Separation of Routing and Scheduling in Backpressure- Based Wireless Networks

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1 Separation of Routing and Scheduling in - Baed Wirele Network The MIT Faculty ha made thi article openly available. Pleae hare how thi acce benefit you. Your tory matter. Citation A Publihed Publiher Seferoglu, Hulya, and Eytan Modiano. Separation of Routing and Scheduling in -Baed Wirele Network. IEEE/ACM Tranaction on Networking 24, no. 3 (June 26): Intitute of Electrical and Electronic Engineer (IEEE) Verion Original manucript Acceed Sun Oct 2 2:26:5 EDT 28 Citable Link Term of Ue Creative Common Attribution-Noncommercial-Share Alike Detailed Term

2 Separation of Routing and Scheduling in -Baed Wirele Network Hulya Seferoglu, Member, IEEE, Eytan Modiano, Fellow, IEEE arxiv: v2 [c.ni] 22 Apr 25 Abtract routing and cheduling, with it throughput-optimal operation guarantee, i a promiing technique to improve throughput in wirele multi-hop network. Although backpreure 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 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. Index Term routing and cheduling, network utility maximization, wirele network. I. INTRODUCTION routing and cheduling ha emerged from the pioneering work in [], [2], which howed that, in wirele network, one can tabilize queue for any feaible traffic by making routing and cheduling deciion baed on queue backlog difference. Moreover, 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 in practical implementation of backpreure in 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, which i the focu of thi paper. In the backpreure framework, each node contruct perflow 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 (note that cheduling Thi work wa upported by NSF grant CNS-95988, ONR grant N , ARO Muri grant number W9NF The preliminary reult of thi paper were preented in part at the IEEE Conference on Computer Communication (INFOCOM), Turin, Italy, April 23. H. Seferoglu wa with the Laboratory for Information and Deciion Sytem (LIDS), Maachuett Intitute of Technology. She i currently with the Electrical and Computer Engineering Department at Univerity of Illinoi at Chicago. hulya@uic.edu. Mail: 37 Science and Engineering Office (M/C 54), 85 South Morgan Street, Chicago, IL, 667. E. Modiano i with the Laboratory for Information and Deciion Sytem (LIDS), Maachuett Intitute of Technology. modiano@mit.edu. Mail: 77 Maachuett Avenue, Room 33-42, Cambridge, MA 239. j Z i (t) (Routing and Scheduling) i Channel State C(t) Z i 2 (t) (a) k Queue Size j U i (t) Diff-Max (Scheduling) i Channel State C(t) U i 2 (t) Diff-Max (Routing) V i,j (t) V i,k (t) (b) Diff-Max k Queue Size Queue Size 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. (a) : Node i contruct perflow queue; Zi and Zi 2, and determine the queue backlog difference at time t; Di,j (t) = Z i (t) Z j (t), D i,k (t) = Z i (t) Z k (t), where {,2}. Baed on the difference a well a the channel tate of the network, C(t), backpreure make joint routing and cheduling deciion. (b) Diff- Max: Node i contruct per-flow queue; Ui and Ui 2 in the network layer, and per-link queue ize; V i,j and V i,k in the link layer, and make routing deciion baed on the queue backlog difference at time t; D i,j (t) = U i (t) Uj (t) V i,j(t), D i,k (t) = Ui (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). algorithm i alo called a max-weight [4]). Although the backpreure framework i conceptually viewed a layered, the deciion of routing and cheduling are made jointly. To better illutrate thi point, let u dicu the following example. 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; Zi (t), Z2 i (t), a well a the queue ize of the other node, i.e., j and k in thi example, and uing the channel tate of the network C(t). In particular, backpreure determine a packet (and it flow) that hould be tranmitted over link i j by = argmax{di,j (t),d2 i,j (t)} uch that {,2}. The deciion mechanim i the ame for link i k. The cheduling algorithm alo determine the link activation policy. In particular, the maximum backlog difference over each link i 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 are made jointly in backpreure, which impoe everal challenge in practice. We elaborate on them next. Routing algorithm are traditionally deigned in the network layer, while the cheduling algorithm are implemented in the

