The Filippov equilibrium and sliding motion in an internet congestion control model

We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.


1.
Introduction. Design of the congestion control of computer networks is receiving increasing attention as the demand for network service bursts in recent decades. One central task of the network algorithm design is to find the optimal scheme to regulate the traffic on the network. The sloppy design of the congestion control algorithm may lead to disastrous consequences [13,20]. Due to the complexity of the network system, undoubtedly it is necessary to look into the problem from a theoretic point of view to make simplifications and provide suggestions on the network design.
To give a mathematical description of the process of delivering data packets, two parallel schemes are developed: the method based on optimization and the continuous flow approximation . The theory of optimization was employed by Kelly [14] to build a framework in which many important issues related to congestion control can be tackled uniformly. A nonlinear utility function is constructed and taken as the objective function to be maximized, subject to a set of linear constraints. For a to study the existence of the equilibrium and the motion near the discontinuous boundary. The rest of the paper is organized as follows. In Section 2, we introduce the underlying model and discuss several basic properties of the solution. In Section 3, the existence of the Filippov equilibrium solution is established for simple network topology. In Section 4, the sliding motion on the discontinuous boundary is investigated and numerical results are shown to provide illustrative examples. Conclusions are summarizes in Section 5.
2. Model for TCP/RED with single user group and single link. Misra et al [19] employed the theory of stochastic differential equation to derive the mathematical model that describes the interaction of TCP and RED and provided some simulation results. Based on the model in [19], we ignore the filtering process for simplicity, and the model describing TCP/RED algorithm can be written as − W (t) W (t − R(q(t)))P (q(t − R(q(t)))) 2R(q(t − R(q(t)))) , where W (t) represents the window size for delivering data packets, q(t) the queue size at the buffer of the router, P the probability of packet marking or dropping, C the transmission capacity, N the number of users. In the user group, all the users share the same system resources and consequently can be considered as identical. Thus, we use a single W (t) to denote the averaged behavior of all users [12,19].
[.] + represents max{., 0} and [.] − represents min{., 0}. R(q(t)) denotes the round trip time or time delay which consists of two parts, namely with the propagation delay τ 0 . Obviously, R(q(t)) is a state-dependent delay. P is a nondecreasing function and takes value in [0, 1]. In [19], P is given as where 0 < P max ≤ 1, 0 < b i < 1 (i = 1, 2) are RED parameters and B is the buffer size of the router. To show the configuration of P , an illustrative example with b 1 = 0.2, b 2 = 0.5, P max = 0.3 and B = 50 is shown in Fig. 1. The model can be briefly interpreted as follows. The equation of the window size W (t) describes the TCP dynamics. Namely, W (t) is increased by one (measured in packets) during each RTT (round trip time), and decreased by an quantity proportional to the current window size and the averaged number of lost packets during the same time period. The motion of q(t) indicates how the active queue management is implemented to TCP. Namely, the rate of change in the queue size is the difference between the number of newly arrived data packets per time unit during each RTT (N W (t)/R(q(t))) and the speed (C) of processing packets by the router.
The stability and dynamics of the above model have been extensively investigated. For example, in [26], the authors showed that there existed multi-attractors in (1) by means of the numerical continuation and provided suggestions for the selection of TCP parameters. However, some basic aspects of (1) are still not clear and become topics of discussion in the following.
. Following the solution definition given in [3], , X(t − τ (X(t)))) where X(t) = (W (t), q(t)) T and f is the right-hand side of (1). Then, with the initial condition in D, X(t) is called a Filippov solution of (1) if a) X(t) is defined on a non-degenerate interval I and absolutely continuous on any closed subinterval of for almost all t ∈ I, whereco represents the closure of the convex hull, B(, ) is the ball centered at the first component with the second component as its radius, and µ represents the Lebesgue measure. We can check that the equilibrium solution of (1) satisfies Definition 1. For the properties of the solutions, we have the following theorem. Theorem 1. All the solutions in (1) with the initial condition in D are nonnegative and ultimately bounded, i.e., 0 ≤ W (t) ≤ ∆ 1 , 0 ≤ q(t) ≤ ∆ 2 for all t > 0, where ∆ 1 and ∆ 2 are constants.
