Diff-Max: Separation of Routing and Scheduling in Backpressure-Based Wireless Networks

Transcription

1 Diff-Max: Separation of Routing and Scheduling in -Baed Wirele Network Hulya Seferoglu and Eytan Modiano Laboratory For Information and Deciion Sytem Maachuett Intitute of Technology {heferog, Abtract The backpreure routing and cheduling, with throughput-optimal operation guarantee, i a promiing technique to improve throughput over wirele multi-hop network. Although the backpreure framework i conceptually viewed a layered, the deciion of routing and cheduling are made jointly, which impoe everal challenge in practice. In thi work, we preent Diff-Max, an approach that eparate routing and cheduling and ha three trength: (i) Diff-Max improve throughput ignificantly, (ii) the eparation of routing and cheduling make practical implementation eaier by minimizing cro-layer operation; i.e., routing i implemented in the network layer and cheduling i implemented in the link layer, and (iii) the eparation of routing and cheduling lead to modularity; i.e., routing and cheduling are independent module in Diff- Max and one can continue to operate even if the other doe not. Our approach i grounded in a network utility maximization (NUM) formulation of the problem and it olution. Baed on the tructure of Diff-Max, we propoe two practical cheme: Diff-ubMax and wdiff-ubmax. We demontrate the benefit of our cheme through imulation in n-2, and we implement a prototype on martphone. I. INTRODUCTION The backpreure routing and cheduling paradigm ha emerged from the pioneering work in [], [2], which howed that, in wirele network where node route packet and make cheduling deciion baed on queue backlog difference, one can tabilize queue for any feaible traffic. Thi eminal idea ha generated a lot of reearch interet. Mot importantly; it ha been hown that backpreure can be combined with flow control to provide utility-optimal operation guarantee [3]. The trength of thee technique have recently increaed the interet on practical implementation of backpreure framework over wirele network, ome of which are ummarized in Section VI. However, the practical implementation of backpreure impoe everal challenge mainly due to the joint nature of the routing and cheduling algorithm, which i the focu of thi paper. In claical backpreure, each node contruct per-flow queue. Baed on the per-flow queue backlog difference, and by taking into account the tate of the network, each node make routing and cheduling deciion. Although the backpreure framework i conceptually viewed a layered, the deciion of routing and cheduling are made jointly. To better illutrate thi key point, let u dicu the following example. Thi work wa upported by NSF grant CNS-95988, ONR grant N , ARO Muri grant number W9NF (a) (b) Diff-Max Fig.. Example topology coniting of three node; i, j, k, and two flow;, 2. Note that thi mall topology i a zoomed part of a large multi-hop wirele network. The ource and detination node of flow and 2 are not hown in thi example, i.e., node i, j, k are intermediate node which route and chedule flow and 2. U i and U 2 i are per-flow queue ize and V i,j and V i,k are per-link queue ize. (a) : Node i determine queue backlog difference at time t; D i,j (t) = U i (t) U j (t), D i,k (t) = U i (t) U k (t), where {, 2}. Baed on thee difference a well a the channel tate of the network, C(t), it make joint routing and cheduling deciion. (b) Diff- Max: Node i make routing deciion baed on the queue backlog difference at time t; D i,j (t) = U i (t) U j (t) V i,j(t), D i,k (t) = U i (t) U k (t) V i,k (t), where {, 2}. Separately, node i make the cheduling deciion baed on V i,j (t), V i,k (t) and C(t). Example : Let u conider Fig. (a) for backpreure operation. At time t, node i make routing and cheduling deciion for flow and 2 baed on the per-flow queue ize; Ui (t), U i 2 (t), a well a the queue ize of the other node, i.e., node j and k in thi example, and uing the channel tate of the network C(t). In particular, the backpreure determine the flow that hould be tranmitted over link i j by = arg max{di,j (t), D2 i,j (t)} uch that {, 2}. The deciion mechanim i the ame for link i k. Note that thi i joint routing (i.e., the next hop deciion) and cheduling (i.e., the flow election for tranmiion). The cheduling algorithm alo determine the link activation policy. In particular, the maximum backlog difference over each link are calculated a; Di,j (t) = D i,j (t) and D i,k (t) = D i,k (t). Baed on Di,j (t), D i,k (t) and C(t), the cheduling algorithm determine the link that hould be activated. Note that the deciion of routing and cheduling (alo named a max-weight algorithm) are made jointly in the backpreure framework, which impoe everal challenge in practice. We elaborate on them next. Routing algorithm are traditionally deigned in the network

2 2 layer, while the cheduling algorithm are implemented in the link layer in current network. However, the joint routing and cheduling nature of backpreure impoe challenge for practical implementation. To deal with thee challenge, [4] implement the backpreure at the link layer, [5] propoe a ytem in the MAC layer. Thi approach i practically difficult due to device memory limitation and trict limitation impoed by device firmware and driver not to change the link layer functionalitie. The econd approach i to implement backpreure in (or below) the network layer, [6], [7], [8]. Thi approach require joint operation of the network and link layer, o that the backpreure framework gracefully work with the link layer. Therefore, the network and link layer hould work together ynchronouly, which may not be practical for many off-the-helf device. Exiting network are deigned in layer, in which protocol and algorithm are modular and operate independently at each layer of the protocol tack. E.g., routing algorithm at the network layer hould work in a harmony with different type of cheduling algorithm in the link layer. However, the joint nature of the backpreure tree joint operation and hurt modularity, which i epecially important in contemporary wirele network, which may vary from a few node network to one with hundred of node. It i natural to expect that different type of network, according to their ize a well a oftware and hardware limitation, may chooe to employ backpreure partially or fully. E.g., ome network may be able to employ both routing and cheduling algorithm, while other may only employ routing. Therefore, the algorithm of backpreure, i.e., routing and cheduling hould be modular. In thi paper, we are intereted in a framework in which the routing and cheduling are eparated. We eek to find uch a cheme where