Distributed Algorithms for Optimal Rate Allocation of. Multipath Traffic Flows

Transcription

1 Distributed Algorithms for Optimal Rate Allocation of Multipath Traffic Flows Koushik Kar, Saswati Sarkar, Ratul Guha, Leandros Tassiulas Abstract We address the question of optimal rate allocation for multipath unicast traffic, with the objective of maximizing the total system utility. First, we consider the problem where the set of paths (routes) between each flow is predetermined, and show how the optimal rates on these paths can be computed. We then consider the problem where the paths are not predetermined, and present an approach that finds the optimal paths as well as the optimal rates on these paths. For both the problems, the algorithms that we develop are completely decentralized, computationally simple, and have a low communication overhead. We demonstrate, through analysis and simulation, that our algorithms converge and attain the maximum system utility. 1 Introduction We consider the problem of optimal rate allocation for multipath traffic in a general communication network, with the objective of maximizing overall system performance. We assume that each traffic flow is associated with a utility function [9, 10], and our goal is to allocate rates (the terms rate and bandwidth are used interchangeably throughout the paper) to these flows such that the K. Kar is with the Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, USA ( koushik@ecse.rpi.edu). S. Sarkar and R. Guha are with the Department of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA ( swati@ee.upenn.edu, rguha@seas.upenn.edu). L. Tassiulas is with the Computer Engineering and Telecommunications Department, University of Thessaly, Greece ( leandros@inf.uth.gr). 1

2 aggregate system utility is maximized. This utility maximization problem can be viewed as a generalized throughput maximization or a generalized fair rate allocation problem, and a wide range of throughput and fairness objectives can be obtained by selecting the utility functions appropriately. For instance, if the utility functions are linear, then the rate allocation objective is to maximize the system throughput. On the other hand, if the utilities are logarithmic, then the optimal rates achieved are proportional fair [9]. We assume that the traffic of any flow can be split over multiple paths. Note that multiple paths can be used for load balancing, thus allowing more efficient use of the network. We develop algorithms for two formulations of the multiple path utility maximization problem. The formulations model the utility maximization problem as convex optimization problems that can be solved using wellknown gradient-search algorithms; these algorithms are however centralized. We present distributed algorithms with low storage and processing complexity that are guaranteed to attain the optimal rate allocations. In Section 2, we assume that the set of paths of each flow is predetermined by some existing multipath routing algorithm, and develop an end-to-end, distributed, iterative flow allocation algorithm that allocates traffic in these paths. We prove that our algorithm converges and attains rates that maximize the overall utility. This algorithm has a low communication overhead and does not require the intermediate routers to maintain per-flow state. Therefore, these algorithms are best used for rate control, once the set of paths is determined. In Section 3, we consider the problem where the set of paths is not determined a priori, and the traffic can be split over all possible paths. In this case, our goal is to determine the jointly optimal set of paths and the rate allocation in these paths that maximize the overall utility. We present a distributed algorithm that combines a hop-by-hop rate control and a depth first search technique for determining the optimal paths and the rates. We prove that the algorithm converges and attains maximum system utility. This algorithm too has a low overhead of computation and communication, and its complexity is polynomial on the problem size, although the number of possible paths of any flow could be exponential. The disadvantage of this algorithm is that it requires per-flow states in 2

3 the routers. Thus, this algorithm attains a more general objective as compared to the previous one, that of joint optimization of routing and flow control, but its implementation is more complex. In Section 4, we demonstrate using simulation that our algorithms attain the optimum rates in an asynchronous environment; note that the analytical guarantees are for distributed but synchronous rate updates. In Section 5, we outline related work on this topic, and compare the previous approaches with our work. We conclude in Section 6. We present the proofs in the appendix. The motivation, derivation and analysis of our algorithms are based on results in non-differentiable optimization theory, mainly those by B.T. Poljak and N.Z. Shor [20] [22]. It is also worth noting that the algorithm presented in Section 3 can solve in a distributed manner the well-known network flow problem [1], as well as its concave and multicommodity generalizations. In the rest of this paper, we consider a network that consists of a set L of unidirectional links, where a link l L has capacity c l (0 < c l < ), a set of nodes, K, and a set J of unicast flows. The source and destination nodes of flow j are denoted by s j and d j, respectively. Each flow j can send its traffic over a set of paths, and has a utility function U j : R + R, which is concave, continuous, bounded, increasing in [0, ) and satisfies the following bounded slope assumption. Assumption 1 (Bounded slope) There exists an A < such that U j (ỹ) A ỹ [0, ) for all j J. Note that the utility functions need not be differentiable. If U j (x) is not differentiable, assumption 1 must hold for all subgradients 1 of U j. 2 Optimal Multiple Path Flow Control We consider the case where the set of paths of each flow j, P j, is established by some multiple path routing algorithm, e.g., [5, 8]. The multiple path flow control problem is to determine the traffic rates on these paths so that the total system utility is maximized. Source-routing schemes [5], [8] 1 A subgradient [22], defined for convex/concave functions, can be viewed as a generalized gradient, and may exist even if the gradient does not exist. 3

4 allow the source to regulate the traffic rates in different paths, and therefore, the algorithm that we propose in this section can be applied to such schemes. In Section 2.1, we formulate the multiple path flow control problem as a convex optimization problem. In Section 2.2, we present an iterative rate update scheme that attains the optimal path rates. In Section 2.3, we describe the analytical convergence guarantees for this algorithm. In Section 2.4, we describe how the algorithm can be implemented in an asynchronous, distributed manner. In Section 2.5, we describe certain salient features of the algorithm and its implementation. 2.1 Problem Formulation Let P = j J P j be the set of all paths for all flows. For any path p P, let L p be the set of links in p. Let P l denote the set of paths of all flows traversing link l L. Associate a rate variable y p with each path p P. Note that the traffic rate of flow j is p P j y p. Our goal is to attain the rates y p that solve the following utility maximization problem: P 1 : maximize U j ( j J p P j y p ), subject to p P l y p c l, l L. (1) y p 0, p P. (2) Constraints (1) ensure that the total rate of traffic in a link cannot exceed the capacity of the link, and (2) ensure that the rate-variables are non-negative. Note that in the problem formulation, for simplicity of exposition, we have not assumed any maximum/minimum constraints on the rates, apart from the obvious non-negativity constraints. However, this formulation and our approach can easily be modified to incorporate additional maximum and minimum rate constraints. 2.2 An Iterative Algorithm We now present an iterative algorithm for the problem P 1. Let y (n) p denote the value of the rate variable y p at the nth iteration. For each link l L, define e (n) l, the link congestion indicator of 4

