THE Internet is increasingly being used in the conduct of

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1 94 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 Global Stability Conditions for Rate Control With Arbitrary Communication Delays Priya Ranjan, Member, IEEE, Richard J. La, Member, IEEE, and Eyad H. Abed, Fellow, IEEE Abstract We analyze Kelly s optimization framework for a rate allocation problem in communication networks and provide stability conditions with arbitrary fixed communication delays. We demonstrate the existence of a fundamental tradeoff between users price elasticity of demand and the responsiveness of resource through a choice of price function. We also show that the stability of the system can be studied by looking at a much simpler discrete time system that emerges from the underlying market structure of the rate control system with a homogeneous delay. We study the effects of nonresponsive traffic on system stability and show that the presence of nonresponsive traffic enhances the stability of system. We also investigate the system behavior beyond stable regime. Index Terms Asymptotic stability, congestion control, time delay system. I. INTRODUCTION THE Internet is increasingly being used in the conduct of commerce, government, and communication. Poor management of congestion in the Internet leads to a degradation in user experience and, potentially, inaccessibility of some parts of the network including popular e-business and government sites. Therefore, the issue of congestion control in the Internet remains a crucial issue. In order to ensure that users do not misbehave in the Internet, which is in the public domain and in a potentially noncooperative environment, researchers have proposed various rate control mechanisms based on some form of pricing mechanism. Kelly [10] has suggested that the problem of rate allocation for elastic traffic can be posed as one of achieving maximum aggregate utility of the users and proposed an optimization framework for rate allocation in the Internet. Here the utility of a user could either represent the true utility or preferences of the user or a utility function that is assigned to the user by the end user rate control algorithm, e.g., Transmission Control Protocol (TCP). Manuscript received December 16, 2003; revised August 11, 2004, and February 17, 2005; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor R. Srikant. This work was supported in part by the National Science Foundation under Grants ANI , ANI , and ANI , by the Defense Advanced Research Projects Agency (DARPA) under Contract IAI-ARPA-6030C, and by the U.S. Army under Contract STTR W911NF-04-C P. Ranjan is with the Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD USA, and also with Intelligent Automation, Inc., Rockville, MD USA ( priya@isr.umd.edu). R. J. La and E. H. Abed are with the Department of Electrical and Computer Engineering and the Institute for Systems Research, University of Maryland, College Park, MD USA ( hyongla@isr.umd.edu; abed@isr.umd.edu). Digital Object Identifier /TNET In the latter case, the selection of the utility function determines the end user algorithm, and the tradeoff between the fairness among the users and system efficiency [1], [11], [15], [20]. Using the proposed framework he has shown that the system optimum is achieved at the equilibrium between the end users and resources. Based on this observation researchers have proposed various rate-based algorithms, in conjunction with a variety of active queue management (AQM) mechanisms, that solve the system optimization problem or its relaxation [10], [17]. The convergence of these algorithms, however, has been established only in the absence of feedback delay initially, and the implications of feedback delay have been left open as well as any tradeoff that may exist between stability and selected utility and price functions. Modeling the communication delay is especially important when the delay is nonnegligible or widely varying with uncertainty, i.e., the delay is not known in advance. A good example of such an environment is multi-hop mobile wireless networks. Tan and Johari [9] have studied the case with homogeneous users, i.e., same round-trip delays and same form of utility functions, and provided local stability conditions in term of users gain parameters and communication delays. In general their results state that the product of gain parameter and communication delays should be no larger than some constant. Similar results have been obtained in [3] and [19] in the context of a single flow and a single resource with more general utility functions and in [2] in the context of a single bottleneck with multiple heterogeneous users. The given stability conditions are similar to those in [9] and state that the product of the delay and gain parameter of end user algorithms needs to be smaller than some constant. This constant depends on the utility function of the users and resource price function in [2] and [3] whereas it does not in [19]. These results, however, focus on characterizing sufficient conditions on the communication delay and gain parameter for stability and, as a result, do not point at a close relationship between system stability and the parameters at the end users and network elements as is done in this paper. Furthermore, the conditions in [2] and [3] are not always easy to verify and may not be necessary, as we will demonstrate in this paper. In this paper, we study the problem of designing a robust rate control algorithm in the presence of communication delays between network resources and end users. However, unlike in the previous studies where the authors give the conditions on the delay and user s gain parameter for stability of the system, we establish a delay-independent stability criterion for a system optimization problem in the presence of arbitrary fixed delays and with any arbitrary gain parameters of end users. Our approach is consistent with the philosophy that network protocols must /$ IEEE

2 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 95 be simple and robust given the complexity and scale of the Internet, and may prove to be more suitable for wireless ad hoc networks where delays are often expected to be unpredictable and widely varying. In particular, the stability conditions given in [2] and [3] may be sufficient to ensure smooth operation of the network when the network is operating normally. However, a network can occasionally experience high congestion and behave unpredictably due to the presence of (a large number of) nonresponsive flows and/or a collapse of a part of network as a result of, for example, a link failure or routing instability. In such a scenario the system may temporarily deviate from the stable regime characterized by the conditions in [2] and [3] because of the increased queueing delay, and the unstable behavior of the end user algorithm can aggravate the congestion level, possibly leading to a congestion collapse. Our approach also provides a fresh way of looking at the issue of communication delay. A natural question that arises in this setting is whether or not it is possible to design a system that is stable with arbitrary communication delays. If possible, what are the necessary and/or sufficient conditions for the stability? A similar problem was investigated in [30]. In addition, what is the impact of the nonresponsive traffic on the system stability? The last question is emerging as an important issue with a growing interest in implementing real-time applications on top of User Datagram Protocol (UDP), which are not as responsive as elastic traffic. Our analysis is based on the invariance-based global stability results for nonlinear delay differential equations [7], [8], [18]. This kind of global stability results are different from those based on Lyapunov or Razumikhin theorems for delay differential equations used in [2], [3], [19], and [28] or from passivity approach [29], and our set up also hints at the structure of emerging periodic orbits (such as their periodicity and amplitude) in the case of loss of stability. Such loss of stability may have important implications for developing practical AQM mechanisms. Generally speaking, our main results can be summarized as follows: 1) There is a close relationship between the stability of delay differential equations that describe network dynamics and that of a simple underlying discrete time map for a homogeneous delay case. In other words, the stability of the delay differential equations can be inferred from that of the underlying discrete time map and vice versa. Since typically a discrete time system is more amenable to analysis and simulation, our results provide a useful tool for studying the stability of a system described by a set of delay differential equations. 