routing i performed independently at the network layer and cheduling deciion are performed at the link layer. The key ingredient of our approach, which we call Diff-Max, are; (i) per-flow queue at the network layer and making routing deciion baed on their difference, (ii) perlink queue at the link layer and making cheduling deciion baed on their ize. Example - continued: Let u conider Fig. (b) for Diff- Max operation. (i) Routing: at time t, node i make routing deciion for flow and 2 baed on queue backlog D i,j (t) and D i,k (t), where {, 2}. Thi deciion i made at the network layer and the routed packet are inerted in the link layer queue. Note that in claical backpreure, routed packet are cheduled jointly, i.e., when a packet i routed, it hould be tranmitted if the correponding link are activated. Hence, both algorithm hould make deciion jointly in claical backpreure. However, in our cheme, a packet may be routed at time t, and cheduled and tranmitted at a later time t+t where T >. (ii) Scheduling: at the link layer, The rationale behind the name of our cheme, i.e., Diff-Max i a follow. Diff mean that the routing part i baed on queue difference, and Max refer to the fact that the cheduling part i baed on the maximum of the (weighted) link layer queue. Finally, the hyphen in Diff-Max i to mention the eparated nature of the routing and cheduling algorithm. link are activated and packet are tranmitted baed on perlink queue ize; V i,j, V i,k, and C(t). The detail of Diff-Max are provided in Section III. Our approach i grounded in a network utility maximization (NUM) framework [9]. The olution decompoe into everal part with an intuitive interpretation, uch a routing, cheduling, and flow control. The tructure of the NUM olution provide inight into the deign of our cheme, Diff-Max. Thank to eparating routing and cheduling, Diff- Max make the practical implementation eaier and minimize cro-layer operation. We alo propoe two practical cheme; Diff-ubMax and wdiff-ubmax. The following are the key contribution of thi work: We propoe a new ytem model and NUM framework to eparate routing and cheduling. Our olution to the NUM problem, eparate routing and cheduling uch that routing i implemented at the network layer, and cheduling i at the link layer. Baed on the tructure of the NUM olution, we propoe Diff-Max. We extend Diff-Max to employ routing and cheduling part, but diable the link activation part of the cheduling algorithm. We call the new framework Diff-ubMax, which reduce computational complexity and overhead ignificantly, and provide high throughput improvement in practice. Namely, Diff-ubMax only need information from one-hop away neighbor to make it routing and cheduling deciion. We propoe a window-baed routing mechanim, wdiffubmax, which implement routing, but diable the cheduling. wdiff-ubmax i deigned for the cenario, in which the implementation of the cheduling algorithm in the link layer i impoible (or not preferable) due to device retriction. wdiff-ubmax make routing deciion on the fly, and minimize overhead. We evaluate our cheme in a multi-hop etting and conider their interaction with tranport, network, and link layer. We perform numerical calculation confirming that Diff-Max i a good a backpreure. We implement our cheme in a imulator; n-2 [], and how that they ignificantly improve throughput a compared to adaptive routing cheme uch a Ad hoc On-Demand Ditance Vector (AODV) []. Finally, we implemented a prototype of wdiff-ubmax on Galaxy Nexu martphone with Android 4. (Ice Cream Sandwich) [2]. The tructure of the ret of the paper i a follow. Section II give an overview of the ytem model. Section III preent the NUM formulation and olution. Section IV preent the deign and development of Diff-Max cheme and their interaction with the protocol tack. Section V preent imulation reult. Section VI preent related work. Section VII conclude the paper. II. SYSTEM OVERVIEW We conider multi-hop wirele network, in which packet from a ource travere potentially multiple wirele hop before being received by their receiver. In thi etup, each wirele node i able to perform routing, cheduling, and flow

3 3 ource node of flow ). The network layer may alo receive packet from the other node and inert them in Ui. The link tranmiion rate i h k,i (t) at time t. h k,i (t) i larger than (or equal to) per-flow data rate over link k i. E.g., we can write for Fig. 2 that h k,i (t) h k,i (t) + h k,i (t) where h k,i (t) i the data rate of flow over link k i. Note that h k,i (t) i the actual data tranmiion rate of flow over link k i, while h k,i (t) i the available rate over link k i, at time t. At every timelot t, Ui change according to the following dynamic. U i (t + ) = max[u i (t) f i,j(t), ] + h j,i(t) Fig. 2. A wirele meh network. The queue at the network and link layer, and the interaction among the queue, inide node i are hown here in detail. Ui and Ui are the network layer queue for flow and, and V i,j and V i,l are the per-link queue for link; i j and i l. Diff-Max algorithm make the routing deciion in the network layer, and the cheduling deciion in the link layer. control. In thi ection, we provide an overview of thi etup and highlight ome of it key characteritic. Fig. 2 how the key part of our ytem model in an example topology. A. Notation and Setup The wirele network conit of N node and L edge, where N i the et of node and L i the et of edge in the network. We conider in our formulation and analyi that time i lotted, and t refer to the beginning of lot t. ) Source and Flow: Let S be the et of unicat flow between ource-detination pair in the network. Each flow S arrive from the application layer to the tranport layer with rate A (t), S at time lot t. The arrival rate are i.i.d. over the lot and their expected value are; λ = E[A (t)], S, and E[A (t) 2 ] are finite. Tranport layer tore the arriving packet in reervoir (i.e., tranport layer per-flow queue), and control the flow traffic. In particular, each ource i aociated with rate x conidering a utility function g (x ), which we aume to be a trictly concave function of x. The tranport layer determine x (t) at time lot t according to the utility function g. x (t) packet are tranmitted from the tranport layer reervoir to the network layer at lot t. 2) Queue Structure: At node i N, there are network and link layer queue. The network layer queue are per-flow queue; i.e., Ui i the queue at node i N that only tore packet from flow S. The link layer queue are per-link queue; i.e., at each node i N, a link layer queue V i,j i contructed for each neighbor node j N (Fig. 2). 