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Appl. Comput. Math. 6 (2007), no.2, pp.192-217

A COOPERATIVE CONTROL APPROACH FOR QUEUE STABILIZATION IN COMMUNICATION NETWORKS SABATO MANFREDI †, § Abstract. This paper is concerned with the analysis, design and validation of a congestion cooperative rate control for communication networks. A sufficient condition for network stability in presence of multiple bottleneck and heterogeneous sources with different time delay is given and it is used for controller parameter design. The stability condition and the resulting cooperative control performance in terms of fairness, link utilization, packet loss and fault tolerance are validated firstly by Matlab/simulink tool and then by discrete packet simulator. Keywords: control of communication networks, Traffic Engineering, pinning congestion control, consensus protocols, network planning, resilience, business continuity.

1. Introduction Todays Internet only provides Best Effort Service by processing traffic as quickly as possible without guarantee any Quality of Service (QoS) [4],[7]. With the rapid increase of demands for Internet service quality it is becoming apparent the business opportunity for the web-companies in developing several service classes will likely be demanded. This service classes may contain single to multiple services with decreasing quality (i.e Gold Service, Silver Service and Bronze Service). Other service class will provide low delay and low jitter services to applications such as Internet Telephony and Video Conferencing. Finally, the Best Effort Service will remain for those customers who only need connectivity. The Internet Engineering Task Force (IETF) has proposed many service models and mechanisms to meet the demand for QoS such as the Integrated Services/RSVP model [5], the Differentiated Services (DS) model [5], MPLS [3], Traffic Engineering [9] and Constraint Based Routing [2]. The introduction of new types of services in the fixed and mobile communication networks underlines as the problem of network congestion control remains a critical issue. In the recent years several methods have been introduced for dealing with the drawbacks of the current congestion control methods, often based on ad hoc control techniques. It has been suggested that feedback strategies more effective are required at intermediate routers to complement the endpoint congestion control. Active Queue Management (AQM) schemes have been proposed in order to deliver preemptively congestion notification to the source for reducing its transmission rate and therefore avoiding buffer overflow. The first aim of AQM schemes is to regulate and stabilize the queue length for efficient resource usage and consistent delay shortening while reducing packet losses. In AQM schemes, a more refined flow control is obtained through a feedback mechanism based on marking 1 (or dropping) packets according to the average queue length. This information when acknowledged by the receiver, allows the transmitter to regulate its transmission rate in accordance with the queue usage. In this direction it has been shown that control theory can offer an invaluable set of tools to improve the performance of existing schemes which can be seen as particular types of feedback control systems [18],[19],[25],[27],[28],[29],[30],[33],[36],[40],[41]. †Sabato Manfredi is with the Faculty of Engineering, University of Naples Federico II, Via Claudio 21, Napoli 80125, Italy. Email: [email protected] §Manuscript received 28 September, 2007 . 1 AQM schemes use either packet dropping or marking (explicit congestion notification (ECN) mechanism [12]) as congestion notification to the sources. Here, we use the term ’marking’ to refer to any action taken by the router to notify the source of oncoming congestion 192

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Other attempts have been made to develop congestion controllers using linear [14], predictive adaptive, fuzzy and neural, and nonlinear control [13]. On the other side the complex network structures, generally modeled as large graphs, have playing an important role in understanding computer network behavior. Examples includes physical connectivity structures such as the Internet’s router-level topology, more logical or virtual maps such as the Internet’s AS-level graphs, overlay networks such as the Web graph or peer-to-peer systems and sensor and/or mobile networks. Connectivity properties cannot in general be dealt with in isolation but have to be viewed within the context of the traffic that traverses these networks and of the dominant protocols that determine how this traffic flows across them. The weight of protocols, topology (i.e. in terms of resilience) and traffic on the network performance and the development of Traffic Engineering (TE) congestion avoidance scheme based on the rate flow control are focus points for assessing QoS and Business continuity in the next generation of network architecture. Multi-agent systems have appeared broadly in several applications including formation flight, clusters of satellites, automated highway systems, and congestion control in communication networks. Distributed control of multi-agent systems and consensus problems have received significant attention in recent years ([6],[14],[24],[15]). One common feature of this research is the sharing of information between agents in order to address a common objective. The aforementioned scenarios justifies the importance and opportunity of introducing cooperative protocols to address agreement problems among communicating dynamic agents in communication network. Despite the successful application of control theory to other complex systems e.g. power, chemical plants the development of network congestion control based on cooperative theoretic concepts is quite unexplored. This in spite of the significant demands placed on the network system over recent years for the delivery of guaranteed performance in terms of quality of service to the users. The control requires only local information exchange between bottleneck nodes. In particular each bottleneck node adjusts sources rate according to both its own congestion level and that of its virtually bottleneck neighbors. Virtually bottleneck neighbors are bottlenecks sharing sources paths. In this paper we have applied a cooperative control approach to ATM networks. In particular we have considered a model of multi-bottleneck ATM network in presence of time-delays in the sources data-flow. A time-delay is due the time elapsed between a rate command signal by a switch controller and the actual time this rate is set. This delay from the control input to the regulated output is the sum of two delays (backward delay τb from controller to source and forward delay τf from source to controller) named the round-trip time delay RTT. We note that the methodology used is general and independent of technology, (i.e. TCP/IP or ATM) and here, as illustrative example, we have considered its application to ATM network environment. The proposed Cooperative Rate Control (shortly CRC) algorithm can be classified as network assisted congestion control and uses queue length information for feedback purpose. CRC operates locally at the router and cooperates with its CRC-router virtually neighbors (in the sense states above) and sends feedback to the sources to regulate their rate. In this direction, here we introduce a new concept in the explicit-rate framework of ATM-network control based on cooperative action between the bottleneck nodes. The basic idea of the proposed strategy is to enhance rate control protocol functionality through bottleneck nodes coordination in order to alleviate and to mitigate congestion effect on the network performance. Most of the rate control approaches to closed-loop congestion control has appeared in the ATM community [13], motivated by the available bit rate (ABR) service, whose input rate is supposed to be explicitly controlled by a closed feedback loop. In particular a feedback signal may be in the form of an explicit rate provided on an end to end basis via resource management (RM) cells. Typically available bandwidth rate (ABR) traffic is not sensitive to service rates nor delays but is sensitive to packet loss so that the throughput of a connection can be decreased as much as necessary, in order to alleviate congestion. The source sends data cells at rate no more than its Allowed Cell Rate (ACR) which varies according to the congestion state of the network. At a connection setup, an Initial Cell Rate

