Distributed Resource Allocation Based on Queue Balancing in Multi-hop Cognitive Radio Networks

Distributed Resource Allocation Based on Queue Balancing in Multi-hop Cognitive Radio Networks Wei Wang, Kang G. Shin and Wenbo Wang Wireless Signal Processing and Network Lab (WSPN) Key Lab o Universal Wireless Communications, Ministry o Education Beijing University o Posts and Telecommunications (BUPT), Beijing 1876, P.R. China Real-Time Computing Laboratory (RTCL) Department o Electrical Engineering and Computer Science University o Michigan, Ann Arbor, MI 819-11, U.S.A. Email: wwsunny@16.com, kgshin@eecs.umich.edu, wbwang@bupt.edu.cn Abstract Cognitive radio allows unlicensed users to access the licensed spectrum opportunistically to enhance the spectrum utilization eiciency. In this paper, the problem o allocating resources (channels and transmission power) in multi-hop cognitive radio networks (CRNs) is modeled as a multi-commodity low problem with the dynamic link capacity resulting rom dynamic resource allocation, which is in sharp contrast with existing low control approaches that assume ixed link capacity. Based on queue-balancing network low control that is suitable or handling dynamically changing spectrum availability in CRNs, we propose a distributed scheme (installed and operational in each node) or optimal resource allocation without exchanging spectrum dynamics inormation between nodes. Considering the power masks, each node makes resource-allocation decisions based on current or past inormation rom neighboring nodes to satisy the throughput requirement o each low. Parameters o these proposed schemes are conigured to maintain the network stability. The perormance o the proposed scheme or asynchronous and synchronous scenarios is analyzed comparatively. We considered both cases o suicient and insuicient network capacity. I. INTRODUCTION Cognitive radio (CR) [1]-[] is an emerging technology or uture wireless communication and networking. CR makes it possible or unlicensed/cognitive users to opportunistically utilize the licensed spectrum when it is not occupied by licensed/primary users. It can overcome the drawback o the current static spectrum allocation policy and improve the spectrum utilization. For multi-hop wireless networks, cross-layer resource allocation [5][6] is a challenging problem; especially, distributed resource allocation is a very hard problem. Joint channel allocation, power control, route selection and congestion control, which aect one another, make the problem even more diicult. In this paper, low control [7] (instead o routing) is adopted or transmission o data rom a source to the corresponding destination. For CR networks, each node has the power mask on every channel to protect primary users, and the spectrum status o channels in a licensed spectrum may change because o the primary users activities, which is known as spectrum dynamics. These characteristics o CR networks introduce new challenges or resource allocation. There have been a number o publications on spectrum allocation [8][9], power control [1][11], and routing [1][13] or CR networks. However, most o them ocus on one o the various aspects o resources allocation, and almost all o existing research on distributed resource allocation requires spread the spectrum dynamics inormation to the nodes all over the CR network. In this paper, we propose a distributed resource allocation scheme that meets end-to-end (EE) throughput demands or multiple sessions in multi-hop CR networks. The proposed scheme considers the power masks or each channel, and can adjust itsel adaptively to accommodate the spectrum dynamics according to the spectrum status and the current link transmission requirement. The proposed distributed scheme is also suitable or asynchronous scenarios, in which all the nodes do not have to execute the scheme at the same time. The main contributions o this paper are summarized as ollows: it Extends the queue-balancing low control rom wired networks with ixed link capacity to wireless networks in which link capacities are dictated by dynamic resource allocation. Especially, or CR networks this scheme is suitable or handling spectrum dynamics in a distributed manner. Proposes a node-level resource-allocation scheme deployed in each node based only on local inormation available to the node. The data rate, power level and channel allocation are determined by the current queue size and adjusted to accommodate the status o channels and the throughput requirement on each link. Investigate the network-level perormance. The resultant parameter conigurations guarantee the network stability i the network has large enough capacity to satisy the throughput requirements o all sessions. The perormance o the proposed scheme in asynchronous scenarios is analyzed and compared with that when all the nodes can execute the algorithm at the same time (i.e., synchronously). The case o insuicient network capacity is also considered. The rest o this paper is organized as ollows. Section II

