rarely concentrate on how to se cognitive radio techniqe to promote the performance of data aggregation in conventional wireless networks, which is the focs of this paper. The main contribtions of or work can be conclded into the following aspects: We employs the cognitive radio capability in wireless networks to accelerate data transmission. Sbseqently, the Minimm Data Aggregation Schedling problem in wireless networks with Cognitive Radio capability (MLDAS-CR) is formalized and investigated. As the first try, the MLDAS-CR problem is formalized as an Integer Linear Programming (ILP) problem. Considering the hardness of solving an ILP, the optimal soltion of its linear programming relaxation is obtained, instead. Sbseqently, a ronding algorithm is employed to obtain a feasible soltion for the ILP from the optimal soltion of the relaxed LP. According to the simlation reslts, we can see that the ILP and LP based method has a good performance, however, it is difficlt to theoretically evalate the soltion. Therefore, a heristic schedling algorithm with garanteed latency bond is presented in or frther investigation. The simlation reslts verify the performance of the proposed soltions. The reminder of this paper is organized as follows: The system model and problem formlation are presented in Section II. In Section III, the proposed soltions are discssed in detail, followed by the performance evalation in Section IV. In Section V, the work of this paper is conclded. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider a dense wireless network co-exists with another wireless network who are willing to temporarily release idle spectrm holes. The objective of this paper is to find ot how cognitive radio capability can contribte to data transmission in conventional wireless networks. Therefore, for the prpose of distingishing this work from existing literatres on conventional CRNs, we refer to the two wireless networks as the wireless network and the Axiliary Network (AN) (the network provides extra spectrm opportnity). A. Network Model Axiliary Network (AN): Consider an AN consisting of m Axiliary Users (AUs) denoted by set U a = {U 1, U 2,..., U m }. The transmission radis and interference radis of an AU are denoted by T a and I a, respectively. Let S a represent the operating spectrm of AUs. Assme the network time is slotted, where the length of a time slot τ a is long enogh for AUs to finish the transmission of a data package. At the beginning of each time slot, AUs make their decisions to stay active (receive or send data) or inactive in the crrent time slot according to the network protocol. The inactive AUs remain silent for the rest time of the crrent slot. Dring each time slot, the active senders follow a two-dimensional Poisson point process X S with density λ. Apparently, the distribtion of the active receiver forms another two-dimensional Poisson point process X R with the same density λ. Wireless Network: The considering wireless network consists of n sers. Let U r = { 1, 2,..., n } denote the set of sers, among which ser b U r wants to get the aggregated information from the network. For i U r, its transmission and interference radii are denoted by T r and I r, respectively. There is a defalt working spectrm S r physically available to all the sers all the time. Each i U r is eqipped with a single, half-dplex cognitive radio, which is capable of accessing the defalt spectrm S r or adapting parameters to access opportnistically appeared spectrm holes on S a. Assme the time in the wireless network is also slotted, where the length of the time slot τ r is long enogh for a ser to monitor the available spectrm conditions and then transmit a data package. De to the radio limitation, a ser can either transmit or receive data, not both, from all directions at one time slot on a specific spectrm. In each time slot t, a ser who has data to send can either operates on S r or opportnistically access S a in a sensing-before-transmission manner, as long as no collision will be cased to both networks. Particlarly, sers in the wireless network have eqal rights to access S r. However, S a can be sed by non-aus if and only if no on-going transmissions in AN will be interrpted. That is, the AUs have absolte priority on S a. Unit Disk Graph Interference Model: In this paper, we consider the Unit Disk Graph (UDG) interference Model, which has been widely sed in existing literatres. Under this model, the interference range and the transmission range of wireless devices are denoted by eqally disks. That is, T a = I a and T r = I r. B. Problem Formlation Definition 2.1: Exterior Collision. At time t, given a link s r ( s, r U r ), where sender s has data to send to receiver r, if the proceeding of this transmission inflences or is inflenced by at least one on-going transmission in AN, it is said that there exists an exterior collision. Let U s and U r be the transmitter and receiver that affect r or affected by s when an exterior collision is occrred. We have U s r T r or U r s I a, and A B is the Eclidean distance