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(ICR), a Minimum Cell Rate and a Peak Cell Rate (PCR) are negotiated. The source begins to send with a rate ICR, and its ACR may vary between MCR and PCR. The ATM forum Traffic Management Specification has specified the structure of the RM (Resource Management) cells which are sent by the source and make the round trip between the source, destination (forward RM) and back to the source (backward RM). According to the degree of the network congestion, the switches may alter the content of RM cells in the two directions (forward or backward). At the arrival of a backward RM cell, the source adjusts its rate according to the congestion indication (CI) bit and the Explicit Rate field of the RM cell. If the value of the ER field in the received RM cell is lower than the current ACR and higher than MCR, then ACR is set to this value. Many papers in the literature deal with the problem of designing the ATM controller at bottleneck nodes dealing with ABR traffic flows ( see [8],[11],[18],[33] and references there in). In particular the paper focus on the stability of the proposed Cooperative Rate Control scheme by introducing a sufficient condition for queue stabilization. The stability condition and the cooperative control performance are validated through Matlab/simulink tool. Then we use ATM packet simulator to demonstrate that the proposed control methodology achieves the desired behavior of the network in more realistic scenarios. In particular: (i)It exhibits stable and robust behavior working over a wide range of network conditions, such as round trip delays (evaluated from 0 to 800 ms RTT), load (5 to 80 sources) without any change in the control parameters; It achieves high utilization with bounded loss performance. (ii) It exhibits good steady-state and transient behavior without observable oscillations and fast rise and quick settling times; (iii) It uses minimal information on the queue length to control system avoiding additional measurements and does not require per connection state information; the control update period is every 2ms, thereby reducing processing overhead; (iv) It achieves high level of max/min fairness both in the steady state and dynamic network conditions without any additional computation or information about bottleneck rates of individual connections; (v) It can be extended to work in an integrated way with different services (e.g., Premium Traffic, Ordinary Traffic, Best Effort Traffic); (vi) it is scalable with respect the network size and it is independent on the technology (i.e. TCP, ATM). The rest of the paper is outlined as follows. In Sec. 2, an ATM multibottleneck model used in this paper is described. Then in Sec. 3, a cooperative control and closed loop sufficient condition are presented and in Sec. 4 the CRC performance issue are outlined. The stability condition is tested by simulation in Sec.5.1. The resulting performance control law are validated and tested through packet numerical simulations in Sec. 5.2. Finally, conclusions and future research are outlined in Sec. 6.

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Figure 1.(a) ATM Network graph

Figure 1.(b) Block diagram of bottleneck node 1.1. Notation. Given a vector x²Cn , xi denotes its i-th component. X = diaggen{x} is a diagonal matrix in Cn×n generated by the vector x and having x as diagonal, while the operator x = diag{X} extracts the diagonal of X. For set V⊂ C, Co(V ) denotes the convex hull of V , while |V | denotes its cardinality. f (X) is the field of values of X defined as {xH Xx, x²Cn×1 , xH x = 1}. λ(X) denotes the eigenvalues of a square matrix X with real eigenvalues, λm and λM denotes its algebraically smallest and largest eigenvalues. σ(M ) denotes the spectrum of a matrix M , s denotes a Laplace complex variable. Let G(N, E, A) be a graph with the set of nodes N , set of edges E ⊆ N ×N , and an adjacency matrix A = {aij } with nonnegative adjacency elements. The set of neighbors of i − th node is defined by Ni = {k²N : aik = 1}. Considering an undirect graph, the degree value di of node i is the number of the neighbors of node i − th (i.e. |Ni |). The Laplacian L matrix is defined by:

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½ PN

i = j; −aij , i 6= j. By the Laplacian definition results: L = D −A with D is the |N |×|N | diagonal matrix having in position i-th the degree value di of the node i-th. ˜ = L + I, with I identity matrix. We define the extended Laplacian as L Considered a network graph consisting of n links and m accessing sources by a specific sourcedestination path (i.e in Figure 1. (a) n = 5 and m = 3), the sources-links interconnections can be described by the routing-matrix: ½ −sτ e i,j , if source j traverses link i; Rij (s) = 0, otherwise. with τi,j denoting the delay of the source j with respect to link i. For sake of notation we −sτ f denotes with Rij (s) the forward routing matrix of elements e fi,j with τfi,j is the forward time b (s) the backward routing matrix of elements delay from source j to link i, and denotes with Rij −sτ e bi,j with τbi,j is the backward time delay from link i to source j. For examples the source S3 in Figure 1. (a) traversing the links 4, 5 and 2 has with respect them the forward delays τf4,3 , τf5,3 and τf2,3 and backward delays τb4,3 , τb5,3 and τb2,3 . Beside the source S3 presents with respect the links listed above the following round trip time delays RT T4,3 = τf4,3 + τb4,3 sec, RT T5,3 = τf5,3 + τb5,3 sec and RT T2,3 = τf2,3 + τb2,3 sec. lij =

j=1,j6=i aij ,

2. An ATM Network model In the recent years various dynamic models have been used by a number of researchers (i.e. [19],[16] to model a wide range of queueing and contention systems. Several variants of the fluid flow model have been extensively used for network performance evaluation and control. Here the main objective is to consider a model having low order complexity which captures the essential dynamics of network behavior suitable for a decentralized cooperative control design. To this aim starting with the fluid queue model of single bottleneck ([11],[33]), we will consider a network by a set of links N = {1, 2, .., n} and sources M = {1, 2, .., m}. In particular let ni source connections accessing the i − th bottleneck node, qi (t) the queue length at the bottleneck