2 3) Flow Rate: Our model optimize the flow rate among different node a well a the flow rate in a node among different layer; tranport, network, and link layer. The tranport layer determine x (t) at time t, and pae x (t) packet to the network layer. Thee packet are inerted in the network layer queue; Ui (auming that node i i the 2 Note that in ome device, there might be only one queue (per-node queue) for data tranmiion intead of per-link queue in the link layer. Developing a model with per-node queue i challenging due to coupling among action and tate, o it i an open problem. + x (t) [i=o()] () where o() i the ource node of flow and i=o() i an indicator function, which i if i = o(), and, otherwie. The data rate from the network layer to the link layer queue i fi,j (t). In particular, f i,j (t) i the actual rate of the packet, belonging to flow, from the network layer queue; Ui to the link layer queue; V i,j at node i. Note that the optimization of flow rate fi,j (t) i the routing deciion, ince it baically determine how many packet from flow hould be forwarded (hence routed) to node j. At every timelot t, V i,j change according to the following queue dynamic. V i,j (t + ) = max[v i,j (t) h i,j (t), ] + S f i,j(t) (2) The link tranmiion rate from i to node j i h i,j (t). A mentioned above h i,j (t) upper bound per-flow data rate; i.e., h i,j (t) S h i,j (t). Note that the optimization of link tranmiion rate h i,j (t) correpond to the cheduling deciion, ince it determine which packet from which link layer queue hould be tranmitted a well a whether a link i activated. B. Channel Model and Capacity Region ) Channel Model: Conider one-hop tranmiion over link l, where l = (i, j), uch that (i, j) N and i j. At each lot t, C(t) i the channel tate vector, where C(t) = {C (t),..., C l (t),..., C L (t)}. C l (t) i the tate of the link l at time t and take value from the et {ON, OF F } according to a ditribution which i i.i.d. over time lot. If C l (t) = ON, packet are tranmitted with rate R l. Otherwie; (i.e., if C l (t) = OF F ), no packet are tranmitted. Γ C(t) denote the et of the link tranmiion rate feaible at time lot t and for channel tate C(t). In particular, at every timelot t, the link tranmiion vector h(t) = {h (t),..., h l (t),...h L (t)} hould be contrained uch that h(t) Γ C(t). 2) Capacity Region: Let (λ ) i the vector of arrival rate S. The network layer capacity region Λ i defined a the cloure of all arrival vector that can be tably tranmitted in the network, conidering all poible routing and cheduling policie [], [2], [3]. Λ i fixed and depend only on channel tatitic characterized by Γ C(t).

4 4 III. DIFF-MAX: FORMULATION AND DESIGN A. Network Utility Maximization In thi ection, we formulate and deign the Diff-Max framework. Our firt tep i the NUM formulation of the problem and it olution. Thi approach (i.e., NUM formulation and it olution) hed light into the tructure of the Diff- Max algorithm. Note that the NUM formulation optimize the average value of the parameter (i.e., flow rate) that are defined in Section II. By abue of notation, we ue a variable, e.g., ϕ a the average value ϕ(t) in our NUM formulation, if both ϕ and ϕ(t) refer to the ame parameter. ) Formulation: Our objective i to maximize the total utility function by optimally chooing the flow rate x, S, a well a the following variable at each node: the amount of data traffic that hould be routed to each neighbor node; i.e., fi,j, the link tranmiion rate; i.e., h i,j. max x,f,h,τ g (x ) S { x,.t. fi,j h if i = o() j,i =, otherwie, i N, S fi,j h i,j, (i, j) L S f i,j = h i,j, S, (i, j) L h Γ. (3) The firt contraint i the flow conervation contraint at the network layer: at every node i and for each flow, the um of the total incoming traffic, i.e., h j,i and exogenou traffic, i.e., x hould be equal to the total outgoing traffic from the network layer, i.e., f i,j. The econd contraint i alo the flow conervation contraint, but at the link layer; the link tranmiion rate; i.e., h i,j hould be larger than the incoming traffic; i.e., S f i,j. Note that thi contraint i inequality, becaue the link tranmiion rate can be larger than the actual data traffic. The third contraint how the relationhip between the network and link layer perflow data rate. The lat contraint how that the vector of link tranmiion rate, h = {h,..., h l,...h L } hould be the element of the available link rate; Γ. Note that Γ i different than Γ C(t) in the ene that Γ i characterized with the lo over each link; p l, l L, rather than the channel tate vector; C(t). The firt and econd contraint are key to our work, becaue they determine the incoming and outgoing flow relationhip at the network and link layer, repectively. Such an approach eparate routing from cheduling, and aign the routing to the network layer and cheduling to the link layer. Note that if thee contraint are combined in uch a way that incoming rate from a node and exogenou traffic hould be maller than the outgoing traffic for each flow, we obtain the backpreure olution [3], [4]. 2) Solution: By relaxing the firt two flow conervation contraint in Eq. (3), we have: L(x, f, h, u, v) = g (x ) + ( u i fi,j S i N S ) h j,i x [i=o()] ( v i,j fi,j h i,j ), S (4) where u i and v i,j are the Lagrange multiplier, which can be interpreted a the repreentative of the network and link layer queue, Ui and V i,j, repectively. 3 The Lagrange function can be re-written a; L(x, f, h, u, v) = (g (x ) u o() x ) + S i N u jh i,j v i,j fi,j + i N S S u i fi,j S v i,j h i,j Eq. (5) can be decompoed into everal intuitive problem uch a flow control, routing, and cheduling. Firt, we olve the Lagrangian with repect to x : ) x = (g ) (u o(), (6) where (g ) i the invere function of the derivative of g. Thi part of the olution i interpreted a the flow control. Second, we olve the Lagrangian for fi,j and h i,j. The following part of the olution i interpreted a the routing. max (u i fi,j u jh i,j) v i,j fi,j f i N S S.t. f i,j = h i,j, i N, j N, S (7) The above problem i equivalent to; max fi,j(u i u j v i,j ) (8) f S Third, we olve the Lagrangian for h i,j. The following part of the olution i interpreted a cheduling. max v i,j h i,j h (5).t. h Γ. (9) The decompoed part of the Lagrangian, i.e., Eq. (6), (8), (9) a well a the Lagrange multiplier; u i and v i,j can be olved iteratively via a gradient decent algorithm. The convergence propertie of thi iterative algorithm are provided in Appendix A. Next, we propoe Diff-Max baed on the tructure of the decompoed olution. 3 Note that u i and v i,j are Lagrange multiplier. Although they are interpreted a the repreentation of the queue ize, they are not actual queue ize, but the function of them. On the other hand, Ui and V i,j are actual queue ize.

