MIMO-Aware Routing in Wireless Mesh Networks

Transcription

1 MIMO-Aware Routing in Wireless Mesh Networks Shan Chu and Xin Wang Department o Electrical and Computer Engineering Stony Brook University, Stony Brook, New York {schu, xwang}@ece.sunysb.edu Abstract Multiple-input and multiple-output (MIMO) technique is considered as one o the most promising emerging wireless technologies that can signiicantly improve transmission capacity and reliability in wireless mesh networks. While MIMO has been widely studied or single link transmission scenarios in physical layer as well as rom MAC perspective, its impact on network layer, especially its interaction with routing has not drawn enough research attention. In this paper, we investigate the problem o routing in MIMO-based wireless mesh networks. We mathematically ormulate the MIMO-enabled multi-source multidestination multi-hop routing problem into a multi-commodity low problem by identiying the speciic opportunities and constraints brought by MIMO transmissions, in order to provide the undamental basis or MIMO-aware routing design. We then use this ormulation to develop a polynomial time approximation solution that maximizes the scaling actor or the concurrent lows in the network. Moreover, we also consider a more practical case where controllers are distributed, and propose a distributed algorithm to minimize the congestion in the network links based on steepest descent ramework, which is proved to provide a ixed approximation ratio. The perormance o the algorithms is evaluated through simulations and demonstrated to outperorm the counterpart strategies without considering MIMO eatures. I. INTRODUCTION In recent years, uncoordinated multi-hop wireless networks, such as ad hoc and mesh networks have been gaining increased research interest and application. On one hand, these networks are known to provide a number o desirable eatures, such as easy deployment and maintenance, robustness to node ailures and extended coverage. On the other hand, they are expected to serve an increased amount o data traic with various demands rom dierent users. However, due to the lack o a central controller and the multi-hop, intererence limited nature o the network, it is extremely diicult and challenging to meet the higher data traic requirements. Thereore, it is critical to introduce some new technology that can enable larger network capacity and higher reliability. Multiple-input multiple-output (MIMO) system has been proven to be able to provide high spectral eiciency and increase channel capacity substantially through multiple spatial channels without need o additional spectrum. With multiple antennas at the transmitter and/or receiver, a MIMO system takes advantage o multiplexing to simultaneously transmit multiple data streams to increase wireless data rate and diversity to optimally combine signals rom dierent transmission streams to increase transmission reliability and range. To meet the high data rate requirements, more and more wireless devices are equipped with multiple antennas. MIMO is prominently regarded as a technology o choice or next generation wireless systems such as IEEE 80.6, IEEE 80.n, and the third and ourth generation cellular systems. It is also being considered or supporting peer to peer applications over an inrastructure ree wireless mesh network. Most existing studies on applying MIMO technique in ad hoc and mesh networks ocus on the physical and MAC layers [] [5]. In wireless ad hoc and mesh networks, routing is an important actor that aects the system perormance. From a network layer s perspective, MIMO nodes provide dierent transmission/receiving capabilities rom conventional single-antenna nodes. A node equipped with multiple antennas could possibly transmit/receive more downlink/uplink streams, which can signiicantly impact the determination o optimal routes or traic transmission. Moreover, the option o dierent MIMO strategies, i.e. spatial multiplexing or diversity with dierent levels o degree o reedom, could urther increase the lexibility o routing decisions in a network with MIMO nodes. Thereore, it is o paramount importance to have the routing scheme to be MIMO-aware in order to ully leverage the beneits brought by MIMO into wireless networks. Some earlier work, i.e. [6], has made an eort in designing heuristic routing algorithms and protocols. However, to the best o our knowledge, there is very limited work that has studied the problem o routing in MIMO-enabled networks rom an optimization perspective and it is still not clear theoretically how much beneit can be achieved by taking advantage o the opportunities and addressing the constraints resulting rom the incorporation o MIMO. In this paper, we study the problem o MIMO-aware routing in wireless mesh networks to leverage the beneits brought by MIMO. Dierent rom previous work, we ormally ormulate the multi-source multi-destination multi-hop routing problem in MIMO-based wireless mesh networks as a multi-commodity low problem to model the end-to-end traic, subject to constraints that model the speciic eatures o MIMO transmissions. We allow more lexible cooperation among nodes. Speciically, nodes in the network can perorm many-to-many transmissions, in which a transmitter node can simultaneously transmit to multiple downstream nodes and a receiver node can simultaneously receive rom multiple upstream nodes, and a transmission path can be established in reerence to dierent MIMO channel modes based on the statistics o the channel conditions and dierent traic demands. With a solid ormulation, we provide a sound theoretical upper bound to serves as the reerence or a practical system design through an approximated centralized algorithm with polynomial time