2) If the user and resource price functions have a stable market equilibrium, which is captured by the underlying discrete time map, then the corresponding dynamical equation for rate control converges to the optimal point in the presence of arbitrary delays. This result essentially shows that stability is related to utility and price functions in a fundamental way. In particular, for a given price function, under certain conditions (to be stated precisely) it is possible to design stable user utility functions such that the ensuing dynamical system converges to the optimal flow irrespective of the communication delay. Similarly, given the user demand function it is possible (under the same conditions) to find a suitable price function to stabilize the system with any arbitrary delay. Conversely, if the underlying market equilibrium is unstable then it is possible to find a large enough delay for which the optimal point loses its stability and gives way to oscillations. In practice, this gives rise to a fundamental tradeoff between the responsiveness of end users and network resources. In other words, given the responsiveness of network resource, there is a limit on how aggressively selfish end users can react to feedback in order to ensure delay-independent network stability. 3) The presence of nonresponsive traffic, which in fact can be thought of as the limiting case of elastic traffic with decreasing responsiveness, improves the system stability. This is consistent with our earlier results that the less responsive users are, the more stable system is. This result contrasts the belief that nonresponsive traffic simply takes away available capacity. It is worth noting that in general characterizing the exact necessary and sufficient conditions for stability with delays is difficult. Hence, our results provide a simple and robust way of dealing with the problem of widely varying feedback delays in communication networks through a clever choice of the users utility functions and price functions. Moreover, our results on nonresponsive flows suggest that if the system is stable with only elastic flows, then the system stability will only be enhanced in the presence of nonresponsive flows. Therefore, when designing the rate control mechanisms, considering only elastic flows is sufficient for system stability, and the changing mixture of elastic and nonresponsive flows in the Internet with time will not affect the system stability. We also study the oscillatory orbits that appear when the system loses stability by explicitly giving the bounds on their amplitude. These bounds are derived from the underlying discrete time map that goes through a period doubling bifurcation with the loss of stability. This paper is organized as follows. Section II describes the optimization problem for rate control. We describe a rate control system model with delays in Section III. Section IV studies the single-flow case with a single resource and presents the general convergence results, which is followed by the multiple heterogeneous users case with general network topologies in Section V. Section VI compares the stability results for heterogeneous delays cases to those of a homogeneous delay case. We apply our results in Sections IV and V to establish the stability condition using a popular class of utility and resource price functions in Section VII. The effects of nonresponsive flows are investigated in Section VIII. Numerical examples are presented in Section IX. We briefly discuss the relationship between the system stability and fairness in Section X. We conclude the paper in Section XI. II. BACKGROUND In this section, we briefly describe the rate control problem in the proposed optimization framework. Consider a network

3 96 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 with a set of resources or links and a set of users. Let denote the finite capacity of link. Each user has a fixed route, which is a nonempty subset of.wedefine a zero-one matrix, where if link is in user s route and otherwise. When the throughput of user is, user receives utility. As mentioned earlier, this utility function could represent either the user s true utility or some function assigned to the user for carrying out the tradeoff between fairness among the users and system throughput [1], [11], [15], [20]. We take the latter view and assume that the utility functions of the users are used to select the desired rate allocation among the users (i.e., the desired operating point of the system), which also determines the end user algorithms as will be shown shortly. The utility is an increasing, strictly concave and continuously differentiable function of over the range. 1 Furthermore, the utilities are additive so that the aggregate utility of rate allocation is. Let and. The rate control problem can be formulated as the following optimization problem: The first constraint in the problem says that the total rate through a resource cannot be larger than the capacity of the resource. Instead of solving (1) directly, which is difficult for any large network, Kelly in [10] has proposed to consider the following two simpler problems. Assume that each user is given the price per unit flow. Given, user selects an amount to pay per unit time,, and receives a rate. 2 Then, the user s optimization problem becomes the following [10]: (1) for all. Furthermore, the rate allocation is also the unique solution to. Assume that every user adopts rate-based flow control. Let and denote user s willingness to pay per unit time and rate at time each resource, respectively. Now suppose that at time charges a price per unit flow of, where is an increasing function of the total rate going through it. Consider the system of differential equations These equations can be motivated as follows. Each user first computes a price per unit time it is willing to pay, namely. Then, it adjusts its rate based on the feedback provided by the resources in the network to equalize its willingness to pay and the total price. For example, in [11] is set to. The feedback from a resource can also be interpreted as a congestion indicator, requiring a reduction in the flow rates going through the resource. For more detailed explanation of (2), refer to [11]. Since we assume that the utility functions of the users are selected to decide the rate allocation amongst the users, under (2) one can see that, in fact, both the users utility functions and resource price functions can be utilized to decide the system operating point. Therefore, the design of rate control algorithms is equivalent to selecting the users utility functions and the price functions of the resources in the network. Kelly et al. have shown that under some conditions on,, the above system of differential equations converges to a point that maximizes the following expression: (2) (3) Note that the first term in (3) is the objective function in the problem. Thus, the algorithm proposed by Kelly et al. solves a relaxation of the problem. The network, on the other hand, given the amounts the users are willing to pay,, attempts to maximize the sum of weighted log functions. Then the network s optimization problem can be written as follows [10]: Note that the network does not require the true utility functions, and pretends that user s utility function is to carry out the computation. It is shown in [10] that one can always find vectors,, and such that solves for all, solves, and 1 Such a user is said to have elastic traffic. 2 This is equivalent to selecting its rate x and agreeing to pay w = x 1. III. NETWORK MODEL WITH DELAYS In this section, we first describe the network model that captures the delays between the network resources and end users under the assumption that the delays are constant. Although in practice the delays are time-varying due to the varying queue sizes, we assume that the variation in the delays due to fluctuating queue sizes is not significant. For example, AQM mechanisms that attempt to either maintain very small queue sizes, e.g., AVQ, or keep the queue sizes around some target queue sizes, e.g., REM, can be well approximated by our model. A simple case of a single flow and a single link with state-dependent time-varying delay is considered in [23], where our results show that the time-varying nature of the delay does not alter the stability conditions. Consider a set of users sharing a network consisting of a set of resources as described in Section II. Let be the set of users traversing resource,