5 5 B. Diff-Max Now, we provide tochatic control trategy including routing, cheduling, and flow control. The trategy, i.e., Diff-Max, which mimic the NUM olution, combine eparated routing and cheduling together with the flow control trategy. Diff-Max: Routing. Node i oberve the network layer queue backlog in all neighboring node at time t and determine; { fi,j(t) Fi max, if Ui = (t) U j (t) V i,j(t) >, otherwie () where Fi max i contant larger than the maximum outgoing rate from node i. According to Eq. (), fi,j (t) packet are removed from Ui (t) and inerted in the link layer queue V i,j (t). Thi routing algorithm mimic Eq. (8) and ha the following interpretation. Packet from flow can be tranmitted to the next hop node j a long a the network layer queue in the next hop (node j) i mall, which mean that node j i able to route the packet, and the link layer queue at the current node (node i) i mall, which mean that the congetion over link i j i relatively mall. Note that if the number of packet in Ui (t) i limited, the packet are tranmitted to the link layer queue beginning from the larget Ui (t) U j (t) V i,j(t). The routing algorithm in Eq. () ue per-link queue a well a per-flow queue, which i the main difference of Eq. () a compared to backpreure routing. The backpreure routing only ue per-flow queue, and doe not take into account the tate of the link layer queue (they do not exit due to formulation). Scheduling. At each time lot t, link rate h i,j (t) i determined by; max h V i,j (t)h i,j (t).t. h(t) Γ C(t), (i, j) L () Thi cheduling algorithm mimic Eq. (9) and ha the following interpretation. The link i j with the larget queue backlog V i,j, by taking into account the channel tate vector; C(t), hould be activated, and a packet() from the correponding queue (V i,j ) hould be tranmitted. We note that thi problem (cheduling or max-weight) i known to be a hard problem, [9], [3]. Therefore, we propoe ub-optimal cheduling algorithm that interact well with the routing algorithm in Eq. (). The cheduling algorithm in Eq. () differ from the claical backpreure in the ene that it i completely independent from the routing. In particular, Eq. () make the cheduling deciion baed on per-link queue; V i,j and the channel tate; C(t), while the claical backpreure ue maximum queue backlog difference dictated by the routing algorithm. A it i een the routing Fig. 3. Diff-Max operation at end-point and intermediate node. and cheduling are operating jointly in backpreure, while in Diff-Max, thee algorithm are eparated. Flow Control. At every time lot t, the flow/rate controller at the tranport layer of node i determine the current level of network layer queue backlog Ui (t) and determine the amount of packet that hould be tranported from the tranport layer to the network layer according to: max x.t. S i=o() S i=o() [Mg (x (t)) U i (t)x (t)] x (t) R max i (2) where Ri max i a contant larger than the maximum outgoing rate from node i, and M i a contant parameter, M >. The flow control part of our olution mimic Eq. (6) a well a the flow control algorithm propoed in [3]. The dicuion on the analyi and performance bound of Diff-Max are provided in Appendix B. IV. SYSTEM IMPLEMENTATION We propoe practical implementation of Diff-Max (Fig. 3) a well a Diff-ubMax, which combine the routing algorithm with a ub-optimal cheduling, and wdiff-ubmax which make routing deciion baed on a window-baed algorithm. A. Diff-Max ) Flow Control: The flow control algorithm, implemented at the tranport layer at the end node (ee Fig. 3), determine the rate of each flow. We implement our flow control algorithm a an extenion of UDP in our imulator n-2 and in our Android tetbed. The flow control algorithm, at the ource node i, divide time into epoch (virtual lot) uch a t i, t2 i,..., tk i,..., where t k i i the beginning of the kth epoch. Let u aume that tk+ i = t k i + T i where T i i the epoch duration. At time t k i, the flow control algorithm determine the rate according to Eq. (2). We conider g (x (t)) = log(x (t)) (note that any other concave utility function can be ued). After x (t k i ) i determined, correponding number of packet are paed to the network layer, and inerted to the network layer queue Ui. Note that there might be ome exceive packet at the tranport layer if ome packet are not paed to the

6 6 network layer. Thee packet are tored in a reervoir at the tranport layer, and tranmitted in later lot. At the receiver node, the tranport protocol receive packet from the lower layer and pae them to the application. 