Proof. For the positivity, q(t) ≥ 0 is guaranteed by the model, thus we only need to show that W (t) ≥ 0 for t > 0. If not, there exists a finite t 0 > 0 such that W (t) ≥ 0 for t ∈ [0, t 0 ] and W (t 0 ) = 0,Ẇ (t 0 ) ≤ 0. However,Ẇ (t 0 ) = 1 R(q(t0)) > 0 which gives a contradiction. According to the model setup, it is clear that q(t) ≤ B = ∆ 2 . We only need to prove that W (t) is bounded above. Assume W (t) is unbounded, then for any positive number V , there must exist a finite t 1 such that W (t 1 ) > V andẆ (t 1 ) ≥ 0.
First, for any t ∈ I 1 = [t 1 − R(q(t 1 )) − R(q(t 1 − R(q(t 1 )))), t 1 ], integrating the first equation in (1) from t to t 1 , we have 2R(q(s − R(q(s)))) P (q(s − R(q(s)))))ds from the second equation of (1), we have t1−R(q(t1))−R(q(t1−R(q(t1)))) dq(s) ds ds which yields a contradiction. This implies that there exists t a ∈ I 2 such that q(t a ) = B. Thenq(t a ) = 0 and q(t) = B for t ≥ t a since we have shown thaṫ q(t) ≥ 0 for t ∈ I 1 . Consequently, q(t 1 − R(q(t 1 )) = B and P (q(t 1 − R(q(t 1 ))) = 1. Therefore,Ẇ The existence of positive equilibrium of (1). Denote the equilibrium of (1) by (W * , q * ) T . We say (W * , q * ) T is desirable if b 1 B < q * < b 2 B. Then we have the following theorem. Proof. Let the right-hand side of (1) be zero, then (W * , q * ) is determined by From the second equation of (5) we have q * = N W * − C τ 0 . Let This means that at the equilibrium, the probability of packet loss or marking, i.e. P (q(t)), is 1, implying even at the steady state, there are excessive data packets in the buffer of the router and consequently the network system is at high risk of congestion. In other words, the congestion control system is not well designed in this case. Therefore, in the above discussion, we don't consider such case by assuming that q < b 2 B at the equilibrium.
To study the stability of the equilibrium (W * , q * ) T , let W (t) = a 1 e λ t + W * , q(t) = a 2 e λ t + q * where a 1 and a 2 are constants and λ is the eigenvalue of the linearized system. From [5], we know that for the terms with state-dependent delay, the local linearization method can be employed by treating the delay as a constant at the equilibrium. Namely, let W (t−R(q(t))) =W (t)+W * , q(t−R(q(t))) =q(t)+q * . Then the linearized system near (W * , q * ) can be written aṡ with In general, we know that, if all the roots of (6) have negative real parts, then (W * , q * ) is locally stable, and is unstable if at least one root has positive real part. Due to the complexity of the system, although we can not provide the explicit conditions to guarantee the stability, we will check this general condition numerically in examples.
3. Dynamic routing and its mathematical description. Now we extend (1) to a general model with single user group and n links, each of which has a transmission delay τ k , capacity C k , buffer size B k , marking parameters b k,1 , b k,2 , P k,max and queue size q k (t), k = 1, 2, · · · , n. The topology of the network is then shown in Fig. 2.
For any particular moment, the user will choose only one of the n links to send The topology of connection with single user group and n links where W (t) is the averaged window size of the user group and L i represents the ith link, i = 1, 2, · · · , n.
data packets, according to the shortest path principle. More specifically, if the RTT of the route by passing through the ith link is the minimum among all the links, then the user chooses the ith link to form a route [22]. Namely, at the moment t =t, the condition for selecting the ith route by the user as the actual route is R(q i (t)) = min {R(q k (t))|k = 1, 2, · · · , n}.
Since the data packets from the source will no longer be accumulated on the other links, we have where k = i. Then the dynamics of the window size of the user and the queue size are governed by where with here R(q i (t)) = τ i + q i (t)/C i and The purpose of designing the dynamic routing algorithm is to provide the users with more available resources to improve the QoS (quality of service). Thus it is important to show the performance of the system when it has accesses to all the available links. For this, we combine all the possibilities of the route selection together and consider the following system of differential equations When n = 2, the system becomes where else.

4.
