Performance Evaluation 52 (2003) 119–132
Optimal flow control and routing in multi-path networks夽 Wei-Hua Wang a,∗ , Marimuthu Palaniswami a , Steven H. Low b,c a
Department of Electrical and Electronic Engineering, ARC Special Research Centre for Ultra-Broadband Information Networks, The University of Melbourne, Melbourne, Vic. 3010, Australia b Department of Computer Science, California Institute of Technology, Pasadena, CA 91125, USA c Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Abstract We propose two flow control algorithms for networks with multiple paths between each source–destination pair. Both are distributed algorithms over the network to maximize aggregate source utility. Algorithm 1 is a first order Lagrangian method applied to a modified objective function that has the same optimal solution as the original objective function but has a better convergence property. Algorithm 2 is based on the idea that, at optimality, only paths with the minimum price carry positive flows, and naturally decomposes the overall decision into flow control (determines total transmission rate based on minimum path price) and routing (determines how to split the flow among available paths). Both algorithms can be implemented as simply a source-based mechanism in which no link algorithm nor feedback is needed. We present numerical examples to illustrate their behavior. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Optimal flow control; Multiple paths network; Optimization; Minimum first derivative path
1. Introduction Flow control can be regarded as a distributed computation over a network to solve an optimization problem [8,9,12–15,19,21]. In this formulation, each source is characterized by a utility function of its transmission rate and the goal is to maximize aggregate utility. Indeed, one can interpret major TCP congestion control protocols, such as Reno [10], Vegas [4], RED [7], and REM [2], within this framework where different protocols are merely different algorithms to solve the same prototype problem with different utility functions [18]. Most of these papers assume that there is a unique path between source and destination, and the issue is to determine the source rate based on network congestion. On the other hand, multi-path routing problem 夽
An earlier version of this paper was presented at the Internet Performance and Control of Network Systems Conference, R.D. van der Mei, F. Huebner-Szaba de Bucs (Eds.), Proceedings of SPIE, vol. 4523, Denver, CO, 21–22 August 2001, The International Society for Optical Engineering. This work was supported by the Australian Research Council through grant A49930405. ∗ Corresponding author. E-mail addresses:
[email protected] (W.-H. Wang),
[email protected] (M. Palaniswami),
[email protected] (S.H. Low). 0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 1 7 6 - 1
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has received significant attention in recent literature, e.g. Refs. [5,6,23], where the issue is to determine efficient loop-free multi-paths. In this paper, we propose algorithms that attempt to jointly optimize flow control and routing when multiple paths are available between source and destination. This problem has been studied in Ref. [13] using a penalty function approach, in Ref. [16] using sliding mode control [22], and in Ref. [11] using a subgradient method. The main obstacle in the multi-path case is that, even if the objective function is strictly concave in the total source rates, it is not strictly concave in path rates. Then the optimal path rates are non-unique and the dual problem becomes non-differentiable. Lagrangian multiplier method generally converges only when the objective function is strictly concave. In the next section, we describe the optimization model. We present two algorithms in Section 3. The first algorithm is derived as the first order Lagrangian method on a modified objective function that has the same optimal solution as the original objective function but apparently better convergence property. The second method is a subgradient-like method based on the idea that only paths that are least congested carry positive flows at optimality. We discuss an implementation method in Section 4 that uses queuing delay to implicit feeds back congestion information. Finally, we present numerical results to illustrate performance of these algorithms in Section 5, and conclude with some limitations of this paper.