describes the system model. Section III applies the queuebalancing low control to CR networks. Section IV analyzes the distributed resource allocation problem by joint rate control, power and channel allocation. Section V investigates the network perormance o the proposed scheme with appropriate parameters. The practical issues are discussed urther in Section VI. The related work and the conclusions are presented in Section VII and Section VIII, respectively. II. SYSTEM MODEL The multi-hop CR network under consideration is assumed to consist o a set o nodes V and a set o links L. Let T (l) and R(l) denote the transmitter and receiver o link l, respectively. Each node is equipped with two radio interaces, one or transmitting data and the other or receiving data. OFDM is assumed to have been deployed in the network, so that multiple channels can be used in each interace. In order to reduce the complexity o resource allocation, several subcarriers are combined to be a channel. The authors o [5] provided a method or estimating the minimum number o channels required or wireless networks. Due to the dierent properties o channels in CR networks, the spectrum can be divided into more channels than that in [5] or lexibility. We will not ocus on how to divide the channels, but assume that a set K o available channels are given. 1 Note that the resource allocation and low control in this paper are only or data transmission, not or control inormation. We assume that there are some other dedicated channels deployed or control inormation. A. Intererence-aware Transmission Model Because o equipment s limited capability, each node has a power constraint P i, so the total power on all the channels should not exceed P i. ω lk P lk < P i (1) T (l)=i k K where P lk is the transmit power at link l on channel k, ω lk is a binary indicator or channel allocation. ω lk = 1() means that channel k is (not) allocated to link l. Let I (l) be the set o links that interere with link l. To avoid the intererence between links, j I (l), ω lk + ω jk 1. The set o interering links can be constructed by either the protocol model based on distance, or the signal to intererence and noise ratio (SINR) threshold model based on the required SINR [1]. Based on the Shannon capacity ormula, the capacity o link l can be written as C l = W k log(1 + ω lkp lk G l ) () I R(l)k k K where W k is the bandwidth o channel k, G l is the path gain or link l, I R(l)k includes the thermal noise and the intererence rom the primary users and other systems. 1 The subcarriers within a channel are considered homogeneous. To protect the communication o the primary nodes, the transmit power o cognitive radio nodes should be restricted. ω lk P lk < Q ik (3) T (l)=i where Q ik is the maximum transmit power o node i on channel k. { Q H Q ik = ik i no primary user () i primary user Q L ik I there is no primary node nearby, the CR nodes can transmit with as much power as they can. Q H ik is the maximum allowed power or node i on channel k because o the equipment s limited capability. I some primary users are discovered via spectrum sensing, the CR nodes should transmit with the power less than a certain threshold Q L ik to avoid an unacceptable level o intererence to the primary receivers. Most o previous work [8][1] considered the binary model o channel availability or cognitive radios, which is a special case o our model i Q L ik is set to. Besides the power mask or protecting the primary users/nodes, another characteristic o CR networks is the spectrum dynamics. The activities o primary nodes aect the channel capacities greatly or CR nodes. B. Traic Flows with Required Throughput We would like to allocate resources (channels and transmit power) to meet the required throughput or each session. There are a set o traic sessions F in the network. Each session is deined by a source node S(), a destination node D(), and the EE throughput demand r. The total throughput over link l must satisy x l C l (5) F where x l is the data rate o session on link l and C l is as deined in Eq. (). From the network low s perspective, the low conservation constraints need to be satisied: x l = r (6) T (l)=i T (l)=s() x l = T (l)=d() R(l)=i x l = (7) x l (i S(), D()). (8) Note, however, that our problem doesn t satisy the low conservation constraints in a strict sense. Each node has a buer to store the data to be orwarded, so it doesn t have to satisy the conservation constraints over a short time duration, but must satisy them in a long-time average sense.

III. QUEUE-BALANCING FOR NETWORK FLOW CONTROL According to the problem ormulation, the problem can be modeled as a multi-commodity low problem. The multicommodity low problem is commonly solved by pricedirective decomposition and the resource-directive decomposition [7]. These methods divide the multi-commodity low problem into single-commodity low problems and ind the paths or each commodity. However, they require a centralized control. Because link capacities are not ixed in our problem setting, but determined by the resource allocation in wireless networks, the nodes should be synchronized to adjust the low and link capacity. The authors o [15] and [16] proposed another method, called queue-balancing low control, or the multi-commodity low problem. It does not choose paths or each commodity, but pushes the data rom sources to the corresponding destinations by using queue potential. Fig. 1 shows a simple example o queue-balancing low control. In order to meet the throughput demands o every session/low in the network, a dynamic (instead o static) queuebalancing algorithm is used or the multi-commodity low problem. There is a queue or each session at both the transmitter and the receiver o each link, as shown in the right-below subigure o Fig. 1. The traic o the sessions enters the network rom the sources and exits the network rom the destinations. The objective o resource allocation is to maximize the total potential decrease or transmitting the data. Because data queuing lets some data remain in the network, to meet the data-rate (throughput) requirement, data needs to be pumped into the network at a rate higher than the required rate. Thereore, the data enters the network at the rate {(1 + ɛ)r }. The potential unction o a queue which belongs to session o size q is set to the same as that in [16]. φ (q) = e α q. (9) I the queue o some source S() is beyond q max, the overtop data is stored at