5 link l at the nth iteration, as e (n) l = 0, if 1, if p P l y (n) p c l, p P l y (n) p > c l. (3) At the nth iteration, link l is considered congested if e (n) l = 1, and uncongested if e (n) l = 0. In the iterative rate update procedure, as described below, the traffic rate on the path of an S-D pair is increased by an amount that is proportional to the derivative of the S-D pair s utility function, and decreased by an amount that is proportional to the number of congested links in the path. Now let us state the rate update procedure formally. Consider a path p of flow j, i.e., p P j. In the following, [ ] + denotes a projection 2 on [0, ). At the nth iteration, the rate variable y p is updated as follows: y p (n+1) = [ y p (n) + λ n ( U j( y (n) ) κ( e (n) p l ) ] +, (4) p P j l L p where κ is a positive constant, and λ n > 0 is the step-size at the nth iteration. If U j is not differentiable, then U j in (4) must be replaced by a subgradient of U j at that point. Note that ( l L p e l ) is the number of congested links in path p. 2.3 Convergence Analysis Let y = (y p, p P ) denote the vector of all path rates. Also, let y (n) denote the value of this vector at the nth iteration. Note that there may be multiple optimal solutions of P 1. Let Y be the set of optimal solutions of P 1. Define the system utility function U : R P + R as U(y) = j J U j( p P j y p ), and let U be the corresponding optimal value. Thus, U = U(y ) for any y Y. Also, let ρ(y, S) = min z S y z denote the Euclidean distance of any point y from any compact set S. Now, we state convergence results under various conditions on the step-sizes. Assume that the sequence of step-sizes {λ n } in (4) satisfies the following criteria: lim λ n = 0, n n=1 λ n =. (5) 2 For any scalar ỹ, [ỹ] + = max(0, ỹ). 5

6 For example, λ n = 1/n satisfies (5). (5). The following theorem shows that our algorithm converges to the optimum if the step-sizes satisfy Theorem 1 Consider the iterative procedure stated in (3)-(4), with the step-sizes satisfying (5). Then, for all κ > A, lim n ρ(y(n), Y ) = 0, and, lim n U(y(n) ) = U. Theorem 1 states that the distance of the rate vector from the set of optimal rates decreases to zero. In the special case where this optimum is unique, the rate vector converges to the unique optimum. The condition n=1 λ n = in (5) can be somewhat relaxed. Theorem 1 holds if the step-sizes λ n satisfy λ n = Λλ n, where 0 < λ < 1 and Λ is a sufficiently large constant. The condition lim n λ n = 0 is however required for convergence, but it may not be possible (due to precision limitations) or efficient (since it could slow down the convergence rate considerably) to decrease the step-size beyond a certain value. Now, if the step-sizes are constant, we can prove the following slightly weaker convergence result. For any compact set S, let Φ r (S) be the set of all points whose distance from S is at most r, i.e., Φ r (S) = {y : ρ(y, S) r}. Theorem 2 Let {y (n) (λ)} denote the sequence of rate vectors defined by (3)-(4) with λ n = λ n. Then, there exists a function r(λ) such that lim λ 0+ r(λ) = 0, and for all κ > A, and λ > 0, lim n ρ(y(n) (λ), Φ r(λ) (Y )) = 0. Theorem 2 states that for a constant step-size, the rate vector converges to a neighborhood around the optimum, and the size of this neighborhood becomes arbitrarily small with decreasing step-size. For a given constant step-size, the size of the neighborhood depends on the parameters of the problem P 1 including the utility functions. We show in Appendix III how r(λ) can be calculated. Theorem 2 also implies that given any neighborhood around the optimum, we can select the constant step-size 6

7 λ to be sufficiently small so that our algorithm converges to the Φ r(λ) (Y ) neighborhood. A similar result holds even when the step-sizes are not constant but converge to some positive value. 2.4 Distributed Implementation We briefly describe how our iterative algorithm can be implemented in a distributed manner. In the iterative algorithm, link l s congestion indicator, e l, can be determined based only on the total traffic rate on the link, which is available locally at the link. The rate update (4) needs the derivative of the utility function and the number of congested links in the path of a flow. Thus, if each source knows the number of congested links in each of the paths it uses, it can update the the rates on its paths, and the iterative algorithm can be executed at the links and the traffic sources using only local information. We discuss how the necessary information can be communicated to the source. We will refer to the node where a link l originates as l s start node, π l, and the node where l ends as l s end node, θ l. Assume that e l is stored and updated at π l. The rate update (4) for each path of a flow is executed at the source of the flow. The source of a flow can periodically send out some congestion probe packets on each path to the destination, and each node π l on the path of a packet can add the congestion indicator bit e l to a particular field in the packet. In this way, the destination of a flow can know the number of congested links on each path of the flow, and can convey it back to the source. The source can then update the paths rates accordingly. In Section 4, we demonstrate the convergence of this distributed implementation in an asynchronous network scenario. 2.5 Discussion Note that the number of bits required to convey the number of congested links on a flow s path is log 2 ( L + 1), where L is the maximum number of links in the path of any flow. Therefore, for most networks, including the internet, a single byte is sufficient to carry this information, as one byte allows 255 links in a flow s path. Thus, the congestion feedback introduces only small overhead. Now, we investigate whether the source can determine the optimal routes using this algorithm, if the routes have not been decided a priori. Now, P j for flow j can include all paths between the 7