2. The optimization problem Consider a network whose links are denoted by L = {1, 2, . . . , L}. Let cl be the capacity of link l ∈ L and c = [c1 , c2 , . . . , cL ]T . Let S = {1, 2, . . . , S} be the set of sources. Each source s has ns available paths or routes from the source to the destination. Let the L × 1 vector Rs,i denote the set of links used by source s ∈ S on its path i ∈ {1, 2, . . . , ns }, whose lth element equals 1 if the path contains link l and 0 otherwise. The set of all the available paths of user s is defined by Rs = [Rs,1 , Rs,2 , . . . , Rs,ns ] and the total paths in the network are defined by a L × R routing matrix R, R = [R1 , R2 , . . . , RS ], where R = n1 + n2 + · · · + nS is the total number of the paths. s For each source s, let xs,i be the rate of source s on path Rs,i , and xs = ni=1 xs,i be the total source rate. Let ms ≥ 0 and Ms ≤ ∞ be the minimum and maximum rates, respectively, i.e., ms ≤ xs ≤ Ms . When source s transmits at a total rate of xs , it attains a utility Us (xs ). We assume that Us : R+ → R is continuous, increasing and strictly concave. Let x = [x1,1 , . . . , x1,n1 , x2,1 , . . . , x2,n2 , . . . , xn,1 , . . . , xn,nS ]T ∈ RR+ be the vector of all path rates of all sources. Our objective is to choose the rates x so as to maximize the total utility s∈S Us (xs ) subject to capacity constraints: max Us (xs ), (1) x
s∈S
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subject to xs =
ns
xs,i ,
ms ≤ xs ≤ Ms , s ∈ S,
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(2)
i=1
Rx ≤ c,
x ≥ 0.
(3)
The constraint (3) says that the total source rates at links l do not exceed the link capacity cl . There exists a unique optimal solution for the source rates xs since the objective function (1) is continuous and the feasible region (2) and (3) is compact. However, the set of path rates xs,i may not be unique since the objective function is not strictly concave in x. Solving (1)–(3) directly is impractical in a real network since the rates are coupled through shared links. The key to a distributed and decentralized solution is to look at the following Lagrangian form and find a saddle-point solution. 2.1. Lagrange multiplier ¯ λ, p, µ) L(x, u) = L(x, λ, n ns ns s = −pT (Rx − c)+µT x Us xs,i + λ¯ s Ms − xs,i − λs ms − xs,i s∈S
=
i=1
Us
s∈S
n s
xs,i −
i=1
i=1 ns
r ps,i xs,i − λ¯ s
i=1
i=1 ns i=1
xs,i +λs
ns
xs,i +pT c+λ¯ T M−λT m+µT x,
i=1
where λ¯ = [λ¯ 1 , λ¯ 2 , . . . , λ¯ S ]T , λ = [λ1 , λ2 , . . . , λS ]T , p = [p1 , p2 , . . . , pL ]T , µ = [µ1,1 , . . . , µ1,n1 , . . . , ¯ λ, p, µ) are all µS,1 , . . . , µS,nS ]T , M = [M1 , M2 , . . . , MS ]T , m = [m1 , m2 , . . . , mS ]T , and u = (λ, r T ¯ nonnegative, and ps,i = p Rs,i . The Lagrange multipliers u = (λ, λ, p, µ) have several simple interprer tations. For example, we may view pl as the price per unit bandwidth at link l, and ps,i as the path price, the sum of all link prices on path Rs,i . Kuhn–Tucker theorem directly provides the optimality condition for our problem. Theorem 1. The optimal solution of the path rates xs,i in problems (1)–(3) must satisfy r Us (xs ) − λ¯ s + λs = ps,i − µs,i ,
(4)
µs,i xs,i = 0,
(5)
λ¯ s (Ms − xs ) = 0,
(6)
λs (ms − xs ) = 0, pl (x l − cl ) = 0,
s ∈ S, i = 1, 2, . . . , ns , l∈L
for some µs,i ≥ 0, λ¯ s ≥ 0, λs ≥ 0, and pl ≥ 0, where source rate xs = which Rl· denotes the lth row of routing matrix R.
(7) (8) ns
i=1
xs,i , link flow x l = Rl· x in
From (4) and (5) we see that, at optimality, the prices on paths Rs,i that carry positive flows xs,i > 0 must be minimum, and hence equal, among all the paths Rs of source s. Moreover, the optimal source
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rates are given by ∗ ∗ s xs,i = [Us−1 (psr )]M xs∗ = ms
and
xs,i = 0
R∗s,i ∈R∗s
∗
r if ps,i > psr , ∗
∗
∗ ∗ r r s where [z]M ms = max(ms , min(Ms , z)), path Rs,i has the minimum path price ps,i = ps , and Rs defines ∗ the set of all minimum price paths Rs,i of source s.