another special overlow buer. The potential unction o an overlow queue o session o size b is b φ (qmax ). This type o parameter coniguration guarantees the stability o a distributed scheme as long as the network capacity is large enough to meet the throughput demands o all sessions, which will be detailed in Section V. The queue-balancing low control in this paper is similar to that in [16], but the latter cannot be applied to multi-hop CR networks directly. Some nontrivial changes are necessary or the ollowing reasons. First, the link capacities change with dynamic resource allocations. Second, in multi-hop wireless networks, it is diicult to synchronously execute a distributed algorithm at dierent nodes. The parameters should be conigured appropriately to guarantee the network stability while considering asynchronous scenarios. Third, additional coordination o nodes is needed because o the complex wireless environment, especially caused by primary users moving in and out o channels. (1+ )r1 S(1) (1+ )r Fig. 1. S() q1,r q,r D() q11,t q1,t link link1 D(1) q1,t q,t An illustration or queue-balancing low control IV. NODE-LEVEL RESOURCE ALLOCATION q11,r q1,r By adopting the queue-balance low control, the problem can be transormed to a resource-allocation problem or each link. Speciically, we need to allocate the channel and power resource or each link so as to maximize the decrease o potential unctions in each time slot. The potential decrease is expressed as δ l = φ (q l,t ) φ (q l,t x l ) +φ (q l,r ) φ (q l,r + x l ) (1) where q l,t and q l,r are the queue size o link l and session at the transmitter and the receiver, respectively. This objective unction balances the size o queues. Because o the exponential potential unction, the network provides higher priority to those sessions with large queue sizes and larger size dierences between queues o the transmitter and the receiver o a link. For allocating resources to maximize the potential decrease, we consider three types o resource allocation, ranging rom small-scale to larger-scale adjustments. Adjust the data rate over a link or each session according to the queue sizes or each session at transmitters and receivers. Allocate the power o the links transmitting rom the same nodes. The power adjustment changes the link capacities, but doesnot aect other links because o intererence-ree channel allocation. Consider the time interval o a time slot as a unit o time, so it is omitted in the rest o this paper or simplicity o expression.

Change the channel allocation to achieve better perormance. To avoid the intererence between links, nearby nodes need to be coordinated or channel allocation. A. Rate Control over a Wireless Link For a link with ixed capacity, the problem is how to allocate the total capacity to the sessions that go through this link, such that the total decrease o potential is maximized. The total potential decrease over link l is δ Ll = δ l (11) F(l) where F (l) is the set o sessions that go through link l. The problem is to maximize the total potential decrease δ Ll subject to the link-capacity constraint Eq. (5). The linkcapacity constraint is a complicating bundle constraint or all the sessions over this link. We decompose this multiple session rate-control problem into multiple single-session rate-control problems by placing cost on the constraint with a Lagrangian multiplier. l = δ l λ l ( x l C l ) (1) F(l) F(l) where λ l is the Lagrangian multiplier or link l. By decomposition, or each session, the objective o low control is to maximize the decomposed Lagrangian unction. l = δ l λ l x l (13) where the optimal value o λ l is the one that satisies F(l) x l = C l. The maximum potential decrease is achieved when the irstorder partial derivative o l with respect to x l is set to. l x l = α e α (q l,t x l ) α e α (q l,r +x l ) λ l =. (1) The data rate o each session on link l can be calculated as x l = 1 ( λ l α ) + e α (q l,r +q l,t ) λ l α ln α e α q l,r. (15) Especially, or the problem with large enough link capacity, the capacity constraint is not tight. For each session, the optimal data rate is x l = q l,t q l,r B. Power Allocation to Nodes (16) The total potential decrease o the links that are used to transmit data rom node i is = = δ l. (17) δ Ni T (l)=i δ Ll T (l)=i F(l) Considering the eect o power allocation or each link l and each channel k on the total potential decrease or node i, the partial derivative is δ Ni = δ L l C l. (18) P lk C l P lk Obviously, a larger potential decrease on channel k over link l can be achieved with a larger power P lk. However, the transmit power is limited by the equipment s capability. How to allocate the limited power P i o node i to every channel o every link transporting data rom this node is the problem we want to solve. The power allocation can be divided into two levels. First, we consider the power allocation to the channels K (l) with a given link power P l. Then, based on the analysis considering a given link power, we allocate the node power to the links to maximize the total potential decrease δ Ni. The water-illing allocation between the channels within a link can achieve the optimal capacity, even i there are power masks or the channels in cognitive radio networks. Theorem 1: For a given set K (l) o channels and allocated power P l on link l, the water-illing power allocation can achieve the optimal perormance. Let Ψ l be the water-illing level o link l, then the power should be allocated as P lk = min{[w k Ψ l I R(l)k /G l ] +, Q T (l)k } (19) where [a] + = max{, a}, and the water-illing level Ψ l is set to satisy P lk = P l. () k K (l) Proo: Considering only the total power constraint or this link k K (l) P lk P l, the optimal solution can be