3 3 between A and B. Definition 2.2: Interior Collision. Let s1 r1 and s2 r2 represent two links in the wireless network, where s1 and s2 are senders, and r1 and r2 are their corresponding receivers. If the concrrent schedling of the two links at some time t leads to a collision, then the collision is called an interior collision. Similarly, when an interior collision is cased, s1 r2 T r or r1 s2 I r can be derived, and vice versa. Based on the network model and definitions, the Minimm Data Aggregation Schedling problem in a wireless network with Cognitive Radio (MLDAS-CR) can be formalized as follows: Given a wireless network denoted by G = (U r, E), where U r = { 1, 2,..., n } is the set of wireless sers, and E is the set of links ( s r E if s r T r ). A ser b U r acts as a base station and desires to obtain aggregated data from the network. Users in U r are eqipped with cognitive radios capable of adapting transmitting parameters as reqired. A defalt working spectrm S r is allocated to U r. An AN consists of m AUs operating on spectrm S a. S a is open to U r if no transmission in the AN is affected. Initially, each ser i U r \{ b } generates a data package d i. For simplicity, let D = {d 1, d 2,..., d n } denote the set of data packages generated in the wireless network, where d i (i b) is the data generated by ser i. An MLDAS-CR problem can be defined as a schedle set S = {S 1, S 2,..., S L }, where each S t (1 t L) is a set of collision-free links in G who are schedled at time slot t. Frthermore, to be an MLDAS-CR, the following constraints are reqired: 1) t (1 t L), neither an exterior collision nor an interior collision is cased by any schedled links in S t. 2) t (1 t L), given two links s1 r1 S t (s 1 r 1 ) and s2 r2 S t (s 2 r 2 ) schedled on either S r or S a, then s 1 s 2, s 1 r 2, s 2 r 1, and r 1 r 2. 3) t 1, t 2 (1 t 1, t 2 L, t 1 t 2 ), if s1 r1 S t1 and s2 r2 S t2, then s 1 s 2. 4) f( L {d st b st b S t }) = f A (D), where d st b is t=1 the data package received by b at time t throgh link st b, f A is the aggregate fnction and f A (D) is the aggregated reslt over D. 5) If t = L, the transmission is s b, and L 1 { s s r S t } { s } { b } = U r. t=1 6) arg min S={S1,S 2,...,S L } L. Constraint 1 shows MLDAS-CR shold be exterior and interior collisions free. No exterior collision garantees that no interference will be cased to AN, and interior collision free avoids extra delay and congestion cased by retransmission in the wireless network. Since i has only one radio, constraint 2 reqires i can either be a sender or receiver at a particlar time slot t, bt not both. Constraint 3 indicates the property of data aggregation, that is, each ser sends its aggregation reslt (aggregated data of its own and data received dring the MLDAS-CR) only once. The data integrity property is ensred by constraint 4, where data received by the base station shold be the aggregated information of the whole network. At the last time slot, the base station shold receive the last transmission from a SU as specified in 5. Constraint 6 denotes that the objective of the MLDAS-CR schedling S is to minimize the total transmission latency. It is known that the Minimm Data Aggregation Schedling (MLDAS) problem in wireless network is NP-hard withot considering the cognitive radio capability. It can be considered as a special case of MLDAS-CR when the AUs in AN are so dense and active that no spectrm holes exist. Therefore, the MLDAS-CR problem is NP-hard. III. SCHEDULING ALGORITHM FOR MLDAS-CR In this section, we first introdce the constrction of a balanced Connected Dominating Set-based tree, which serves as the roting tree dring the data aggregation process. Sbseqently, two schedling algorithms for MLDAS-CR are discssed in detail. A. Constrction of a Balanced Roting Tree Given a graph, a Connected Dominating Set (CDS) is a connected component with the property that for every vertex on the graph, it is either