The sliding motion of (13). Following the result given in Theorem 2, we know under certain conditions, there exist two desirable equilibrium points in (13), namely, X 1 * = (W 1 * , q 1 * 1 , q 1 * 2 ) T = (W 1 * , q 1 * 1 , 0) T and X 2 * = (W 2 * , q 2 * 1 , q 2 * 2 ) T = (W 2 * , 0, q 2 * 2 ) T . This makes sense because each single link is supposed to work well separately to be considered as a potential resource. In addition, there is a possibility that the two links are utilized simultaneously, as shown in Fig. 3, where the motion of (13) is governed by the joint contribution of F 1 (X(t)) and F 2 (X(t)) by noticing that both q 1 (t) and q 2 (t) are not zero in the long time evolution. This indicates  From the time history of q 1 (t) and q 2 (t), it is clear that the links are used simultaneously since q 1 (t) and q 2 (t) are not zero at any moment. This suggests that the two vector fields may be combined in some manner.
that both of the links are utilized which implies R(q 1 (t) = R(q 2 (t)) is satisfied. This observation suggests that for this case, the motion of X(t) is actually restricted to the hyperplane defined by Σ = {(W, On Σ, the motion of the system is determined by neither F 1 (X(t)) nor F 2 (X(t)) but some combination of them. Rewrite the system as follows where K(X) is the vector field that governs the evolution of the system in Σ and is to be determined. In the following, we will focus our attention on the study of the dynamics around Σ. For X ∈ Σ, let where ·, · is the standard scalar product, H X (X) represents the normal vector of Σ which is According to the Filippov convex method [9,16], the crossing region Σ c ⊂ Σ and sliding region Σ s ⊂ Σ are defined as Σ c ={X ∈ Σ|σ 1 (X)σ 2 (X) > 0}, Σ s ={X ∈ Σ|σ 1 (X)σ 2 (X) < 0}.
Then the dynamics of the system restricted to Σ s is called sliding motion. It is clear that on both sides of Σ s , the orbits are attracted to or repelled from Σ s . Indeed, if σ 1 (X) > 0, σ 2 (X) < 0, the sliding region is stable in the normal direction of Σ, while σ 1 (X) < 0, σ 2 (X) > 0, the sliding region is unstable. At the points in Σ c , the orbits of the system cross Σ because the vector fields on both sides are pointing in the same direction. To find the exact expression of K(X), we employ the Filippov construction [9] which associates K(X) with the combination of F 1 (X) and F 2 (X). Namely, for any X ∈ Σ, let K(X) = λF 1 (X) + (1 − λ)F 2 (X).
When the vector field K(X) is perpendicular to the normal of Σ, that is, we can obtain a specific coefficient λ so that the evolution of the system can be restricted to Σ,λ Noticing thatλ = 1 1−σ1(X)/σ2(X) , when σ 1 (X)σ 2 (X) < 0 ⇒ 0 <λ < 1 and σ 1 (X)σ 2 (X) > 0 ⇒ 1 <λ orλ < 0. Thus we have 4.1. The existence of the pseudo-equilibrium of (14). Particularly, we are curious about whether there will be steady state in (14) for which both of the links are utilized. This corresponds a special type of sliding motion, namely, the equilibrium on the sliding region.
Definition 2.( [16]) If there exist 0 < λ < 1 andX * ∈ Σ such that K(X * ) = 0 where K(.) is given by (15), thenX * is a pseudo-equilibrium of (14). It is reasonable to assume that at the pseudo-equilibrium, the queue size of the two links is neither too big nor too small. According to the physical interpretation of the model, if the queue size of the ith linkis is so small that P i (.) = 0, then it is clear that the network resources are not fully utilized and consequently the utility of the user is not to be maximized. In turn, if the queue size is big enough to yield P i (.) = 1, then the system load is too heavy and consequently the network is at high risk of congestion. In either case, the steady state is not well designed and even if the stability of the pseudo-equilibrium is guaranteed, the network system can not provide the best service.
As we already know that the pseudo-equilibriumX * Parallel to the result in Theorem 2, we have and m = min{ then (14) with (15) has a desirable pseudo-equilibrium.

SHU ZHANG AND YUAN YUAN
Substituting (18) to (17)(a), then solving (17) is reduced to finding the root for L 2 (W ) = 0. Obviously, L 2 (W ) is a continuous and decreasing function for W ∈ (M, m). From L 2 (M ) > 0 and L 2 (m) < 0, we claim that L 2 (W ) = 0 must have a unique root in (M, m). Therefore, b i,1 B i < q * i < b i,2 B i for i = 1, 2, which implies that the pseudo-equilibrium is desirable.

4.2.
