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Where the his, the lis, the fijs, the bys, the cijs, the dis and m are defined such as in equations (5)-(7) and (17).

These equations say that Wi population has multiplicative increase with coefficient of hi, but this growth will be inhibited by two factors: predation by P and competition with Wj.

### 4.1 Illustrative Example

According to the equations (18)-(20), we can write the congestion control regime of the test network of the Fig. 2, in the form of equations (21)-(29).

dW1=W1(1-0.9P1-0.05P2-0.05P3-0.05P4-0.02q-(1.2W1+W2+W3+W4)/50) (21)

dW2 —W2(1-0.05p -0.9P2 -0.05P -0.05P4 -0.02q-(W1 + 1.2W2+W3+W4)/50) (23)

dW3—W3(1-0.05P1 - 0.05P2 -0.9P3 - 0.05P4 - 0.02q- (W1+W2+L2W3+W4)/50) (25)

dW4=W4(1-0.05P1-0.05P2-0.05P3-0.9P4-0.02q-(W1+W2+W3+L2W4)/50) (27)

In this case, again Pis are the marked packet counts in the flow i, q is length of queue in the congested router, and Wi is congestion window size for source i. As mentioned in the previous section, in order to establish a stably operated and fairly shared network the effect of self-inhibitive action must be larger than inhibitive action by others. Hence, in Equations (25-33) any bti and cti are several times (in this example 18 times) larger than other bij and cij respectively.

In order to simulate the test network and assess its behavior, we solve the equations (21)-(29) numerically, again using Matlab 7.1. We use the following initial state in which each source has different window size.

P1(0)= P2(0)= Ps(0)= P4(0)=0.1, q(0)=1, W(0)= 1, W2(0)=2, W3(0)=4, W4(0)=6

Fig. 9 illustrates the behavior of the sources sharing the bottleneck link. It shows the time curves of the congestion window size for any source. After a transient phase, all of the Wis converge to a constant value. Fig. 10 shows the evolution of marked packet counts in the congested router. The throughputs of bottleneck link, has been given in Fig. 11.a. This throughput refers to aggregate loads of all of the sources in the bottleneck link (W1 + W2+W3+W4). In Fig. 11 .b we can find the mn MVidinv (Il : - ■

Jbe=

o 30 40 GO

a. Evolution of Wj

a. Evolution of Wj

a. Evolution of W3

C4r*fi«*tlen Window n' Siuwn 4

C4r*fi«*tlen Window n' Siuwn 4

a. Evolution of W3

a. Evolution of W4

Fig. 9. The sources behavior

a. Evolution of Pj

b. Evolution of P2

a. Evolution of Pj b. Evolution of P2

Fig. 10. Marked packets counts trace queue size of congested router. This queue size achieved through the dq = w1 + w2 + w3 + w4 - min(50, q + w1 + w2 + w3 + w4). In order to reference to the results of Figs. 9-12, we note that:

1. As we can see in Fig.s 9, 10 and 12, the source rates and the link prices (marking probability) track more stable behavior in compare to the predator-prey approach. In their steady state, there is no oscillation, and the speed of convergence to this steady state is more accelerated than predator-prey approach.

Tabel 2. Average throughput and packets mark rate of Hybrid model

Source 1 Source 2 Source 3

Source 4

Average throughput (pkt/RTT) Average marked packets count (pkt/RTT)

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