2) Routing: The routing algorithm, implemented at the network layer of each node (both the end and intermediate node) (ee Fig. 3), determine routing policy, i.e., the next hop() that packet are forwarded. The firt part of our routing algorithm i the neighbor dicovery and queue ize information exchange. Each node i tranmit a meage containing the ize of it network layer queue; Ui. Thee meage are in general piggy-backed to data packet. The node in the network operate on the promicuou mode. Therefore, each node, let u ay node j, overhear a packet from node i even if node i tranmit the packet to another node, let u ay node k. Node j read the queue ize information from the data packet it receive or overhear (thank to operating on the promicuou mode). The queue ize information i recorded for future routing deciion. Note that when a node hear from another node through direct or promicuou mode, it claifie it a it neighbor. The neighbor node of node i form a et N i. A we mentioned, queue ize information i piggy-backed to data packet. However, if there i no data packet for tranmiion for ome time duration, the node create a packet to carry queue ize meage and broadcat it. The econd part of our routing algorithm i the actual routing deciion. Similar to the flow control algorithm, the routing algorithm divide time into epoch; uch a t i, t 2 i,..., t k i,..., where t k i i the beginning of the kth epoch at node i. Let u aume that t k+ i = t k i + T i where T i i the epoch duration. Note that we ue t k i and T i intead of tk i and T i, becaue thee two time epoch do not need to be the ame nor ynchronized. At time t k i, the routing algorithm at the network layer check Ui k (t i ) Uj k (t i ) V i,j (t k i ) for each flow. Note that Uj k (t i ) i not the intantaneou value of Uj at time t k i, intead it i the latet value of Uj heard by node i before t k i. Note alo that V i,j (t k i ) i the per-link queue at node i, and thi information hould be paed to the network layer for routing deciion. According to Eq. (), f i,j (t k i ) i determined, and f i,j (t k i ) packet are removed from Ui and inerted to the link layer queue V i,j at node i. Note that the link layer tranmit packet from V i,j only to node j, hence the routing deciion i completed. The routing algorithm i ummarized in Algorithm. Note that Algorithm conider that there are enough packet in Ui for tranmiion. If not, the algorithm lit all the link j N i in decreaing order, according to the weight; Ui k (t i ) Uj k (t i ) V i,j (t k i ). Then, it begin to route packet beginning from the link that ha the larget weight. 3) Scheduling: The cheduling algorithm in Eq. () aume that time i lotted, and determine the link that hould be activated and the (number of) packet that hould be tranmitted at each time lot. Although there are time-lotted ytem implementation, and alo recent work on backpre- Algorithm The routing algorithm at node i for packet from flow at lot t k i. : for j N i do 2: Read the network layer queue ize information of neighbor: U j (t k i ) 3: Read the link layer queue ize information: V i,j(t k i ) 4: if U i (t k i ) U j (t k i ) V i,j(t k i ) > then 5: f i,j(t k i ) = F max i 6: ele 7: f i,j (t k 8: Remove f i,j(t k i 9: Pa f i,j (t k i i ) = ) packet from U i ) packet to the link layer and inert them to V i,j ure implementation over time-lotted wirele network [8], IEEE 82. MAC, an aynchronou medium acce protocol without time lot, i the mot widely ued MAC protocol in the current wirele network. Therefore, we implement our cheduling algorithm (Eq. ()) on top 82. MAC (ee Fig. 3) with the following update. The cheduling algorithm contruct per-link queue at the link layer. Node i know it own link layer queue, V i,j, and etimate the lo and link rate. Let u conider that p l and R l are the etimated value of p l and R l, repectively. p l i calculated a one minu the ratio of correctly tranmitted packet over all tranmitted packet in a time window over link l. 4 Rl i calculated a the average of the recent (in a window of time) link rate over link l. V i,j, p i,j, and R i,j are piggy-backed to the data packet and exchanged among node. Note that thi information hould be exchanged among all node in the network ince each node i required to make it own deciion baed on global information. Alo, each node know the general topology and interfering link. The cheduling algorithm that we implemented mimic Eq. (). Each node i know per-link queue, i.e., V l, etimated lo probabilitie, i.e., p l, and link rate, i.e., Rl, for l L a well all maximal independent et, which conit of link that are not interfering. Let u aume that there are Q maximal independent et. For the qth maximal independent et uch that q =,..., Q, the policy vector i; π q = {πq,..., πq, l..., πq L }, where πq l = if link l i in the qth maximal et, and πq l =, otherwie. Our cheduling algorithm elect q th maximal independent et uch that q = arg max q { l L V l( p l ) R l πq}. l Node i olve q a one of the parameter; V l, p l, Rl change l L. If, according to q, node i decide that it hould activate one of it link, then it reduce the contention window ize of 82. MAC o that node i can acce the medium quickly and tranmit a packet. If node i hould not tranmit, then the cheduling algorithm tell 82. MAC that there are no packet in the queue available for tranmiion. Note that we update 82. MAC protocol o that we can implement the cheduling algorithm in Diff-Max. The cheduling algorithm i ummarized in Algorithm 2. Note that Algorithm 2 i a hard problem, becaue it reduce 4 Note that we do not ue intantaneou channel tate C l (t) in our implementation, ince it i not practical to get thi information. Even if one can etimate C l (t) uing phyical layer learning technique, C l (t) hould be etimated l L, which i not practical in current wirele network.