2 complexity, and we also propose a distributed algorithm with provable eiciency to balance the traic over the network and control the network congestion. The rest o the paper is organized as ollows. Section II discusses the related work and we introduce the system model in Section III. We deine the constraints or MIMO-aware routing and ormally ormulate the problem in Section IV. Section V presents an alternative ormulation o the problem to acilitate a centralized polynomial time approximation solution. Then in Section VI, we propose a distributed approximated solution to solve the joint routing and MIMO channel assignment problem. Finally, we provide the simulation results in Section VII and conclude the paper in Section VIII. II. RELATED WORK The application o MIMO technique in wireless mesh and ad hoc networks has gained increased attention in recent years. Although a number o eorts have been made in developing eicient MAC schemes to enable MIMO communications in ad hoc networks [] [5], there is very limited research on designing routing algorithms and protocols to speciically consider the MIMO eatures. The authors in [6] propose a routing protocol or ad hoc networks that leverages the various characteristics o MIMO links. A routing algorithm with QoS provisioning is presented in [7] to exploit the multiplexing gain and intererence cancelation properties o MIMO antennas. The studies in [6], [7], however, are purely heuristic. In [8], the authors consider the problem o jointly optimizing power and bandwidth allocation at each node and multi-hop/multi-path routing in a MIMO-based ad hoc network. Routing is modeled as a simpliied constraint there and the speciic eatures o MIMO transmissions have not been taken into account in establishing routing path. Meanwhile, multi-channel multiradio ad hoc networks have also drawn great research interest [9], [0], []. As MIMO system shares some similarity as multi-channel multi-radio system, those works provide a good reerence or our design. However, dierent rom ixed requency channels, MIMO transmission has more modes and there are more complicated constraints to be considered. III. SYSTEM MODEL We consider a ixed wireless mesh network where nodes are peers to each other. Nodes in the network are equipped with antenna arrays to acilitate MIMO communications. With multiple antennas at the transmitter and/or receiver, spatial multiplexing can be used to transmit multiple independent data streams between a node pair. At the receiver, each antenna receives a superposition o all o the data streams. In a rich scattering environment where the transmission channels or dierent streams are dierentiable and independent, i.e., orthogonal, an intended receiver node can separate and decode its received data streams based on their unique spatial signatures. This achieves the multiplexing gain that can provide a linear increase (in the number o antenna elements) in the asymptotic link capacity. As the transmission quality could be very dierent or multiple spatial paths, r =3 r =3 Fig.. s S 3/4 3/4 v d d r =3 r =3 s S 3/3 3/3 v 3/3 3/3 Illustration o routing in a MIMO-based network. spatial diversity may be exploited to improve transmission reliability. There are dierent types o diversity techniques. When channel inormation is unavailable, dependent streams can be transmitted on dierent antenna elements over multiple time slots and improve transmission quality through space time coding. With adequate channel inormation, a subset o antennas that can transmit signals at better quality can be selected or transmissions through selection diversity, which is shown to outperorm space-time coding []. Although some recent studies have been perormed at the physical and MAC layer to address the challenges o leveraging MIMO advantages in networking, we believe that it is very important or the network layer to be aware o the speciic characteristics o MIMO nodes and make more intelligent routing decisions. Based on the eatures o MIMO strategies, the array o antennas in each node can be grouped to orm dierent MIMO channels, with dierent number o antennas and dierent achievable channel capacity. Figure illustrates the advantage o using MIMO-aware routing. In a network o ive nodes, node s and s have 3 units o traic demand or d and d respectively. The end-to-end paths o the two traic lows both have to go via node v. As each node is equipped with two antennas in this network, we assume that the channel using two antennas has the capacity 4 and the channel using one antenna has the capacity 3 or each link. As node v can receive two independent data streams at most, a conventional routing strategy can route at most 3 units o traic or either s or s as shown in Figure, using the -antenna channel. As a better alternative, MIMO-aware routing can adaptively select the set o MIMO-channels to route the traic, so that all the 6 units o traic demand can be satisied as in Figure, by having each link use the -antenna channel. Due to the speciic eatures o MIMO-based network, we have the ollowing issues to consider in the design. Link capacity. Although in conventional networks the link capacity generally depends on network topology and channel conditions, in MIMO-based networks, it also depends on the size o the antenna arrays o nodes. For a transmission link between a node pair, the link capacity can be chosen rom a set o varied capacities o dierent antenna combinations and strategies. Moreover, more than one combination may be used simultaneously to orm several MIMO channels. The actual capacity o each MIMO channel can be estimated on a periodic basis and the statistics is used in routing decision. Link channel assignment. As an antenna array has limited size, the number o simultaneously used antenna combinations should not exceed the available number o d d