obtained by the Lagrange multiplier method, which is similar to the proo in [18]. P lk = [W k Ψ l I R(l)k /G l ] +. (1) The partial derivative o C l with respect to P lk can be calculated rom Eq. (). C l W k G l =. () P lk P lk G l + I R(l)k It is obvious rom Eq. () that the derivative is non-negative, meaning that C l will not decrease with the increase o P lk. For the total link-power constraint and the power mask constraints or each channel, it is necessary or the optimal solution that at least one o the constraints is tight or each channel. Thereore, the optimal solution is to allocate the maximum power subject to two constraints, and Eq. (19) ollows. Based on the water-illing power allocation between the channels within a link, each channel belongs to one o three types o states, as shown in Fig.. State 1: No power is allocated because o large I R(l)k /G l, W k Ψ l I R(l)k /G l State : Power is allocated according to the water-illing level but not the power mask, < W k Ψ l I R(l)k /G l < Q T (l)k State 3: Power is allocated according to the power mask. W k Ψ l I R(l)k /G l Q T (l)k

Fig.. QT(l)k k Pkl IR(l)k/Gl Three states or power allocation between the channels within a link Let Kl act be the set o channels in State whose power is determined by the water-illing level or link l. The capacities o the channels in Kl act change when the link power P l changes. According to Theorem 1, the partial derivative o C l with respect to the allocated power P l or link l is C l P l = 1 Ψ l k K act l W k. (3) For the link power allocation, Eq. (18) can be transormed as δ Ni P l = δ L l P l = δ L l C l C l P l () δ Ll / C l is just the Lagrangian multiplier λ l in the last subsection, and C l / P l is given by Eq. (3). δ Ll / P l is non-negative and decreases with the increase o P l. To achieve the maximum potential decrease δ Ni, the optimal method or power allocation between links is to allocate the node power to the links such that the derivative δ Ll / P l is equal or all the links transmitting data rom the same node. C. Intererence-ree Channel Allocation 1) Eect o Changing Channel Occupation: In order to allocate channels eiciently, we irst estimate the potential change as a result o adding or subtracting a channel or each link. Let C l,k be the capacity o link l when the set o channels on link l is K (l)\k i k K (l), or K (l) k i k / K (l). Using the water-illing power allocation, we can obtain C l,k. Let δl l (C) denote the optimal potential decrease over link l achieved by calculating x l as Eq. (15) when the link capacity is C. Finding an optimal λ l,k, which is the Lagrangian multiplier or the case when ω lk changes, we can calculate δl l (C l,k ). This way, the change o the potential decrease is δl l (C l,k ) δl l (C l ). However, δl (C l,k) indicates the potential change, only considering the allocated power or link l ixed at P l. By having the transmitter nodes adjust the power allocation adaptively, the potential decrease can be improved. It is diicult to calculate the exact improvement o the potential decrease, because the channel allocation on other links o this node may also change. Let Ni in be the number o incoming links and the number o outgoing links o node i. We can then estimate the change o the potential decrease as δ L l (C l ) δ L l (C l,k ) N T out (l) p l p l (5) + 1. N out i N out T (l) The above expression represents the eect o adaptive power allocation between the links that have the same transmitter node as link l. Based on the analysis in the last subsection, δl l (C l )/ p l is the same or all the links o a node. I the channel allocation causes a larger dierence between δl l (C l,k )/ p l and δl l (C l )/ p l, the adaptive link power allocation can make a larger improvement. On the other hand, the improvement is larger i the node has more outgoing links, because the power allocation between links has more degrees o reedom. It is deined in the system model that a set o links I (l) have conlict with link l. We also deine N (L l ) as the nodes that are the transmitters o the links in I (l), and N (N i ) as the set o nodes within the interering range o node i. The number o the interering links I (l) o link l is N I l = N in T (l) + N out R(l) + i N (T (l)) N in i + i N (R(l)) N out i 1. (6) The irst two terms are to guarantee that the channel is used or at most one o the node s links. For this purpose, besides the two terms, there are also the links transmitting rom T (l) and the links receiving at R(l), which are included in the third and ourth terms. Because the allocation o a channel or link l means that other Nl I 1 links can t use this channel. Although this channel may achieve dierent perormance on all the interering links because o dierent power masks, the utility divided by Nl I can give an estimation o potential change in terms o the average o all the interering links. Let U lk denote the achieved utility on the potential i link l is assigned channel k. Considering both the change o the potential decrease with ixed link power and the eect o adaptive link power allocation, i k / K (l), U lk = N out T (l) I k K (l), ( δ Ll (C l,k ) δ L l (C l ) ) δ L l (C l ) p l δ L l (C l,k ) p l (N out N I l (N out T (l) + 1). (7) T (l) U lk = + 1) (δ l (C l) δl (C l,k)) (8) NT out (l) N l I δ L (C l ) l p l δ L (C l,k ) l p l. ) Coordination Between Links: When channel allocation to links is optimal or each channel, so is the channel allocation in the network. We can thus consider channels individually by restricting at most one channel change on a link at a time. Allocation o a single channel can be modeled as a weighted independent set problem, which is NP Complete [19]. So, we propose a greedy channel-allocation protocol, achieving suboptimality.

