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c. Evolution of W3 d. Evolution of W4

Fig. 5. Behavior of the sources c. Evolution of W3 d. Evolution of W4

Fig. 5. Behavior of the sources

This table shows that in spite of inequality of initial states, all sources enjoy same amount of bandwidth of the bottleneck link and the proposed model satisfies the fairness metric.

Performance-In addition to the prevention of congestion collapse and concerns about fairness, a third reason for a flow to use end-to-end congestion control can be to optimize its own performance regarding throughput, utilization, and loss. Throughput can be measured as a router-based metric of aggregate link throughput, as a flow-based

Fig. 6. Marked packets counts trace

metric of per-connection transfer times, and as user-based metrics of utility functions. It is a clear goal of most congestion control mechanisms to maximize throughput, subject to application demand and to the constraints of the other metrics.

a. Evolution of aggregated traffic of the sources on the link a. Evolution of aggregated traffic of the sources on the link

b. Evolution of queue length in congested router (q) Fig. 7. Aggregated traffic and queue trace

In most occasions, it might be sufficient to consider the aggregated throughput only. In this example, simulation results show that the aggregated traffic is 41.7348 pkt/RTT. This throughput refers over 80% utilization on the bottleneck link. According to the report of AT&T research group, utilization levels in Internet links are routinely around 70% during the peak hours [36]. So, the achieved utilization in this model is good enough. Fig. 7.b, on the other hand, shows that if we set the queue capacity of the congested router around 20 packets then we roughly wouldn't have any packet loss. The small size of queue length, also, leads to low jitter and low delay for those connections that share this queue capacity.

Stability-In fact, instability can cause three problems. First, it increases jitters in source rate and delay and can be detrimental to some applications. Second, it subjects short-duration connections, which are typically delay and loss sensitive, to unnecessary delay and loss. Finally, it can lead to under-utilization of network links if queues jump between empty and full [4].

From the differential equation viewpoint, one usually looks at a steady state point and wants to know what happens if one starts close to it. A fixed point x0 of f(x) is called stable if for any given neighborhood U(x0) there exists another neighborhood V (x0) c U(x0) such that any solution starting in V (x0) remains in U(x0) for all t > 0. A stability analysis about the steady state is equivalent to the phase plane analysis. Typically, analysis of linear stability can be carried out using community matrix (in ecological context) and its eigenvalues examination [31], but in order to simplify we

a. Closed (Wi,Pi) phase plane b. Closed (W2,P2) phase plane
Fig. 8. Closed phase plane trajectory for (Wi, Pi)

use the numerical solution of equations (8)-( 16) and use phase-plane approach for stability discussion. Fig. 8 shows the closed (Wi, Pi) phase plane trajectories.

Let us examine a little more carefully the results given in Fig. 8. First, note that the direction of time arrows in Fig. 8 is clockwise. This is reflected in Fig. 5 and Fig. 6. Thus all of the trajectories converge roughly around the point (10.4,1.0), and so have a plausible stability near this point. This stability is similar to spiral stability in the context of ordinary differential equations [32].

Fig. 7.a, on the other hand, shows an interesting behavior of the aggregated traffic. This figure. shows that the aggregated traffic has decreasing oscillation level and converges to 43 pkt/RTT. Convergence of aggregated traffic is more obvious than any individual source rate in Fig. 5. This means that because of a global coordination between the sources, the overall behavior of the network has more stability than any of the sources. Fig. 7.b shows that the queue length of congested router approaches to less than 5 packets. Hence, not only the evolution of queue length has a stable regime but also short queue length leads to decrement of delay and jitter in the source rates.

4 Hybrid Approach: Congestion Control Using Combination of Competition and Predator-Prey Models

In order to define a complex system precisely, we should consider all of the involved processes and relation among them. By this motivation, we consider the competition effects between the network users and develop a new equation that addresses another perspective of congestion control problem. Essentially the network users are competitive in the sense that they want to dominate network resources in order to maximize the individual's QoS (Quality of Service) during its communication. In analogy it can be said that "all of the w, species that share the bottleneck link incorporate in inter- specie and intra-specie competition in order to maximize their own share of link bandwidth". Using equations (3)-(4), this competition can be described by the equation (17).

Where the his are the linear birth rates and the lis are the carrying capacities. The fjS measure the competitive effect of Wj on Wi.

In order to develop a hybrid mathematical model that include both predation and competition effects we combine the equations (5), (6), (7) and (17). This integration can be described by equation (18)-(20).

dt dq dt

0 0

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