3 antennas o the node, which is known as transmitter degree constraint in scheduling. Meanwhile, or simultaneous transmissions rom multiple spatial channels, the set o antennas used by dierent spatial channels should not overlap, which we name it as antenna compatibility constraint. Also, as dierent antenna combinations have dierent capacities, it is important to determine which antenna combination to use when a route is determined. Intererence consideration. With multiple antennas, a node can receive data streams while canceling intererence streams concurrently, and the total number o received streams depends on the antenna size, which is described as receiver degree constraint in scheduling. From the perspective o routing, the antenna size can be also regarded as a measure o a node s capability o concurrent data receiving and intererence cancellation. Multi-path routing. As nodes are endowed with manyto-many transmission capability by multiple antennas, it is beneicial to incorporate multi-path routing or end-toend lows in order to better exploit multi-path diversity and maximize throughput. While using multi-path may lead to the problems regarding packet re-ordering and loss recovery, these issues have been studied in literature work on multi-path routing and are beyond the scope o this paper. In this paper, we ocus on routing traic between dierent source/destination pairs and corresponding MIMO mode selection. The problem o scheduling the low in a speciic time slot is beyond the scope o this paper. IV. PROBLEM FORMULATION Based on the system model described in Section III, we ormally ormulate the MIMO-aware routing problem in wireless mesh networks as an optimization problem. In order to model end-to-end traic, we use a multi-commodity low model or the routing o data packets across the network. That is, source nodes may send dierent data to their intended destination nodes through multi-path and multi-hop routing. A. Graph representation We represent the multi-hop wireless network via node topology graph G =(V,E E I,F), where V is the set o nodes in the network, F is the set o data lows to be routed, E is the set o directed edges between nodes that can transmit data rom one to the other and E I is the set o directed edges which indicate the intererence rom a transmitter to nodes within its intererence range during data transmission. To be more speciic, given a data link e E, t(e)/h(e) is used to represent the transmission/receiving end o the link e, and there is a directed edge rom t(e) to h(e). In the network, there is a set o sources s, which send data to a set o destinations d, with the end-to-end rate demand vector r. Assume the rate vector has F< V ( V ) components. Each source-destination pair between which there is a traic request is termed as a commodity. Let s()/d() represent the source/destination node or commodity, and r() represent the low that has to be routed rom s() to d(). B. Channel and low constraints Deine the concept MIMO channel (MC) as the MIMO spatial channel over a link that uses a designated set o antennas and corresponds to a speciic MIMO operation mode. Denote the set o MIMO channels over link e as MC(e), and each element MC i (e) MC(e) has a size, denoted as m t i (e) or the transmitter node and m h i (e) or the receiver node, which is the number o antennas used or constructing this MIMO channel at the two ends o the link e. By using dierent sizes, a set o MIMO channels can be constructed to take advantage o spatial multiplexing and/or spatial diversity. Note that, with the calculation o antenna weights at transmitters and transmit over eigen-modes o the channel, MIMO channels can be considered orthogonal [3]. Suppose node v V has antennas. Each MIMO channel (e, i) is associated with a set o antennas o node t(e), which is indicated using the parameter u i,a,e (a =,...,Nt(e) ant). u i,a,e =i and only i the MIMO channel i over link e uses antenna a o node t(e) or transmission. For example, as in Figure, node v has two antennas a 0 and a, which can be used to compose dierent MIMO channels or the link e = v d. Channels 0 and, i.e. (e, 0) and (e, ), both use one antenna, so m t 0(e) = m t (e) =m h 0(e) =m h (e) =, and thus u 0,0,e = u,,e = and u 0,,e = u,0,e =0. Channel and 3, i.e. (e, ) and (e, 3), are constructed by transmitting simultaneously over antennas a 0 and a,sou,0,e = u,,e = u 3,0,e = u 3,,e = and m t (e) =m t 3(e) =. I the MIMO transmission strategy used or channel is spatial multiplexing, i.e., independent data streams are transmitted simultaneously rom the two antennas, the receiver has to use at least two antennas or successul decoding, thereore m h (e) =. Alternatively, i space-time coding, i.e. Alamouti code [4], is used or the transmission over channel 3, the receiver only needs one antenna or decoding so m h 3(e) =. Note that the values o m h i (e), mt i (e) and u i,a,e are easy to obtain o-line and are static or each node. N ant v Each data link e has capacity c i (e) on MIMO channel i, and there is an estimated capacity or a given MIMO channel over a link or an estimation period. The set o MCs and the values o c i (e) can be saved as a look-up table and updated in each estimation period according to the topology/channel condition variations. The length o the period should be properly determined so that the value o c i (e) can correctly relect the actual link condition. We use x i (e) to denote the low on channel i over data link e that carries the data o the end-to-end low session, and deine g i (e) = x i (e)/c i(e) as the utilization o MIMO channel i over link e or all lows. A necessary condition or rate vector r to be achievable is the existence o link low x i (e) that satisies the ollowing low conservation constraints:

4 e E in (v) i MC(e) e:t(e)=s() i MC(e) e:h(e)=d() i MC(e) x i (e) = x i (e) =r(), ; () x i (e) =r(), ; () x i (e),, v s(),d(); (3) where E in (v) and E out (v) are incoming and outgoing edges o node v in the set E. Asthelow capacity constraint, each link should satisy: x i (e) ci(e), e, i MC(e), (4) which can be simpliied as g i (e), e, i MC(e). While constraints ()-(4) are conventional or low problems, the use o MIMO technique imposes new constraints. Even though the degree constraints introduced in Section III are generally ormulated in MAC layer, they actually have a signiicant impact over routing in the network layer. In order to address these constraints, we irst present them with linklow variables in each time slot, and then translate them into end-to-end rate variables or routing purpose. Let I e,i,τ be the indicator variable that has value i and only i channel i is active over link e at time slot τ. Note that the channels over outgoing edges o v in E are considered active i there are data transmissions rom node v, and the channels over incoming edges o v in the set E and E I are considered active i there are data transmissions and intererence transmissions to v respectively. To satisy the degree constraint at the transmitter side, the number o antennas used by the active outgoing edges o a node v must be no larger than its number o antennas Nv ant in each time slot τ: m t i(e)i e,i,τ N ant v, v. (5) Similarly, corresponding to the receiver s degree constraint, the total number o antennas that is required to decode the receiving transmissions, including data and intererence transmissions, should not exceed the receiving capability o the node. Thereore, we have: e E in (v) E in I (v) i MC(e) m h i (e)i e,i,τ N ant v, v. (6) Suppose routing is perormed or each T time slots. Adding these sets o equations or all the T time slots and dividing by T results in the constraints: e E in (v) E in I (v) i MC(e) m t i(e)g i(e) N ant v, v; (7) m h i (e)g i(e) N ant v, v; (8) where g i (e) is the ractional link utilization or channel i over link e. Speciically, g i (e) = x i (e) c = i(e) T τ T I e,i,τ or all e and i. In addition, each node only has a limited number o antennas, and an antenna cannot be used or transmission over dierent MIMO channels simultaneously. To address this antenna compatibility constraint, we use the indicator variable u i,a,e introduced earlier to represent the constraint as ollows: u i,a,e I e,i,τ, τ,v,a. (9) e E out(v) i MC(e) Similarly as in (7)(8), adding these sets o equations or all the T time slots and dividing by T,wehave: u i,a,eg i(e), v, a. (0) C. Optimization ormulation So ar, we have derived the set o constraints or a easible low or routing data packets in a MIMO-based mesh network. There are many dierent objectives o interest that can be solved using an optimization ramework. Based on the constraints, we ormulate the routing problem in the orm o a concurrent low problem, where the desired rate vector is scaled and the objective is to determine the maximum scaling actor λ that satisies the necessary conditions. In this way, the airness in the resource allocation over lows can be ensured. The resulting linear program (LP) is given below: Subject to: e:t(e)=s() i MC(e) e E in (v) i MC(e) max λ, () e:h(e)=d() i MC(e) x i (e) = e E in (v) E in I (v) i MC(e) x i (e) =λr(), ; () x i (e) =λr(), ; (3) x i (e),, v s(),d(); (4) m t i(e)g i(e) N ant v, v; (5) m h i (e)g i(e) N ant v, v; (6) u i,a,eg i(e), v, a; (7) 0 g i(e),x i (e) 0, e, i MC(e); (8) where equations ()-(4) are low conservation constraints, equations (5)-(6) stand or the routing constraints as the result o using MIMO antenna arrays, and equation (7) is used to meet the antenna compatibility constraint. So ar, we have presented a straightorward ormulation with low variables.

5 V. THE CENTRALIZED ALGORITHM The optimization problem ormulated in section IV is linear and can generally be solved by linear optimization algorithms, i.e. simplex method. However, the complexity is still an important concern. In this section, we ollow the work in [5] and [9], and develop a ully polynomial time approximation algorithm using primal-dual algorithm, which is simple to implement and thereore can be potentially applied in a practical wireless network. In order to acilitate the solution, we irst reormulate the problem using edge-path ormulation and generalize the constraints, which is amenable to the development o the algorithm, then we describe the primal-dual algorithm to solve the optimization problem and obtain the maximum scaling actor λ. A. Edge-path reormulation First, note that the set o constraints (5)-(7) share a similar ormat in that each o them concerns a speciic set o link/mimo-channel pairs, so it is possible to generalize them into a simpler orm or easier reormulation. Suppose there are L sets {Q j } composed o link/mimo-channel pairs that are as deined in constraints (5)-(7), then each o these constraints can be stated in the orm as ollows: α i(e)g i(e) β(q j),j =,,...,L, (9) (e,i) Q j where α i (e) and β(q j ) are constants associated with the above constraints. For example, or a node v, constraint (5) concerns the set Q j = {(e, i) e E out (v )&i MC(e)}, the corresponding constants are then α i (e) = {m t i (e) e E out (v )&i MC(e)} and β(q j ) = Nv ant. In this way, although the number o constraints as described in (5)-(7) remains the same, they are generalized into a single ormula (9). In order to have an approximate solution, we irst reormulate the problem into an edge-path ormulation, so that the multi-commodity lows are represented as positive LPs. Let P represents the set o all possible simple paths composed o link/mimo-channel pairs or the commodity. For a path P P that is rom s() to d(), letx(p ) be the amount o low on this path, constraints ()-(4) are then translated to: x(p )=λr(),. (0) P P Furthermore, x i (e), the total amount o low on channel i over link e is given by: x i(e) = x(p ), (e, i). () P P,(e,i) P As g i (e) =x i (e)/c i (e), equation (9) becomes: P P,(e,i) P α x(p ) i(e) β(q j),j =,...,L. c (e,i) Q i(e) j () In this constraint, link/mimo-channel pairs that are both on path P and in set Q j, i.e. (e, i) P Q j, are examined. Consider a single path x(p ) P, rom (), we have (e,i) P Q j c i(e)/α i(e) P P (e,i) Q j α i (e),(e,i) P x(p ) c i(e) β(q j ). Thereore, x(p, j) =β(q j )( (e,i) P Q j c i(e)/α i(e) ) is the maximum amount o low on path P allowed by Q(j). In summary, the edge-path ormulation o the constraints in the original optimization problem is restated as ollows: P P,(e,i) P x(p ), β(q (e,i) Q j)c i(e)/α i(e) j j =,,...,L; (3) x(p )=λr(), ; (4) P P x(p ) 0, P P,. (5) B. Primal-dual solution According to the weak duality property, the objective value o any easible solution o the minimization problem gives an upper bound on the optimal objective o the dual maximization problem. Following [5] and [9], we ormulate the dual o the LP problem and develop a ully polynomial time approximation algorithm using a primal-dual algorithm. Let y(j) be the dual variables or each set Q j, and z() be the dual variable or the rate scaling constraints in (5). The dual o the LP problem is then as ollows: min y(j), (6) j Subject to: (e,i) P j:(e,i) Q j α i (e)y(j) β(q j ) c i(e) z(), P P, ; (7) r()z() ; (8) y(j) 0,j =,...,L. (9) The dual problem is essentially an assignment o lengths to link/mimo-channel pairs, such that j y(j) is minimized. The proposed primal-dual algorithm is given in algorithm. The algorithm initially assigns a weight o δ to all sets Q j, and then proceeds in phases. In each phase we route r() units o low rom s() to d() or each commodity, and a phase ends when all the F commodities are routed. For each commodity, Ther() units o low rom s() to d() are sent via multiple iterations, as in lines 5-3. In each iteration, j:(e,i) Q y(j)/β(q each pair (e, i) is assigned with a length j j) c i(e)/α i(e), and a corresponding shortest path path P rom s() to d() that minimizes the sum o the length is determined by a shortest path algorithm, i.e., the Dijkstra s algorithm. Among the sets in {Q j } that have the intersection with P, we compare their values o maximum allowable low x(p,j), and the one with the minimum value u = min j:(e,i) P &(e,i) Q j x(p,j) is the amount o the low that can be sent on P in this iteration. Moreover, since r() units o low have to be sent or commodity in each phase, the actual amount o low sent is the lesser o u and the remaining amount o low r to

6 Algorithm Centralized Routing 0: Initialize: : y(j) =δ, j {,...,L} and b =0 : while j y(j) < do 3: or =,,...,F do 4: r = r() 5: while r>0 do 6: Assign each pair (e, i) with length l i(e) = j:(e,i) Q j α i (e)y(j)/β(q j ) 7: Find the c i (e) shortest length or each edge: l(e) = min i MC(e) l i(e) 8: Compute the shortest path P rom s() to d() based on {l(e)} 9: Find the bottleneck capacity u =min j:(e,i) P &(e,i) Q j x(p,j) 0: δ =min{r, u}, r r δ : x i(e) x i(e)+δ, (e, i) P : y(j) y(j)(+ δ x(p,j) i) P &(e, i) Q j 3: end while 4: end or 5: b b + 6: end while x 7: ρ =max i (e) j (e,i) Q j c i (e) 8: Output λ = b ρ make up r() in this phase. Once a low is sent via a path, the weights o the sets {Q j } associated with the link/mimochannel pairs that carry the low is updated, as in line. The algorithm then alternates between sending low along shortest paths and adjusting the length o the link/mimo-channel pairs along which low has been sent until an optimal solution is reached. The complexity o the primal dual algorithm mainly lies in solving a sequence o shortest path problems. Following [5], it can be shown that by choosing δ and appropriately, the solution can get as close to the optimum solution as desired at the expense o increasing running time, as in the ollowing remark. Remark I: The algorithm computes a ( ) 3 optimal solution to the scaling actor o the maximum concurrent low problem in time polynomial in F, L, V and /, where F is the number o commodities, L is the number o constraining sets, and V is the number o nodes. VI. THE DISTRIBUTED ALGORITHM The primal-dual algorithm in the previous section gives an upper bound on the achievable maximum concurrent throughput. In many practical wireless mesh networks, it is important to develop a distributed algorithm, where the computing o routes is perormed in a distributed manner to approach a global optimization objective. It is thereore more practical to use an alternative objective unction, i.e. to optimally distribute the end-to-end traic into dierent paths and link/channel pairs thus balance the load and control the congestion o the network. In this section, we ollow [6] and derive a distributed version o the MIMO-aware routing algorithm in wireless mesh networks that can achieve ast convergence to the near-optimum solution. We assume that each commodity