8 source and destination of j, and the rate y p for each path can be computed using this algorithm. The optimal paths will consist of all paths p for which y p > 0, and the source would send data at rate y p in each path p in the optimal set. The shortcoming of this approach is that the storage and processing complexity at each source is linear in the number of paths of the flow, and there can be exponential number of paths for each flow. Therefore, this algorithm is best used for determining the optimal flows in the multiple paths, once the paths are computed. We would later show that in each network there is at least one optimal set of paths that has at most L paths for each flow, where L is the set of links in the network. Thus, the storage and processing complexity at each source is linear in L if the paths are computed apriori. 3 Joint Optimization of Routing and Flow Control We consider the problem of jointly optimizing the paths and the flows on these paths. In Section 3.1, we formulate this joint optimization problem as a generalization of the multicommodity flow optimization problem. In Section 3.2, we present a scheme that iteratively solves the generalized multicommodity flow optimization problem, and describe its convergence guarantees. In Section 3.3, we present a depth first search based scheme that uses the solution of the multicommodity flow problem to computes the optimal paths and the flow allocations on the paths. In Section 3.4, we describe how the optimal paths and flows can be computed in an asynchronous, distributed manner. In Section 3.5, we describe certain salient features of the algorithm and its implementation. 3.1 Problem Formulation Since the traffic of a flow j can be routed through any set of paths, for each link l, we consider a variable x lj that denotes the traffic rate a flow j on link l. Let I k and O k respectively denote the set of incoming and outgoing links at node k. Then, our goal is to obtain the x lj, for all l and j, that solve the following utility maximization problem: P 2 : maximize U j ( x lj x lj ), j J l O sj l I sj 8

9 subject to x lj = x lj, k K \ {s j, d j }, j J, (6) l I k l O k x lj c l, l L, (7) j J x lj 0, l L, j J. (8) Note that the term l O sj x lj l I sj x lj represents the net flow of flow j leaving the source node s j (and arriving at the destination node d j ), and is therefore the traffic rate of flow j. The constraints (6) represent the fact that incoming traffic must be equal to the outgoing traffic at each node (except the source and destination nodes of the flow) in the network. Constraints (7) are the link capacity constraints, and (8) are the non-negativity constraints on the rates. 3.2 An Iterative Algorithm Next, we present an iterative algorithm for the problem P 2. Any node that is not the source or destination node of flow j is an intermediate node for S-D pair j. Let x (n) lj denote the value of the rate variable x lj at the nth iteration. For each link l L, define the link congestion indicator of link l at the nth iteration, ε (n) l, as ε (n) l = 0, if 1, if j J x(n) lj c l. j J x(n) lj > c l. (9) Now, for each intermediate node k K \{s j, d j } for each S-D pair j J, define the node congestion indicator of node k for S-D pair j at the nth iteration, ν (n) kj, as ν (n) kj = 0, if 1, if 1, if l I k x (n) lj l I k x (n) lj l I k x (n) lj = l O k x (n) lj, > l O k x (n) lj, < l O k x (n) lj. (10) For an S-D pair j, intermediate node k is balanced if ν kj = 0, congested if ν kj = 1, and underutilized if ν kj = 1. Now, we motivate the rate update algorithm. An S-D pair s rate on a link is decreased if the link is congested, the link s start node is under-utilized, or the link s end node is congested. Similarly, an S-D pair s rate on a link is increased if the link is under-utilized, the link s start node is congested, 9

10 or the link s end node is under-utilized. Also, if a link s start node (end node) is the source node of an S-D pair, then the S-D pair s rate on the link is increased (decreased) according to the derivative of the S-D pair s utility function. Next we state the rate update procedure formally. Let κ be a positive constant, and λ n > 0 be the step-size at the nth iteration. The update procedure at iteration n for x lj for flow j in link l is x (n+1) lj = [ x (n) lj [ x (n) lj [ x (n) lj + λ n ( U j ( l O sj x (n) lj + λ n ( U j ( l O sj x (n) lj + λ n ( κ(ε (n) l l I sj x (n) lj ) κ(ε(n) l + ν (n) θ l j ) ) ] +, if l O s j, l I sj x (n) lj ) κ(ε(n) l ν (n) π l j ) ) ] +, if l I s j, + ν (n) θ l j ν(n) π l j ) ) ] +, In the above expression, if l O dj, then the term ν (n) π l j l I dj, then the term ν (n) θ l j should be interpreted as zero. otherwise. (11) should be interpreted as zero. Similarly, if We investigate the convergence of the iterative algorithm for various conditions on the step-sizes. Let x = (x lj, l L j, j J) denote the vector of all rates, and x (n) denote the value of this vector at the nth iteration. Also, let X be the set of optimal solutions of P 2. Theorem 3 Consider the iterative procedure stated in (9)-(11), with the step-sizes satisfying (5). Then, there exists a κ 1 <, such that for all κ > κ 1, lim n ρ(x(n), X ) = 0. Theorem 4 Let {x (n) (λ)} denote the sequence of rate vectors defined by (9)-(11) with λ n = λ n. Then, there exists a κ 1 < and a function r(λ) such that lim λ 0+ r(λ) = 0, and for all κ > κ 1, lim n ρ(x(n) (λ), Φ r(λ) (X )) = 0, λ > Computation of the Optimal Paths and the Flows on these Paths The iterative algorithm computes the optimal rates of the S-D pairs on each link. Every node k can now route packets of each S-D pair j in accordance with the S-D pair s optimal rates x lj on the outgoing links l of the node. The problem in this approach is that the packets may loop in the network if the optimal solution allocates flows in cycles. 10