3. Algorithms In this section, we present two distributed algorithms for the solution of problem (1)–(3). When the objective function is not strictly concave, such as ours, it is well known that a first order Lagrangian algorithm may oscillate. 3.1. Algorithm 1 The idea of Algorithm 1 is to apply the first order Lagrangian method to the following modified objective function: n ns s 1 max Us xs,i − (9) (xs,i − x¯s,i )2 , x≥0,x≥0 ¯ 2 i=1 s∈S s∈S i=1 ¯ is a maximizer of (4) subject to the constraints (2) and (3), where x¯s,i is an augmented variable. If (x, x) then x must also be an optimal solution of the original problem (1)–(3). This is because s at optimality, x = 2x¯ so that the added non-positive term is zero. With the non-positive term − s∈S ni=1 (1/2)(xs,i − x¯s,i ) , the modified objective function becomes strictly concave in x for a fixed x, ¯ and strictly concave in x¯ for a fixed x. It is, however, not strictly concave in (x, x). ¯ Based on the Arrow–Hurwicz gradient method [1, pp. 154–165], we have the following optimization algorithm: r xs,i (t + 1) = [(1 − γ )xs,i (t) + γ x¯s,i (t) + γ (Us (xs (t)) − λ¯ s (t) + λs (t) − ps,i (t))]+ , ns x¯s,i (t + 1) = (1 − γ )x¯s,i (t) + γ xs,i (t), xs (t + 1) = xs,i (t + 1),
λ¯ s (t + 1) = [λ¯ s (t) − γ (Ms − xs (t))]+ , pl (t + 1) = [pl (t) + γ (x l (t) − cl )]+ ,
i=1
λs (t + 1) = [λs (t) + γ (ms − xs (t))]+ ,
where γ > 0 is a small step size, and [z]+ = max{0, z}, x l = Rl· x is the aggregate source rate at link r l, and ps,i = p T Rs,i is the path price for routing Rs,i . Then we have the following synchronous flow control algorithm for multiple paths. Algorithm 1. • Source s’s algorithm: At times t = 1, 2, . . . , source s: r (1) Receives from the network the path prices ps,i (t) = pT (t)Rs,i for all its paths Rs,i , i = 1, 2, . . . , ns .
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(2) Updates the path rate xs,i (t + 1), its optimal estimation x¯s,i (t + 1), i = 1, 2, . . . , ns and the source rate xs (t + 1): r (t))]+ , xs,i (t + 1) = [(1 − γ )xs,i (t) + γ x¯s,i (t) + γ (Us (xs (t)) − λ¯ s (t) + λs (t) − ps,i ns x¯s,i (t + 1) = (1 − γ )x¯s,i (t) + γ xs,i (t), xs (t + 1) = xs,i (t + 1). (10) i=1
(3) Computes the new Lagrange multipliers λ¯ (t + 1) and λ(t + 1) for the next step λ¯ s (t + 1) = [λ¯ s (t) − γ (Ms − xs (t))]+ ,
λs (t + 1) = [λs (t) + γ (ms − xs (t))]+ .
(11)
(4) Communicates the new flow rate xs,i (t + 1) to all the links contained in paths Rs,i . • Link l’s algorithm: At times t = 1, 2, . . . , link l: (1) Receives flow rates xs,i (t) for all paths Rs,i that contain link l. (2) Computes a new price pl (t + 1) = [pl (t) + γ (x l (t) − cl )]+ , where x l = Rl· x. (3) Communicates new prices pl (t + 1) to all the sources whose path Rs,i contains link l. From (11), x¯s,i (t) is a low-pass version of xs,i (t). If the algorithm converges, then (xs,i (t) − x¯s,i (t)) will converge to zero. By subtracting (11) from (10), we see that either xs,i (t) → 0 or r (t) → 0 Us (xs (t)) − λ¯ s (t) + λs (t) − ps,i
as t → ∞. This is the Kuhn–Tucker condition, and hence the limit point must be optimal. When source s has only a single path, then the source algorithm simplifies to s xs (t + 1) = [xs (t) + γ (Us (xs (t)) − psr (t))]M ms ,
(12)
which is a ‘smoothed’ version of the algorithm in [20]. A popular utility function is Us (xs ) = as log xs . Then (12) becomes Ms as r xs (t + 1) = xs (t) + γ . (13) − ps (t) xs (t) ms As observed in [19], since Us (xs (t)) = as /xs (t) is large when xs (t) is small, (13) can lead to severe rate and queue oscillation. To damp the oscillation, (13) can be modified to s xs (t + 1) = [xs (t) + α(as − psr (t)xs (t))]M ms ,
where α is a new step size. It is a discrete version of the primal algorithm in [13]. The same modification can be applied to the multi-path case where (10) is modified to r (t))xs (t))]+ . xs,i (t + 1) = [(1 − γ )xs,i (t) + γ x¯s,i (t) + α(as − (λ¯ s (t) − λs (t) + ps,i