4 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 97 Fig. 1. Network model with delays. Under this model the price of resource at time depends on the rates of the users at time due to the delay from the senders to the resource. The feedback signal generated by the resource price functions is then delayed by before sender receives it. In this paper, we are interested in studying the stability of the system given by the set of delay differential equations in (4). In particular, our goal is to find necessary and/or sufficient conditions on the utility and resource price functions that will ensure the convergence of,, to the solution of (3) regardless of the delays and.wefirst describe the utility and price functions that are of primary interest to us in this paper. We are interested in the class of users utility functions of the form i.e.,. We assume that,, are strictly increasing and continuous. The feedback information from the resources to user, which is typically carried by acknowledgments (ACKs), is delayed due to link propagation and transmission delays. The rate of a user is typically constrained due to the finite receiver buffer size. This is to prevent packet losses at the receiver buffer when the sender transmits packets at a rate faster than the receiver can process the packets from the buffer. 3 Therefore, throughout the paper we assume that there exists a finite upper bound on rate such that. However, can be made arbitrarily large. In addition, there is a lower bound on the rate since the user needs to probe the congestion level of the network by continually transmitting packets, although this lower bound can be arbitrarily small. For instance, in the example of TCP the transmission rate of a connection cannot be smaller than one packet size divided by the round-trip time of the connection. We denote this lower bound by. Hence, throughout the paper we assume that user s rate and that when the rate of a flow reaches the upper bound (lower bound ), the time derivative of flow is given by (, resp.). For all and let denote the forward delay user s packets experience before reaching resource from the sender and the reverse delay of the feedback signal from resource to user. This is shown in Fig. 1. If, we assume that. Suppose that the links in are arranged in the order user s packets visit, where. Here denotes the cardinality of.define,, i.e., the total round-trip delay before the receipt of the ACK of a packet. Under this general model, the end user dynamics are given by where 3 This is the function of the original flow control mechanism of TCP. (4) (5) In particular, has been found useful for modeling the utility function of TCP algorithms [12]. This class of utility functions has been used extensively in engineering literature [10], [12]. Also, it has been used to carry out a tradeoff between system throughput and fairness among the users [1] in a cellular network. With the utility functions of the form in (6) one can easily show that the price elasticity of demand decreases with as follows. Given a price per unit flow, the optimal rate of the user that maximizes the net utility is given by. The price elasticity of demand, which measures how responsive the demand is to a change in price, is defined to be the percent change in demand divided by the percent change in price [26]. In our case, the price elasticity of demand is given by Therefore, one can see that the price elasticity of demand decreases with, 4 i.e., the larger is, the less responsive the demand is. The class of resource price functions that we are interested in is of the form where, is some positive constant, and is the capacity of resource. However, can be replaced with any positive constant, e.g., virtual capacity in AVQ, so that the price function can be dynamically adjusted based on the current load using the virtual capacity as the control variable [13]. Throughout this paper we assume that unless stated otherwise. This kind of marking function arises if the resource is modeled as an queue with a service rate of packets per unit time and a packet receives a mark with a congestion indication signal if it arrives at the queue to find at least packets in the queue. The parameter is used to change the shape of the price function. The larger is, the more convex and responsive the price function is. 4 When comparing the price elasticity, typically the absolute value of (7) is used. (6) (7) (8)

5 98 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 Although these are the utility and resource price functions of interest to us, in the following sections we first establish more general stability results. Then, we apply these results to the system with the user utility and resource price functions of (6) and (8), respectively, in Section VII. Before providing the stability conditions in the following sections, we first comment on the existence and uniqueness of a solution of (4). Suppose that both users utility functions and resource price functions are continuously differentiable and Lipschitz continuous on appropriate domains. Then, a solution of (4) exists and is unique from Theorem 2.3 in [6, p. 44] if the initial function is continuous. Throughout this paper, we implicitly assume that users utility functions and resource price functions are continuously differentiable. The Lipschitz continuity condition will be made explicitly on the functions shortly. IV. SINGLE-FLOW SINGLE-RESOURCE CASES In this section, in order to provide the insight behind our approach we begin with a simple case where a single flow traverses a single resource. The rate control problem of (3) in this case is given by the following simple optimization problem [10]: where. We study the stability of (12) in place of (11). For, this equation can be seen as following singular perturbation (13) of general nonlinear difference equation with continuous argument given by,, where and in the context of (12). Under certain natural invertibility conditions on, it leads to a much studied equation [25] (14) where. For the solution of (14) to be continuous for, along with the continuity of and initial condition, a so-called consistency condition is required [8], [25]. It turns out that a great deal about the asymptotic stability of (13) can be learned from the asymptotic behavior of the following difference equation, with denoting the set of nonnegative integers [7]: where is the rate, is the utility of the user when it receives a rate of, is the price per unit flow charged by the resource when the rate is, and is the capacity of the resource. The proposed end user algorithm in the absence of delay is given by the following differential equation [11]: (9) (10) where is the price per unit time the user is willing to pay,, and is a gain parameter. In this section we will study the stability of the system in (10) with general utility functions that satisfy a set of assumptions to be stated shortly. Following the end user algorithm given in [11], we assume that. Now, suppose that there is no forward delay from the sender to the resource, and the congestion signal generated at the resource is returned to the sender after a fixed round trip delay. In the presence of delay the interaction is given by the following delay differential equation: (11) After normalizing time by and replacing, (11) becomes (12) The concave utility functions and resource price functions assumed in [10] and [11] do not satisfy the assumptions in [7], and hence we cannot directly apply their results. However, the general approach used in the paper for establishing the stability can be extended to study the convergence of (11). In the following subsections, we first establish general convergence results for one dimensional case described here and a bound on the range of the rate when the system loses its stability. A. Convergence Results In this subsection, we establish the conditions for convergence of the system in (11) with an arbitrary communication delay and a gain parameter. We make the following substitutions: and Note that is simply user s willingness to pay at time based on its rate, and is the total price charged by the resource when the rate traversing it is. We first make the following assumptions on the functions and. Assumption 1: (i) The function is strictly decreasing with for all ; (ii) the function is strictly increasing in ; and (iii) both and are Lipschitz continuous on. Assumption (i) implies that the marginal utility decreases faster than from the definition of. Hence, a change in price per unit flow does not cause a large change in user demand, and a user demand is relatively insensitive to price changes, i.e., user demand is inelastic. It also guarantees the existence of. Therefore, the convergence of to implies the convergence of to.