11 Consider an S-D pair j and the directed sub-graph G j (j s optimal flow graph) of the network that consists of the links l where x lj > 0. Note that under the optimal flow allocation, j s packets will traverse only G j. If G j is a directed acyclic graph (DAG), then j s packets will not traverse in cycles. Note that there exists at least one optimal flow allocation for which G j is a DAG. This follows from the following observation: if the minimum x lj amongst the links in a cycle is subtracted from the x lj of the links in the cycle, then we obtain another flow allocation that does not have the cycle and satisfies the link capacity, flow conservation and non-negativity constraints. Since the removal of this cycle does not alter the net flow originating from j s source, the new flow allocation attains the optimal utility as well. The challenge is to ensure that the iterative approach results in an optimal allocation for which G j is a DAG for each flow j. Consider the case where the utility derivatives are lower-bounded by a constant a > 0, i.e., such that U j (ỹ) a ỹ [0, ) for all j J. In this case, we solve the problem using our approach with an additional fictitious flow traversing each link; each such flow has a utility function U(x) = (ax)/2. In the solution obtained, cycles are avoided, and the choice of the utility functions for the fictitious flows ensures that the optimum total utility of flows 1,..., J do not change due to the introduction of these fictitious flows. The fictitious flows however increase the computation and storage complexity in the network. An alternative and more general approach is to explicitly compute the optimal paths and the rates on the paths, and route according to the computed values. For each flow j, in G j, a path p 1j is computed between s j and d j using depth first search or breadth first search. Path p 1j is an optimal path for j. The minimum x lj among the links in p 1j is the optimal rate for j on p 1j ; this minimum value is subtracted from the rate value x lj for each link l in P 1j. After the subtraction, each link with residual rate of zero is removed from G j, resulting in residual graph G 1j. The procedure is repeated with the residual rate values and G 1j, resulting in another optimal path p 2j and an optimal rate for j on p 2j. The procedure terminates when no more paths between s j and d j exist in the residual graph. Note that there can be at most L optimal paths for j as a link is removed from the residual graph every time a new optimal path is obtained. The complexity of computing each path, the rates 11

12 on each path, and the residual graph is O( K + L ). Thus, for each flow j J, the complexity of the whole procedure is O ( L ( K + L )). 3.4 Distributed Implementation The iterative algorithm is inherently distributed as it can be executed at the nodes and the links using only information about adjacent nodes. The congestion indicator of each link can be updated if the total traffic rate at the link is known (see (9)). The congestion indicators at each node can be updated if the rates of the flows traversing the node are known (see (10)). The rate of a flow on a link can be updated if the congestion indicator of the link and the congestion indicators of the start and end nodes of the link are known (see (11)). Assume that link l s start node, π l, stores and updates the link congestion indicator for l, ε l, and rate variable x lj for each flow j. Also a node k stores and updates the node congestion indicator ν kj for each flow j. Then it is straightforward to see that to implement the updates procedures in (9)-(11), each node only needs to know the values of the rate and congestion indicator variables stored at its neighboring nodes in the previous iteration. These distributed update procedures can also be carried out asynchronously. Note that with measurement-based estimation of rates, only the congestion indicators need to be exchanged amongst neighboring nodes. Since a congestion indicator is only a single bit of information, the communication overhead of this algorithm is quite small. As mentioned before, if required, the optimal paths and the optimal rates on those paths can be computed using an iterative DFS or BFS. Both DFS and BFS can be implemented in a distributed manner. 3.5 Discussion Since a node congestion indicator variable ν kj can only be 0, 1, or 1, communicating a node congestion indicator variable requires only 2 bits. Moreover, with measurement-based traffic rate estimation at the nodes, the overhead of rate communication between neighboring nodes can be avoided. Thus the communication overhead of this algorithm is fairly small. 12

13 s s 4 5 s 2 s 3 4 d 4 d d 3 d 2 Figure 1: An example network (Each edge represents a bidirectional link with a capacity of 5 MBps in each direction. s i and d i respectively denote the source and destination of flow Fi, i = 1, 2, 3, 4.) Note that the complexity of storage and computation at any node k is O(( I k + O k ) J), irrespective of the number of optimal paths. The storage and processing complexity at a node is however proportional to the number of flows traversing the node. Maintaining per-flow state is feasible in virtual private networks (VPNs) and intranets, but not in backbone routers since a large number of flows traverse the backbone routers. In backbone routers, state aggregation can be used to reduce the overhead of these additional flow states. The optimization problem P 2 represents a generalized multi-commodity flow problem with concave utility functions [1]; note that the multicommodity flow problem is usually defined with linear objective functions. Thus this algorithm can be used to solve the generalized multi-commodity flow problem in a distributed way; standard solutions of the multicommodity flow problem are centralized [1]. Since the network flow problem is a special case of the multicommodity flow problem, this algorithm also provides a distributed solution to the network flow problem. 4 Simulation Results Simulations carried out on various network topologies/scenarios confirm that distributed implementation of our algorithms achieve the optimal rates in an asynchronous time-varying network environment. In this section, we present a few representative examples to demonstrate this fact. Since the nature of the simulation results for both the cases of our end-to-end and hop-by-hop 13

14 algorithms are similar, we only present the results for the former case. Figure 1 shows the network that we consider. First, let us consider the multipath rate control algorithm as outlined in Section 2. We assume that there are 4 flows sharing the network, denoted by F1, F2, F3, F4, and the flows respectively use 2, 3, 2, 1 predetermined paths. The utility functions of all flows are ln(1 +x) (where x is expressed in MBps), and the step-size is constant. Figure 2, which shows some rate plots in the time window 0-45 secs, demonstrates the convergence of our algorithm in the particular example considered. We assume that flows F1, F2 and F3 are already present in the network at time t=0, and have converged into the optimal rates. Flow F4 arrives at t=15 secs, and flow F3 leaves at t=30 secs. Note that the sudden changes in the optimal rates at t = 15, 30 secs are due to the arrival/departure of flows. In the experiments, λ = 0.05, and κ = 2. Our algorithm was implemented in a distributed manner, as described in Section 2.4, and the rate notification and congestion probe packets are sent out sources periodically after every msec. Observe that in the plots in figure, the computed rates do not exactly converge to the optimal rates, but fluctuate rapidly, remaining close to the optimal rates. The thickening of the rate plots are due to these small but rapid fluctuations around the optimal values. Recall that in Section 2.3, we argued that we need step-sizes close to zero in order to guarantee exact convergence. If the step-size is constant, but small, as in the case of the plots in Figure 2, then we can only guarantee that our algorithm achieves rates that are close-to-optimal. When the total traffic is close to the link capacity, the link congestion indicator fluctuates between 0 and 1, as can be expected from intuition. This causes the path rates to fluctuate, like those seen in Figure 2. Smaller step-sizes cause smaller fluctuations, but also result in lower convergence speeds. Thus the choice of the stepsize is a trade-off between the convergence speed and the magnitude of fluctuations. In the figure, the step-size has been chosen appropriately, based on this trade-off. In practice, a flow could choose large step-sizes initially (to ensure fast convergence), and reduce the step-sizes once it detects that its rate is fluctuating around the same mean value (to reduce fluctuations when the rates are close to the optimal values). 14