7 7 Algorithm 2 Diff-Max cheduling algorithm at node i. : if V l, p l, or R l i updated uch that l L then 2: Determine q uch that q = arg max q { l L V l( p l ) R l π l q } 3: if (i, j) uch that π (i,j) q =, j N i then 4: Reduce 82. MAC contention window ize and acce the medium 5: Tranmit a packet from V i,j according to FIFO rule 6: ele 7: Tell 82. MAC that there are no packet in the queue available for tranmiion to maximum independent et problem, [9], [3]. Furthermore, it introduce ignificant amount of overhead; each node need to know every other node queue ize and link lo rate. Due to the hardne of the problem and overhead, we implement thi algorithm for mall topologie over n-2 for the purpoe of comparing it performance with ub-optimal cheduling algorithm, which we decribe next. B. Diff-ubMax Diff-ubMax i a low complexity and low overhead counterpart of Diff-Max. The flow control and the routing part of Diff-ubMax i exactly the ame a in Diff-Max. The only different part i the cheduling algorithm, which ue 82. MAC protocol without any change. When a tranmiion opportunity arie according to underlying 82. MAC at time t, then the cheduling algorithm of node i calculate weight for all outgoing link to it neighbor. Let u conider link i j at time t. The weight i ω i,j (t) = V i,j (t)( p i,j ) R i,j. Baed on the weight, the link i choen a; l = arg max i ω i,j (t). Thi deciion mean that a packet from the link layer queue V l i choen according to FIFO rule and tranmitted. Note that thi cheduling algorithm only perform intra-cheduling, i.e., it determine from which link layer queue, packet hould be tranmitted, but it doe not determine which node hould tranmit, which i handled by 82. MAC. Diff-ubMax reduce the complexity of the algorithm and overhead ignificantly. In particular, each node i calculate and compare weight ω i,j (t) for each neighbor node. Therefore, the complexity i linear with the number of (neighbor) node. The overhead i alo ignificantly reduced; each node need to know the queue ize of only it one-hop away neighbor. C. wdiff-ubmax wdiff-ubmax i an extenion of Diff-ubMax for the cenario that link layer operation and data exchange (between the network and link layer) are not poible due to wifi firmware or driver retriction or may not be preferable. Therefore, wdiff-ubmax doe not employ any cheduling mechanim, but the routing and flow control. The flow control algorithm i the ame a in Diff-Max. Yet, the routing algorithm i updated a explained in the next. Eq. () require per-flow queue a well a per-link queue for routing deciion. If per-link queue are not available at the network layer, thee parameter hould be etimated. wdiff-ubmax, window-baed routing algorithm, implement Eq. () by etimating per-link queue ize. In particular, the routing algorithm end a window of packet, and receive acknowledgement (ACK) for each tranmitted packet. The ACK mechanim ha three function: (i) carrie per-flow queue ize information, (ii) provide reliability, i.e., packet which are not ACKed are re-tranmitted, (iii) etimate per-link queue ize. The algorithm work a follow. At time t k i, the window ize for link i j i W i,j (t k i ), the average round trip time of the packet i RT T i,j, and the average round trip time of the packet in the lat window i RT T i,j (t k i ). If Ui k (t i ) Uj k (t i ) > and RT T i,j (t k i ) < RT T i,j, then W i,j (t k i ) i increaed by. If Ui k (t i ) Uj k (t i ) > and RT T i,j (t k i ) > RT T i,j, then W i,j (t k i ) i decreaed by. If none of the packet in the lat window i ACKed, W i,j (t k i ) i halved. After W i,j (t k i ) i determined, f i,j (t k i ) i et to W i,j (t k i ) and f i,j (t k i ) packet are paed to the link layer. wdiff-ubmax, imilar to Diff-ubMax, reduce computational complexity and overhead ignificantly a compared to Diff-Max. A. Numerical Simulation V. PERFORMANCE EVALUATION We firt imulate our cheme, Diff-Max a well a claical backpreure in an idealized time lotted ytem in our inhoue imulator. The imulation reult how that Diff-Max perform a good a the claical backpreure. Next, we dicu the imulation etup and reult in detail. We conider triangle and diamond topologie hown in Fig. 4(a) and (b). In the triangle topology, there are two flow between ource; S, S 2 and receiver; R, R 2, repectively. S i originated from node A and end at node B, and S 2 i originated from node A and end at node C. In the diamond topology, there are two flow between ource; S, S 2 and receiver; R, R 2, repectively. S i originated at node A and end at node B, and S 2 i originated at node A and end at node D. In both topologie, all node are capable of relaying packet to their neighbor. The imulation duration i lot, and each imulation i repeated for eed. Each lot i either on ON or OF F tate according to the lo, which i.i.d over lot and uniformly ditributed at each lot. Fig. 5 how throughput v. the lo for the triangle topology. The lo i only over link A c. Fig. 5(a) how the total throughput of the two flow, i.e., the one from S to R and S 2 to R 2, while Fig. 5(b) and Fig. 5() preent individual throughput of flow from S to R and S 2 to R 2, repectively. A it i een, both the total throughput and individual throughput of Diff-Max cheme i equal to the backpreure. Thi obervation i confirmed for different lo cenario and the diamond topology in Fig. 6, 7, 8. B. n-2 Simulation In thi ection, we imulate our cheme, Diff-Max, DiffubMax, wdiff-ubmax a well a claical backpreure in the n-2 imulator []. The imulation reult how that Diff- Max, Diff-ubMax and wdiff-ubmax ignificantly improve throughput a compared to the adaptive routing cheme; Ad hoc On-Demand Ditance Vector (AODV) []. Next, we preent the imulator etup and reult in detail.

8 8 (a) Triangle topology (b) Diamond topology (c) Grid topology Fig. 4. Topologie ued in imulation. (a) Triangle topology. There are two flow between ource; S, S 2 and receiver; R, R 2, i.e., from node A to B (S - R ) and from node A to C (S 2 - R 2 ). (b) Diamond topology. There are two flow between ource; S, S 2 and receiver; R, R 2, i.e., from node A to B (S - R ) and from node A to D (S 2 - R 2 ). (c) Grid topology. 2 node are randomly placed over 4 3 grid. An example node ditribution and poible flow are illutrated in the figure Throughput of flow; S R.5 Throughput of flow; S 2 R Lo over link A C Lo over link A C Lo over link A C (a) v. lo (b) Throughput of flow from S to R v. lo (c) Throughput of flow from S 2 to R 2 v. lo Fig. 5. Triangle topology hown in Fig. 4(a). The lo i over link A C. (a) (um of the throughput of flow from S to R and S 2 to R 2 ) v. lo. (b) Throughput of flow from S to R v. lo. (c) Throughput of flow from S 2 to R 2 v. lo Throughput of flow; S R.5 Throughput of flow; S 2 R Lo over all link Lo over all link Lo over all link (a) v. lo (b) Throughput of flow from S to R v. lo (c) Throughput of flow from S 2 to R 2 v. lo Fig. 6. Triangle topology hown in Fig. 4(a). The lo i over all link. (a) (um of the throughput of flow from S to R and S 2 to R 2 ) v. lo. (b) Throughput of flow from S to R v. lo. (c) Throughput of flow from S 2 to R 2 v. lo Throughput of flow; S R.5 Throughput of flow; S 2 R Lo over link A B Lo over link A B Lo over link A B (a) v. lo (b) Throughput of flow from S to R v. lo (c) Throughput of flow from S 2 to R 2 v. lo Fig. 7. Diamond topology hown in Fig. 4(b). The lo i over link A B. (a) (um of the throughput of flow from S to R and S 2 to R 2 ) v. lo. (b) Throughput of flow from S to R v. lo. (c) Throughput of flow from S 2 to R 2 v. lo. ) Setup: We conidered two topologie: diamond topology hown in Fig. 4(b); and a grid topology hown in Fig. 4(c). In the diamond topology, the node are placed over 5m 5m terrain. Two flow are tranmitted from node A to node B and D. In the grid topology, 4 3 cell are placed over a 8m 6m terrain. 2 node are randomly placed to the cell. In the grid topology, each node can communicate with other node in it cell or with the one in neighboring cell.