We deine three types o messages, INFO, REQ, OCCUPY, which contain the inormation o channel index, link index, and the corresponding channel-allocation utility. Note that or a given link l, messages are sent to the transmitters N (L l ) o all the links that interere with link l. For simplicity, the messages can be sent to the nodes within two times o the interering range rom node T (l), such that all the nodes in N (L l ) can receive the messages. The ollowing part describes the inormation exchange process at link l or allocating channel k. Algorithm 1 Distributed Channel Allocation 1: (time period 1) : i k(l ) = k then 3: send INFO message : else i didnot receive INFO then 5: send INFO message 6: else i U lk > U lk or all U lk rom received INFO then 7: send REQ message 8: end i 9: i U lk > U lk or all U lk rom received INFO and REQ then 1: use channel k and send OCCUPY message 11: else 1: stop using channel k 13: end i Theorem : The proposed distributed coordination o channel allocation yields intererence-ree channel allocations. Proo: Consider only one channel k in this proo. Suppose there exist two links l and m which interere with each other. Without loss o generality, U lk > U mk. I link l sends INFO or REQ messages, then the channel will not be allocated to link m. I channel k is allocated to link m, link l should send neither INFO nor REQ messages. As the proposed coordination scheme, link l does not transmit INFO messages only i link l intereres with another link j, and link l does not transmit REQ messages only i U lk < U jk or link l has requested the allocation o another channel. In such cases, the channel will not be allocated to link l. Thereore, at most one o the interering links can house the channel, so the channel allocation is intererence-ree. By repeating the proposed distributed coordination o channel allocation until all the nodes either transmit or receive INFO messages during the irst step, the resultant channel allocation is not optimal, but is a maximum set in the sense that the transmitting link set is not contained by any other transmitting set or each channel. V. NETWORK-LEVEL RESOURCE ALLOCATION A. Node-based Distributed Algorithm Based on the above analysis, we propose a node-based distributed algorithm or joint low control, channel allocation and power control. Step 1:Adjust the queue sizes or each node. Step :Broadcast the control inormation needed or resource allocation. Step 3:Pre-determine the resource allocation and broadcast the INFO and REQ messages, i necessary. Step :Determine the channel allocation strategy and the corresponding power and data rate or each session. Exchange OCCUPY messages. In Step 1, add (1 + ɛ)r to the queue size at the source nodes. At the destinations, decrease the size o the queues o the corresponding sessions to. Balance the queue sizes within each node or all o its sessions. In Step, nodes transmit the control inormation to the nodes which transmit data to it. The inormation includes the noise and the inerence rom other systems I ik or each channel, the queue size or each session q l,r or receiver links and the number o the transmitting links and the channelquality eedback G l. In addition, the nodes need to transmit the link number Ni in and Ni out to the nodes N (i). In Step 3, we restrict at most one REQ message transmitted or each link, because the channel occupation utilities are calculated by considering only one channel allocation change based on the current status, although channels are allocated individually as described in the last section. Channel k is chosen i the dierence between U lk and the maximum utility o the received messages is the largest or all channels. In Step, the channel allocation is determined according to the INFO and REQ messages. Based on the allocated channels or each link, the optimal power allocation or the links and the rate control scheme within each link are deployed as described in Section IV.A and IV.B. B. Perormance In order to evaluate the perormance o the proposed algorithms, an ad-hoc secondary network is employed or dynamic simulation in multi-hop CR networks. secondary users are distributed randomly in a 3km 3km area, and sessions in this network have the data rate Mbps, 5Mbps, 8Mbps and 1Mbps, respectively. The parameter ɛ in queue balancing scheme is set to.. 1 channels are considered, each o which has a bandwidth o 5MHz. The path loss is calculated based on the distance using a ree space propagation model as P L = P L d/d, where P L = 7dB and d = 1km. The maximum transmit power is 1dBm and the thermal noise power is 1dBm. Each primary user occupies a channel randomly. To protect the primary users, the power masks are set to guarantee the intererence at the primary users to be less than 1dBm. Figs. 3 and show the stability o queue size and EE throughput, respectively, when there is suicient network capacity. When a new session begins, a period o time is needed to spread packets to the nodes all over the network to build the queue system. Ater this period, the size o queues would not always increase and the required throughput can be achieved i the network capacity is large enough. The condition or network stability is provided in [16]. It can also be observed that the curves converge ater 15 time slots

Queue sizes at source nodes 8 7 6 5 3 1 session 1 session session 3 session Total EE throughput (Mbps) 3 5 15 1 5 Proposed Random 5 1 15 5 3 Time 6 8 1 Number o channels Fig. 3. Stability o the queue sizes o source nodes Fig. 5. Total EE throughput v.s. number o channels EE throughput (Mbps) 16 1 1 1 8 6 session 1 session session 3 session 5 1 15 5 3 Time Fig.. Stability o EE throughput Total EE throughput (Mbps) 3 5 15 1 proposed (5 channels) 5 random (5 channels) proposed (1 channels) random (1 channels) 6 8 1 Number o primary users Fig. 6. Total EE throughput v.s. number o primary users in this coniguration. Thereore, we use the perormance ater 15 time slots as steady-state perormance. Note that we use the average value within a 1 time slot window rather than the value just at this time slot. Fig. 5 presents the total EE throughput perormance o the proposed scheme. The random spectrum allocation and average power & rate allocation are considered as the baselines or the comparison. The result shows the