Figure 2.(a) Block diagram of cooperating i-th and k-th bottleneck nodes;

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Figure 2.(b) Block diagram of CRC control system node and ri,j (t) be the data flow rate of the j − th source, an ATM network dynamic model is given by: X q˙i (t) = ri,j (t − τfi,j ) − ci (t) j²S¯i

for i²N and j²S¯i = {s²M : s across the link i − th} and ci (t) is the rate at which data is sent out from the node. The rates ri,j (t) will be assigned to the sources j-th by a feedback controller ui,j located at the bottleneck i-th resulting in the following closed loop model depicted in Figure 1. (b): q˙i (t) =

X

ui,j (t − RT Ti,j ) − ci (t)

j²S¯i

with RT Ti,j is the round trip time of the source j-th w.r.t link i-th. We note that the command rates ui,j should satisfy the constrain on the aggregate available rate ui computed byPthe controller. So if ui,j = ki,j ui , ki,j are non negative controller parameters to be fixed so that j²S¯i ki,j ≤ 1. Let us assume that the source sends packets according to the minimum rate value among the rate values assigned by the links along the path of its flow (i.e. um = mini ui,j with i²Bj = {l²N : l is a bottleneck for the source j}). Because the minimum operation is taken over a finite number of links, there should exist at least one link such that ri,j = um . Therefore, each flow i has at least one bottleneck. Analyzing the network model we do the following assumptions:. A.1. We assume that the sources are persistent until the closed-loop system reaches steady state meaning that the source always has enough data to transmit at the allocated rate. A.2. We assume all nodes to be bottleneck so we can assume ci (t) = ci with ci to be the i − th link capacity. 3. Cooperative Rate Control (CRC) In what follows we will present a cooperative rate control action and a sufficient stability condition for ATM networks. For sake of clarity T ¯ we introduce the set of virtually bottleneck ¯ neighbors of i − th node as Ni = {k²N : Si Sk 6= ∅, aik = 1}. Virtually bottleneck neighbors are bottlenecks sharing sources virtual paths. For example referring to Figure 1.(a) links 1 and 2 share the virtual path of the source S1 and so they are virtually bottleneck neighbors. In the same way (2, 5) (2, 3) and (4, 5) are two by two one step virtually neighbors. Theorem 1 Consider n-link m-source P ATM network described by (2). Then chosen the cooperative control action ui,j (t) = ki,j k²Ni S{q0 } (qk (t) − qi (t)) + kf cˆi (t) with j²S¯i :

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a) the network is globally asymptotically stable if π ki < ¯ , (1) 2|Si |RT TM i λM ˜ kf cˆi (t) is a feed∀i²N with RT TM i = max{RT Ti,j , j²S¯i }, λM maximum eigenvalue of L, forward action for link capacity compensation with gain kf and link capacity estimation cˆi (t). b) the network asymptotically converges to the set point value q0 . In what follows without loss of generality we assume: A.3. Absence of background traffic and for assumption A.2 results cˆi (t) = ci . Notice that cf f i = kf cˆi (t) in ui,j acts as a feedforward control action for compensating link capacity ci (t) and here we consider the case of perfect compensation. This does not constrain the message we would give here. Indeed the violation of the previous assumptions can be overcome by introducing an opportune link output rate predictor, object of future research. So in what follows we will not consider the design of cf f i neglecting it in the proof. A.4. As in ([11],[33]) ki,j = ki ∀j²S¯i . In so doing all sources sharing a common bottleneck receive the same command signal ui,j = ki ui , ∀j²S¯i . Proof Proof a) In order to prove the stability condition (1) we recast the ATM network physical system in terms of feedback control scheme. To this aim starting from the bottleneck i − th scheme in Figure 1.(b) and considering that the sources behavior replies the controller rate command ri,j (t) = ui,j (t − τbi,j ), we obtain for the bottleneck i the equivalent control scheme in Figure 2. (a). The CRC control action ui,j at link i − th regulates the sources rate according its level of congestion qi and the level of congestion qk , k²Ni of virtually bottlenecks neighbors as depicted in Figure 2.(a). Substituting ui,j in the closed loop equation (2) and assuming perfect link capacity feedforward compensation from A.3, results: X X q˙i (t) = ki,j (qk (t − RT Ti,j ) − qi (t − RT Ti,j )). j²S¯i

k²Ni

S {q0 }

Considering the assumption A.4 and separating the constant term in q0 , we obtain: q˙i (t) =

X j²S¯i

q˙i (t) = ki

ki (

X

(qk (t − RT Ti,j ) − qi (t − RT Ti,j )) + q0 − qi (t − RT Ti,j ));

k²Ni

X X ( (qk (t − RT Ti,j ) − qi (t − RT Ti,j )) − qi (t − RT Ti,j )) + ki |S¯i |q0 . j²S¯i k²Ni

P ˜ Notice that k²Ni (qk (t) − qi (t)) − qi (t) represents the i − th element of product −Lq(t) with T q(t) = [q1 , .., qn ] meaning the vector of network queue lengths at the time t. Introduced the diagonal nxn matrix T ˜ R(s) = diaggen{diag{Rf (s)Rb (s)}}