in the CDS or has some one-hop neighbor in the CDS. This property makes CDS qite sitable for serving as roting infrastrctre in wireless networks. However, it is not the case that an arbitrary CDS-based roting tree is efficient for the data aggregation application. In this paper, a Balanced CDS-based Roting Tree (BRT) is employed for the prpose of distribting transmission workload evenly, redcing the delay of sers with large degree, and then accelerating the aggregation process. Definition 3.1: 2-norm. Given a vector X = (x 1, x 2,..., x n ), the 2-norm of X is defined as: X 2 = n i=1 x i 2. According to [17], given a vector X as defined in Def. 3.1, X 2 can be sed to measre the balance among all variables x i (1 i n). Let vector W = (w 1, w 2,..., w n ) denote the workload, where w i represents the load allocates to i. Then, W 2 can be sed to measre how balance the workload is distribted among sers in U r. Especially, the smaller W 2, the more balance of workload allocation. Initially, i, w i = 0. The constrction of BRT can be described as follows: Step 1: Set the layer of b as 0, and bild a Breadth First Search (BFS) tree rooted at b. Then, search the BFS tree from root to leaves, by layer, mark all the sers who form a maximal independent set BLACK. Step 2: Start at the 2nd layer, mark the parent of BLACK nodes GRAY. Sbseqently, for each GRAY node, find a BLACK node from the same layer or one pper layer to be its parent. Dring this process, pdate W for BLACK nodes according to their nmber of GRAY children. If mltiple choices are available to a GRAY node, then the BLACK gives minimm W 2 after allocation will be chosen as its parent. Step 3: In the last step, the nmarked WHITE nodes are balanced allocated to BLACK nodes. In order to obtain a BRT, a

4 4 WHITE node accepts the BLACK in its one-hop neighborhood who can minimize W 2 as its parent. The details are illstrated in Alg. 1. Finally, from root to leaves, each node pdates its layer according to its parent s layer. Algorithm 1: Balanced Allocation inpt : The tree gets from step 2 otpt: BRT 1 for each w i do 2 w i = 0; 3 for each nmarked node i do 4 mark in WHITE; 5 check all neighbors in BLACK denoted as set NB(i); 6 if j NB(i) and the allocation of i to j achieves the minimm increase of W 2 then 7 set j as i s parent; 8 pdate w j accordingly. Since the maximm nmber of WHITE nodes is n 1, and the nmber of black nodes in a white node s one-hop neighborhood is no more than 5 (Lemma 4), therefore, the rnning time of Alg. 1 is O(n). Based on the constrction of BRT, the following lemma holds: Lemma 1: All the BLACK sers are in even layers. All GRAY sers are in odd layers. Each GRAY ser has a BLACK parent and at least one BLACK child. The BLACK and GRAY sers form a CDS. Any WHITE ser is leave on the BRT and has a BLACK parent. Particlarly, in order to intitionally show links on BRT, we transfer G to G B = {U r, {E B, E}}, where E B contains the links on the BRT. For simplicity, we only consider the directed links from children to parents in E B, while ignore links in the opposite direction. The reason is that data in the network is aggregated only from bottom (children) to top (parent) on the BRT. B. Schedling Algorithm Based on LP In this sbsection, a mathematical model is employed to formalize the MLDAS-CR problem. According to the formalization, a schedling algorithm based on Linear Programming (LP) is discssed in detail. We define { two schedling variables Rij t and At ij as: 1, if link Rij t = i j is schedled on S r at time t 0, otherwise and, { 1, if link A t ij = i j is schedled on S a at time t 0, otherwise Variable Y ij is sed to indicate the AUs activity arond link i j, where { 1, if schedle Yij t = i j at t case exterior collision 0, otherwise To be specific, at time t, Yij t = 1 if there is at least one receiver in AN active in i s interference range or one sending activity is detected within j s transmission range at t. Let Q t i = 1 indicate that i has obtained data from all its children at t, Otherwise, Q t i = 0. Apparently, the Q variable of any WHITE node on the BRT is 1. De to the constraint of half-dplex radio, a ser can active as a sender or receiver bt not both on a particlar spectrm at a particlar time. Therefore, at a specific time t, for an arbitrary ser a on BRT, it may keep silent or play one role on S a or S r, that is: Rai t + Ria t + A t ai + A t ia 1 a i E B i a E B a i E B a i E B (1) On the other hand, to prevent re-transmission and nnecessary energy consmption