The attractivity of the sliding region of (14). After proving the existence of the pseudo-equilibrium, we now study the attractivity of the sliding region of (14). On the discontinuous boundary Σ, from τ 1 + q1 C1 = τ 2 + q2 C2 and 0 ≤ q i ≤ B i for i = 1, 2, it is easy to see that In fact, we have Lemma 2. If Then there exists the sliding region on Σ in (14). It should be noticed that when τ 1 + B1 C1 = τ 2 or τ 2 + B2 C2 = τ 1 , Σ contains only the point (q 1 , q 2 ) T = (B 1 , 0) T or (q 1 , q 2 ) T = (0, B 2 ) T . This implies that the switch between the two links is inactivated and τ 1 + B1 C1 = τ 2 or τ 2 + B2 C2 = τ 1 can not guarantee the existence of the sliding region. Thus, in the following, we assume τ 1 + B1 C1 > τ 2 and τ 2 + B2 C2 > τ 1 in Σ. In most of the real-world situations, there is a link with highest priority among all the accessible links. In other words, the propagation delay of this link is the smallest among all the links, so it is reasonable to let τ 1 = τ 2 . Let Then based on Lemma 2, we have the following result.
Therefore, σ 1 (X)σ 2 (X) < 0 and Σ s is locally attractive. For a pseudo-equilibrium to be stable, the following three conditions should be satisfied [16]: a) There exists such pseudo-equilibrium; b) The sliding region is attractive; c) The pseudo-equilibrium is stable in Σ s . From the results given in Theorems 3 and 5, we know that the sliding region of (14) is locally attractive and under certain conditions, the system possesses a pseudoequilibrium. Although the theoretical analysis for the stability of the pseudoequilibrium in Σ s is out of the scope of this article, we can carry out the numerical investigation on the original switch system (13) when we choose τ 1 = 0.11, τ 2 = 0.13, N = 10, C 1 = 200, C 2 = 150, B 1 = 50, b 1,1 = b 2,1 = 0.2, b 1,2 = b 2,2 = 0.95, P 1,max = P 2,max = 0.4 and B 2 = 30 in Fig. 4 and B 2 = 15 in Fig. 5. For both groups of parameters, the conditions in Theorems 3 and 5 are satisfied, implying the pseudo-equilibrium does exist and the sliding region is locally attractive. As shown in Fig. 4(a), for B 2 = 30, all the eigenvalues of the linearized system around the pseudo-equilibrium of the Filippov vector field have negative real parts, indicating the pseudo-equilibrium is stable in Σ s . When we reduce the value of B 2 to B 2 = 15 ( Fig. 5(a)), one pair of eigenvalues crosses the imaginary axis and therefore the pseudo-equilibrium becomes unstable, as a consequence of a bifurcation similar to the Hopf bifurcation. Even though the sliding region is locally attractive, the system is oscillatory in Σ, as shown in Figs. 5(b)-(d), which is different from the dynamical system without switch.

5.
Conclusion and discussion. In this paper, we consider some basic problems for a TCP/RED congestion control model, which is described by a state-dependent delayed system with discontinuous right-hand side function. We prove that all the solutions of the system are bounded, and under some conditions on the parameters, there exists a unique positive equilibrium in the system. For a TCP/RED network system with a single user group and two optional links, we have shown that the sliding motion always exists and is locally stable in a relatively simple framework. This result deserves particular attention. The sliding motion can be realized only under the Filippov vector field constructed by convexly combining the individual vector fields with coefficients selected from (0, 1). However, in real-world applications, the switch of the system is represented by Boolean variables which jump between 0  and simulation (b)-(e) by XPP-AUT [8] for (13) as B 2 = 30. Initial conditions: W (0) = 6, q 1 (0) = 11, q 2 (0) = 7. (a) shows the distribution of the real and imaginary parts of the eigenvalues oḟ X(t) = K(X(t)), λ =λ given by (16). The real parts of all the eigenvalues are negative and consequently the pseudo-equilibrium is stable in Σ s which is confirmed by the time history plots (b), (c) and (d). (e) shows that the dynamics of the system is restricted to Σ, in other words, Σ s is locally attractive. shows the distribution of the eigenvalues ofẊ(t) = K(X(t)). The maximum of the real parts of the eigenvalues is positive and consequently the pseudo-equilibrium is unstable in Σ s which is confirmed by the time history plots (b), (c) and (d). (e) shows that the dynamics of the system is restricted to Σ, implying the local attractivity of Σ s . and 1. Therefore, even an equilibrium in the sliding region is stable, the system could be oscillatory. In other words, the resources have been consumed but the stability is not improved remarkably. Some may argue that the continuous switch between individual systems, which is unlikely to be realized in the real-world cases, is a reason for the phenomenon observed in the mathematical model. However, the attractivity of the sliding region implies that the trajectory of the system has a tendency to move towards the sliding region and oscillates, no matter at what period the routing table is updated. This may suggest that it is necessary to modify the current routing switch algorithm.