P with j²S¯i e−sRT Ti,j on the diagonal position i-th, defined K = diaggen{k} the controller gains matrix, P (s) = diaggen{ 1s } the queue process then the controlled network physical system reduces to the feedback control system in Figure 2.(b) with q0 is a scalar constant denoting the reference queue length and ¯s = [|S¯1 |, .., |S¯n |]T is the vector of the number of the sources

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traversing the links. So the return ratio transfer function of closed loop system (2) is: ˜ ˜ H(s) = K R(s)P (s)L. Beside results: ˜ L) ˜ ⊂ f(K RP ˜ )f(L) ˜ = Co{ki r˜i pi }Co{λ(L)} ˜ ⊆ Co{ki P ¯ σ(H(s)) ⊂ f(K RP j²Si

e−sRT Ti,j s

}[λm , λM ].

Indeed being the argument matrices normal the first and the second inclusions follow from the spectral containment and field values properties, the next equality follows from the normality −sRT Ti,j P property [32]. We note that the real part of the set Co{ki j²S¯i e s } is lower limited by 2 the point −ki RT TM i |S¯i | π setting RT Ti,j = RT TMi , ∀j²S¯i and sRT TMi = jw π2 . So if (1) holds than H(jw) do not intersect (−∞, −1] for all w and for the Generalized Nyquist criterion [37] the closed loop system is global asymptotically stable. This completes the proof a). Proof b) Considered the sensitivity function So (s) of the closed loop system in Figure 2, the steady-state value of the error for the final value theorem is: ˜ −1¯sq0 = 0 lim e(t) = lim So (s)qo = lim s(I + H(s))−1 L t→∞

s→0

s→0

˜ invertible being qo (s) = K a stabilizing controller for proof a), So (s) = 0, H(0) and L matrices 2. This completes the proof of Theorem 1. We note that CRC operates locally at the router and cooperates with its CRC-router virtually neighbors and sends feedback to the sources to regulate their rate. So in the network of Figure 1. (a) for example the virtually neighbors links 1 and 2 cooperate one with each other for setting the controlled source rate. q0 s ,

4. CRC Performance issue Robustness and performance tradeoff Observing the condition (1), we note that the network stability depends on the controller gain ki , sources round time delay and on the virtually interconnection topology by the largest extended Laplacian eigenvalue λM . This evidences as the upper bound on the admissible round trip time-delay in the network is inversely proportional to the largest eigenvalue λM of the virtual information flow graph. From Gerˇ sgorin theorem λM is upper bounded by 2dM with dM is the maximum degree of the nodes of the virtual network. Hence (1) becomes: ki < 4|S¯ |RTπT d . i Mi M This means that networks with nodes traversed by relatively high number of sources cannot tolerate relatively high communication time delays. On the other hand any arbitrary large timedelay can be tolerated, by setting a sufficiently small controller gain ki . Hence there is a tradeoff between robustness of a protocol to time-delays and its performance to take into account for network design problem. Remark 1 Recently great attention has been posed in studying graph topologies and their effects on the network of dynamical systems by introducing model of complex networks (i.e. small-world model, scale-free model [26],[35],[39]). The issue of controlling complex dynamical networks are becoming a focal topic within the control systems community. Recently, the technique of feedback pinning has been applied to control large-scale dynamical networks for suppressing spatiotemporal chaos [22]. We remark as topologies with hub nodes (i.e. scale free network) are more conservative w.r.t. regular topologies. This outlines as the topology can affect the stability, as soon as network performance (i,e. transient response, failure node tolerance). We 2Indeed they are strictly diagonally dominant matrices and for Gerˇ sgorin theorem are invertible

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remark that the stability condition can be manipulated in order to substitute the dependence in λM with one in the estimation of dM (information available by link state algorithm) reducing the conservativeness of (1). Remark 2 The convergence of the network to the set point q0 is guaranteed if the logical communication topology between virtually bottleneck neighbors is connected and if we have pinned at least one node at a set-point. Also the velocity of convergence of the protocol depends on which node/nodes is/are pinned. We can see the proposed scheme as a pinning congestion control approach. Steady State Analysis and link utilization We define the set of flows bottlenecked at link i, Sib = {j²M : ri,j = ui,j } and the set of all flows b(j) b(j) Sinb not bottlenecked at link i and traversing it by S¯i −Sib = {j²M : ri,j = ui,j and ui,j < ui,j } P with bj ²Bj is some bottleneck for flow j. At the steady state from (2) results: j²S¯i ri,j = ci (t) implying that the capacity at the link i is fully utilized. Therefore results: X j²Sib

ui,j +

X

b

ui,jj = ci (t).

(2)

j²Sinb

Assuming at the steady state equal share ui,j = usi of the resource between the bottlenecked sources, results: P ci (t) − j²Si ui,j s nb ui = . |Sib | b