from transmission collision, a schedled link cannot interrpt or be interrpted by any on-going transmission in both networks. That is, the schedling of link s r E B cannot reslt in exterior or interior collision. For a particlar time t, to avoid interior collision, InEq. (2) is reqired. R t sr + i NB( r ), i j E B R t ij 1 (2) and, on the other hand, to avoid interference with the activities in AN, A t sr + A t ij 1 (3) i NB( r), i j E B where NB( i ) is the set of one-hop neighbor of i in the wireless network. According to InEq. (2) and (3), the following constraint specifies the property that a confliction-free schedling plan shold have. Rsr t + A t sr + (Rij t + A t ij) 1 (4) i NB( r ), i j E B Frthermore, link s r E B can be schedled on S a if and only if S a is available, i.e., A t sr 1 Y t sr (5) Since a node needs to wait for all its children for aggregating data, for s r E B, R t sr Q t s, A t sr Q t s (6) With the prpose of ltimately tilizing spectrm holes on S a and redcing spectrm competition on S r, the tility fnction of schedling s r E B at t is defined as: f t sr = R t sr + (αd s + βl s + 1)A t sr (7) where d s and l s are the degree and layer of s, respectively. Two variables α and β are sed to adjst the weight of the two properties according to demand. Based on the above constraints, we can conclde the MLDAS-CR problem as:

5 5 Maximize sbject to 1 L L fsr t t=1 s r E B Rai t + Ria t a i E B i a E B + A t ai + a i E B R t sr + A t sr + A t sr 1 Y t sr R t sr Q t s A t ia 1 a i E B i NB( r ), i j E B (R t ij + A t ij) 1 schedled on S a, and links with R sr = 1 shold be schedled on S r. Then, we can obtain a data aggregation schedling for Algorithm 2: Ronding Algorithm inpt : Optimal soltion from LP otpt: Feasible soltion for ILP 1 Sort the inpt by non-descending order denoted as L = { s1 r1, s2 r2,...}; 2 while L = do 3 for the ordered nmarked links in L do 4 if Y sr = 0 then 5 mark s r, set A sr = 1, R sr = 0; 6 for all the links conflict with s r in L do 7 mark s r, set A s r = R s r = 0; A t sr Q t s 1 t L, R sr {0, 1}, A sr {0, 1}, L N where L is assmed to be the length of the schedling time. Even thogh the introdced objective fnction and constraints formalize the MLDAS-CR problem to a 0-1 Integer Linear Program (ILP), we are in a dilemma to find a schedling plan based on the formalization. The major difficlty we are facing is that the activity Ysr t for link s r E B at time t is npredictable. There is no way we can get the information of Ysr t ntil time t. Therefore, instead of solving the problem considering continos time, we switch to find optimal schedling for each time slot. That is, at a particlar time t, given Ysr t for any s r E B, how can we make the best decision so that we can get the maximm nmber of links schedled? In this case, we only care abot links in E B denoted as E B that have not been schedled yet. For simplicity, in the description below, t is removed from the sperscript. Then we have: Maximize sbject to f sr s r E B R ai + R ia a i E B i a E B + A ai + a i E B R sr + A sr + A sr 1 Y sr R sr Q s A sr Q s R sr {0, 1}, A sr {0, 1} A ia 1 a i E B i NB( r), i j E B (R ij + A ij ) 1 Solving the ILP for time t is still at least NP-hard. However, a Linear Program (LP) is polynomial-time solvable. Therefore, a natral choice is to derive an LP by relaxing the constraints R sr {0, 1}, A sr {0, 1} to 0 R sr, A sr 1. Instead of solving the ILP, the optimal soltion of LP can be obtained by an LP solver. After that, a ronding algorithm (as shown in Alg. 2) is employed to get a feasible soltion for the ILP. Based on the otpt of Alg. 2, all the links with A sr = 1 can be 8 else 9 mark s r, set A sr = 0, R sr = 1; 10 for all the links conflict with s r in L do 11 mark s r, set A s r = R s r = 0; MLDAS-CR by iteratively solving the LP problem according to the dynamic network condition, the details are presented in Alg. 3. Algorithm 3: SLP (Schedling based on LP) inpt : G B = {U r, {E B, E}} otpt: Schedle S = {S 1, S 2,..., S L } 1 t = 0; 2 for each ser i U r do 3 if i is WHITE then 4 set Q t i = 1; 5 else 6 set Q t i = 0; 7 while E B is not empty do 8 t++; 9 for each link s r E B do 10 sense spectrms, and set Ysr t accordingly; 11 solve the formlated LP; 12 call Alg. 2; 13 for each Rsr t = 1 or A t sr = 1 do 14 if Rsr t = 1 then 15 schedle s r on S r; 16 else 17 schedle s r on S a ; 18 S t = S t { s r }; 19 remove s r from E B ; 20 for each ser i U r do 21 pdate Q t i accordingly; C. Schedling with Expected Delay Garantee As shown in Section III-B and IV, a feasible schedling policy based on the LP can be derived and its performance is good. However, it is difficlt to theoretically show that