6 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 99 It is easy to verify that the utility and resource price functions given in (6) and (8) satisfy the above assumption. From the definition of,wehave (15) Clearly, (15) satisfies Assumption 1. Assumption 1 allows us the following change of coordinate: (16) Let. Clearly, under Assumption 1. Using this substitution in (16), we get the following form: (17) We study (17) and show that there is a close correspondence between invariance and global stability properties of the discrete time map (18) and those of (17). In particular, we will prove that if has a stable fixed point then (17) will have a uniformly constant solution for all possible time delays if the initial function s range is contained in the immediate basin of attraction of this fixed point. The proofs are based on the invariance property of the underlying map and the monotonicity of function. Note that the map is strictly decreasing because is strictly decreasing and is strictly increasing under Assumption 1 and a composition of a strictly increasing function and a strictly decreasing function is a strictly decreasing function. Assumption 2: Suppose that is a closed invariant interval under, i.e.,. In particular, let be compact. We denote by the Banach space of continuous functions mapping the interval into. Let, where, and. Under this assumption, we have invariance for the solution of (17) for all time and for all. The invariance property of the solutions, which is stated below (Theorem 1), ensures that they stay positive and bounded by the initial set. Theorem 1: (Invariance) If, the corresponding solution satisfies for all. Proof: The proof is given in Appendix I. The next theorem considers the case when the map has an attracting fixed point, where is the solution to (9), with immediate basin of attraction for any. Let. Then, the following theorem holds. Theorem 2: (Stability) For any and, wehave. Proof: The proof is given in Appendix II. The above theorem tells us that if the initial function lies in, then converges to and, hence, the rate converges to the solution of (9) regardless of the value of and. Therefore, it establishes a strong convergence result in the presence of a communication delay without any conditions on the value of and. This contrasts the results obtained in [3] and [19]. B. Linear Instability In the previous subsection, we have demonstrated that the system in (11) converges with an arbitrary delay and gain parameter under Assumptions 1 and 2 if the initial condition lies in the specified invariant set. In this subsection we study the case where the map defined by (18) loses stability and goes through a period doubling bifurcation with its eigenvalue, where is an unstable fixed point of the map. We describe how the instability of the underlying discrete time map is translated to the instability of delay differential equation in (17). Assuming that the map given by (18) is locally smooth, it is possible to find conditions for linear instability of the fixed point of the map and that of constant function for the delay differential equation in (17). In order for to be locally asymptotically stable for all, the following variational equation should have its zero solution stable: because (19) where and. Now in order to determine the stability of, we can apply the following well-known results [21]. (1) Eq. (19) is stable for all only if and (20) (2) In the case when the above condition in (20) is violated, we still have stability for some values of time delays : and (21) For our case, it follows from (19) that is always negative, which holds due to the fact that is always positive. The second condition is crucial to the stability of (17). Clearly, for the case (period doubling condition for the map ), the linear stability condition given by (20) is violated, and for a large enough the constant solution will not be stable. Thus, we know that in an unstable situation solutions will be more complex than a constant function, but will stay within the interval they initially start from due to the invariance results given by Theorem 1. It is worth noting that (21) also yields conditions similar to those in [3], [9], and [19] that when (20) is violated, the system is stable if the product of is less than or equal to some

7 100 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 constant. This is because both and are proportional to from the definition of. The following theorem provides bounds for the rate in case of instability. Theorem 3: Let be the basin of attraction for a period two orbit of the underlying discrete time map and and be the period two fixed points such that, and. Suppose that the initial condition, and is the solution of (17). Then, for all sufficiently small there exists a finite such that for all. Proof: The proof is given in Appendix III. The above theorem essentially gives bounds for the interval which will contain the solution asymptotically. Thus, our basic conclusion is that solutions either converge to the fixed point or oscillate around it. The upper limit on period two fixed points is given by physical constraints on flow rate, e.g., link capacity and receiver buffer size, while the lower limit is determined by choices of utility and price functions and hence the map [22]. where, (23) The function takes the delayed feedback information and provides the total price of user at time. In order to write (22) in a simpler form, we first define V. GENERAL NETWORKS WITH HETEROGENEOUS DELAYS In this section, we extend the results in Section IV to general network models described in Section III. Although the basic idea used in establishing the stability conditions is the same, the notation becomes less manageable due to the fact that the end user algorithm utilizes feedback information at different points of time [see (4) and (5)] and these values must be explicitly kept track of. Recall that and,,, denote the forward delay from sender to resource and the reverse delay from resource to sender, respectively. They are assumed to be zero if.define for all. Similarly as in the single-flow, single-resource case, we make the following assumption on,, and the resource price functions,. Assumption 3: (i) The function is strictly decreasing with for every and for all, (ii) the price functions are strictly increasing in for all, and (iii) is Lipschitz continuous on and is Lipschitz continuous on, where and are the assumed lower and upper bounds on,, respectively. This assumption ensures that the state dependent gain matrix is a positive definite matrix and that the inverse functions,, exist. Let and. We define. Note that the end user update rule at time does not depend on,. Thus, limits the history of the rate vector we need to maintain for the delay differential equations in (4). In order to facilitate our analysis, we follow similar steps as Section IV and rewrite (4) in terms of,. (22) and,. Using these variables we can write (22) in the following matrix form: where with (24) is the state dependent diagonal gain matrix, and is user s total price based on the delayed rates of the users. Hence, it is clear from (24) that users attempt to reach an equilibrium point where their willingness to pay equals their prices. In order to establish the stability of the system (24) with multiple users of heterogenous utility functions we need to prove the global convergence in a multi-dimensional space. Our approach extends the basic approach used by Verriest and Ivanov [27]. The basic idea behind this approach is to use invariance and continuity properties of the map defined earlier and find a sequence of bounds using convex sets, which converges to the singleton with the solution to (3). Hence, the convergence is derived from the map, which provides the bounds for the trajectories of the delay differential system. Let.Wedefine the invariance and a fixed point of the map as follows. Definition 1: A set is said to be invariant under the map defined in (24) if whenever, i.e., and for all. A vector is said to be a fixed point of if. The invariance of the map can be interpreted as follows. Suppose that users willingness at time as well as time delayed values, and, belong to the set. Then, the invariance of the set implies that stays within the set. Similarly, a fixed point means that if, for all and, then remains constant at. One can verify that if is a fixed point of, then is a solution

8 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 101 to (3) from (22) and (23), i.e.,, where is a solution to (3). We now state the assumption under which the asymptotic stability of (24) is established. Assumption 4: Multidimensional map has a fixed point, and there is a sequence of closed, convex product spaces,, such that and, where denotes the interior of. Let be a subset of initial functions and a solution of (24) constructed using an initial function. Theorem 4: (Asymptotic Stability) All solutions starting with initial function converge to as for all nonnegative and. Proof: The proof is provided in Appendix IV. Theorem 4 tells us that the attracting fixed point of the map is stable in the set. Since this tells us that as. Our stability results consider the case where arbitrary delays,, are allowed. However, if the delays are finite and upper bounded by some constant, then the system may be stable without satisfying our conditions stated in this subsection, and less stringent stability conditions may be sufficient to ensure stability if the upper bounds on the delays are known beforehand. VI. COMPARISON WITH HOMOGENEOUS DELAY CASE In this section we discuss the relationship between a homogeneous delay system where every user has the same feedback delay, and a heterogeneous delay system discussed in Section V. We first describe the system where all users have the same feedback delay, all of which lies in the reverse path from the receiver to the sender. Eq. (4) can now be written in a much simpler form: (25) where is the common feedback delay of the users. Following the same steps in Section V we can write (25) in the following simple matrix form: where, and. Here one can easily see the dynamic market structure in which every user updates its willingness to pay to its total price in each iteration based on the current values of willingness to pay of all users. Suppose that the multi-dimensional map has some fixed point, and that there is a sequence of closed, convex product spaces, such that and. Then, it is shown that if the initial function, i.e., for all, then for all [22]. This result tells us that there is a close correspondence between the stability of the discrete time system in (27) and that of (26). A key observation to be made here is the following: Consider a system consisting of a set of users and resources that satisfy Assumption 1. First, suppose that the users have the same delay and the assumption in the previous paragraph holds for some sequence of,, and hence the system is stable provided that the initial function lies in. Then, one can easily verify from the monotonicity of the functions that even when the users have heterogeneous delays, the same sequence,, satisfies Assumption 4 and, hence, the system is stable if the initial function belongs to. This observation is intuitive as follows: Suppose that in the homogeneous delay case, the delay of the flows is, and in the heterogeneous delays case, the delays of the flows satisfy for all. Then, since the communication delays of the users in the heterogeneous delays case are no larger than of the homogeneous delay case, one would expect the system with heterogeneous delays to be stable if the system with homogeneous delay is stable. However, since our stability results for homogeneous delay case hold for any arbitrary [22], one should expect the system with heterogeneous delays to be stable irrespective of,, as well. This observation provides a powerful tool for studying the stability of a system given by delay differential equations (4) with arbitrary delays. It tells us that the stability of the system can be studied by studying the stability of the homogeneous delay system, which in turn can be studied using the simple discrete time system given by (27). Therefore, the study of (4) reduces to that of the simple discrete time system in (27). where (26) Note that since every user has the same delay, the multi-dimensional nonlinear map has the domain (as opposed to in Section V). We can define the following discrete time map: (27) VII. APPLICATION TO AN EXAMPLE UTILITY AND RESOURCE PRICE FUNCTIONS In this section, we illustrate how our results in the previous sections can be used to establish the stability condition of the rate control problem described in Section II with users utility functions and resource price functions given by (6) and (8), respectively. A. Heterogeneous Users With General Networks In this subsection, we consider a general network where users utility and resource price functions are of (6) and (8), respectively. Making use of the observation stated in Section VI,