This means that to the flows not bottlenecked at link i are assigned data rates ui,jj and the remaining unused capacity is fairly distributed to flows bottlenecked at link. Fairness In a shared environment the throughput for a source depends upon the demands by other sources. The most commonly criterion for the correct share of bandwidth for a source in a network environment is the so called max-min allocation [23]. It provides the maximum possible bandwidth to the source receiving the least among all contending sources. Notice that max-min allocation is both fair and efficient in the sense that all sources get an equal share on every link and that each link is utilized to the maximum load possible. In what follows for sake of nation we denotes ri,j with rj . P Definition 1 A rate vector < r1 , ..., rm > is feasible if rj ≥ 0, ∀j²M and j²S¯i rj ≤ ci . Definition 2 : A vector r is max-min fair if it is feasible, and for each j²M and feasible fair rate r¯ for which rj < r¯j , there is some j 0 with rj ≥ rj 0 > r¯j 0 . Definition 3 : A steady-state rate rs vector is a vector whose components are the steady-state rates of the flow controlled connections. Max-min fair rate vector is such that for every rate ri , any attempt to increase ri must result in a decrease of another rate rj , for which ri ≥ rj to maintain feasibility. The following theorem shows that max-min fairness is achieved by the proposed algorithm. Theorem 2 The vector rs of the steady-state rates of the flow controlled connections by the proposed cooperative rate control achieves max-min fairness. Proof Assuming that all CRC algorithms at the links in the network satisfy (1), each flow has at least one bottleneck link and the rate vector at steady state is feasible holding the (2) since feasibility is a necessary condition for stability. If we increase the rate value of flow j which is bottlenecked at some link i while maintaining feasibility, we should reduce the data rate of flow j 0 6= j that traverses link j. Since rjs0 ≤ rjs for all j 0 ²S¯i by the definition of bottleneck link, we are reducing the rate value of flow j 0 . So the steady-state rate vector complies with Definition 2, and the Theorem 2 follows.

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5. Controller validation Now, we shall seek to validate the effectiveness of the CRC Controller derived above and compare its performance with respect to other ATM congestion controller schemes. To this aim, after validating the CRC controller in Matlab/Simulink, we used the NISTHFC ATM network simulator [17],[20],[21] (in the follows shortly NIST) for taking into account also the effects of discretization and nonlinear nature of network behavior. NIST is packet network simulator developed to provide a means for researchers and network planners to analyze the behavior of ATM networks . In particular we compare the CRC performance with respect to standard ATM rate controller scheme ERICA and EPRCA of which we give a brief description. The ERICA (Explicit Rate Indication for Congestion Avoidance) algorithm [6] tries to achieve a fair and efficient allocation of the available bandwidth to contending sources. The basic idea in ERICA is to monitor, at each switch, the incoming cell rates of each ABR traffic source, and compare the aggregate ABR traffic demand to the desired target utilization U for ABR traffic sources (typically U = 0.95 of available ABR capacity in LAN environment, and U = 0.90 of available ABR capacity in WAN environment). If the aggregate demand is less than the target load, then traffic source rates can be increased. If the aggregate demand exceeds the target load, then traffic source rates must be decreased. A scaling factor, the ratio of actual load to desired load, is used to control the gradient of rate adjustments for the sources. ERICA estimates source rates by counting incoming cells over an averaging interval. EPRCA [38] is an enhancement of an older version namely PRCA (Proportional Rate Control Algorithm) which is based on the congestion indication. EPRCA introduces the estimation of the available bandwidth and uses the queue length as a control parameter. More precisely, it ¯l and Q¯u in order to determine the degree of the congestion. uses two congestion thresholds: Q If the congestion is detected, EPRCA has the capability to reduce selectively the rate of all the sources that have an ACR (allowed cell rate) larger than MACR. The latter represents the mean over all the ACRs. Each time an RM cell is received by the switch, EPRCA allocates the rate to the corresponding connection depending on the queue length: If the queue length is smaller ¯l , the switch detects no congestion and the content of the RM cell is not affected. If the than Q ¯l and Q¯u , the switch considers itself as congested and updates the queue length is between Q ER field of the RM cell in order to reduce the source rate to ER value. if the current cell rate (CCR) of the source is larger than MACR, then the rate of the source shall be reduced to ER. Sources that have a current cell rate smaller than ER will not be affected. Finally, if the queue length exceeds Q¯u , then the switch is in heavy congestion and reduces selectively the rate of all the sources to ER. 5.1. CRC stability condition validation. In this session we want to illustrate the effectiveness of the CRC control in the simple scenarios in Figure 3.(a) with three sources accessing to bottlenecks switches (SWi ). The buffer size is set to 10 cells while the queue set point is 5 cells.

Figure 3. (a) Network scenarios;

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Figure 3. (b) Nyquist plot of i − th channel of H(s); We shall validate the Theorem 1 and the proposed CRC strategy by matlab/simulnk simulations. We consider a first case with RT TMi = 0.13 sec, ki = 0.9 , i²{1, 2, 3} satisfying stability condition (1). The stabilization action of the controller is shown in Figure 3.(b) by Nyquist plot of i − th channel of H(s). In Figure 4.(a) we observe the effectiveness of CRC control guaranteeing set-point regulation, queue balancing, no packet loss and link utilization with a good transient response time. Then we have considered the case of network parameters not satisfying theorem 1 condition (ki = 1.5, RT TMi = 0.5). As shown in Figure 4.(b), the the network presents a damped behavior with performance degradation. We simulate the aforementioned network scenarios by NIST simulator. Figures 5.(a) shows the switches queue length confirming the good CRC performance in terms of queue stabilization, set-point regulation, transient response and queue balancing. Then we validate also the worsening of network performance when the the condition (1) is not satisfied Figures 5.(b). Matlab and NIST simulations confirm theorem 1. 5.2. CRC Performance evaluation. Simulations, unless otherwise stated, refer to a multiple bottlenecked topology in Figure 6.(a) with switch running the CRC scheme connected by a link with a capacity of 155Mb/s. The round trip propagation delay for the flows is 215 ms. The target queue length q0 is set to 180 cells (60% of queue occupancy) while the maximum buffer size is 300 cells. ABR sources have minimum bit rate 100Mbit/s and maximum bit rate 160Mbit/s. The sampling period of the CRC control scheme is 2ms. For all other schemes the controller parameters, the sampling frequency, such as the rest of unspecified parameters, are fixed to values recommended in the original papers and in NIST simulator. We report below a variety of numerical simulations and experiments in multibottleneck scenarios. Namely, we investigate (i) nominal case (ii) the robustness of CRC to network parameter uncertainties as load and roud trip time variations; (iii) dynamic behaviour in presence of VBR (Variable Bit Rate) background and Cross traffics (iv) the CRC resilience and disaster recovery capacity in presence suddenly fault of a router. This assures Business Continuity that is particularly wished in Internet Service quality environment. We evaluate the performance calculating link utilization, packet loss and JAN index. The latter index quantifies how much the allocation is unfair with respect to the max-min one [23]. 5.3. Nominal case. We consider the CRC rate algorithm in the nominal case. Figure 7. shows that the algorithm presents no overshoot, no packet loss, reduced queue variance and a fast set-point convergence. Also the queue balancing is guaranteed.