6 6 how well the feasible soltion is. In this sbsection, we focs on schedling algorithm with expected delay garantee. Meanwhile, the algorithm shold be easy to implement. According to Lemma 1, links in E B can be classified into three types: the sender s is WHITE and the receiver r is BLACK, s is BLACK and r is GRAY, and s is GRAY and r is BLACK. For simplicity, let l wb, l bg, and l gb denote the three kinds of links, respectively. Definition 3.2: Interior Interference Link Set. Given s r E B, the Interior Interference Link Set (IILS) of s r, denoted as I sr, is defined as all the links in E B active on S r or S a which will case interior collision if s r is schedled on the same spectrm (S r or S a, accordingly). Frthermore, Isr is defined as the interference degree of link s r, which is the total nmber of links that may interfere with s r in the wireless network. Given link set L, the conflict graph of L denoted by C[L] is an ndirected graph on L in which there is an edge between two links if they cannot be schedled simltaneosly withot collision. Then, given a link set L, the interference degree of s r is eqal to its degree on C[L], and I sr is s r s one-hop neighbor on C[L]. The algorithm (Alg. 4, Alg. 5), which based on the BRT constrcted in Section III-A, can be conclded into two stages: Stage 1: All the links of type l wb are schedled. A link s r in E B is said ready to be schedled if the sender s has received data from all of its children. Since the WHITE sers have no children, they are ready for transmission. Firstly, links of type l wb are sorted in a non-decreasing order according to their interference degrees. For simplicity, let L wb = { s1 r1, s2 r2,..., sw rb } denote the set of sorted links. Sbseqently, a first-fit schedling policy is employed to schedle the sorted links in L wb. To be specific, links in L wb are considered in order from s1 r1 to sw rb. For a link si ri, check its spectrm availability and collision stats, and then schedle the link whenever a transmission opportnity exist. Alg. 4 shows this stage in detail. Stage 2: Iteratively schedle links of type l bg and l gb. Let E B denote the set of nschedled links in E B after Stage 1. The schedling condcts iteratively, where the ready l bg links are schedled in even iterations, and the ready l gb links are schedled in odd iterations. Dring each iteration, ready links are sorted in non-decreasing order according to their interference degree. After that, a first-fit schedling policy is applied to arrange the schedling plan. The detailed psedocode is shown in Alg. 5. According to Alg. 5, we can see that each iteration may consist of several time slots, and the length of different iterations may be different. That depends on the reqired time for schedling the ready links nder consideration. In the following part, we analyze the latency of the proposed algorithm. Lemma 2: The expected nmber of spectrms available to a link in the wireless network is 1 + e πλ(t 2 r +T 2 a ), where T r and T a are the transmission radis for sers in the wireless network Algorithm 4: -S1 (First-Fit Schedling Stage 1) inpt : G B = {U r, {E B, E}} otpt: Schedle S = {S 1, S 2,..., S t} 1 t = 0; 2 Sort links in E B of type l wb in non-decreasing order according to their interference degree; 3 L wb = { s1 r1, s2 r2,..., sw rb } denote the set of sorted links; 4 while L wb do 5 t++; 6 for each link in L wb do 7 if E si r i = then 8 schedle si ri on S a; 9 else if I si r i = then 10 schedle si ri on S r ; 11 for each schedled link si ri do 12 S t = S t { si ri }; 13 remove si ri from E B and L wb ; Algorithm 5: -S2 (First-Fit Schedling Stage 2) inpt : G B = {U r, {E B, E}} otpt: Schedle S = {S t+1, S t+2,..., S L } 1 t = t + 1, iter = 0; 2 while Not all schedled do 3 if iter%2 = 0 then 4 Sort ready links in E B of type l bg in non-decreasing order according to their interference degree; 5 let L bg = { s1 r1, s2 r2,..., sb rg } denote the set of sorted links; 6 while L bg do 7 for each link in L bg : from s1 r1 8 if E si r i = then 9 schedle si ri on S a ; 10 else if I si r i = then 11 schedle si ri on S r; 12 for each schedled link si ri do 13 S t = S t { si ri }; 14 remove si ri from E B ; 15 t++; to sb rg do 16 else 17 repeat step 4 to 15 bt replace links of l bg with l gb ; 18 iter++; and AN, respectively. Given a connected graph G = {V, E}, let G[U] denote a sbgraph of G indced by U V, and δ are the maximm and minimm degree of G, respectively. The indctivity of G is defined as δ (G) = MAX U V (G[U]). Lemma 3: Given an non-decreasing ordering O =< o 1, o 2,..., o n >, let d i denote the degree of o i, it is proved that a first-fit coloring policy in smallest-degree-last ordering ses at most 1 + δ colors, where δ = MAX 1<i<n d i [6]. Lemma 4: Let C represent a disk of radis r, and U is a