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3.2. Algorithm 2 Algorithm 1 is derived by applying the Lagrange first order method to a modified objective function that has the same optimal solution as the original objective function. In this section, we present another algorithm based on a subgradient method. It decomposes the source algorithm into a flow control problem which determines the total source rate, and a routing problem which decides how to split the total rate among a set of least congested paths. Recall that (4) and (5) imply that at optimality, only those paths that have the minimum price carry a positive flow. This is the idea of minimum first derivative path discussed in, e.g., Ref. [3]. Indeed, (4)–(7) ∗ imply that if psr is minimum path price among Rs , then the optimal total source rate xs∗ is given by ∗ ∗ ∗ r s xs,i xs∗ = = [Us−1 (psr )]M and xs,i = 0 if ps,i > psr , (14) ms R∗s,i ∈R∗s
∗
∗
r = psr , and R∗s defines the set of all minimum price where path R∗s,i has the minimum path price ps,i ∗ paths Rs,i of source s. The condition (14) suggests a way to adapt the total source rate to congestion, but it does not specify how the total rate should be split among the available paths. A naive approach is to simply split it evenly only along paths that have the least current price. This algorithm, however, does not converge, e.g., when multiple paths have different link capacity. We present a routing strategy, based on the idea of Bertsekas and Tsitsiklis [3], that has a better convergence property. Algorithm 2 uses the same link algorithm as in Algorithm 1 to update the link prices. The source algorithm is decomposed into two decisions, flow control and routing. Flow control at source s: ∗
s xs (t + 1) = [Us−1 (psr (t))]M ms ,
∗
psr (t) =
min
i=1,2,...,ns
r ps,i (t). ∗
Hence, at each step t + 1, source s sends at a rate xs (t + 1) determined by the minimum path price psr (t). Routing at source s: ∗
r • xs,i (t + 1) = [xs,i (t) − γ (ps,i (t) − psr (t))]+ for all i = 1, 2, . . . , ns . ∗ • Pick any Rs,j that has the minimum price and set its rate to + xs,j (t + 1) = xs (t + 1) − xs,i (t + 1) . i=1,...,j −1,j +1,...,ns
Hence, at each step t + 1, the rates on all paths that cost more than the minimum are reduced by an amount proportional to the excess price, and the rate on one of the minimally priced paths is increased, so that the new rates on all paths sum to the new total source rate determined in the flow control decision. Alternatively, the aggregate excess traffic from all paths with non-minimum prices can be distributed uniformly among paths with minimum price. Both alternatives can be regarded as approximations of a subgradient at the current point—if paths with rates on all non-minimum paths were set to zero and all their traffics are shifted to paths with minimum price, however, distributed, then the resulting (cl − x l (t)) is the subgradient of the dual problem. Assigning all excess traffic to a single minimum-priced path is simpler to implement. When the algorithm converges, only paths with the minimum price will carry positive flows.
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Algorithm 2. • Link l’s algorithm: At times t = 1, 2, . . . , link l: (1) Receives flow rates xs,i (t) for all paths Rs,i that contain link l. (2) Computes a new price pl (t + 1) = [pl (t) + γ (x l (t) − cl )]+ , where x l = Rl· x. (3) Communicates new prices pl (t + 1) to all the sources s whose path Rs,i contains link l. • Source s’s algorithm: At times t = 1, 2, . . . , source s: r (t) = pT (t)Rs,i for all its paths Rs,i , i = (1) Receives from the network the path prices ps,i ∗ r (t). 1, 2, . . . , ns , and decides the minimum path price psr (t) = mini=1,2,...,ns ps,i (2) Updates the source rate xs (t + 1): ∗
s xs (t + 1) = [Us (psr (t))]M ms .
