Complex Dynamics in Communication Networks (Understanding by Ljupco Kocarev, Gabor Vattay

By Ljupco Kocarev, Gabor Vattay

Laptop and verbal exchange networks are between society's most vital infrastructures. the net, specifically, is a big international community of networks with vital keep an eye on or management. it's a paradigm of a fancy procedure, the place complexity may possibly come up from diverse assets: topological constitution, community evolution, connection and node variety, and /or dynamical evolution. this can be the 1st booklet solely dedicated to the hot and rising box of nonlinear dynamics of TCP/IP networks. It addresses either scientists and engineers operating within the common box of conversation networks.

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1 and assume that the system has evolved long enough in time to have the form immediately after a reset L (1 − r)l δ0 (x − lΔt) + m ˜ n (r, L)δ0 (x − D) ρ(x, t) = r l=0 where the mass m ˜ n (r, L) at the rate bound x = D is n m ˜ n (r, L) = r l=L+1 (1 − r)l + (1 − r)n+1 ∞ 0 ρ(x, 0) dx. Lemma 2. If we jitter the duration of the ith congestion epoch by i /N for i a uniformly distributed value in [− , ], then the source rates change by max{(1 − r) i /N, r i /N }. Proof. The reader is asked to verify that if Δti = Δt + i /N and Δti+1 = Δt − (1 − r) i /N , then before the (i + 1)st reset, the first rate population is at rate Δt−(1−r) i /N while all the others are at rate kΔt+ i /N −(1−r) i /N = Dynamics of Congestion Control 29 kΔt + r i /N .

Furthermore, the initial conditions x0 and y0 affect only the rate of convergence of Δtn . Next we show that if the source with the smaller rate is always reset, the system never synchronizes and tends to a “deadlocked” steady-state where one source sends at rate C and prevents the second from sending at all. This result is a form of chronic congestion and an extreme form of unfairness! Theorem 6. If the source with the smaller rate is always reset when the aggregate rate exceeds capacity C, then the aggregate rate tends to the capacity C as t → ∞ with the larger source sending at rate C and the smaller source contirbuting nothing to the total rate.

But it seems that randomized drop-policies such as RED do little to reduce the periodic behavior. On the contrary, both our model and simulations suggest that RED makes the periodic behavior more pronounced, more stable, converge faster, and cause lower overall link utilization. On one hand it is of value to know that a simple analytical model 46 Anna C. Gilbert (that captures only the bare characteristics of TCP) correctly predicts periodic behavior. On the other hand, it is disconcerting to discover that periodic behavior arises so easily and is so difficult to avoid.

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