9 Throughput of flow; S R.5 Throughput of flow; S 2 R Lo over all link Lo over all link Lo over all link (a) v. lo (b) Throughput of flow from S to R v. lo (c) Throughput of flow from S 2 to R 2 v. lo Fig. 8. Diamond topology hown in Fig. 4(b). The lo i over all link. (a) (um of the throughput of flow from S to R and S 2 to R 2 ) v. lo. (b) Throughput of flow from S to R v. lo. (c) Throughput of flow from S 2 to R 2 v. lo. Four flow are generated randomly. We conider CBR traffic. CBR flow tart at random time within the firt 5ec and are on until the end of the imulation which i ec. The CBR flow generate packet with interarrival time.m. IEEE 82.b i ued in the MAC layer (with update for Diff-Max implementation a explained in Section IV). In term of wirele channel, we imulated a Rayleigh fading channel with average channel lo rate, 2, 3, 4, 5%. 5 We have repeated each ec imulation for eed. The channel capacity i M bp, the buffer ize at each node i et to packet, packet ize are et to B. We compare our cheme; Diff-Max, Diff-ubMax, and wdiffubmax with AODV, in term tranport-level throughput. The Diff-Max parameter are et a follow. For the flow control algorithm; T i = 8m, Ri max = 2 packet, M = 2. For the routing algorithm; T max i = m, Fi = 4 packet. 2) Reult: Fig. 9, preent imulation reult in n-2 imulator over diamond and grid topologie for different lo rate. Fig. 9(a) how the reult for the diamond topology. The lo rate i over the link between node A and B. Diff-Max perform better than the other cheme for the range of lo rate. The reaon i that Diff-Max activate link baed on perlink queue backlog, lo rate, and link rate. On the other hand, Diff-ubMax, wdiff-ubmax, and AODV ue claical 82. MAC, which provide fairne among the competing node for the medium, which i not utility optimal. When the lo rate over link A B increae, the total throughput of all the cheme reduce a expected. A it can be een, the decreae of our cheme; Diff-Max, Diff-ubMax, wdiffubmax i linear, while the decreae of AODV i quite harp. The reaon i that when AODV experience lo over a path, it delete the path and re-calculate new route. Therefore, AODV doe not tranmit over loy link for ome time period and trie to find new route, which reduce throughput. Fig. 9(b) elaborate more on the above dicuion. It how the throughput of two flow A to B and A to D a well a their total value when the lo rate i % over link A B. A it can be een, the rate of flow A B i very low in AODV a 5 We conider the lo rate in the range up to 5%, becaue recent tudie of IEEE 82.b baed wirele meh network [6], [7], have reported packet lo rate a high a 5%. compared to our cheme, becaue AODV conider the link A B i broken at ome period during the imulation, while our cheme continue to tranmit over thi link. Let u conider Fig. 9(a) again. Diff-ubMax and wdiffubmax improve throughput ignificantly a compared to AODV thank to exploring route to improve utility (hence throughput). The improvement of our cheme over AODV i up to 22% in thi topology. Alo, Diff-ubMax and wdiffubmax have imilar throughput performance, which emphai the benefit of routing part and the effective link layer queue etimation mechanim of wdiff-ubmax. Fig. 9(a) alo how that when lo rate i 5%, the throughput improvement of all cheme are imilar, becaue at 5% lo rate, link A B become very inefficient, and all of the cheme tranmit packet motly from flow A to D over path A C D and have imilar performance at high lo rate. Fig. 9(b) how the reult for the grid topology. The throughput improvement of our cheme i higher than AODV for all lo rate in the grid topology and higher a compared to the improvement in the diamond topology, e.g., the improvement i up to 33% in the grid topology. The reaon i that AODV i deigned to find the hortet path, but our cheme are able to explore interference free path even if they are not the hortet path, which i emphaized in larger topologie. C. Android Prototype We conider a cenario in which a group of martphone collaborate in the ame geographical area. In our etting, we ue four Android 4. [2] baed Galaxy Nexu phone, and configure them to operate in ad-hoc mode over Wifi. We implement our wdiff-ubmax cheme (flow control and routing) a an extenion of UDP ocket. We firt conider a cenario in which two phone (A and B) are connected to each other. Phone A tranmit 4MB audio file to phone B. The tranmiion time for wdiff-ubmax wa 6ec which i comparable with it TCP counterpart, which wa 4ec. Thi example how the efficiency of our algorithm a an extenion of UDP, which caue packet loe or too long tranmiion time. In the econd cenario, we placed/eparated phone to be able to create a topology imilar to the diamond topology hown in Fig. 4(b). In thi etup, phone A tranmit 4MB

