Appendix B. Scalability

This appendix is dedicated to a mathematical analysis, even if approximative, of the scalability of a mail management model such as the one we describe in this document. Such model is used on a geographically distributed network of services that do not support client redirection (in our case SMTP and POP/IMAP).

In this model, messages (connections) arrive with the same probability on each of the servers available, and are redirected to the final destination server through an internal channel (in our case, a VPN).

The hypothesis of geographical (non local) distribution is that this private channel uses up the same resources as the incoming traffic. It is reasonable to think that this channel has some possibility to compress the traffic that goes through it, and we will note as c the compression factor.

Let us then suppose we have an incoming traffic of i (the figure should be some bandwidth unit) and let us name N the number of servers: if the probability of the message to be redirected is

then for each server the incoming traffic will be and it will generate an outgoing traffic of on the private channel

Each server will receive a traffic of o_s from the other N - 1 servers, so that the total incoming traffic (including the incoming traffic from the private channel) is:

For N tending to infinite, the value of i_tot tends to:

The meaning of this equation is that such a traffic model (i.e. with internal redirection) is linearly scalable with the number of installed server N: in fact reverting the equation we find out that once we fix the total availability of bandwidth of each server, the total bandwidth capacity of the system is directly proportional to N. In the case, for example, of mail traffic, the compression coefficient c can assume very low values, thus minimizing the redirection overhead.

Another way to see the last result is that, considering the bandwidth of each server as generally set (technically and economically), this is split up (in the limit of N very big), on each box, to half traffic for internal redirections and half traffic for incoming external traffic.