Restricted overlay routing

Abstract

Overlay routing has been widely used in Internet, mainly because of user demands for security, reliability, bandwidth, etc. Many approaches use overlay routing by deploying certain number of overlay nodes and expose them to users. Thus, users have much more flexibility in specifying their end-to-end paths. But these benefits also bring some troubles, e.g., more serious congestions may happen on some links due to the selfish nature of network. In this paper, we adopt ROR (restricted overlay routing) which will not expose all overlay nodes to users, but provide limited number of them while considering overall performance. Different users may get different overlay nodes, and they can still make their choice according to their own demands. We carry out substantial simulations under various topologies and parameters, we show that we can achieve better performance, i.e., lower Min-max link utilization.

Appendix

1.1 Convergence property of ROR routing

Here we will show the convergence property of ROR routing without a rigorous proof.

The basic assumption is that the background traffic won’t be under violent oscillation. If not, the network will be unpredictable.

Assumption 1

The traffic of a incoming or outgoing flows is very small compared to the total traffic between some source and destination. That is, $\triangle demand(s,t)\rightarrow 0,\forall (s,t)\in V\times V$. This is reasonable for a large ISP network whose total traffic is very large.

Each time a flow enters, $\triangle demand(s,t)>0$, and a flow leaves leads to $\triangle demand(s,t)<0$. As a flow is assigned only one overlay node, such that $o(d_{s,t,v}^{overlay})$ changes with variation of $\triangle o(d_{s,t,v}^{overlay})$ for some v.

Assumption 2

Each flow leaves with equal probability, which is not controlled by operator. It means that, if $\triangle demand(s,t)<0$, then $o(d_{s,t,v}^{overlay})$ changes ($\triangle o(d_{s,t,v}^{overlay})<0$) with probability of $\frac{\triangle o(d_{s,t,v}^{overlay})}{\underset{q\in V}{\sum }\triangle o(d_{s,t,q}^{overlay})}$.

If some flow enters into the network, we assign an overlay node v whose $g(d_{s,t,v}^{overlay})>0$. After that the real traffic vector between s and t becomes:

$$\begin{aligned} (r(d_{s,t,1}^{overlay}),r(d_{s,t,2}^{overlay}),\ldots r(d_{s,t,v}^{overlay})+\triangle demand(s,t),\,\ldots r(d_{s,t,n}^{overlay})); \end{aligned}$$

While the optimal traffic vector becomes:

$$\begin{aligned} (o(d_{s,t,1}^{overlay})+\triangle o_{1},\,o(d_{s,t,2}^{overlay})+\triangle o_{2},\,\ldots o(d_{s,t,n}^{overlay})+\triangle o_{n}), {\text{where}} |\triangle o_{i}|\le \triangle demand(s,t); \end{aligned}$$

Thus the gap between real traffic and optimal traffic is:

$$\begin{aligned}&(g'(d_{s,t,1}^{overlay}),\,g'(d_{s,t,2}^{overlay}),\,\ldots g'(d_{s,t,v}^{overlay}),\,g'(d_{s,t,n}^{overlay}))=\\&(g(d_{s,t,1}^{overlay})+\triangle o_{1},\,g(d_{s,t,2}^{overlay})+\triangle o_{2},\,\ldots g(d_{s,t,v}^{overlay})-\\&\triangle demand(s,t)+\triangle o_{v},\,g(d_{s,t,n}^{overlay})+\triangle o_{n}). \end{aligned}$$

We can get$-2\triangle demand(s,t)\le \triangle o_{v}-\triangle demand(s,t)\le 0$, and as $\triangle demand(s,t)$ is quite small, so $2\triangle demand(s,t)<g(d_{s,t,v}^{overlay})$, that is, $\triangle demand(s,t)-\triangle o_{v}<g(d_{s,t,v}^{overlay})$. Thus we can infer that $0<g(d_{s,t,v}^{overlay})-\triangle demand(s,t)+\triangle o_{v}\le g(d_{s,t,v}^{overlay})$. With $\underset{0<i\le n}{\sum }\triangle o_{i}=\triangle demand(s,t)$, finally, we can get $\underset{q\in V}{\sum }|g'(d_{s,t,q}^{overlay})|\le \underset{q\in V}{\sum }|g(d_{s,t,q}^{overlay})|$, which means ROR routing will draw near to optimal routing when new flow comes in.

If some flow leaves, there is a probability that $\underset{q\in V}{\sum }|g'(d_{s,t,q}^{overlay})|$ will increase or decrease. But remember that $\sum \triangle demand(s,t)=0$, for a long time scale when network is stable. Thus most of the time, ROR routing will get close to optimal routing, while away from optimal routing sometimes. In this way, it will come close to optimal after a long time period.