(3) Updates the path rate xs,i (t + 1) on path Rs,i : ∗
r (t) − psr (t))]+ . xs,i (t + 1) = [xs,i (t) − γ (ps,i
(4) Picks any path Rs,j that has the minimum price and set its rate to: + xs,j (t + 1) = xs (t + 1) − xs,i (t + 1) . i=1,...,j −1,j +1,...,ns
(5) Communicates all the new flow rate xs,i (t + 1) to links l contained in paths Rs,i . 4. End-to-end implementation Both Algorithms 1 and 2 require communication between sources and links: source s must obtain the r sum ps,i (t) of link prices in its paths Rs,i for i = 1, 2, . . . , ns , and link l must obtain the aggregate source l rate x (t). Note that x l (t) is the sum of transmission rates at the source and is not the aggregate input flow rate observable at link l, because queuing and multiplexing at upstream links distorts the flow of s from xs (t) at the source into some other fluid flow. To eliminate the need for source-to-link communication, link l can measure the local aggregate input rate xˆ l (t) and use that to approximate the aggregate source rate x l (t), as suggested in Ref. [17]. Hence the link algorithm (10) is modified to pl (t + 1) = [pl (t) + γ (xˆ l (t) − cl )]+ , where xˆ l (t) is the aggregate input rate measured by link l at time t. In the reversed direction, a marking scheme is proposed in [2] to communicate path prices to sources using a single bit. In this section, we propose a way to communicate path prices implicitly, i.e., a source deduces the aggregate price on each path from observed round trip time (RTT) without the need for explicit feedback from links. The idea is to use a different step size β/cl at each link l: + β l pl (t + 1) = pl (t) + (xˆ (t) − cl ) . (15) cl
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Since queuing length bl (t) evolves according to bl (t + 1) = [bl (t)(xˆ l (t) − cl )]+ , multiplying both sides by β/cl , we have + β β l β bl (t + 1) = bl (t) + (xˆ (t) − cl ) . cl cl cl
(16)
Comparing (16) with (15), we see that link price at time t is proportional to the current queuing delay ql (t) := bl (t)/cl at link l: pl (t) = β
bl (t) = βql (t). cl
The price on path Rs,i is then proportional to the end-to-end queuing delay at time t: r ps,i (t) = pT (t)Rs,i = βq T (t)Rs,i ,
where q(t) = [q1 (t), q2 (t), . . . , qL (t)]T . Given end-to-end propagation delay PDrs,i , the end-to-end queuing delay q T (t)Rs,i , and hence the aggregate price, can be deduced from the RTTrs,i (t) observed at the source: r ps,i (t) = pT (t)Rs,i = β(RTTrs,i (t) − PDrs,i (t)).
In practice, the PDrs,i can be estimated by the minimum RTTrs,i (t) observed so far. The idea of using queuing delay as a congestion measure and extracting it from RTT has been used in Refs. [4,15,20,21]. 5. Numerical examples In this section we present numerical results on two network topologies. In the first example both Algorithms 1 and 2 converge; we also show the behavior of Lagrangian method applied to the original objective function as opposed to the modified objective function used in Algorithm 1 and the behavior of the recently proposed algorithm by Kar et al. [11]. Example 1. Consider the following simple network which consists three unidirectional links labeled L1, L2 and L3 with capacities c = (1, 2, 3) as shown in Fig. 1. There is a single source with the utility function U1 (x1 ) = log x1 . Its total rate x1 is upper bounded by 5 and it routes its flow along two paths (L1 → L3) with path rate x1,1 and (L2 → L3) with path rate x1,2 . We have run both Algorithms 1 and 2 on this network, with the modification (12) and step sizes α = 0.1, β = 0.02 and γ = 0.02. The results are shown in Figs. 2 and 3 for Algorithms 1 and 2, respectively. For
Fig. 1. Network topology of Example 1.