3 2 link layer. However, the joint routing and cheduling nature of backpreure impoe challenge for practical implementation. To deal with thee challenge, [5] implement backpreure at the link layer, and [6] propoe update to 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 the network layer, e.g., [7], [8], [9], which require joint operation of the network and link layer o that backpreure implemented in the network layer operate gracefully with the link layer functionalitie. Thu, 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 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 the routing operate in the network layer and the cheduling i implemented in the link layer. The key ingredient of our framework, which we call Diff-Max, are; (i) per-flow queue at the network layer and making routing deciion baed on their difference, (ii) per-link 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). Thi deciion i made at the network layer and the routed packet are inerted into the link layer queue. Note that in 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 the deciion jointly in backpreure. However, in Diff-Max, 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, link are activated and packet are tranmitted baed on per-link 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 []. The olution decompoe into everal part with an intuitive interpretation, uch a routing, Note that Diff mean that the routing i baed on the queue difference, and Max refer to the fact that the cheduling i baed on the maximum of the (weighted) link layer queue. Finally, the hyphen in Diff-Max i to mention the eparation of the routing and cheduling. cheduling, and flow control. The tructure of the NUM olution provide inight into the deign of our cheme, Diff- Max. By eparating routing and cheduling, Diff-Max make the practical implementation eaier and minimize cro-layer operation. The following are the key contribution of thi work: We propoe a new ytem model and NUM framework to eparate routing and flow 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 how that the determinitic verion of Diff-Max optimize utility, and we conjecture that it tochatic verion atifie tability and utility optimality. We extend Diff-Max to employ routing and intra-node cheduling, but diable inter-node cheduling. 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. Furthermore, we how that the determinitic verion of Diff-ubMax provide utility optimality for the network with pre-determined inter-node cheduling. We propoe a window-baed routing cheme, wdiffubmax, which implement routing, but diable the cheduling. wdiff-ubmax i a heuritic developed baed on Diff-Max and Diff-ubMax, and it i deigned for the cenario, in which the implementation of the cheduling in the link layer i impoible (or not deirable) e.g., due to device retriction. wdiff-ubmax make the routing deciion on the fly, and reduce overhead. We evaluate our cheme in a multi-hop etting and conider their interaction with tranport, network, and link layer. 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) [2] and Detination-Sequenced Ditance-Vector Routing (DSDV) [3]. The tructure of the ret of the paper i a follow. Section II give an overview of the ytem model. Section III preent the Diff-Max formulation and deign. Section IV preent the development and implementation detail of Diff-Max cheme. Section V preent imulation reult. Section VI preent related work. Section VII conclude the paper. II. SYSTEM OVERVIEW In thi ection, we provide an overview of the ytem model for eparation of routing and cheduling. We alo provide background on the backpreure framework o that we can make a connection and comparion between our cheme and backpreure throughout the ret of the paper. A. Separation of Routing and Scheduling We conider multi-hop wirele network, in which packet from a ource travere potentially multiple wirele hop

4 3 k x h k,i if i=o() h k,i U i h k,i ' h k,i ' U i m x ' if i=o(') Routing (network layer) i f i,l f i,j ' f i,l ' f i,j h i,l V i,l V i,j h i,j l Scheduling (link layer) Fig. 2. A wirele meh network. The queue at the network and link layer a well a the interaction among the queue inide node i are hown in detail. Ui and U i are the network layer queue for flow and, and V i,j and V i,l are the per-link queue for the link; i j, i l. The routing algorithm operate in the network layer, the cheduling i implemented in the link layer. before being received at their detination. In thi etup, each wirele node i able to perform routing, cheduling, and flow 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. Setup: We conider a wirele network which conit of N node and L edge, where N i the et of node and L i the et of edge. 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 an application layer to a tranport layer with rate A (t), S at time lot t. The arrival are i.i.d. over the lot and their expected value are; λ = E[A (t)], ande[a (t) 2 ] are finite. The tranport layer tore the arriving packet in reervoir (i.e., tranport layer per-flow queue), and control the flow. In particular, each ource i aociated with rate x and 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), and x (t) packet are tranmitted from the tranport layer reervoir to the network layer at lot t. 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 nodei N that only tore the 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 a neighbor node j N (Fig. 2). 2 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 ource node of flow ). 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. h i,l h i,j j The flow rate from the network layer to the link layer queue i fi,j (t). In particular, f i,j (t) i the flow 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/routed to node j. The link tranmiion rate from i to j i h i,j (t). Note that h i,j (t) bound per-flow data rate; i.e.,h i,j (t) S h i,j (t). E.g., h k,i (t) h k,i (t)+h k,i (t) in Fig. 2 where h k,i (t) i the flow rate of flow over link k i. 