15 F1 4.5 F Rate(Mbps) 2.5 Rate(Mbps) Time(ms) x Time(ms) x F F Rate(Mbps) 2.5 Rate(Mbps) Time(ms) x Time(ms) x 10 4 Figure 2: Convergence of computed rates. (The straight lines represent the optimal rates.) 5 Related Work In recent literature, several different rate control algorithms have been proposed for the case when each flow uses a single, predetermined path. In [15], Low et al. propose an algorithm based on the dual approach for the same problem. In [2], the authors suggest a randomized marking based implementation of the algorithm in [15], that uses only one bit for the network congestion feedback. In [10], the authors propose both primal and dual algorithms that solve approximate versions of the same problem. Another related, but different, approach is proposed in [13], in which the user adjusts its rate based on the proportion of marked packets or end-to-end (measurable) losses. In [14], the authors present a window-based based flow control approach for this problem. Here the 15

16 users choose some weights and the window-based flow control scheme, on convergence, allocates rates that are proportionally fair with respect to those weights. The weights are updated in such a way that the algorithm finally converges to the optimal rates. In [11], the authors present a simple algorithm for the same problem where the user adjusts its rate based on the number of congested links. Like the approach taken in this paper, the algorithm in [11] is also based on non-differentiable optimization methods, and has certain similarities with the single-path version of the first algorithm presented in this paper. It is important to note that all of the above-mentioned algorithms are endto-end flow control algorithms. As already mentioned, the multipath case of the utility maximization problem have not been adequately addressed in the literature. Most of the approaches for the singlepath case of the problem, as mentioned above, require strict concavity of the objective function in order to guarantee convergence. However, in the multipath case, the overall objective function may not be strictly concave, even if the individual user utility functions are strictly concave (consider the objective function of P 1 ). This is one of the reasons why extending these approaches to the multipath case becomes difficult, and in fact, direct extensions of these algorithms do not provide convergence guarantees. An alternative approach to this single-path rate allocation problem, for the special case of linear utility functions, is presented in [3]. In [10], the authors present both primal and dual algorithms that attempt to solve approximations of the multipath utility maximization problem. These are also generalizations of the algorithms for the single-path case of the problem, presented in the same paper. However, no formal convergence result is stated in the paper for the multipath case of the problem. The authors do show that the value of the approximate objective function (that they are interested in maximizing) is increasing in time, but that does not guarantee convergence to the optimal solution. In [16], the authors present a dual based approach to this problem, which is based on the algorithm in [15]. However, the authors do not provide any guarantees on the convergence of the algorithm to optimality. These previous approaches do not scale with increasing number of paths, and require the flows to keep track of the different paths it uses. Therefore, these approaches cannot be used to solve the problem of joint routing and rate control. This necessitated the development of the hop-by- 16

17 hop algorithm presented in this paper, based on the multicommodity flow formulation. Distributed solutions of network flow and multicommodity flow problems have been proposed in the network optimization literature (see [4, 7]). The problems addressed in this regard typically fall into two categories. The first one is the standard network flow problem. However, this problem does not have the multi-flow aspect that we have in our problem. The second kind of problems are traffic routing problems based on multicommodity flow formulations. However, these problems are only concerned with the optimal routing of flows, and does not have the rate control aspect of our problem. Moreover, these existing approaches are concerned with minimizing a cost function of the link load, whereas we are concerned with maximizing the aggregate utility of the flows. These factors make our problem considerably different from the previously-addressed problems. Also note that the techniques used to develop the algorithms in this paper are also significantly different from the dual-based methods used in previous approaches. Note that if the there is upper limit on the number of paths that a flow can use, then the problem of joint optimization of routing and rate assignment is quite different from the problem addressed in this paper; in that case, the problem can be shown to be NP-hard. A special case of this problem, for the case of a ring network, is addressed in [17]. 6 Conclusion In this paper, we addressed the problem of optimization based rate allocation for multipath flows. We first considered the case where the set of paths (routes) between each flow is predetermined, and developed an end-to-end algorithm that achieves the optimal rates. We then addressed the case where the paths are not predetermined, and presented a hop-by-hop algorithm algorithm jointly optimizes the paths as well as the rates on these paths. Our algorithms are decentralized, scalable, and can be used for bandwidth provisioning and traffic management in large-scale communication networks. 17

18 References [1] R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice-Hall, [2] S. Athuraliya, S. Low, D. Lapsley, Random Early Marking, Proceedings of the First International Workshop on Quality of future Internet Services (QofIS) 2000, Berlin, Germany, September [3] Y. Bartal, J. Byers and D. Raz, Global Optimization using Local Information with Applications to Flow Control, Proc. of the 38th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp , [4] D. P. Bertsekas, Network Optimization: Continuous and Discrete Models, Athena Scientific, [5] J. Chen, P. Druschel, D. Subramanian, An Efficient Multi-Path Forwarding Method, Proceedings of Infocom 1998, March [6] D. P. Bertsekas, Nonlinear Programming, Athena Scientific, [7] D. P. Bertsekas, J. N. Tsitsiklis, Parallel and Distributed Computation: Numerical Methods, Athena Scientific, [8] J. Chen, P. Druschel, D. Subramanian, A Simple, Practical, Distributed Multi-path Routing Algorithm, TR98-320, July 1998, Rice University. [9] F. P. Kelly, Charging and Rate Control for Elastic Traffic, European Transactions on Telecommunications, vol. 8, no. 1, 1997, pp [10] F. Kelly, A. Maulloo, D. Tan, Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability, Journal of Operations Research Society, vol. 49, no. 3, 1998, pp