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Fig. 2. Simulation results of Example 1 using Algorithm 1, α = 0.1, β = 0.02 and γ = 0.02.
both algorithms, source rate x1 and path rate x1,1 and x1,2 converge to the optimal point (3, 1, 2), and the path prices converge to U1 (3) = 1/3. Fig. 4 shows the behavior of Algorithm 1 when γ = 0. This corresponds to a first order Lagrangian method on the original objective function (1). As expected, the algorithm does not converge. This motivated the modification that leads to Algorithm 1. Fig. 5 shows the behavior of Kar’s algorithm in [11], with a κ = 1 and step size λ = 0.01. As expected, it converges to a limit cycle in a small neighborhood of the optimal. Example 2. The network consists six unidirectional links labeled L1, L2, . . . , L6 as shown in Fig. 6. Link capacities are c = (20, 25, 20, 60, 60, 60). It is shared by two sources 1 and 2. Source 1 with a total rate of x1 uses the paths: (L1 → L4 → L5) with rate x1,1 and (L2 → L4 → L5) with rate x1,2 . Source 2 with a total rate of x1 uses the paths: (L2 → L4 → L6) with rate x2,1 and (L3 → L4 → L6) with rate
Fig. 3. Simulation results of Example 1 using Algorithm 2, β = 0.02 and γ = 0.02.
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Fig. 4. Simulation results of Example 1 using Algorithm 1, α = 0.02, β = 0.1 and γ = 0.
Fig. 5. Simulation results of Example 1 using Kar’s algorithm, λ = 0.01, and κ = 1.
Fig. 6. Network topology of Example 2.
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x2,2 . Source s have utility functions Us (xs ) = as log(xs ) and minimum and maximum source constraints ms and Ms . The simulation proceeds in three stages: • Stage 1: t = 0 → 1000: a1 = 10, a2 = 20, neither source has rate constraints, i.e., ms = 0 and Ms = ∞. • Stage 2: t = 1001 → 2000: Source 2 increases a2 to 50. • Stage 3: t = 2001 → 3000: Source 1 adds minimum source rate requirement m1 = 30. Fig. 7 shows the behavior of Algorithm 1, with the step sizes α = 0.1, β = 0.1 and γ = 0.1. • Stage 1: Source rates x1 (t) and x2 (t) climb rapidly and converge to the optimal rates (20, 40), and their path rates converge to an equilibrium. Also the path prices converge to 0.5, the value of U1 (20) and U2 (40).
Fig. 7. Simulation results of Example 2 using Algorithm 1, α = 0.1, β = 0.1 and γ = 0.1.
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Fig. 8. Simulation results of Example 2 using Algorithm 2, β = 0.1 and γ = 0.2.
• Stage 2: When source 2 increases its a2 to 50, source rate x2 (t) increases and exhausts the bandwidth of both links L2 and L3 to achieve a maximal rate 45. Source 1 transmits its flow x1 (t) only across L1 r and has a rate 15. Its path rate x1,2 (t) drops to 0. The path price p1,1 (t) converges to U1 (15) = 2/3 and the other path price converges to U2 (45) = 10/9. • Stage 3: Since a minimal rate requirement of 30 is added to source 1, x1 (t) increases from 15 to 30, and x2 (t) drops to 30. All path prices converge to a value U2 (30) = 5/3, and path rates converge to their equilibria. Fig. 8 shows the simulation results with Algorithm 2 using β = 0.1 and γ = 0.2. The source and path prices converge to the same values as in Fig. 7 for Algorithm 1. The path rates also converge, but the path rates in stages 1 and 3 are slightly different from those of Fig. 7. This confirms that optimal path rates are generally non-unique.