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. At every time lot t, Ui change according to the following dynamic. Ui(t+) max[ui(t) fi,j(t),]+ h j,i(t) +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. Note that () i an inequality, becaue the actual amount of flow rate of flow over link j i may be lower than h j,i (t) a there may not be enough packet from flow in the link layer queue at node j. At every time lot t, V i,j change according to the following queue dynamic. V i,j (t+) max[v i,j (t) h i,j (t),]+ Sf i,j(t) (2) Note that (2) i an inequality a the number of packet in U i (t) may be lower than f i,j (t). Channel Model: At lot t, C(t) i the channel tate vector, wherec(t) = {C (t),...,c l (t),...,c L (t)}, wherel repreent the edge uch that l = (i,j), (i,j) L and i j. We aume that C l (t) i the tate of link l at time t and take value from the et {ON, OF F} according to a probability ditribution which i i.i.d. over time lot. If C l (t) = ON, packet can be tranmitted with rate R l. Otherwie; (i.e., if C l (t) = OFF), packet cannot be tranmitted uccefully. Let Γ C(t) denote the et of the link tranmiion rate feaible at time lot t for channel tate C(t) accounting for interference among the wirele link. In particular, at every time lot 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). Stability Region: Let (λ ) be the vector of arrival rate S. The network tablity region Λ i defined a the cloure of all arrival rate 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 and interference.

5 4 k k,i x if i=o() k,i ' m Z i k,i ' k,i ' Z i x ' if i=o(') Routing & Scheduling i i,l ' i,l l i,j i,j ' i,j i,l ' i,j i,l ' Fig. 3. ytem model in a wirele meh network. Per-flow queue inide node i are hown in detail. Zi and Zi are the per-flow queue for flowand, repectively. The backpreure routing & cheduling algorithm operate jointly over the per-flow queue. B. Background on In thi ection, we provide background on backpreure o that we can make a connection and comparion between our cheme and backpreure throughout the ret of the paper. We conider a imilar ytem model a in the previou ection. Fig. 3 how the key part of backpreure ytem model in an example topology. At node i N, there are per-flow queue. The per-flow queue Zi i the queue at node i N that only tore the packet from flow S. The flow rate x (t) i determined at time t, and the correponding number of packet are inerted in the network layer queue; Zi (auming that node i i the ource node of flow ). The flow rate from node i to node j for flow i ξi,j. Note that the optimization of flow rate ξi,j (t) i both the routing and cheduling deciion, ince it baically determine how many packet from flow hould be forwarded/routed over which link. Thu, at every time lot t, Zi change according to the following dynamic. Zi(t+) max[zi(t) ξi,j(t),]+ ξj,i(t) +x (t) [i=o()] (3) The backpreure cheme operate on per-flow queue Z i and make routing and cheduling deciion baed on the following algorithm. : Routing & Scheduling: At each time lot t, the rate ξi,j (t) i determined by; max ξi,j(t)(z i(t) Z j(t)) ξ i S.t. ξ(t) Γ C(t), (4) routing and cheduling algorithm in (4) tabilize the network and average queue backlog ize are bounded [], [2]. Moreover, it ha been hown that backpreure can be combined with flow control to provide utility-optimal operation guarantee [3]. j III. DIFF-MAX: FORMULATION AND DESIGN A. Network Utility Maximization In thi ection, we formulate and deign Diff-Max. Our firt tep i the NUM formulation of the problem and it olution. Thi approach hed light into the tructure of the Diff-Max algorithm. 3 ) Formulation: Our objective i to maximize the total utility by optimally chooing the flow rate x, a well a the amount of data traffic that hould be routed to each neighbor node; i.e., fi,j, and the link tranmiion rate; i.e., h i,j. max g (x ) x,f,h S.t. fi,j h j,i = x [i=o()], i N, S fi,j h i,j, (i,j) L S fi,j = h i,j, S,(i,j) L h Γ. (5) 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 the flow conervation contraint 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 an inequality, becaue the link tranmiion rate can be larger than the actual data traffic. The third contraint give the relationhip between the network and link layer per-flow data rate, and the lat contraint require 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 repreent long-term average rate rather than intantaneou rate. The firt two contraint are key to our work, becaue they determine the incoming and outgoing flow relationhip at the network and link layer, repectively. Thi approach eparate routing and 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 [4], [5]. 2) Solution: Lagrangian relaxation of the firt contraint give the following Lagrange function: 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 ), (6) (i,j) L S 3 NUM 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.