9 102 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 we only consider the case with a homogeneous delay described in Section VI. In this case, the underlying discrete time map from (27) is given by B. A Single User With A Single Resource In this subsection, we consider the special case of a single user traversing a single resource to explain some of the intuition behind our stability results. Using the utility function and resource price function of (6) and (8), respectively, (11) can be written as (29) and the underlying discrete time difference equation is given by Note that is strictly decreasing in each of,. We define for all, and assume that users are ordered by increasing, i.e.,. Let be the unique solution of the optimization problem in (3). We assume that, where the matrix is the routing matrix described in Section II. A sufficient condition for this is that if for all. Suppose that, where denotes the Cartesian product of is some finite positive constant larger than one, and positive constant that satisfies is a and (28) where. Lemma 1: Suppose that for all.define, where. Then, any such that satisfies (28). Proof: The proof is provided in [22]. We assume that satisfies the condition in Lemma 1. Now, for,wedefine odd even Lemma 2: Suppose that for all. Then,, and. Proof: The proof is provided in [22]. Theorem 5: Suppose that for all. If the initial function lies in, then produced by (4) converges to asymptotically for all and. Proof: The theorem follows from Lemma 2 and Theorem 5 in [22]. Now note that as increases, goes to. Hence, since (resp. ) can be made arbitrarily small (resp. large), we can see that starting from any arbitrary rate vector, the rates converge to asymptotically from the above results. This discrete time system has a fixed point (30) (31) and the market equilibrium price is given by. The market equilibrium price can be obtained from. This expression of equilibrium flow shows that increases with decreasing (for a fixed value of ). The eigenvalue at this fixed point, which is interestingly independent of the fixed point, is (32) Suppose that. Then, the fixed point is locally attracting. In fact, can be shown to be globally stable as follows. According to the Sharkovsky cycle coexistence ordering [24] the most general condition for the fixed point to be globally attracting is that the second iteration of the map does not have a fixed point in relevant state interval other than, and is locally attracting. This in turn implies the global delay independent stability of (29) from Theorem 2. Hence, our results completely characterize the stability condition in terms of the parameters of the utility function and resource price function, revealing the relationship between them. The tradeoff between user s price elasticity of demand characterized by the parameter and the responsiveness of resource price function captured by the parameter can also be seen from the definition of the discrete time map. The derivative of evaluated at the equilibrium is given by (33) by the chain rule, where. Here the first term in the right-hand side of (33) gives the slope of the resource price function as a function of the flow rate, while the second term is a function only of user s utility function, namely

10 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 103. Therefore, (33) clearly describes the tradeoff between user s price elasticity of demand and the responsiveness of resource price function for stability. VIII. EFFECTS OF NONRESPONSIVE TRAFFIC Some applications in the Internet, such as real-time streaming, cannot react to congestion as fast as responsive flows and hence adopt a different transport layer protocol that adapts to congestion state much slower. Since their reaction times are much larger than those of responsive flows, for modeling purposes in the time scale of interest they can be approximated as nonresponsive flows, whose rates do not vary. We model this nonresponsive traffic as a deterministic constant bit rate source instead of stochastic noise that has been used to model the aggregate traffic of short-lived (responsive) flows. In this section, we analyze the effect of the presence of deterministic nonresponsive flows on system stability. As nonresponsive traffic can be assumed to have no dynamics but contributes to aggregate rate presented to the resource, the dynamics of the model given by (11) need to be modified. Results for stochastic nature of nonresponsive traffic will be reported elsewhere. Here we only consider a single-flow, single-resource case but similar results can be shown for multiple heterogeneous user case. Suppose that we denote the aggregate rate of the nonresponsive flows by. Then, the underlying discrete time system is described by Fig. 2. Fig. 3. Topology of the example network. User rate evolution. (34) The unique fixed point for this system can be computed as the solution of the following equation: (35) It is clear from (35) that as the amount of nonresponsive traffic increases, the solution of (35) decreases, which is intuitive. We look at the eigenvalue of (34) to study the effects of nonresponsive traffic on stability. Lemma 3: The eigenvalue of the system in (34) at the fixed point is increasing in. Lemma 3 states that the stability of the system improves with. A similar observation has been made in the context of TCP-RED [16]. The proof of the lemma is provided in [22]. IX. NUMERICAL EXAMPLES In this section, we present simple numerical examples to illustrate some of our results presented in the previous sections. A. General Networks In this subsection, we consider a simple network consisting of two links. The first link is shared by users 1 and 3, while the second link is share by users 2 and 3. This is shown in Fig. 2. The capacities of the links are set to. The utility functions are of the form in (6) with and, and the resource price functions are of (8). We select two different sets of parameters,, 2, to create both a stable system and an unstable system. The initial function is given by for all. The gain parameters are set to for all users. 1) Stable System: The resource price parameters for the first case are set to. Since for all users, the system is stable. One can solve the optimization problem in (3) and show that the solution is given by. The feedback delays are given by unit times. Here we assume that all delays lie in the reverse path from the receivers to the senders. Fig. 3 plots the evolution of user rates according to (4). As one can see the user rates converge to the optimal rates. 2) Unstable System: In the second case, the resource price parameters are increased to. One can easily see that the stability conditions in Theorem 5 do not hold. We increase the feedback delays by tenfold from the previous case, i.e., unit times. Fig. 4 shows the unstable behavior of the system as the user rates show no sign of settling down, resulting in large oscillations in the rates. X. IMPLICATIONS ON FAIRNESS In practice, achieving high utilization at bottlenecks is an important issue. We have suggested that this may be achieved by dynamically adjusting one of control parameters at the routers