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5.4. Robustness to network parameters uncertainties. We begin with the investigation of the robustness of the controllers schemes in the presence of variations of (i) the load N ; (ii) the round trip propagation delay Tp . 5.4.1. Robustness to load variations. We consider the bottleneck topology introduced above and repeat the simulation for different values of the sources, N , varying from 5 to 80. For each value of the load, we record the link utilization and the Jan index for each switch under the action of different controllers. As shown in Figures 8. , the EPRCA and ERICA control schemes present the worsening of performance when the load increases. Differently, CRC scheme presented in this paper achieves a good queue stabilization with packet loss reduction and max-min fair link utilization also in presence of load variations. This is very important since the controller not only avoids congestion but also fully utilizes the available resources, even for demands considerably exceeding the available link capacity. 5.4.2. Robustness to variations of the round trip propagation delay. Keeping the same topology introduced above, we now consider variations of the the average round trip propagation delay between 0.1 and 0.8 s. In Figure 9, the performance indexes are reported as a function of the round trip time variations. Also in this situation, the CRC scheme shows the best performance when compared with other controllers. This is due to the better CRC queue stabilization and therefore queueing and jitter delays reduction. ERICA and EPRCA presents high queue standard deviation implying variable round trip time for the sources and also low queue utilization if the queue goes frequently to zero. 5.4.3. Presence of VBR and CROSS traffics. An important aspect is the possible presence on the network of background sources, whose traffic clearly affects the overall network dynamics. In this section, we study the effects of adding two VBR flows (depicted in Figure 6.(b) to ABR traffic. Those can be seen as variations of link capacities advertised from ABR flows caused by background traffic. Finally we have evaluate the CRC queue dynamics in presence of crosstraffic. We note that this affects the communication delay of protocol control information in the reversing path. In the both cases (Figures 10-11 and Figures 12-13.) the CRC assures better queue set point regulation, no packet loss and disturbance rejection with respect to VBR traffic. 5.4.4. Rerouting and node fault scenarios. We select the network test configuration shown in Figure 14 and set all sources always have cells to transmit. For better underline the congestion phenomena we have considered switches with buffer size of 1000cells, q0 = 200 cells, propagation delays q1,1 -q1,2 and q2,1 − q2,2 equal to 0.650s, delay q1,1 -q2,2 equal to 0.750s. We have considered two groups S 1 , S 2 of 5 and 4 sources. After 20 seconds the source group S 1 switches its transmission destinations from D1 to D2 . The chosen configuration allows easy to represent scenarios where two different intra o infra domain subnetworks suddenly interact for example due to re-routing strategy purposes (i.e TE, ATM and TCP based routing strategy), and/or fault of router q1,2 and/or switching in destinations uploading from sources S 1 . From Figures 15 we note the fast recovery action of CRC controller with no the packet loss and avoiding the overload of switch q2,2 . This is mainly due to the cooperative nature of CRC controller that permits to switch q1,1 of knowing the congestion level of q2,2 . Differently, as appears in Figure 16, the non-cooperative ERICA and EPRCA algorithms do not dealing with this scenarios with resulting network collapse due to congestion chain effects of subnetworks q2,1 − q2,2 . Beside Figure 17.(a)-(b) shown as CRC assures fairness both in nominal and dynamic critical condition and also in the presence of large disparity in distance from the switches (i.e. far S1 and local S2 source groups respect to q2,2 ). Hence, the numerical analysis confirms that, when compared to other existing schemes, the CRC strategy guarantees the best regulation performance in the presence of unwanted delays, load and disturbances. Notice that the proposed controller ensures that the network is scalable in the sense that conditions in the local interactions also guarantee that the entire arbitrary