6 5 where u i and v i,j are the Lagrange multiplier. The Lagrange function can be re-written a; L(x,f,h,u,v) = (g (x ) u o() x )+ S i N u j h i,j v i,j fi,j + i N S (i,j) L S u ifi,j S (i,j) L v i,j h i,j (7) can be decompoed into everal intuitive ub-problem uch a flow control, routing, and cheduling. Firt, we olve the Lagrangian with repect to x : ) x = (g (u ) o(), (8) where (g ) i the invere function of the derivative of g. Thi part of the olution can be interpreted a the flow control. Second, we olve the Lagrangian for fi,j and h i,j. The following part of the olution can be interpreted a the routing. max f (u ifi,j u jh i,j) i N S (i,j) L S v i,j fi,j.t. fi,j = h i,j, i N,j N, S (9) The above problem i equivalent to; max fi,j f (u i u j v i,j) () (i,j) L S Third, we olve the Lagrangian for h i,j. The following part of the olution can be interpreted a the cheduling. max v i,j h i,j h (i,j) L (7).t. h Γ. () The decompoed part of the Lagrangian, i.e., Eq. (8), (), () and the Lagrange multiplier; u i and v i,j can be olved iteratively via a gradient decent algorithm. The convergence propertie of thi olution to the utility optimal operating point are provided in Appendix A. Next, we deign Diff-Max baed on the tructure of the NUM olution. B. Diff-Max Now, we provide a 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. Diff-Max: Routing: Node i determine fi,j (t) according to; max f.t. fi,j (t)(ũ i (t) Ũ j (t) V i,j(t)) i S i S fi,j (t) Fmax i (2) where Fi max i contant larger than the maximum outgoing rate from node i, N i i the et of node i neighbor, and Ũ i (t) i the network layer virtual queue. According to (2), fi,j (t) packet are removed from Ũi (t) and inerted to the link layer queue V i,j(t). Thi routing algorithm mimic () 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. If the number of packet in Ui (t) i maller than the routing variable calculated by (2), 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 (2) ue per-link queue a well a per-flow queue, which i the main difference of (2) a compared to the backpreure routing. The backpreure routing only ue per-flow queue, and doe not take into account the tate of the link layer queue, which do not exit in the tandard backpreure formulation. Scheduling: At each time lot t, the link rate h i,j (t) i determined by; max h (i,j) L V i,j (t)h i,j (t).t. h(t) Γ C(t), (i,j) L (3) (3) mimic () and ha the following interpretation. The link i j with the larget queue backlog V i,j, taking into account the channel tate vector, hould be activated, and a packet() from the correponding queue, i.e., V i,j, hould be tranmitted. Note that the cheduling in (3) i known to be a difficult problem [], [4]. Therefore, in Section IV, we propoe ub-optimal, low-complexity cheduling algorithm that interact well with the routing algorithm in (2). The cheduling algorithm in (3) differ from backpreure in the ene that it i completely independent from the routing. In particular, (3) make the cheduling deciion baed on the per-link queue; V i,j and the channel tate; C(t), while backpreure ue maximum queue backlog difference dictated by the routing algorithm. A it i een, the routing 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 number of packet that hould be paed 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 (4) where Ri max i a contant larger than the maximum outgoing rate from node i, and M i a finite contant, M >. The flow control in our olution mimic (8) a well a the flow control algorithm propoed in [3].