10 Total Throughput Diff ubmax wdiff ubmax AODV Throughput Throughput from A to D Throughput from A to B Total Throughput Total Throughput Diff ubmax wdiff ubmax AODV Lo Probability Diff ubmaxwdiff ubmax AODV Lo Probability (a) Diamond topology. v. average lo rate (b) Diamond topology. Throughput of different policie (c) Grid topology. v. average lo rate Fig. 9. v. average lo rate for different policie and in two different topologie. (a) v. average lo rate in diamond topology. (b) Total and per-flow throughput for different policie when the average lo rate i et to % in the diamond topology. (c) v. average lo rate in grid topology. audio file to phone D either uing phone B or C a a relay. We firt conider TCP connection over the path A B D and configure phone B o that it drop relaying packet after ec tranmiion. A expected, TCP connection fail when B top relaying packet. On the other hand, wdiff-ubmax continue tranmiion even after B top, by relaying packet uing phone C, and complete the tranmiion in 4ec. VI. RELATED WORK and follow-up work. Thi paper build on backpreure, a routing and cheduling framework over communication network [], [2], which ha generated a lot of interet in the reearch community [5]; epecially for wirele and-hoc network [8], [9], [2], [2], [22], [23]. Alo, it ha been hown that backpreure can be combined with flow control to provide utility-optimal operation guarantee [3], [22]. Thi paper follow the main idea of backpreure framework, and reviit it conidering the practical challenge that are impoed by the current network. implementation. The trength of the backpreure framework have recently increaed the interet on practical implementation of backpreure over wirele network. Multi-path TCP cheme i implemented over wirele meh network [6], where TCP flow are tranmitted over multiple pre-determined path and packet are cheduled according to backpreure cheduling algorithm. At the link layer, [4], [5], [24], [25] propoe, analyze, and evaluate link layer backpreure-baed implementation with queue prioritization and congetion window ize adjutment. The backpreure framework i implemented over enor network [7] and wirele multi-hop network [8], which are alo the mot cloe implementation to our. Our main difference are that; (i) we conider eparation of routing and cheduling to make practical implementation eaier, (ii) we deign and analyze a new cheme; Diff-Max, (iii) we imulate and implement Diff- Max over n-2 and android phone. and Queue. According to backpreure framework, each node contruct per-flow queue. There i ome work in the literature to tretch thi neceity. For example, [26], [27] propoe uing real per-link and virtual per-flow queue. Such a method reduce the number of queue required in each node, and reduce the delay. Although thi approach reduce the backpreure framework to make routing deciion uing virtual queue and cheduling deciion uing the real per-link queue by decoupling routing and cheduling, it doe not eparate routing from cheduling. Therefore, thi approach require trong ynchronization between the network and link layer, which i difficult to implement in practice a explained in Section I. VII. CONCLUSION In thi paper, we propoed Diff-Max, a framework that eparate routing and cheduling in backpreure-baed wirele network. Diff-Max improve throughput ignificantly. Alo, the eparation of routing and cheduling make practical implementation eaier by minimizing cro-layer operation and it lead to modularity. Our deign i grounded on a network utility maximization (NUM) formulation of the problem and it olution. Simulation in n-2 demontrate the performance of Diff-Max a compared adaptive routing cheme, uch a AODV. 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Titikli, Parallel and Ditributed Computation: Numerical Method, New Jerey: Prentice-Hall, 989. APPENDIX A Lagrange Multiplier. The Lagrange multiplier; u i and v i,j are calculated uing gradient decent: u i (t + ) ={u i (t) α t [ fi,j(t) h j,i(t) x (t) i=o() ]} + v i,j (t + ) ={v i,j (t) + β t [ S f i,j(t) h i,j (t)]} + (3) where t i the iteration number, α t and β t are the tep ize of the gradient decent algorithm, and the + operator make the Lagrange multiplier poitive. Convergence. The convergence of the olution et, Eq. (6), (8), (9), (3) follow directly from the convergence of convex optimization problem through gradient decent [4], [28]. In particular, if lim t α t =, t= α t = and lim t β t =, t= β t =, then the olution converge, i.e., lim t x(t) x =. Numerical Calculation. We confirm the convergence of our olution through numerical calculation. Fig. how total throughput (um of the two flow) v. iteration number graph for the triangle topology hown in Fig. 4(a) and for different lo probabilitie, i.e.,.2,.5, and.8. It i een that the total throughput converge to the optimal value for all lo rate. We repeated the ame imulation for the diamond topology whoe reult are hown in Fig.. Thee reult alo confirm the convergence of our olution. APPENDIX B Let u recall the queue dynamic in Eq. () and Eq. (2): Ui (t + ) max[ui (t) fi,j(t), ] + h j,i(t) + x (t) [i=o()] (4) V i,j (t + ) [V i,j (t) h i,j (t), ] + S f i,j(t) (5) Now let u conider a virtual queue Z i (t); Zi (t + ) max[zi (t) fi,j(t), ] + fj,i(t) + x (t) [i=o()] (6) In the following, we conider a variant of Diff-Max algorithm uing the virtual queue Z i (t) intead of U i (t).6 Diff-Max with virtual queue: Routing. Node i oberve the network layer queue backlog in all neighboring node at time t and determine; { fi,j(t) Fi max, if Zi = (t) Z j (t) V i,j(t) >, otherwie (7) Scheduling. At each time lot t, link rate h i,j (t) i determined by; max V i,j (t)h i,j (t) h.t. h(t) Γ C(t), (i, j) L (8) Theorem : If channel tate are i.i.d. over timelot, and the arrival rate λ, S are interior to the capacity region Λ, then Diff-Max with virtual queue tabilize the network and the total average backlog i bounded. Proof: We firt define the Lyapunov function a; L(H(t)) = Zi (t) 2 + V i,j (t) 2, (9) i N S i N 6 Note that we conider in thi ection that arrival rate are inide the capacity region. The extenion of the analyi provided in thi ection i traightforward when the arrival rate are outide the capacity region by uing an appropriate flow control algorithm. Some remark regarding thi iue are added at the end of thi ection.