E eW

1 Active Queue Management.

Whereas the parameters are defined by:

ai is the growth rate of Wi in the absence of P and q. bji is the decrement rate per encounter of Wi due to Pj. r is the decrement rate per encounter of Wi due to q. di is the decrement rate of Pi in the absence of W, that we set it to 0.8B/k. cij is the efficiency of turning predated Wj into Pi. m is set to Min(B, q+Z Wi ). ej is the efficiency of turning predated Wj into q that we set it to 1.

In the network context Pis are rate mismatch i.e. difference between input rate and target capacity dt and q refers to buffer occupancy. The sources use equations (5) to computes its own congestion window size (TCP) and routers use equations (6)-(7) to adjust their Pis and q based on aggregated rates.

3.1 Illustrative Example

In this section, we illustrate the proposed model through of an example. We use a four-connection network, as given in Fig. 2. The network has a single bottleneck link of capacity 50 pkts/RTT, shared by 4 sources. All other links have bandwidth of 100 pkt/RTT. All flows are long-lived, and sources always are active It is expected that the proposed method is applicable to solve networks that are more complicated.

According to the proposed model in (5)-(7), we can write the congestion control regime of the test network in the form of equations (8)-(16). According to mathematical biology, 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 [33]. Hence, in equations (8)-(16) any bii and ca are several times (in this example 18 times) larger than other bij and cij respectively.

dW

= wj(1-0.9p -0.05P2 -0.0513 -0.05P4 -0.02q)

(8)

dP

=pj(0.9W + 0.05W + 0.05W + 0.05W4 -10)

(9)

dW2

= W2(1-0.05p -0.9P2 -0.0513 -0.05P4 -0.02q)

(10)

dP2

= P2(0.1W + 0.9W2 + 0.1W3 + 0.1W4-10)

(11)

dW3

= W3(1-0.05p -0.05P2 -0.9P3 -0.05P4 -0.02q)

(12)

dP3

=p3(0.05W + 0.05W2 + 0.9W3 + 0.05W4 -10)

(13)

dW4

= w4(1-0.05p -0.05p -0.05P3 -0.9P4 -0.02q)

(14)

dP4

=p4(0.05W + 0.05W2 + 0.05W3 + 0.9W4 -10)

(15)

dq

= Wj + W2 + W3 + W4 -min(50, q + W, + W2 + W3 + W4)

(16)

In order to simulate the test network and assess its behavior, we solve the equations (8)-(16) numerically using Matlab 7.1. We use the following initial state in which each source has different window size:

W1(0)= 1, W2(0)=2, W3(0)=4, W4(0)=6, P1(0)= P2(0)= P3(0)= P4(0)=0.1, q(0)=1

Fig. 5 illustrates the behavior of the sources sharing the bottleneck link. It shows the time curves of the congestion windows size. Fig. 6 illustrates the behavior of the congested link and shows the evolution of marked packets counts. The throughputs of bottleneck link, has been given in Fig. 7.a. This throughput refers to the aggregated loads of all the sources in the bottleneck link. In Fig. 7.b we can find the queue size of the congested router. Evaluation of proposed model is done from the following perspectives.

Fairness- There are several ways of defining and reaching fairness in a network, each one leading to a different allocation of link capacities [34, 35]. We are talking here about fairness in the sharing of the bandwidth of the bottleneck link regardless of the volume of resources consumed by a connection on the other links of the network. This kind of fairness is called, in the literature, the max-min fairness. Other types of fairness however exist, where the objective is to share not only the resources at the bottleneck, but also the resources in other parts of the network.

According to the simulation results in Fig.s 5 and 6, the average throughput and packets mark counts for each source can be summarized as in the table 1:

Table 1. Average throughput and packet mark rate of predator-prey mode
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