19 [11] K. Kar, S. Sarkar, L. Tassiulas, A Simple Rate Control Algorithm for Maximizing Total User Utility, To appear in Proceedings of Infocom 2001, April [12] K. Kar, S. Sarkar, L. Tassiulas, Optimization Based Rate Control for Multirate Multicast Sessions, To appear in Proceedings of Infocom 2001, Anchorage, USA, April [13] S. Kunniyur, R. Srikant, End-to-End Congestion Control Schemes: Utility Functions, Random Losses and ECN Marks, Proceedings of Infocom 2000, March [14] R. La, V. Anantharam, Charge-Sensitive TCP and Rate Control in the Internet, Proceedings of Infocom 2000, March [15] S. Low, D. E. Lapsley, Optimization Flow Control, I: Basic Algorithm and Convergence, IEEE/ACM Transactions on Networking, vol. 7, no. 6, December [16] W. H. Wang, S. Palaniswami, S. Low, Flow control in networks with multiple paths, Proceedings of SPIE ITCom, Denver, August [17] J. Wang, L. Li, S. Low, J. Doyle, Can TCP and Shortest Path Routing Maximize Utility, Proceedings of Infocom, San Francisco, April [18] W. H. Wang, S. Palaniswami, S. Low, Flow control in networks with multiple paths, Proceedings of SPIE ITCom, Denver, August [19] J. Moy, OSPF Version 2, STD 54, RFC 2328, April [20] B. T. Poljak, A General Method of Solving Extremum Problems, Soviet Math Doklady, vol. 8, no. 3, 1967, pp [21] R. T. Rockafellar, Convex Analysis, Princeton Univ. Press, [22] N. Z. Shor, Minimization Methods for Non-differentiable Functions, Springer-Verlag,

20 [23] F. A. Tobagi and W. K. Noureddine, Back-Pressure Mechanisms in Switched LANs Carrying TCP and Multimedia Traffic, IEEE Globecom 99, Symposium on High-Speed Networks, December [24] S. Vutukury and J.J. Garcia-Luna-Aceves, MPATH: a loop-free multipath routing algorithm, Elsevier Journal of Microprocessors and Microsystems 24 (2000), pp [25] W.T. Zaumen, J. J. Garcia-Luna-Aceves, Loop-free Multipath Routing Using Generalized Diffusing Computations, Proceedings of Infocom 1998, March Appendix I: Subgradients and their properties Definition 1 [22] (Subgradient and Subdifferential) Consider a convex and continuous function f defined on a convex set F R k. Then a vector w 0 R k is called a subgradient of f at a point x 0 F if it satisfies f(y) f(y 0 ) (w 0, y y 0 ) y F The subdifferential of f at y 0 F, denoted by f(y 0 ), is the set of all subgradients of f at y 0, i.e., f(y 0 ) = {w 0 R k : f(y) f(y 0 ) (w 0, y y 0 ) y F } In general, subgradient at a point may be non-unique. However, if f(y 0 ) exists, then f(x 0 ) = { f(x 0 )}. Next we state some properties of subgradients (see Theorems 1.12 & 1.13 of [22]), which will be useful in our analysis. Lemma 5 Let I be a finite index set. Let f i, i I, be convex, continuous functions defined on a convex set F. Let y 0 F, and w i0 f i (y 0 ), i I. (a) Let f(y) = i I a if i (y), where a i 0, i I. Then i I a iw i0 f(y 0 ). (b) Let f(y) = max i I f i (y). Define Ĩ(y) = {i I : f i(y) = f(y)}. Then w i0 f(y 0 ), for all i Ĩ(y 0). 20

21 In terms of subgradients, the optimality condition is as follows (Theorem 1.11 of [22]): Lemma 6 Let f be a convex, continuous function defined on a convex set F. Then an interior point y 0 of F is the minimum point of f in F if and only if 0 f(y 0 ). Appendix II: Proof of Theorem 1 We will first state a lemma that would be used in the proof of Theorem 1. For each l L, define g l : R R + R as g l (y) = p P l y p c l. Thus the capacity constraint for link l can be simply written as g l (y) 0. Now consider the following problem P : maximize U j ( y p ) κ max{0, g l (y)}, subject to y p 0 p P p P j j J l L where κ is a non-negative constant. Now define a function Ũ : R R + R as Ũ(y) = j J U j( p P j y p ) κ l L max{0, g l(y)}. Thus P is the problem of maximizing Ũ(y) subject to y 0. Let Ỹ denote the set of optimal solutions of P, and Ũ be the corresponding optimal value. Lemma 7 If κ > A, then Ỹ coincides with Y. Proof: Define Y L = {y : g l (y) 0 l L}. Thus the set of link constraints can be simply written as y Y L. Consider a point ỹ / Y L. Therefore, there exists a l L, such that g l(ỹ) > 0. Choose any p P l. Then from the properties in Lemma 5, Ũ(y) y ỹ p A κ < 0. Therefore, Ũ(ỹ), the set of subgradients of Ũ at ỹ, can not incude the zero vector. Therefore, from Lemma 6, ỹ can not be an optimal solution of P. Therefore all optimal solutions of P must belong to Y L. However, for any y Y L, the values of the objective function of P and P are equal. Therefore any optimal solution of P is an optimal solution of P, and vice versa. Therefore, for κ > A, Ỹ = Y. The above result is fairly intuitive. Comparing problems P and P, we see that the link constraints in P have been transferred to the objective function in P. The term κ max{0, g l (y)} can be interpreted as the penalty associated with the violation of the capacity constraint of link l. Thus the above lemma 21