is associated with an agent. The multiple agents make parallel routing decisions without coordination with each other. The only accessible global inormation or each agent is a common clock and the utilization level o the network edges. The objective is to route r() amount o low or low rom s() to d() or all F, possibly along several paths, such that the maximum ratio o the total low routed along an link/mimo-channel pair (e, i) to its capacity is minimized. In other words, we aim to distribute the traic evenly in the network and hopeully no (e, i) would be congested or overloaded. Recall that the utilization o a MIMO-channel over an edge e is previously deined as g i (e) = x i (e) c, and i(e) the objective is to minimize max (e,i) g i (e). The distributed scheduling scheme is described in Algorithm. Throughout the algorithm, x i (e) and l i (e) are the current low value and length o (e, i) or the agent o commodity respectively, and x is the amount o low that has been routed in the current phase or commodity. Algorithm Distributed Routing 0: Initialize: : Set x i (e) ci(e)/f, x = c i(e)/f and l i (e) = c i (e) k x i (e)/(ci(e)) or each link/mimo-channel pair (e, i) and commodity : or N p =(logk)/ phases do 3: For each commodity, doinparallel: 4: while x <r()/n p do 5:. subroutine: PRE-CHECK 6:. Deine the capacities c i (e) = x i (e)/ log k i the channel pair (e, i) is not yet tagged 7: 3. Find the shortest path P rom s() to d() under the current length unction {l i (e)} 8: 4. Compute a blocking low x(p ) under capacities {c i (e)} along the shortest path P 9: 5. Δx =min{x(p ),r()/n p x } 0: 6. x i (e) x i (e)+δx and l i (e) = c i (e) k x i (e)/(ci(e)), (e, i) P, x x +Δx : end while : x =0, 3: end or Similar to the centralized algorithm, the distributed algorithm is also based on the steepest descent ramework as in [5]. Let k be the number o (e, i) pairs in the network, obviously k O( E ). The algorithm goes though N p = (log k)/ phases. A low o amount r()/n p is routed or each commodity in each phase and a easible solution is derived at the end. For each phase, the process is urther divided into steps, as in the while loop in line 4-. In each step, each commodity perorms in parallel to route a raction o its own low. Dierent rom the case in the centralized algorithm, we have the additional problem o how to eiciently perorm concurrent routing in a distributed scenario, as concurrent attempts to route on a shortest path may lead the path to be no longer shortest and result in

7 Algorithm 3 Subroutine: PRE-CHECK : or Nodes v {v} do : i Constraint (5) is not satisied then 3: Tag (e, i) {(e, i) e E out(v ), i MC(e)} 4: end i 5: i Constraint (6) is not satisied then 6: Tag (e, i) {(e, i) e E in(v ) E I in(v ), i MC(e)} 7: end i 8: i Constraint (7) is not satisied or antenna(s) {a} o node v then 9: Tag (e, i) {(e, i) a {a},u i,a,e =} 0: end i : end or : or Nodes v that is connected to any node in {v} by any edge e E E I do 3: i Constraint (6) is not satisied then 4: Tag (e, i) {(e, i) e E in(v ) E I in(v ), i MC(e)} 5: end i 6: end or unpredictable oscillations. To handle this problem, a special approach is to guarantee the so-called step-size constraint, that the length increase o any link/channel pair (e, i) can be no larger than an raction. Throughout the algorithm, we initially route a tiny amount o low o all commodities on all link/mimo-channel pairs, and later increase the low multiplicatively. Although the initial pre-low may not even satisy the low conservation constraints, as its total capacity is o the actual capacities, it has eect on the optimality only to the extent. In each step, the algorithm computes the shortest path based on the current length unction, and determines the blocking low along the path, which is the maximum amount o low that can be routed in the path under the capacity constraint c i (e) or each (e, i). By computing the blocking low, it saturates at least one edge on the path, which eectively reduces the number o steps. In order to make the solution easible, especially or MIMObased networks, we revisited the constraints in equations (5)- (7). Denote {v} as the set o nodes that are in the augmenting paths o the previous step. Recall that g i (e) = x i (e) c can i(e) be regarded as a measure o congestion o channel i over link e. Thereore, a PRE-CHECK step is added at the beginning o each step, as in algorithm 3, so that each node that is included in the augmenting paths o the last step examines i it still satisies constraints (5)-(7). I either (5) or (6) is not satisied, the node can no longer accept extra load, so its incident edges are set to have capacity 0 or all the possible MIMO channels. I (7) is not satisied, it indicates that some o the antennas, say a, o the node is ully-loaded, so MIMO channels that have u i,a,e =are set to have capacity 0. We also check the nodes that are connected to nodes in the augmenting paths by edges in E E I, in order to account or the intererence rom the lows in the augmenting paths. We irst prove that the algorithm can achieve a ( + O()) approximation. The analysis proceeds as in [6], but is slightly dierent since a link can be associated with several MIMO channels in our algorithm. Denote Φ as the potential o the network: Φ= (e,i)(k / ) gi(e). (30) Assume the optimum value o max (e,i) g i (e) is, so the optimum value o Φ satisies Φ k /. Consider phase p and step t,letl (t) and l (t) be the length o the shortest path at the beginning and the end o the step or commodity. In each step, each commodity simultaneously augments its low along certain paths. Suppose commodity augments low Δx i (e) along (e, i), the total additional low is Δx i (e) = Δx i (e). We irst calculate the overall increase in Φ due to the augmentation along (e, i): ΔΦ(e) =k xi(e)/ci(e) (k Δxi(e)/ci(e) ) (3) k x i (e)+δx i (e) c i (e) Δx i(e) log k c i (e) (3) = Δx i (e)log k k gi(e) /, (33) c i (e) where g i (e) is the utilization actor ater the augmentation step. The inequation (3) is derived rom the inequality e a ae a by letting a = Δxi(e)logk c i(e). Note that c i(e) kgi(e) / is the length o (e, i) in the next step, denoted as l i (e). Based on the above inequation, the total increase in Φ at the end o this step is: ΔΦ (e,i) P log k Δx i = log k log k (e) l i (e) (34) Δx l (t) (35) Δx ( + )l (t), (36) where P is the shortest path ound on line 7 o Algorithm, Inequation (34) is derived directly rom (33), equation (35) is derived rom the act that the blocking low values on all edges o the path P are the same and equal to Δx, and l (t) = (e,i) P l i (e) is the length o the shortest path. Inequation (36) is rom the step-size constraint which ensures that the length increase o each edge can be at most an raction o the original length, i.e. l i (e) ( + )l i (e). Note that l (t) l (p) where l (p) is the length o the shortest path at the end o phase p. As we route r()/n p = r()/ log k amount o low or each commodity in each phase, we can estimate the change in the potential during phase p as ollows: Φ(p) Φ(p ) log k Δx (t)( + )l (t) (37) t ( + ) r()l (p). (38) Denote the optimum solution to the problem as {x i (e) }.Note that l i (e) or all is the same at the end o phase p, denoted as l i (e) p. For each (e, i), since x i (e) c i (e),wehave:

8 Traic demand Traic demand Number o MIMO Channels Number o MIMO Channels Fig.. Grid topology: impact o traic demand. Fig. 4. Grid topology: impact o the number o MIMO channels Number o Flows Number o Flows Traic demand Traic demand Fig. 3. Grid topology: impact o the number o lows. Fig. 5. Random topology: impact o traic demand. Φ(p) = (e,i) c i(e)l i(e) p (e,i) x i (e) l i(e) p r()l (p), (39) as l (p) is the shortest path length rom s() to d() and the total low amount is r(). Combining (38) and (39), we have Φ(p) Φ(p )/( ( + )). Initially, g i (e) =0 or all (e, i), so Φ(0) = k. As N p = log k/, we have log k/ Φ(N p ) k( + ) k O() k / k O() Φ. It can be proved that an k O() -approximation o Φ yields a ( + O()) approximation o max (e,i) g i (e). Once the algorithm runs to the end, we can get the solution with {x i (e)}, which actually includes the end-to-end routes and the corresponding link/mimo-channel pairs or each low commodity. From [6], it can be shown that the convergence time o the proposed algorithm is bounded and essentially linear in the maximum path length o the network. We then arrive at the ollowing remark. Remark II: The distributed algorithm achieves an +O()- approximation to the optimum solution. The while loop can ends in O(L(log k log(f/))/ 4 ) steps, where L is the largest number o edges in a path. VII. PERFORMANCE EVALUATION In this section, the perormance o our proposed algorithms is evaluated through simulations. Our goal is to veriy that by adaptively selecting a set o MIMO channels or each link subject to MIMO constraints, the MIMO-aware routing can achieve better perormance under dierent network settings, compared with the reerence non-mimo-aware routing strategy which does not have the lexibility to switch between channels and always uses the MIMO-channel with the highest capacity or each link. The evaluated perormance metrics are the objectives o the proposed algorithms, namely the scaling actor λ (which is also a measure o achievable throughput) or the centralized algorithm and the maximum utilization actor g = max (e,i) g i (e) or the distributed algorithm. We generate both grid and random topologies, and run simulations with dierent parameter settings. In each evaluated network, a node is equipped with an array o antennas to acilitate MIMO transmission. For a link with Nt ant /Nh ant antennas at transmitter/receiver ends, we consider up to Ñ = min{nt ant,nh ant } MIMO channels are available to the link, with each MIMO channel corresponding to one MIMO operation mode and having the degree-o-reedom value ranging rom to Ñ. The channel capacity value is estimated by averaging over a sequence o ading coeicients and set as an empirical parameter or each topology setting. We generally assume all the nodes have the same number o antennas N ant to show the perormance improvement by MIMO-aware routing, and we also present the perormance under dierent values o N ant. The traic in the network is modeled by two parameters: the number o lows F and the demand o each low r(). For simplicity, all lows are assumed to have the same demand, whose value is normalized to the capacity o MIMO-channel with the degree-o-reedom. The constants δ and used in the algorithms are set as empirically derived values. The deault values o N ant, F and r() are 4, 5 and 0.5 respectively, and the network has 30 nodes i not otherwise speciied. For the clarity o comparison, results are normalized with regard to the minimum value in each igure. We irst study the perormance in a grid topology. Consider a 5 6 grid topology with 30 nodes and each node has at most 4 neighbors. Dividing the grid into our quadrants and the our nodes centered in each quadrant are set as sinks or lows. The destination o a node is the sink node that is closest to it. As traic demand increases in igure, MIMO-aware routing consistently obtains a larger value o scaling actor λ (up to 5% higher) and a smaller maximum utilization actor g (up to 55% lower) than that or non-mimo-aware routing. With an increased number o lows, MIMO-aware routing improves