7 7 (a) Collision with link l bg (b) Collision with link l gb Fig. 1. Example of Collision. set of points with mtal distance at least 1, then the nmber of points with mtal distance at least 1 on the disk is pper bonded by 2π 3 r 2 + πr + 1, that is, U C 2π 3 r 2 + πr + 1 [6]. Lemma 5: The expected latency for -S1 is pper 5 bonded by, where is the maximm degree 1+e πλ(t r 2+T a 2 ) of G. Proof: According to Alg. 4, in stage 1, all links of type l wb are schedled. The algorithm employ a firstfit schedling based on the ordering of links interference degree. Let L wb denote the set of l wb links. Then, the latency is pper bonded by δ (C[L wb ]) + 1 (Lemma 3), where δ (C[L wb ]) is the indctivity of L wb s conflict graph. Frthermore, δ (C[L wb ]) = MAX s r L wb I wi b i, which is the maximm degree of C[L wb ]. Assme the abstraction of l wi b i is the vertex with maximm degree in C[L wb ], where the degree is eqvilent to the nmber of links that conflict with l wi b i. Based on the network model and interference model specified in Section II-A, link s r that cannot be schedled simltaneosly with l wi b i have the property that l wi r T r or l bi s T r. Since only l wb links are schedled in stage 1, for any conflicting link l wj b j, we have l wi l bj T r or l bi l wj T r, where l wi l bj T r contains the BLACK sers in wi s onehop neighborhood, and l bi l wj T r specifies bi s onehop WHITE neighbors. According to Lemma 4, a WHITE ser may have at most 5 BLACK one-hop neighbors, where one of them is its parent based on the constrction of the BRT. If denotes the maximm degree of G and G B, then, for each BLACK node, it has at most WHITE neighbors. Therefore, I wi b i = 5 1. According to Lemma 3, 5 spectrms are needed to color the links. Since we only have 1 + e πλ(t 2 r +T 2 a ) spectrm available, the iteration we need to finish the schedling is 5 1+e πλ(t 2 r +T 2 a ). Lemma 6: The expected latency for -S2 is at most 44D+1, where D is the diameter of the wireless network. 1+e πλ(t r 2+T a 2 ) Proof: The proof of stage 2 is similar to Lemma 5, we concentrate on finding the indctivity of the conflict graph for the link set in each iteration. Since the algorithm performs iteratively, the latency can be derived based on the following two propositions. Proposition 1: The latency for the even iteration is at most e πλ(t r 2+T a 2) Proposition 2: The latency for the odd iteration is pper bonded by if 1+e πλ(t r 2+T a 2) bi b, and, 1+e πλ(t r 2+T a 2) otherwise. Finally, based on the constrction of the BRT, the maximm nmber layer of the BRT is pper bonded by 2D, where D is the diameter of the wireless network (the hops between the farthest two sers in the wireless network). According to Alg. 5, Case 1 and case 2 will alternatively rn at most D iterations, respectively. Therefore, the expected latency for stage 2 is 44D+1. 1+e πλ(t r 2+T a 2) Theorem 1: The expected latency for the proposed first-fit 5 +44D+1 schedling algorithm is, where and D are 1+e πλ(t r 2+T a 2 ) the maximm degree and diameter of the wireless network, respectively. The proof of theorem 1 can be directly derived from Lemma 5 and Lemma 6. IV. PERFORMANCE EVALUATION In this section, we evalate the performance of or proposed schedling algorithms with respect to different network parameters. To keep consistency with Section II, the same notations are sed in this section. To be specific, let m, n denote the nmber of AUs and wireless sers, respectively; T a (respectively, T r ) is the transmission radis of AUs (respectively, wireless sers). At each time slot, the active senders and receivers in the AN follow Poisson Distribtion with density λ. The network configration is initially set p as: A = , m = 100, n = 0, T a = T r = 1.5, and λ = 0.3. For simplicity, the tility fnction for LP is defined as f t sr = R t sr A t sr, where sers are encoraged to se the axiliary spectrm if allowed. In the simlation, in order to verify the inflence of different parameters on the proposed algorithms, we adjst one of the parameters per time while keep the rest nchanged. Particlarly, the performance of or proposed algorithms are compared with the SAS algorithm. SAS is a seqential aggregation schedling algorithm based on CDS- aggregation tree and first fit coloring schedling algorithm proposed in [6]. It is the algorithm we can find in existing literatre with the best latency bond 15R + 4, where R is the network radis and is the maximm degree. For comparison, in SAS, we assme only S r is available. The reslts are shown in Fig. 2, where the impacts of AUs are evalated in Fig. 2(a) and Fig. 2(c), and the inflence