improvement o the proposed scheme. Fig. 6 shows the EE throughput with a dierent number o primary users. With more primary users, the transmit powers o secondary users are limited more strictly, and less network capacity is achieved. When there are 1 channels, the proposed scheme can guarantee the required throughput o all sessions even i there are 1 primary users. With larger number o primary users, the proposed scheme can achieve more improvement, because it can allocate the channels to the secondary links which is ar away rom the primary users occupying them. C. Asynchronous Scenarios So ar, the proposed distributed scheme has been based on an assumption that all the nodes execute this procedure at the same time. We need to consider the case when execution o the resource allocation procedure is not synchronized (i.e., asynchronous scenarios). In such a case, our algorithm in each node is similar to that in the synchronous case, but uses the inormation about other nodes last time when they made resource-allocation decisions. This use o old inormation has the ollowing eects. The queue size inormation is not accurate, which may cause some error in allocating resources to maximize the potential decrease. Nodes have to wait or one period when they want to use a new channel such that all the other interering nodes stop using this channel. 1) Inaccurate Queue Size Inormation: I we use the queuesize inormation received during the last period, the actual potential decrease may not be the same as the estimated value. For the obsolete queue-size inormation o receivers, i the queue size received last time is larger than the current actual

queue size, the estimated potential decrease is larger than that o using current inormation. In the worst case, no data is transmitted out o R(l) and the queues at the transmitters o all the links whose receiver is R(l) have the largest queue size q max. The maximum possible queue size o receiving links at +q l,r R(l) is qmax because o Eq. (16). The numbers o the queues or transmission and reception links in a node are the same. So, the actual queue size is at most q max +q l,r + q l,r. (9) The upper-bound error o the receiver s queue size is q max q l,r. I the queues in this network can grow unboundedly, the network becomes unstable. Let L be the length o the longest low path in the network and F be the number o sessions. Considering the error in queue-size inormation, an appropriate parameter coniguration is needed to guarantee the network stability i the network transmission capacity is large enough. Theorem 3: The network is stable using the proposed asynchronous scheme with ɛ α = 16Lr ln L (3) 3ɛ q max = 1 F (1 + ɛ) ln( ) (31) α ɛ(1 ɛ) i the network has the capacity to transmit (1+ɛ)r or each low. Proo: See Appendix A. From Theorem 3, we can see that α is smaller and q max is larger than those in the synchronous case in [16]. Without synchronization, larger buers are needed in network nodes, and more data may be queued up in the network. ) Overhead o Channel Re-allocation: The other eect o asynchrony is the waiting or a channel re-allocation. I a link wants a new channel to be allocated, it has to wait or one time slot and then grab the channel in order to guarantee that all the interering links have stopped using the channel, as shown in Fig. 7. Let t w l 1l be the waiting time when link l grabs the channel on link l 1. Considering the channel-switch overhead, i link l wants to grab a new channel away rom link l 1, the link l s channel accommodation utility U lk should be divided by (1 + t w l 1l ) or normalization, and then compares it with U l1k. During the waiting time, the channel switch will succeed i it gains a larger normalized utility rom the messages received rom others. However, it is possible that some larger accommodation utilities may appear during the waiting, such that the channel can t be used or a long time. To avoid this situation, i any o the links which are holding the channel stops using the channel, meaning that the channel is in the switching mode, the link stops transmitting REQ messages or this channel in one time slot even i it has a larger normalized channel holding utility. I an OCCUPY message rom link l was received in the last time slot but is not received in the Link 1 Link Link 3 Total Fig. 7. waste time channel occupation occupation request An example asynchronous channel switch current time slot, the node can know that link l stopped using the channel. Consider a link conlict graph, in which links are the vertexes and there are edges between the interering links. The ollowing theorem gives an upper bound o resource waste because o waiting or a channel switch. Theorem : Let N be the vertex number o the maximum clique in the link conlict graph, the portion o time wasted by the proposed scheme or the asynchronous case is at most N 1 N 1. Proo: Any two vertices in a clique have an edge to connect with each other. The links in a clique interere with each other and only one o them can transmit at a time. Consider the smallest period rom the time the channel is held by a link to the time that the channel returns to this link. No link uses the channel more than once, so this period is the smallest. The total waiting time in this period is t w = t w l 1l + t w l l 3 + t w l 3l +... + t w l (n 1) l n. (3) Because t w l il j + t w l jl i = T, t w = nt (t w l l 1 + t w l 3l + t w l l 3 +... + t w l nl (n 1) ). (33) The second term o the right side o the above equation is just the waiting time the links hold the channel as the opposite sequence rom l n to l 1, which is at least T seconds, so t w n 1 (n 1)T and the portion o waiting time is at most n+(n 1). For the clique with N vertices, the waiting time portion is at N 1 most N+(N 1). In a graph containing a maximum clique o N vertices, or the same period as the above analysis, the wasted time is also (n 1)T, but it is possible that more than one link hold the channel at the same time. Thereore, the waiting time is less than N 1 N 1. Considering the link capacity at each time slot, the capacity loss during the waiting is less than that at the next time slot which has the largest channel holding utility in that slot. Corollary 1: The average capacity in the asynchronous case is more than 1/ o that in the synchronous case.