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interconnection is robustly stable. The validation in more realistic scenarios by NIST outlines that the proposed CRC is implementable. It should be noted that the methodology used is general and independent of technology, as for example TCP/IP or ATM. Generically, we have used the terms packet for both IP packets and ATM cells, and switch for ATM switch and IP routers. The feedback signal must be communicated to the sources for action. Several approaches may be adopted including, namely RM cells in an ATM setting, in TCP by modifying the receiver window field in the TCP header of a packet sent by receiver to source or using Explicit Congestion Notification (ECN); using implicit feedback, as timeout due to lost packet, or even more sophisticated schemes, such as adaptive binary marking. Anyway here we left the implementation details of the feedback signalling scheme and for a simulative evaluation of the proposed CRC we have used explicit feedback, provided by updating special fields in packets (RM cells in an ATM setting ([13])) in order to communicate virtual neighbors queues information. We would remark the proposed scheme can be a generic scheme for handling multiple differentiated classes of traffic, using an integrated dynamic CRC congestion control approach. For example, suppose to differentiate each class namely in Premium (requiring strict guarantees of delivery, within given delay and loss bounds), Ordinary (cannot tolerate loss of packets but can however tolerate queueing delays) and Best effort (requiring no guarantees on either loss or delay) services. The control objective should be of allocating primarily traffic to Premium Class and to fairly redistribute the available capacity leftover to Ordinary traffic by cooperative CRC scheme. In the same way should be allocated the left-over capacity from Ordinary traffic to the Best effort traffic. This assures a dynamic balanced re-distribution of leftover capacity from higher to lower priority classes optimizing network resources utilization. 6. Conclusions and future work We have discussed the problem of rate control in Network resource management. By using multi-bottleneck ATM model to describe the dynamics of the heterogeneous sources and the switches, we have proposed an ATM cooperative rate control that: a.) stabilizes the ATM network once chosen the controller parameter according to condition (1); b.) balances the queue length at a desiderate set point, reducing packet loss and improving link utilization c)is robust to load and round trip time variations,d) guarantees max-min fair allocation e) has resilience capacity in presence of sudden events (i.e. forced re-routing, fault of node, etc..). The implementation issue of controller has been assessed by using an ATM packet network simulator. We would remark that the CRC approach is independent on the particular technology and here, as illustrative example, we have considered its application to ATM network environment. Future direction of research includes the introduction of link capacity estimator (when does not hold the assumption on the perfect feedforward compensation of link capacity ), the extension of CRC controller to TE-MPLS environment and its validation, presence of differentiate class of services, experimental validation. References [1] Abraham,S.P., Kumar,A., A stochastic approximation approach for max-min fair adaptive rate control of ABR sessions with MCRs, 17th Annual Joint Conference of the IEEE Computer and Communications Societies. Vol 3, 1998 INFCOM, 1998. [2] Albert, R. A. ,and Barabasi,A. L. Scaling and percolation in the small-world network model, Phys. Rev., pp. 60, 7332.7344., 1999. [3] Albert, R. A., and Barabasi,A. L. Statistical mechanics of complex networks, Rev. Mod. Phys., vol, 74, pp: 47-97, 2002. [4] Altman, E.,Basar,T., Srikant, R. Multi-user rate-based flow control with action delays: a team-theoretic approach., In Proceedings of 36th Conference on Decision and Control, 1996. [5] Ataslar,B., Iftar,A., Kalyanaraman,S., Kang,T., Ozbay,H. and Quet,P.H.Rate based flow controllers for communication networks in the presence of uncertain time varying multiple time delays, Automatica, vol.38, pp. 917-928, 2002.

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[6] ATM Forum Trafficc Management, AF-TM-0056.000. The ATM Forum Traffic Management Spec. Vers. 4.0, ,April 1996. [7] Awduche, D., et al., Requirements for Traffic Engineering over MPLS,, Internet draft, draft-ietf-mpls-trafficeng.00.txt, Oct. 1998. [8] Belhumeur,P.N., T. Eren, and A. S. Morse, Coordination of groups of mobile agents using nearest neighbor rules, Proc. of the IEEE Conference on Decision and Control, 2002. [9] Biberovic,E., Iftar, A., and Ozbay, H., A solution to the robust flow control problem for networks with multiple bottlenecks, in Proc. of the 40th IEEE Conference on Decision and Control, Orlando, FL, U.S.A., Dec. 2001, pp. 2303–2308. [10] Blanchini, F.,and Lo Cigno, R. and Tempo, R. Robust rate control for integrated services packet networks, Proc. IEEE/ACM Transactions on Networking, Vol. 10, N. 5, pp. 644-652, 2002. [11] Boyd,S.,and Xiao,L., Fast linear iterations for distributed averaging, Proc. of the Conference on Decision and Control, 2003. [12] Braden, R., et al., Resource ReSerVation Protocol (RSVP) - Version 1 Functional Specification, RFC 2205, Sept. 1997. [13] Braden, R., Clark,D., and Shenker,S., Integrated Services in the Internet Architecture: an Overview, Internet RFC 1633, June 1994. [14] Braden, R., et al.,An Architecture for Differentiated Services, RFC 2475, Dec. 1998. [15] Cavendish,D., Gerla,M., Mascolo, S. A control theoretical approach to congestion control in packet networks, IEEE/ACM Transactions on Networking, Vol 12, No. 5, 2004. [16] Crawley, E. et al., A Framework for QoS-based Routing in the Internet,, RFC 2386, Aug. 1998. [17] Desoer, C. A., and Yang, Y. T.. On the generalized Nyquist stability criterion, IEEE Transactions on Automatic Control,1980 [18] Ferguson, P. and Huston, G., Quality of Service, ,Wiley, 1998. [19] Hollot,C. V., Misra,V., Towsley,D., Gong, W., Analysis and Design of Controllers for AQM Routers Supporting TCP Flows, IEEE Transactions on Automatic Control, Vol 47, no. 6, pp. 945-959, 2002. [20] Horn, R. A., and Johnson,C. R., Topics in Matrix Analysis, Cambridge University Press, 1995. [21] Jadbabaie,A., Lin,J., and Morse, S. A., Closing ranks in vehicle formations based on rigidity, IEEE Trans. on Automatic Control, vol. 48, no. 6, pp. 988-1001, 2003. [22] Jain, R., Kalyanaraman, S., Goyal, R., Fahmy, S., and Viswanathan,R., ERICA Switch Algorithm: a Complete Description, AF-TM 96- 1172, August 1996. [23] Jain,R., The art of compouter systems performance analisys, Jhon Wiley Sons,New York, 1991. [24] Keshav, S. A control-theoretic approach to flow control , SIGCOMM Comput. Commun. Rev., vol. 25, no. 1, pp. 188-201, 1995. [25] Larry L. Peterson, Bruce S. Davie, Computer Networks: a system approach, Morgan Kaufnann 2003. [26] Li, X., Wang,X. F., andChen, G. , Pinning a complex dynamical network to its equilibrium, IEEE Trans. Circ. Sys.-I. 2004. [27] Manfredi, S., Bernardo,M. di, Garofalo, F., ”Robust Output feedback Active Queue Management Control in TCP Networks”, Conference on Decision and Control, Bahamas, Vol 1, pp:1004-1009, 2004. [28] Manfredi, S., Bernardo,M. di, Garofalo,F., Reduction-based Robust Active Queue Management Control, Control Engineering Practice, Vol 15, pp. 177-186, 2007. [29] Munyas, I., and Iftar,A., H-infinity based flow control for ATM networks with multiple bottlenecks, in Proc. of the IFAC World Congress 2005, Prague, Czech Republic, July 2005. [30] Munyas, I., Yelbasi, O., and Iftar, A., Decentralized robust flow controller design for networks with multiple bottlenecks, in Proc. of the the European Control Conference, ambridge, U.K., Sep. 2003. [31] Nist ATM/HFC Network Simulator, http: //w3.antd.nist.gov/Hsntg/prd− atm-sim.html [32] Ptsilliders, A., Ioannoau, P., Lestas,M., Rossides, L.Adaptive Nonlinear Congestion Controller for a Differentiated-Services Framework, IEEE/ACM Transactions on Networking, Vol. 13, n. 1, 2005 [33] Quet, P.F., and Ozbay,H., On the Design of AQM Supporting TCP Flows Using Robust Control Theory,IEEE Transactions on Automatic Control, Vol 49, no. 6, pp. 1031-1036, 2004, [34] Ramakrishnam, K., and Floyd, S., A proposal to add Explicit Congestion otification (ECN) to ip, RFC 2481, 1999. [35] Roberts, R., Enanched PRCA (Proportional Rate-Control Algorithm), AF-TM 94- 0735R1, August 1994. [36] Rosen,E., Viswanathan,A., and Callon,R. , Multiprotocol Label Switching rchitecture, Internet draft, draftietf-mpls-arch-01.txt, Mar. 1998. [37] Saber, R.O., Murray,R.M., Consensus Problems in Networks of Agents with witching Topology and TimeDelays, IEEE Transactions on Automatic Control, Vol. 49, No. 9, 2004. [38] Simulation Study of ABR Service over IEEE 802.14 MAC, IEEE PROJECT 802.14. Cable TV Protocol Working Group, 1997. ttp: //www.cs.virginia.edu/papers/97 − 011.pdf