9 Number o Flows 0 Fig. 7. Fig Number o Flows Random topology: impact o the number o lows. 3 4 Number o MIMO Channels Number o MIMO Channels Random topology: impact o the number o MIMO channels. λ up to 5% and reduces g up to 50% as in igure 3. The results show that by being aware o the MIMO constraints and adaptively selecting MIMO channels, the traic demand in the network can be better served. We can urther observe rom igure 4 that the advantage o MIMO-aware routing is even more signiicant with the increase o the number o MIMOchannels, as the improvement o λ and g increases rom 0% to % and 3% to 45% respectively, when the number o MIMO channels in each link increases rom to 4. With more MIMO channels, there are more options or perorming more lexible routing. Figures 5-7 show the perormance o our routing algorithms in random topologies. A random topology is generated by populating nodes randomly in a grid. The transmission range is set as 00, and each topology generated is ensured to be connected. For each low, the source and destination are randomly selected rom the set o nodes in the network. Each data point is obtained by averaging over 0 dierent random topologies. In igure 5, a 33% increase in λ and a 45% decrease in g are achieved with increasing traic demand. As the number o lows in the network increases in igure 6, MIMO-aware routing outperorms its counterpart by up to 45% higher λ and 3% lower g. The results in random topologies are consistent with that in the grid topology and demonstrate that being MIMO-aware is an eective way to leverage MIMO beneits and improve routing perormance. The better perormance also exists or dierent number o MIMO-channels, as in igure 7. VIII. CONCLUSIONS As a promising technology to improve transmission capacity and reliability in wireless mesh networks, MIMO has been studied extensively in physical and MAC layers, but has not drawn much attention rom network layer s perspective. In this paper, we propose the concept o MIMO-aware routing and investigate how it can urther leverage the advantages brought by MIMO. We irst present constraints that capture the characteristics o MIMO transmissions, and mathematically ormulate the MIMO-enabled multi-source multi-destination multi-hop routing problem into a multi-commodity low problem. We then propose a centralized algorithm to provide an approximated solution to achieve maximum concurrent low in the network, as well as a distributed algorithm that minimizes the maximum congestion o link/mimo-channels. The perormance o our algorithms is evaluated through simulations with varied traic demands, number o lows, and available MIMO channels. The results demonstrate that our MIMOaware routing algorithms signiicantly outperorm the routing scheme that does not consider MIMO transmission eatures and constraints in all the test scenarios. The results in this paper provide a basis or our uture work on practical MIMOaware routing protocol design. REFERENCES [] M. Hu and J. Zhang. MIMO ad hoc networks: Medium access control, saturation throughput, and optimal hop distance. Journal o Communications and Networks, Special Issue on Mobile Ad Hoc Networks, pages , 004. [] M Zorzi, J Zeidler, A. Anderson, B. Rao, J. Proakis, A.L. Swindlehurst, M. Jensen, and S. Krishnamurthy. Cross-layer issues in MAC protocol design or MIMO ad hoc networks. IEEE Wireless Communication Magzine, (4):6 76, August 006. [3] K. Sundaresan, R. Sivakumar, M. Ingram, and T-Y. Chang. A air medium access control protocol or ad-hoc networks with MIMO links. In Proc. IEEE INFOCOM 004, June 004. [4] S. Chu and X. Wang. Opportunistic and cooperative spatial multiplexing in MIMO ad hoc networks. In Proc. ACM MobiHoc 008, pages 63 7, May 008. [5] S. Chu and X. Wang. Adaptive and distributed scheduling in heterogeneous MIMO-based ad hoc networks. In Proc. IEEE MASS 009, pages 7 6, Oct 009. [6] K. Sundaresan and R. Sivakumar. Routing in ad-hoc networks with MIMO links. In Proc. IEEE ICNP 005, pages 85 98, 005. [7] A. Gkelias, F. Boccardi, C.H. Liu, and K.K. Leung. MIMO routing with QoS provisioning. In Proc. ISWPC 008. [8] J. Liu, Y.T. Hou, Y. Shi, and H. Sherali. Cross-layer optimization or mimo-based wireless ad hoc networks: Routing, power allocation, and bandwidth allocation. IEEE J. Select. Areas o Commun., (6):93 96, August 008. [9] M. Kodialam and N. Thyaga. Characterizing the capacity region in multi-radio multi-channel wireless mesh networks. In Proc. ACM Mobicom 05, pages 73 87, May 005. [0] M. Alicherry, R. Bhatia, and L. Li. Joint channel assignment and routing or throughput optimization in multi-radio wireless mesh networks. In Proc. ACM Mobicom 05, pages 58 7, May 005. [] X. Lin and S. Rasool. A distributed joint channel-assignment, scheduling and routing algorithm or multi-channel ad-hoc wireless networks. In Proc. IEEE Inocom 07, pages 58 7, May 007. [] Y. Pan and S. Aissa. Perormance analysis o selective space-time coding and selection diversity under perect and imperect CSI. In Proc. PIMRC 005, pages , September 005. [3] D. Tse and P. Viswanath. Fundamentals o Wireless Communication. Cambridge University Press, May 005. [4] S. M. Alamouti. A simple transmit diversity technique or wireless communications. IEEE J. Select. Areas Commun., pages , October 998. [5] N. Garg and J. Koenemann. Faster and simpler algorithms or multicommodity low and other ractional packing problems. In Proc. FOCS 98. [6] B. Awerbuch, R. Khandekar, and S. Rao. Distributed algorithms or multicommodity low problems via approximate steepest descent ramework. In Proc. SODA 07.