Queue size 6 5 3 1 Node 1 Node Node 3 Node Node 5 EE thorughput (Mbps) 3 1 session 1 session session 3 session 1 3 5 Time 5 1 15 Time Fig. 8. Change o queue sizes or spectrum dynamics Fig. 9. EE throughput in case o insuicient network capacity without any drop mechanism D. Spectrum Dynamics One o the advantages o the proposed scheme is the adaptation to spectrum dynamics. In multi-hop CR networks, because o the presence o primary users, the channel condition varies within the licensed spectrum. On the channels the primary users occupy, the nearby CR users cannot use the power beyond Q L ik to protect the primary users communication. Using the queue-balancing low control, the algorithm need not control the data low to the links with better channel situation when making a resource allocation, but only consider the queue sizes that are aected by the spectrum dynamics and represent the channel situation. The proposed scheme is more adaptive to spectrum dynamics and requires only local inormation, which is essential or distributed control. The ollowing theorem gives the requirement o the transmission capability o the network. In spite o the varying channel situation, as long as the average EE throughput can meet the requirement, the network stability can be guaranteed. Theorem 5: I the network can transmit more than +ɛ ɛ r or each session, then the network stability can be guaranteed by adjusting the conigurable maximum queue size. Proo: See Appendix B. Fig. 8 gives an example or the change o queue size i the capacity o one o the links changes. Our simulation scenario is a simple 5-link chain, and the session low goes rom node 1 to node 6. Each link has 1Mbps capacity, and the capacity o link changes rom 1Mbps to 5Mbps at time slot 3. I the capacity o a link decreases, the queue sizes o the nearby nodes increase. This way, the resource allocation schemes at the nodes which are ar away rom the changed link can be adapted using the spectrum dynamics inormation without exchanging messages. E. Insuicient Network Capacity The analysis in this paper is based on the assumption that the network capacity is large enough or handling sessions requirements. In case the throughput requirements o sessions are higher than the network capacity, admission EE throughput (Mbps) 8 7 6 5 3 1 session 1 session session 3 session 5 1 15 Time Fig. 1. EE throughput in case o insuicient network capacity with a drop mechanism control is necessary. Estimating the network capacity is in general a diicult problem [][3][], but we can estimate the network congestion according to the queue sizes in the network, especially the sizes o the overlow queues in the source nodes. For example, i the size o an overlow queue increases with a speed larger than 1 r or consecutive time slots, the low s requirement is not met, so this low should be dropped to return its network resources. I the network capacity is not large enough to transmit all traic, some adjustment at the sources o each session is necessary. The adjustment method depends on the properties o the sessions services. When the overlow queue is too large at the traic sources, the source nodes should decrease the data rate into the network, i possible, or reject some sessions i the throughput requirements are strict. Figs. 9 and 1 show the perormance comparison o the cases with or without session drop process. The simulation coniguration is the same as in Section V.B, except the bandwidth o each channel is.1mbps. Without any drop mechanism, only session 1 can achieve the required throughput.

When a drop mechanism is deployed, the required throughputs o both session 1 and session are provided by dropping session 3 and. VI. PRACTICAL ISSUES We now discuss some practical issues associated with the proposed scheme. Data remaining in the network: The queue-balancing method allows a certain portion, ɛ, o the data to stay in the network or a long time or orever. Thereore, this method is suitable only or long-lived sessions. Channel coding is a way to correct the lost inormation. Interleaving can avoid bursty continuous errors to ensure the perormance o data-correctness approaches []. Estimation o power masks: CR nodes can obtain the threshold Q L ik in several ways. I the primary receiver is a transmitter as well, Q L ik can be obtained based on the received signal strength o primary users. For a dummy primary receiver, the power mask can be estimated by restricting the intererence at the edge o the service range o the primary transmitter. For example, the IEEE 8. Working Group [1] proposed a power-masking scheme by considering the distance rom a CR node to the TV transmitter. Estimation o L: In order to prevent the data o a session rom spreading to all the nodes o the network, the maximum number o hops rom the sources to the destinations should be restricted. The author o [16] provided a method or inding L heuristically by trying dierent values. Limited buer size: The required maximum queue size to guarantee the network stability is given or the proposed scheme. I the buer size o each node is limited and not enough or the required value, the required capacity or network stability would be larger than that in our analysis. Not only the potential decrease but also the queue sizes in the limited buers should be considered when making resource allocations. VII. RELATED WORK In [1] and [], the multi-commodity low problem was investigated or wireless networks rom an inormation-theoretic perspective. They adopt a combinational intererence model to avoid intererence between links, making the power control unaect the perormance o other nodes. Some bounds o the network capacity are derived. Based on the multi-commodity low model, several publications ocus on the cross-layer resource-allocation problem in wireless networks. The authors o [6] investigated the routing and resource allocation in wireless networks. It is assumed that the link capacity is only a unction o local resource allocation, but they did not consider the spectrum reuse at dierent links that are ar apart rom each other. In [5], routing, spectrum and power are controlled jointly or wireless networks. However, they control the spectrum allocation to ind a conlict-ree combination a priori. When the spectrum situation changes at any node in the network, this scheme must work again or the whole network, and the ollowing routing and power control have to iterate rom the initial state, which cannot work well or CR networks with spectrum dynamics. For CR networks, there are only a ew existing schemes or cross-layer joint resource allocation. In [5] and [6], centralized and distributed schemes are proposed or CR networks, respectively. Both o them only considered dierent available channels at each node or CR. By contrast, we considered more characteristics o CR. The power mask model is considered as a general case o the channel availability model. Spectrum dynamics, which are seldom analyzed in the previous research on resource allocation, are considered in this paper. With the queue-balancing low control method, the proposed distributed resource-allocation scheme can deal with spectrum dynamics well by using local queue inormation only. VIII. CONCLUSIONS In this paper, the resource-allocation problem in multihop CR networks is modeled as a multi-commodity low problem. To solve this problem, the queue-balancing low control method is proposed. Considering the characteristics o CR, we extend the queue-balancing to multi-hop CR networks with varying link capacity and dynamic spectrum conditions. Using the queue-balancing ramework, we analyzed distributed resource allocation. The data rate, power and channel allocation are determined by the local queue size and adjusted to relect the status o channels and the throughput requirement on each link. The optimal rate control or each session on a link is derived irst. Power allocation at nodes is divided into two levels, which are the power allocation between links and the water-illing power allocation or the channels within a link. Coordination between links or channel allocation is achieved via some control messages according to the estimated channel holding utilities or each channel on each link. Based on the analysis on resource allocation, a node-based distributed algorithm is proposed or joint low control and resource allocation. The parameters in the proposed scheme are conigured to achieve the network stability in the asynchronous case. The perormance degradation caused by the waiting or a channel switch is evaluated and compared with the synchronous case. Based on the queue-balancing scheme, the resource allocation or spectrum dynamics can be adjusted only by using local queue inormation. The adaptation to the spectrum dynamics is also investigated. In the insuicient network capacity cases, the drop mechanism can reject some o the session to satisy the throughput requirement o the remaining sessions. For the implementation o the proposed scheme, we consider several practical issues as well. ACKNOWLEDGMENTS The work reported in this paper was supported in part by NSF under Grants CNS-51998 and CNS-7159, China Scholarship Council, Intel Corporation, Nokia Corporation and BUPT PhD Innovation Foundation.

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Hou, A distributed optimization algorithm or multi-hop cognitive radio network, Proc. o IEEE Inocom 8, to appear, Apr. 8 APPENDIX A PROOF OF THEOREM 3 To prove the network stability, meaning that the queue sizes or each session at all nodes are inite, the basic idea is the total potential unction decreases i the queue size is large enough. When the potential unction is inite, the queue size must be inite. In the resource-allocation process at every time slot, there are three parts that change the potential o the network. 1) Add (1 + ɛ)r data into the network rom the sources. ) Balance the queues or each session at each node. 3) Transmit data through links to decrease the potential and remove the data out o the network rom the corresponding queues at the destinations. For the irst part, because the potential unction is a monotonous increasing convex unction, the increase o the potential o session is at most (1 + ɛ)r φ (s i), where s i is the source queue size ater adding the traic. For the second part, balancing the queues or each session will never increase the potential because o the exponential potential unction. For the third part, when the data rate o session over link l is x l, considering the error o the queue size at the receiver e l, the potential decrease δ l is δ l = φ (q l,t ) φ (q l,t x l ) +φ (q l,r + e l ) φ (q l,r + e l + x l ) (3) Using Taylor series to decompose φ (q l,t ) and φ (q l,r + e l + x l ), the above expression can be bounded urther as δ l x l φ (q l,t ) x l φ (q l,t ) x l φ (q l,r + e l ) x l φ (q l,r + e l + x l ) x l φ (q l,t ) x l φ (q l,t ) x l φ (q l,r ) x l e l φ (q l,r + e l ) x l φ (q l,r + e l + x l ) (35) Consider the potential decrease along a path, the size o the queue at the receiver o one link is equal to that at the transmitter o the next hop link, because the queue sizes are balanced or each session at each node. This way, x l φ (q l,r) can cancel x l φ (q l,t ) o the next hop link. Deine qj max as the maximum value o the sizes o the queues along path j, and L (j) as the link set o path j. The total potential decrease along the path is at least x l φ (qj max ) x l φ () l L (j) l L (j) l L (j) x l e l φ (q l,r + e l ) x l φ (q l,t ) x l φ (q l,r + e l + x l ) (36) The path j has the maximum length L. Only considering the path rom the node with the largest queue size to the