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[39] Watts and Strogatz, Collective dynamics of small world networks, Nature, pp. 393, 440.442, 1998. [40] Yan, P., and Gao ,Y., and Ozbay,H., A Variable Structure Control Approach to Active Queue Management for TCP with ECN, IEEE Transactions on Control System Technology, Vol. 13, pp. 203-215, 2005. [41] Zhang,H., Yang, O. W., and Monftah,H., Design of robust congestion controllers for ATM networks, in Proceedings of the IEEE INFOCOM ’97, pp. 302-309.

Figure 4.Matlab validation - Time evolution of the queue length at the Switches [Cell]: (a) ki = 0.9, RT TMi = 0.13 sec;

Figure 4. Matlab validation - Time evolution of the queue length at the Switches [Cell]: (b) ki = 1.5, RT TMi = 0.5 sec;

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Figure 5.Nist validation - Time evolution of the queue length at the Switches [Cell]: (a) ki = 0.9, RT TMi = 0.13 sec; (b) ki = 1.5, RT TMi = 0.5 sec

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Figure 6.Nist validation - Time evolution of the queue length at the Switches [Cell]: (a) ki = 0.9, RT TMi = 0.13 sec; (b) ki = 1.5, RT TMi = 0.5 sec;

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Figure 7. NIST Experiment: (a) Multibottleneck Network scenarios; (b) VBR disturbance

Figure 8.NIST Experiment - Time evolution of the queue length at the Switches [Cell]: CRC

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Figure 9.Nist Experiment. Static Load variations: switches link utilization and Jain index

Figure 10.Nist Experiment. Static RTT variations: Switches link utilization and packet loss

Figure 11. Nist Experiment. Presence of VBR traffic: Time evolution of the queue length at the Switches [Cell]: CRC

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Figure 12. Nist Experiment. Presence of VBR traffic: Time evolution of the queue length at the Switches [Cell]: (a) ERICA; (b) EPRCA;

Figure 13. Nist Experiment. Presence of CROSS traffic: Time evolution of the queue length at the Switches [Cell]: CRC

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Figure 14. Nist Experiment. Presence of CROSS traffic: Time evolution of the queue length at the Switches [Cell]: (a) ERICA; (b) EPRCA;

Figure 15. Nist Experiment: Representative Network Scenarios.

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Figure 16. Nist Experiment. Representative Network scenarios: time evolution of the queue length for CRC [Cell].

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Figure 17. Nist Experiment. Representative Network Scenarios: time evolution of the queue length [Cell] : (a) ERICA; (b) EPRCA;

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Figure 18. Nist Experiment. Representative Network Scenarios: (a) Transmission rate of controlled sources; (b) Dynamic evolution of Jain index;

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SABATO MANFREDI - received Master degree (”Laurea”) in Electronic Engineering from University of Napoli ”Federico II”, Napoli, Italy, 2001. In 2004 he receives the Ph.D degree in Control Engineering from University of Naples Federico II. During 2004/2005 he spent some time as guest researcher at the ”Feedback Control Lab” - Department of Electrical and Computer Engineering - University of Massachusetts, Amherst. From 2005 he is Assistant Professor of Automatic Control with the Department of Computer Science and Systems, Engineering Faculty of University of Napoli ”Federico II”, Italy. His research interests include the modeling, analysis, and control of communication networks with a particular emphasis on the robust congestion control of Internet and consensus protocols in overlay networks. He also works on the modeling and control of complex systems and underwater apparatus.

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