11 104 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 Fig. 4. User rate evolution of an unstable system. to select the desired utilization, e.g., parameter in (8). Kunniyur and Srikant [14] have proposed a dynamic mechanism that utilizes a virtual queue associated with each link. The idea behind this approach is to adapt the virtual queue capacity in order to maintain certain desired utilization at the bottlenecks. This allows the end users using the algorithm in (2) to solve the SYSTEM problem of maximizing the aggregate utility of the users in (1). A similar idea was also used in [4]. It is easy to verify that the stability conditions do not change with different choices of,. Hence, with increasing the system stability can be improved without sacrificing the utilization by adjusting,, at the routers. We explain the effect of the selection of s using an example. Consider the example shown in Fig. 5(a), where there are three users that share two links in the network. The capacity of both links is assumed to be one. Suppose that all users have the same utility function,. The optimal rates that solve the SYSTEM problem in (1) for a given are plotted in Fig. 5(b). Since, we only plot and. As one can see, as increases,,, converge to 0.5, which is the max-min fair allocation. This can be easily explained by the closed form solution of the problem. After a little algebra, one can show that, where. Hence, as,wehave and. 5 One can also attain similar results for general multiple bottleneck cases, i.e., the solutions to the SYSTEM problem converge to the max-min fair allocation as. This suggests that both (max-min) fairness among the users and system stability improve as increases. XI. CONCLUSION We have studied the stability of a rate control mechanism in the presence of an arbitrary delay and showed that the system stability is determined by the interaction of underlying utility 5 Note that the limit of solutions to the SYSTEM problem with a # 0 is the solution to the NETWORK problem with w = w for all i 2 I, i.e., proportionally fair allocation [10]. Fig. 5. Example. (a) Topology. (b) Plot of x (a). and price functions. We have illustrated that a discrete time framework arises as a natural tool to study the dynamics of delayed rate control schemes. In addition, we have demonstrated a fundamental tradeoff between users price elasticity of demand and the responsiveness of the resources. We have proved that when the users utility or resource price function is too responsive in relation to the other, it leads to network instability. We have explicitly characterized the stability condition for a class of utility functions and resource price functions. Furthermore, we have shown another tradeoff between the global stability of system and the utilization of the resource. These results offer some guidelines for jointly designing the end users algorithms and AQM mechanisms at the routers in the presence of communication delays between end users and network elements. We have also demonstrated that the presence of nonresponsive flows enhances the stability of the system. Since a nonresponsive flow can be viewed as a limiting case of a flow with decreasing price elasticity of demand, this result is intuitive. This implies that when designing a rate control mechanism, only elastic flows need to be considered for system stability. Therefore, our stability conditions are sufficient for system stability even with nonresponsive traffic in the network. We believe that understanding the effects of delays in a network is important for emerging network applications. These applications include remote instrument control, telerobotic actuation, and computational steering. As they are envisioned to be deployed over a large area, thorough understanding of the role of potentially large delays in networked control systems over a wide-area network is crucial. Large delays and delay jitter in the

12 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 105 network can destabilize an otherwise stable system, resulting in instrument damages or erroneous execution of automated actuation. We are currently working to extend the framework proposed in this paper to study the stability of future networked applications with arbitrary delays in the underlying network. APPENDIX I PROOF OF THEOREM 1 We prove the theorem by contradiction. Suppose that the claim is not true and there exists such that for some. Let be the first time when solution leaves, i.e., for all there exists such that (36) First, assume that. In this case, for every right-hand neighborhood of, i.e.,,, we can find a point,, such that and. However, since from (36), we have such that and. However, as for all,wehave from (17). This contradicts the earlier assumption that. The other case where leaves the interval from the left end can be handled similarly. Now assume that. In particular, let. We claim that is decreasing for all, where is the first time such that. We first argue that by contradiction. Suppose that for all. From (17), for all, we have because. Then, there exists a limit due to the Bolzano Weierstrass theorem [5] which says that every strictly decreasing sequence which is bounded from below has a limit. As is not a fixed point of map,. This tells us from (17) that from (17) and Assumption 2 that is invariant under, i.e.,. This contradicts the earlier assumption that. Similarly, suppose that and the trajectory exits from the left end of the interval. Then, for every interval,, we can find,, such that and. However, because, we have from (17) and Assumption 2. This, however, contradicts the assumption. Hence, the theorem follows. APPENDIX II PROOF OF THEOREM 2 Before proving the theorem, we will first state a lemma which is the key to the proof of Theorem 2. Lemma 4: Suppose that a closed interval is mapped by into itself. If neither of the endpoints of the interval is a fixed point, then for every there exists a finite such that for all. Proof: From Theorem 1, it is clear that for all. The claim here is that after certain time it will be limited by. We consider two cases. First, assume that. Then it can be shown that for all by contradiction. Suppose that this is not true, and let be the first time when leaves the interval. In particular, assume that it leaves the interval through the right end, i.e., every time interval, where, contains a point such that. The interval also contains a point for all sufficiently large. This implies that as, which is a contradiction, because this means that crosses for some finite. Hence,. Now, we can apply the argument used in the first case, i.e.,, to the system with and for all to prove that for all. The other case where can be handled similarly. Now we provide the proof for Theorem 2 using Lemma 4. For any,define and. Clearly,. Let be the smallest closed invariant interval containing which is a subset of. Then, from the existence of a stable fixed point of the map, and. Using the invariance result and Lemma 4 repeatedly one can find a sequence of,, such that for all,. The convergence now follows from the assumption that converges to. APPENDIX III PROOF OF THEOREM 3 The proof follows a similar argument used in the proof of Theorem 2. For any, define and. Clearly,. Let be the smallest closed invariant interval containing which is a subset of. Then, from the existence of period two fixed points for the map and the assumption that is the basin of attraction for a period two orbit of the map,wehave and. Using invariance and Lemma 4 in the proof of Theorem 2 repeatedly, one can show that for all sufficiently large, the trajectory will lie in, and the proof is complete.