7 6 Application Tranport Layer Flow Control Network Layer / IP Routing Scheduling MAC U i (t) V i,j (t) U i (t) V i,j (t) Network Layer / IP Routing Scheduling MAC V i,j (t) U i (t) V i,j (t) Application Tranport Layer Flow Control Network Layer / IP Routing Scheduling MAC U i (t) V i,j (t) Algorithm The routing algorithm at node i at lot t k i. : for j N i, S 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: fi,j (t k i ) = 5: {j, } = argmax [i, S] {U i (t) U j (t) V i,j(t)} 6: fi,j (t k i ) = Fi max 7: Remove fi,j (t k i ) packet from Ui 8: Pa f i,j (t k i ) packet to the link layer and inert them in V i,j Fig. 4. Source Intermediate Receiver Diff-Max operation at end-point and intermediate node. IV. IMPLEMENTATION DETAILS We propoe practical implementation of Diff-Max (Fig. 4) 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. 4), determine the rate of each flow. We implement our flow control algorithm a an extenion of UDP in the n-2 imulator. 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 thekth epoch. Let u aume thattk+ 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 (4). 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, a correponding number of packet are paed to the network layer, and inerted to the network layer queue Ui. Packet that are not forwarded to the network layer 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 (ee Fig. 4), determine routing policy, i.e., the next hop() that packet are forwarded to. 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 in 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 that 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, the node create a packet to carry queue ize information 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 andt i intead of tk i andt i, becaue thee two time epoch do not need to be the ame nor ynchronized. At time t k i, the routing algorithm 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 k t i, but the latet value of Uj heard by node i before k t i. Note alo that V i,j (t k i ) i the per-link queue at nodei, and thi information hould be paed to the network layer for routing deciion. According to (2), fi,j k (t i ) i determined j N i, S, and fi,j k (t 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; U i (t k i ) U j (t k 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 (3) aume that time i lotted. Although there are time-lotted ytem implementation, and alo recent work on backpreure implementation over time-lotted wirele network [9], IEEE 82. MAC, an aynchronou medium acce protocol without time lot, i the mot widely ued MAC protocol in current wirele network. Therefore, we implement our cheduling algorithm (formulated in (3)) on top of 82. MAC (ee Fig. 4) 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 probability 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 linkl. 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, 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.

8 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 = argmax [ 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 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 (3). 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,...,π q,...,π l q 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 = argmax [ q] { l L V l( p l ) R l πq l }. Node i calculate q whenever one of the parameter; V l, p l, Rl change. 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 in order to complement Diff-Max cheduling, the 82. protocol ha to be lightly modified. The cheduling algorithm i ummarized in Algorithm 2. Note that Algorithm 2 i a hard problem, becaue it i reduced to maximum independent et problem, [], [4]. Furthermore, it introduce ignificant amount of overhead; each node need to know every other node queue ize and link lo rate. Due to the complexity 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 the 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 = argmax [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-node 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 ha everal nice feature. We how in Appendix A that the determinitic verion of Diff-ubMax provide utility optimality for the network with pre-determined inter-node cheduling uch a CSMA/CA. Furthermore, DiffubMax 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 only of it one-hop away neighbor. C. wdiff-ubmax wdiff-ubmax i a heuritic deigned a 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 Wi-Fi