6. Conclusion In this paper, we propose two flow control algorithms for networks with multiple paths between source–destination pairs. Algorithm 1 is a first order Lagrangian method on a modified objective function
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that has the same optimal solution as the original objective function but has a better convergence property. Algorithm 2 is based on the idea that, at optimality, only paths with the minimum price carry positive flows, and naturally decomposes the overall decision into flow control (determines total transmission rate based on minimum path price) and routing (determines how to split the flow among available paths). Both algorithms can be implemented as simply a source-based mechanism in which no explicit link algorithm nor explicit feedback is needed. We have presented numerical examples to illustrate their behavior. The deterministic fluid model in this paper ignores many important details of a real network, the foremost of which is delay. Delay generally makes the system more prone to instability, and necessitates a smaller step size γ [19]. Another important feature that is ignored is randomness in the network, due to random arrival and departure of persistent flows, due to non-rate-adaptive flows (e.g., UDP) or short-duration flows that are not effectively controlled end-to-end, due to fluctuations in the capacity and availability of links and nodes, and due to changes in routing decisions. The inclusion of these randomness in the model will provide additional insights to the first-order description of network behavior the current deterministic fluid model furnishes. Acknowledgements The authors are very grateful to the anonymous reviewers for their valuable suggestions on and corrections of the manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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[15] R. La, V. Anantharam, Charge-sensitive TCP and rate control in the Internet, in: Proceedings of the IEEE INFOCOM, March 2000. [16] A. Lagoa, H. Che, Decentralized optimal traffic engineering in the Internet, SIGCOMM Computer Communication Review, October 2000. [17] S.H. Low, Optimization flow control with on-line measurement, in: Proceedings of the 16th International Teletraffic Congress, Edinburgh, UK, June 1999. [18] S.H. Low, A duality model of TCP flow controls, in: Proceedings of the ITC Specialist Seminar on IP Traffic Measurement, Modeling and Management, September 2000. http://netlab.caltech.edu. [19] S.H. Low, D.E. Lapsley, Optimal flow control, I: Basic algorithm and convergence, IEEE/ACM Trans. Networking 7 (6) (1999) 861–874. [20] S.H. Low, L. Peterson, L. Wang, Understanding Vegas: a duality model, J. ACM 49 (2) (2002) 207–233. [21] J. Mo, J. Walrand, Fair end-to-end window-based congestion control, IEEE/ACM Trans. Networking 8 (5) (2000) 556–567. [22] V.I. Utkin, Sliding Modes in Control Optimization, Springer, Berlin, 1992. [23] S. Vutukury, J.J. Garcia-Luna-Aceves, MPATH: a loop-free multipath routing algorithm, Microprocess. Microsyst. 24 (2000) 319–327. Wei-Hua Wang received his B.Eng. from Xi’an Jiaotong University, P.R. China in 1992, M.Eng. from Northeastern University, P.R. China in 1995 and from Nanyang Technological University, Singapore in 1997, all in Electrical Engineering. From 1997 to 1999, he was a lecturer at the Department of Electronic Engineering, Fudan University, P.R. China. Since 1999, he has been a Ph.D. candidate at the Department of Electrical and Electronic Engineering, the University of Melbourne, Australia. His research interests include flow control in Internet, nonlinear systems and robust control theory.
Marimuthu Palaniswami obtained his B.E. (Hons) from the University of Madras, M.Eng. Sc. from the University of Melbourne, and Ph.D. from the University of Newcastle, Australia. He is an Associate Professor at the University of Melbourne, Australia. His research interests are in the fields of computational intelligence, computer networks, nonlinear dynamics, intelligent control and Internet computing. He has published more than 180 conference and journal papers in these topics. He was an Associate Editor of the IEEE Tran. on Neural Networks and is on the editorial board of a few computing and electrical engineering journals. He served as a Technical Program Co-chair for the IEEE International Conference on Neural Networks, 1995 and has served on the programme committees of a number of international conferences. His invited presentations include several key note lectures and invited tutorials, in the areas of Machine Learning, Bio Medical engineering, and Control. He has completed several industry sponsored projects for National Australia Bank, Broken Hill Propriety Limited, Defence Science and Technology Organisation, Integrated Control Systems Pty Ltd, and Signal Processing Associates Pty Ltd. He has also been supported with several Australian Research Council Grants, Industry Research and Development grants and industry research contracts. He was also a recipient of foreign specialist award from the Ministry of Education, Japan.
Steven H. Low received his B.S. degree from Cornell University and Ph.D. from the University of California-Berkeley both in electrical engineering. He was with AT&T Bell Laboratories, Murray Hill, from 1992 to 1996, and was with the University of Melbourne, Australia, from 1996 to 2000, and is now an Associate Professor at the California Institute of Technology, Pasadena. He was a co-recipient of the IEEE William R. Bennett Prize Paper Award in 1997 and the 1996 R&D 100 Award. He is on the editorial board of IEEE/ACM Transactions on Networking. He has been a guest editor of the IEEE Journal on Selected Area in Communications and on the program committee of many conferences. His research interests are in the control and optimization of communications networks and protocols. His home is netlab.caltech.edu and email is
[email protected].