22 states that when the penalty associated with constraint violations is sufficiently large, the optimal solution set of the unconstrained problem P becomes the same as that of P. Proof of Theorem 1: We will first show that lim n ρ(y (n), Ỹ ) = 0. Choose an arbitrary δ > 0. Let δ = (δ/2). For any ɛ > 0, define D ɛ as D ɛ = {y : y 0, Ũ(y) Ũ ɛ }. It follows from Theorem 27.2 of [21] that there exists an ɛ = ɛ(δ ) > 0 such that D ɛ {y : ρ(y, Ỹ ) δ } (12) Consider an n for which y (n) / D ɛ. Therefore, Ũ(y(n) ) < Ũ ɛ. Choose any ỹ Ỹ. Note that the rate update procedure for the receiver nodes, as stated in (4), can be compactly described as: y (n+1) = [ y (n) +λ n v (n) ] +, where v (n) is a subgradient of Ũ(y(n) ), and [ ] + denotes a projection on the non-negative orthant. Since v (n) Ũ(y(n) ), and using the definition of a subgradient (Definition 1), we obtain (v (n), y (n) ỹ ) Ũ(y(n) ) Ũ(ỹ ) < ɛ (13) From Assumption 1, it is easy to see that v (n) is upper-bounded. Let v (n) Ã for all n. Using these facts, and (13), we obtain, y (n+1) ỹ 2 = [ y (n) + λ n v (n) ] + ỹ 2 y (n) + λ n v (n) ỹ 2 (14) = y (n) ỹ 2 + λ 2 n v (n) 2 + 2λ n (y (n) ỹ, v (n) ) < y (n) ỹ 2 + Ã2 λ 2 n 2ɛλ n (15) Note that (14) follows from the fact that ỹ 0 (use projection theorem). Since λ n 0, λ n (ɛ/ã2 ) when n is sufficiently large. For all such n, from (15), we get y (n+1) ỹ 2 < y (n) ỹ 2 ɛλ n (16) Now, for the sake of contradiction, let us assume that there exists a N ɛ < such that y (n) / D ɛ for all n N ɛ. Therefore, there exists N ɛ N ɛ be such that (16) holds for all n N ɛ. Summing up 22

23 the inequalities obtained from (16) for n = N ɛ to N ɛ + m, we obtain N y (Nɛ+m+1) ỹ 2 < y (Nɛ) ỹ 2 ɛ +m ɛ λ n (17) n=n ɛ which implies that y (Nɛ+m+1) ỹ as m, since λ n diverges. This is impossible, since y (Nɛ+m+1) ỹ 0. Hence our assumption was incorrect. Hence, there exists an infinite sequence n 1,ɛ < n 2,ɛ < n 3,ɛ <... such that y (n i,ɛ) D ɛ for all i = 1, 2, 3,... This implies that there exists an i 1 such that (16) holds for all n n i1,ɛ. Also, since λ n 0, there exists and i 2 such that λ n (δ /Ã) for all n n i 2,ɛ. Let i = max(i 1, i 2 ). We show that ρ(y (n), Ỹ ) δ for all n n i,ɛ. Pick any n n i,ɛ. There can be three cases: Case 1 : n = n j,ɛ for some j i : In this case, y (n) D ɛ. From (12), it trivially follows that ρ(y (n), Ỹ ) δ < δ. Case 2 : n = n j,ɛ + 1 for some j i : In this case, y (n) = y (n j,ɛ+1) = [ y (n j,ɛ) + λ nj,ɛ v (n j,ɛ) ] +. Thus y (n) y (n j,ɛ) = [ y (n j,ɛ) + λ nj,ɛ v (n j,ɛ) ] + y (n j,ɛ) y (n j,ɛ) + λ nj,ɛ v (n j,ɛ) y (n j,ɛ) = λ nj,ɛ v (n j,ɛ) Ãλ n j,ɛ δ (18) From (18) and the fact that ρ(y (n j,ɛ), Ỹ ) δ (Case 1), we get ρ(y (n), Ỹ ) ρ(y (n j,ɛ), Ỹ ) + y (n) y (n j,ɛ) δ + δ = 2δ = δ (19) Case 3 : n j,ɛ + 1 < n < n j+1,ɛ for some j i : Note that y (n ) / D ɛ for all n satisfying n j,ɛ < n < n j+1,ɛ. From (16), it follows that y (n +1) ỹ < y (n ) ỹ. Summing up these inequalities obtained for n = n j,ɛ + 1 to n 1, we obtain y (n) ỹ < y (nj,ɛ+1) ỹ. Since this inequality holds for all ỹ Ỹ, hence ρ(y (n), Ỹ ) < ρ(y (nj,ɛ+1), Ỹ ). Since ρ(y (nj,ɛ+1), Ỹ ) δ (Case 2), it follows that ρ(y (n), Ỹ ) δ. From cases 1, 2, 3, if follows that ρ(y (n), Ỹ ) δ for all n n i,ɛ. By virtue of the arbitrariness of δ, it follows that lim n ρ(y (n), Ỹ ) = 0. Now, from Lemma 7, it follows that if κ > A, then lim n ρ(y (n), Y ) = 0. 23