firmware or driver retriction or may not be deired. Therefore, wdiff-ubmax doe not employ any cheduling mechanim, but only the routing and flow control. The flow control algorithm i the ame a in Diff-Max. Yet, the routing algorithm i updated a explained next. Per-flow queue a well a per-link queue are required in (2) to make the routing deciion. If per-link queue are not available at the network layer, the routing deciion may not be efficient a there may be (uncontrolled) congetion in the link layer queue. In order to make the routing deciion efficiently, we propoe a heuritic called wdiff-ubmax. The main idea behind wdiff-ubmax i to react to link layer congetion, while till implementing (2). To achieve thi, wdiff-ubmax employ acknowledgement (ACK) mechanim and ue an additive increae/multiplicative decreae (AIMD) algorithm. wdiff-ubmax label each packet with a timetamp at the network layer. When a packet i received by the next hop, an ACK packet, echoing the timetamp of the packet, i tranmitted back to the previou hop. The network layer of the previou hop receive ACK and determine round trip time (RTT) for each packet. RTTi,j k (t i ) i the average round trip time of the ACK received in the lat lot (i.e., at lot t k i ), and RTTi,j i the average round trip time of the packet. wdiff-ubmax keep a window ize Wi,j k (t i ) for link i j and flow at lot t k i. At each lot t k i, the routing parameter fi,j k (t i ) i et to Wi,j k (t i ) and fi,j k (t i ) packet are paed to the link layer. wdiff-ubmax determine the window ize according to AIMD a explained next. If Ui k (t i ) Uj k (t i ) > and RTTi,j k (t i ) < RTTi,j, then Wi,j k (t i ) i increaed by. Note that, in thi cae, perlot RTT i maller than the average RTT, which mean that congetion level in the link layer i low, and there i a poitive queue backlog difference between the two node. Thu, more packet can be tranmitted over thi link, o W i,j (t k i ) i i ) > and i ) i decreaed by ince increaed. On the other hand, if U i (t k i ) U j (t k RTT i,j (t k i ) > RTT i,j, then W i,j (t k

9 8 the link layer congetion level i high, and le packet hould be tranmitted over thi link. If none of the packet in the lat lot i ACKed, thi mean that congetion i very high, and packet are dropped. In thi cae, Wi,j k (t i ) i halved o that the number of packet over link i j could be reduced harply. After Wi,j k (t i ) i determined, fi,j k (t i ) i et to Wi,j k (t i ) and fi,j k (t i ) packet are paed to the link layer. Note that wdiff-ubmax, imilar to Diff-ubMax, reduce computational complexity and overhead ignificantly a compared to Diff- Max. A. Numerical Simulation V. PERFORMANCE EVALUATION We imulate Diff-Max a well a the tandard backpreure in an idealized time lotted ytem. In particular, we conider triangle and diamond topologie hown in Fig. 5(a) and (b). In both topologie, there are two flow; S R and S 2 R 2, and all node are capable of forwarding packet to their neighbor. The imulation duration i lot, and each imulation i repeated for eed. Each lot i in the ON or OFF tate according to an i.i.d. random proce with a given lo probability. The utility function in thee imulation i log utility, i.e., g (x (t)) = log(x (t)). Fig. 6 how the throughput and total utility (aggregated over per-flow utilitie) v. the lo probability for the triangle topology when the link A C i loy. A can be een, the throughput and the total utility of Diff-Max i equal to that of backpreure. Similar reult are oberved in Fig. 6 for the ame etup when all link are loy. Note that the total utilitie in Fig. 6 and Fig. 7 are negative a we employ log utility in thee imulation. A can be een, the throughput and utility of Diff-Max i equal to that of backpreure. The ame reult are hown for the diamond topology in Fig. 8 and Fig. 9. Thee reult how that Diff-Max achieve the ame throughput and utility a backpreure. B. n-2 Simulation In thi ection, we imulate our cheme, Diff-Max, DiffubMax, wdiff-ubmax uing the n-2 imulator []. The imulation reult how that our cheme ignificantly improve throughput, utility, and delay performance a compared to Ad hoc On-Demand Ditance Vector (AODV) [2], and Detination-Sequenced Ditance-Vector Routing (DSDV) [3] routing cheme. Next, we preent the imulator etup and reult in detail. ) Setup: We conidered three topologie: the diamond topology hown in Fig. 5(b); a grid topology hown in Fig. 5(c), and a random topology. 