13 106 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 14, NO. 1, FEBRUARY 2006 APPENDIX IV PROOF OF THEOREM 4 We first show that the invariance property of the map also implies the invariance property of (24). Then, we prove that Assumption 4 implies the stability of (24) as well. We use the following three lemmas. Lemma 5: (Invariance) Suppose that is a closed, convex, invariant product space under. Then, for any initial function the resulting from (24) belongs to the set for all. Proof: We prove the lemma by contradiction. Suppose that the lemma is false. Then, there exist some initial function and,, such that.define for every interval where such that Then, there is such that for all, where, there exists,, such that. Here denotes the projection of the th component of. We assume that leaves the interval through the right end, i.e.,. Then, for all there exists,, such that and. This, however, leads to a contradiction as follows. From (22) we have because and, and, hence, is less than or equal to. This contradicts the earlier assumption that. The other case that leaves through the left end, i.e.,, can be shown to lead to a similar contradiction. Therefore, the lemma follows. Lemma 6: Fix,. Let be an open product space that contains and whose closure is contained in, i.e.,. Suppose that the initial function. Then, there exists a finite,, such that, for all,. In order to prove the lemma, we first prove the following coordinate-wise invariance. Lemma 7: (Coordinate-Wise Invariance) If for some, then for all. Proof: Suppose that the lemma is not true, and there exists at which or. We assume that is the smallest such time, i.e., where is the boundary of the set, and. Then, we can find such that for all,. This implies that for all from (22) because and, thus,, leading to a contradiction. A similar argument can be used for the case. Now let us proceed with the proof of Lemma 6. Proof: Suppose that the lemma is false. Then, from Lemma 7 there exists such that for all,. We show that this leads to a contradiction. Suppose that for all. Since with, there exists some positive constant such that for all sufficiently large from (22). This, however, implies that as, contradicting the assumption that for all. A similar contradiction can be shown when we assume for all. This completes the proof of the lemma. Lemma 8: Let be a closed, invariant product space and an open product space that contains and whose closure is contained in, i.e.,. Suppose that the initial function and for some. Then, for all. Proof: The lemma follows directly from Lemma 7. The proof of the theorem can now be completed as follows. By repeatedly applying Lemmas 5, 6, and 7, one can find a sequence of finite,, such that for all. The theorem now follows from Assumption that. REFERENCES [1] R. Agrawal, A. Bedekar, R. J. La, and V. G. Subramanian, C3wpf scheduler, in Proc. ITC, 2001, pp [2] T. Alpcan and T. 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14 RANJAN et al.: GLOBAL STABILITY CONDITIONS FOR RATE CONTROL WITH ARBITRARY COMMUNICATION DELAYS 107 [20] J. Mo and J. Walrand, Fair end-to-end window-based congestion control, IEEE/ACM Trans. Netw., vol. 8, no. 5, pp , Oct [21] S. Niculescu, E. I. Verriest, L. Dugard, and J. Dion, Stability and robust stability of time delay systems: a guided tour, Stability and Control of Time-Delay System, vol. 228, pp , [22] P. Ranjan, R. J. La, and E. H. Abed, Trade-offs in rate control with communication delay, ISR, Univ. Maryland, College Park, Tech. Rep., [23], Global stability in the presence of state-dependent delay for rate control, in Proc. CISS, Oahu, HI, [24] A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak, and V. V. Fedorenko, Dynamics of one Dimensional Maps. Boston, MA: Kluwer Academic, [25] A. N. Sharkovsky, Y. L. Maistrenko, and E. Y. Romaneko, Difference Equations and Their Applications. Boston, MA: Kluwer Academic, Translated from Russian. [26] H. R. Varian, Intermediate Microeconomics. New York: Norton, [27] E. I. Verriest and A. F. Ivanov, Robust stability of systems with delayed feedback, Circuits, Syst. Signal Process., vol. 13, no. 2 3, pp , [28] Z. Wang and F. Paganini, Global stability with time-delay in network congestion control, in Proc. IEEE Conf. Decision and Control, Las Vegas, NV, 2002, pp [29] J. Wen and M. Arcak, A unifying passivity framework for network flow control, in Proc. IEEE INFOCOM, 2003, pp [30] L. Ying, G. E. Dullerud, and R. Srikant, Global stability of Internet congestion controllers with heterogeneous delays, in Proc. American Control Conf., 2004, pp Priya Ranjan received the B.Tech. degree from the Indian Institute of Technology, Kharagpur, in 1997 and the M.S. and Ph.D. degrees from the University of Maryland, College Park, in 1999 and 2003, respectively, all in electrical engineering. He is currently a Research Associate with the Institute for Systems Research, University of Maryland, and a Research Scientist with Intelligent Automation, Inc., Rockville, MD. He has published in the areas of nonlinear dynamical systems, power electronics, modeling and analysis of logistic systems, and communication networking. Richard J. La (S 98 M 01) received the B.S.E.E. degree from the University of Maryland, College Park, in 1994, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Berkeley in 1997 and 2000, respectively. From 2000 to 2001, he was a Senior Engineer in the Mathematics of Communication Networks group at Motorola. Since August 2001, he has been on the faculty of the Department of Electrical and Computer Engineering, University of Maryland, College Park. His research interests include resource allocation in communication networks and application of game theory. Dr. La is a recipient of a National Science Foundation (NSF) CAREER award. Eyad H. Abed (S 80 M 83 SM 93 F 02) received the S.B. degree from the Massachusetts Institute of Technology, Cambridge, in 1979, and the M.S. and Ph.D. degrees from the University of California, Berkeley, in 1981 and 1982, respectively, all in electrical engineering. He has been with the Department of Electrical and Computer Engineering, University of Maryland, College Park, since 1983, where he is currently a Professor of electrical and computer engineering and Director of the Institute for Systems Research. His research interests center on the control of nonlinear dynamical systems and applications. Dr. Abed was a recipient of the Presidential Young Investigator Award, the O. Hugo Schuck Best Paper Award, the Senior Fulbright Scholar Award, the Alan Berman Research Publication Award, and several research and teaching awards from the University of Maryland. He currently serves on the Advisory Editorial Board of Nonlinear Dynamics.