24 Appendix III: Proof of Theorem 2 Define a function Ũ(y) as in the proof of Theorem 1, i.e., Ũ(y) = j J U j( p P j y p ) κ l L max{0, p P l y p c l }. Consider the problem of maximizing Ũ(y) subject to y 0. Let Ỹ be the set of optimal solutions for this problem, and Ũ be the corresponding optimal value. Since κ > A, Ỹ = Y, and Ũ = U (use Lemma 7 in Appendix II). Let L be the maximum number of links on any flow s path. Now define the set D(λ) as follows D(λ) = { y 0 : Ũ(y) Ũ 2κ 2 L2 P λ } (20) Now define r(λ) as follows Now define r(λ) as r(λ) = max ρ(y, Ỹ ) (21) y D(λ) r(λ) = r(λ) + κ L P λ (22) It is easy to show that r(λ), and hence D(λ), is bounded. Moreover, from Theorem 27.2 of [21], it follows that r(λ) 0 as λ 0+. Therefore, from (22), it follows that r(λ) 0 as λ 0+. The proof of Theorem 2 is along the same lines as the proof of Theorem 1. The proof is provided below, for the sake of completeness. Proof of Theorem 2: Consider an n for which y (n) / D(λ). Therefore, Ũ(y(n) ) < Ũ 2κ 2 L2 P λ. Choose any ỹ Ỹ. Note that the rate update procedure for the receiver nodes, as stated in (4), can be compactly described as: y (n+1) = [ y (n) + λv (n) ] +, where v (n) is a subgradient of Ũ(y(n) ), and [ ] + denotes a projection on the non-negative orthant. Since v (n) Ũ(y(n) ), and using the definition of a subgradient (Definition 1), we obtain (v (n), y (n) ỹ ) Ũ(y(n) ) Ũ(ỹ ) < 2κ 2 L2 P λ (23) From Assumption 1, it is easy to see that v (n) is upper-bounded as v (n) κ L P for all n. Using these facts, and (23), we obtain, y (n+1) ỹ 2 = [ y (n) + λv (n) ] + ỹ 2 24

25 y (n) + λv (n) ỹ 2 (24) = y (n) ỹ 2 + λ 2 v (n) 2 + 2λ(y (n) ỹ, v (n) ) < y (n) ỹ 2 + κ 2 L2 P λ 2 2κ 2 L2 P λ 2 = κ 2 L2 P λ 2 (25) Now, for the sake of contradiction, let us assume that there exists a N < such that y (n) / D(λ) for all n N. Therefore, there exists N N be such that (25) holds for all n N ɛ. Summing up the inequalities obtained from (25) for n = N to N + m, we obtain y (N+m+1) ỹ 2 < y (N) ỹ 2 mκ 2 L2 P λ 2 (26) which implies that y (N+m+1) ỹ as m. This is impossible, since y (N+m+1) ỹ 0. Hence our assumption was incorrect. Hence, there exists an infinite sequence n 1 < n 2 < n 3 <... such that y (ni) D(λ) for all i = 1, 2, 3,... This implies that there exists an i such that (25) holds for all n n i. We show that ρ(y (n), Ỹ ) r(λ) for all n n i. Pick any n n i,ɛ. There can be three cases: Case 1 : n = n j for some j i : In this case, y (n) D(λ). From (20)-(22), trivially follows that ρ(y (n), Ỹ ) r(λ) < r(λ). Case 2 : n = n j + 1 for some j i : In this case, y (n) = y (nj+1) = [ y (nj) + λv (nj) ] +. Thus y (n) y (n j) = [ y (n j) + λv (n j) ] + y (n j) y (n j) + λv (n j) y (n j) = λ v (nj) κ L P λ (27) From (27) and the fact that ρ(y (n j), Ỹ ) r(λ) (Case 1), we get ρ(y (n), Ỹ ) ρ(y (nj), Ỹ ) + y (n) y (nj) r(λ) + κ L P λ = r(λ) (28) Case 3 : n j + 1 < n < n j+1 for some j i : Note that y (n ) / (λ) for all n satisfying n j,ɛ < n < n j+1. From (25), it follows that y (n +1) ỹ < y (n ) ỹ. Summing up these 25

26 inequalities obtained for n = n j + 1 to n 1, we obtain y (n) ỹ < y (nj+1) ỹ. Since this inequality holds for all ỹ Ỹ, hence ρ(y (n), Ỹ ) < ρ(y (nj+1), Ỹ ) r(λ) (from Case 2). From cases 1, 2, 3,, if follows that ρ(y (n), Ỹ ) r(λ) for all n n i. Since Y = Ỹ, it follows that lim n ρ(y (n), Φ r(λ) (Y ) = 0. Appendix IV: Proof Outlines of Theorems 3 and 4 Then problem P 2 can alternatively be posed as follows, by replacing the equality constraint in (6) as two inequality constraints: P 2 : maximize U j ( x lj ) j J l O sj j subject to x lj l I kj x lj l I kj x lj k K j \ {s j, d j } j J (29) l O kj x lj k K j \ {s j, d j } j J (30) l O kj x lj c l l L (31) j J l x lj 0 l L j j J (32) Note that in P 2, the constraints are linear and hence there is no duality gap, and lagrange multipliers exist. Let µ be a lagrange multiplier vector of P 2. For all k K j \ {s j, d j } for all j J, define functions h kj and h kj as h kj (x) = l I kj x lj l O kj x lj and h kj (x) = l O kj x lj l I kj x lj (= h kj (x)), respectively. Also, for each l L, define a function g l as g l (x) = j J l x lj c l. Now consider the following problem: P 2 : maximize U j ( max{0, g l (y)} j J x lj ) κ l O sj j l L κ max{0, h kj (x) κ j J k K j \{s j,d j } j J subject to x lj 0 l L j max{0, h kj (x)} k K j \{s j,d j } where κ is a non-negative constant. It follows from Theorem 4.2 of [22] that if κ > µ, then the sets of optimal points of P 2 and P 2 are the same. 26

27 Let Ṽ (x) represent the objective function of P 2, and Ṽ be its optimal value. Now note that the rate update procedure in (9)-(11) can be compactly stated as: x (n+1) = [ x (n) + λ n v (n) ] +, where v (n) is a subgradient of Ṽ (x(n) ), and [ ] + denotes a projection on the non-negative orthant. Now following the steps in the proof Theorem 1 (Appendix I) (eqn. (13) onwards), we can show that distance of the vector x (n) from the set of optimal points of P 2 (which is also the set of optimal points of P 2, by our previous argument), converges to zero. The proof of Theorem 4 is very similar to the proof of Theorem 2, and is omitted for brevity. In this case, r(λ) can be expressed in the same way as in (20)-(22), by replacing Ũ(y), Ũ and Ỹ by Ṽ (y), V and X respectively. 27