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 in the cell. In the grid topology, each node can communicate with other node in it cell or with the one in neighboring cell. Four flow are generated randomly. In the random topology, 2 node are randomly generated and located over a 8m 8m terrain according to a uniform ditribution. Ten flow are generated, and tranmitted between randomly elected node. We conider CBR flow, which tart at random time within the firt 5ec and remain on until the end of the imulation which i ec. The CBR flow generate packet with interarrival time.m. IEEE 82. i ued in the MAC layer (with update for Diff-Max implementation a explained in Section IV). 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 Mbp, 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 and DSDV in term of tranport-level throughput, total utility (added over all flow), and average delay (averaged over all packet and flow). We employ log utility in our imulation, i.e., g (x (t)) = log(x (t)). On the other hand, packet delay i meaured at the tranport layer. Let r,k be the time that the kth packet of flow i received by the tranport layer at the receiver ide, and t,k be the time that the ame packet i een by the tranport layer at the tranmitter ide. Then, the packet delay i r,k t,k. 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 i = m, Fmax i = 4 packet. 2) Reult: Fig. (a) how the imulation reult for the diamond topology, where only link A B i loy. Diff-Max perform better than the other cheme for the range of lo rate, ince Diff-Max activate the link baed on the perlink queue backlog, lo rate, and link rate. On the other hand, Diff-ubMax, wdiff-ubmax, AODV, and DSDV ue claical 82. MAC. When the lo rate over link A B increae, the total throughput of all the cheme decreae a expected. A can be een, the decreae in our cheme; Diff-Max, Diff-ubMax, wdiff-ubmax 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. On the other hand, DSDV perform better than AODV at low lo rate thank to keeping track of multiple route and exploiting a new route when one become loy. Yet, it i wore than AODV at higher lo rate, a it require more packet exchange among node at high lo rate, which conume higher bandwidth and reduce throughput. Diff-ubMax and wdiff-ubmax improve throughput ignificantly a compared to both AODV and DSDV thank to exploring route to improve throughput. The improvement of our cheme i up to 22% and 2% over AODV and DSDV, repectively. Alo, Diff-ubMax and wdiff-ubmax have imilar throughput performance, which emphaize the benefit of the routing part and the effective link layer queue etimation mechanim of wdiff-ubmax. Fig. (a) alo how that when lo rate i 5%, the throughput improvement of all cheme (except DSDV) are 5 We conider the lo rate in the range up to 5%, becaue previou tudie of IEEE 82.b baed wirele meh network [6], [7], have reported packet lo rate a high a 5%.

10 9 R B S R S 3 R B S S 2 A R 2 D R 2 S 4 R 3 S 2 S S 2 A R 2 C C R 4 (a) Triangle topology (b) Diamond topology (c) Grid topology Fig. 5. 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 2.5 Total utility Lo probability over link A C Lo probability over link A C Lo probability over link A C (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Fig. 6. Numerical reult for the triangle topology hown in Fig. 5(a). The lo i over link A C. (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Throughput of flow; S R.5 Throughput of flow; S 2 R 2.5 Total utility Lo probability over all link (a) Throughput of S R flow Lo probability over all link (b) Throughput of S 2 R 2 flow Lo probability over all link (c) Total utility. Fig. 7. Numerical reult for the triangle topology hown in Fig. 5(a). The lo i over all link. (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Throughput of flow; S R.5 Throughput of flow; S 2 R 2.5 Total utility Lo probability over link A B Lo probability over link A B Lo probability over all link (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Fig. 8. Numerical reult for the diamond topology hown in Fig. 5(b). The lo i over link A B. (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Throughput of flow; S R.5 Throughput of flow; S 2 R 2.5 Total utility Lo probability over all link Lo probability over all link Lo probability over all link (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility. Fig. 9. Numerical reult for the diamond topology hown in Fig. 5(b). The lo i over all link. (a) Throughput of S R flow. (b) Throughput of S 2 R 2 flow. (c) Total utility.