8 withotcr AUs' active density (a) Impact of AUs active density withotcr AUs' transmission radis (b) Impact of AUs transmission radis withotcr The nmber of AUs (c) Impact of AUs poplation withotcr RUs' transmission radis (d) Impact of RUs transmission radis withotcr The nmber of RUs (e) Impact of RUs poplation Fig. 2. Performance Evalation.

9 9 of RUs are tested in Fig. 2(d) and Fig. 2(e). For simplicity, is sed to represent the algorithm proposed in Section III-B, refers to the first-fit schedling algorithm introdced in Section III-C, and withotcr refers to the SAS algorithm. The performance of and is compared with the performance of withotcr. According to the reslts shown in Fig. 2, the performance of the proposed algorithms otperform the comparison algorithm in all aspects, which clearly shows the advantage of cognitive radio capability. Since and seek transmission opportnity on both S a and S r, the delay for the two algorithms is shorter than withotcr which only relies on S r. Particlarly, generates the schedling plan based on an optimm algorithm, which achieves a better time performance compared with. Frthermore, becase the schedling of withotcr has nothing to do with S a, so that changes on the AN does not affect the time performance of withotcr. The performance of the three algorithms with respect to the change of AUs active density is evalated in Fig. 2(a). With the increasing of active AUs density, more senders and receivers are active in the AN at each time slot, which reslts in a higher risk of exterior collisions. In order to avoid collision, the nmber of links which are schedled on S a at the same time slot decreases, therefore, more time is reqired to finish the data aggregation. Fig. 2(b) shows that the agment of AUs transmission radis cases a longer schedling delay. The reason is that, a larger transmission radis forms a bigger interference range. The increased interference range prevents more sers schedled on S a if an AU is active at a particlar time, hence, leads to more delay. Similar to the above two scenarios, AUs poplation has a negative inflence on the time performance of and (as shown in Fig. 2(c)). The growth of AUs poplation introdces more active AUs into AN, so that enlarge the effect of exterior collisions, which reslts in more delay in the end. We verify the inflence of RUs transmission radis and poplation in Fig. 2(d) and Fig. 2(e), respectively. In Fig. 2(d), we can see that the latency increases with the increasing of RUs transmission radis. The reason is similar to the inflence of AUs transmission radis. It has no relation with the exterior collision, however, abot the interior collision instead. The change of transmission radis may inflence BRT, however, the increased transmission radis has more negative effect on enlarging the interference range of RUs, which reslts in an redction of the nmber of RUs that can be schedled concrrently. Therefore, more time is needed. Since the exterior interference which comes from AN and the interior interference that comes from other wireless sers are inevitable, so that the nmber of RUs that can be schedled collision-free at each time slot is limited. Therefore, the latency increases with the growth of RUs poplation. Particlarly, both RUs transmission radis and poplation affect the network condition on S r, that is why we can see the same trend on,, and withotcr. V. CONCLUSION In this paper, we investigate the Minimm Data Aggregation Schedling in wireless networks with Cognitive Radio capability (MLDAS-CR) problem. As the first try, a schedling algorithm based on Integer Linear Programming (ILP) and Linear Programming (LP) is proposed. Since getting the optimal soltion of an LP is time and resorce consming, another efficient algorithm based on a balanced roting tree is presented. Theoretical analysis shows that the later proposed algorithm has a garanteed latency bond. 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