11 Total Throughput Diff ubmax wdiff ubmax AODV DSDV Average Delay Diff ubmax wdiff ubmax AODV DSDV Lo Probability (a) Total throughput Lo Probabilty (a) Average delay. Throughput Throughput from A to D Throughput from A to B Total Throughput Delay Delay from A to D Delay from A to B Total Delay 5 2 Diff ubmaxwdiff ubmax AODV DSDV Diff ubmaxwdiff ubmax AODV DSDV (b) Per-flow throughput. Fig.. Diamond topology. (a) Total throughput (in kbp) v. average lo rate in the diamond topology. (b) Per-flow (a well a total) throughput of different policie when the average lo rate i et to %. (b) Per-flow delay. Fig.. Diamond topology. (a) Average delay (in ec) v. average lo rate in the diamond topology. (b) Per-flow (a well a total) delay of different policie when the average lo rate i et to 5%. 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. DSDV i wore becaue it require more packet exchange to keep the routing table, which wate reource. Fig. (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 can be een, the rate of flow A B i very low in AODV a compared to our cheme, becaue AODV conider link A B to be broken at ome period during the imulation, while our cheme continue to tranmit over thi link. Although DSDV outperform AODV, our cheme are till better than it in term of throughput thank to exploring route to improve throughput. Fig. (a) how the delay v. lo probability for the diamond topology, where only the link A B i loy. A can be een, Diff-Max introduce higher delay a compared to Diff-ubMax and wdiff-ubmax, becaue Diff-Max can delay packet tranmiion from ome queue depending on their occupancy. In other word, Diff-Max tranmit packet from the node with larger queue ize, which may delay ome packet ignificantly. On the other hand, Diff-ubMax and wdiff-ubmax tranmit packet from the link layer queue baed on 82. MAC cheduling, which reduce delay. On the other hand, the delay performance of Diff-ubMax and wdiff-ubmax i comparable to and better than AODV and DSDV for all lo rate, which how that our algorithm are quite efficient in term of delay. The delay performance of DSDV i high and increae with lo rate a DSDV hould update it routing table periodically and needed, which increae packet delay [3]. Fig. (b) how per-flow and total delay of each algorithm when the lo rate over link A B i 5%. A can be een, the delay of each flow i very large in DSDV while the delay performance of other algorithm are comparable. Fig. 2 preent the reult for the grid topology. In thi cenario, one third of the link, which are choen randomly, are loy with % lo rate. Fig. 2(a) how the total throughput of our cheme a well a AODV and DSDV. Although the throughput performance of our cheme are better than AODV, the total throughput of DSDV i lightly better than our cheme. The reaon i that DSDV treat ome flow (with longer path) unfairly, and they do not get much (or even any) opportunity to tranmit. Since the flow with horter path can tranmit mot of the time, the total throughput of DSDV become better. On the other hand, Fig. 2(b) how that the total utilitie of Diff-ubMax and wdiff-ubmax are better than DSDV. It i expected a our cheme are deigned to maximize the total utility in (4). In other word, even though the total throughput of DSDV may be higher at ome cenario, the total utility, which we maximize, of Diff-ubMax and wdiff-ubmax i higher. Fig. 3 preent the imulation reult for the ame grid topology etup. Fig. 3(a) how the total utility v. average lo rate for our cheme a well a AODV and DSDV. Our cheme Diff-ubMax and wdiff-ubmax ignificantly improve the total utility a compared to both AODV and DSDV. Fig. 3(b) conider the ame etup, and preent average delay v. average lo rate. A can be een, DiffubMax and wdiff-ubmax ignificantly improve delay a compared to AODV and DSDV. Fig. 4 preent the imulation reult for the random topology. In thi cenario, one third of the link, which are choen randomly, are loy with a rate between % to 5%. Fig. 4(a) how the total utility v. average lo rate reult. A can be een, our cheme ignificantly improve the total utility a compared to AODV and DSDV. The improvement in thi cenario i higher a compared to the grid topology ince there are more routing opportunitie that can be exploited in thi random topology cenario.