Here, devices have fixed capacity, but packet sizes are random. So, we still have a capacity constraint for the links, but we accept that it won’t be possible to ensure that we stay within those limits all the time, and seek instead to minimise the probability that the limits are exceeded, while keeping throughput as high as possible.

An important result is Chernoff’s Bound: . The proof is very straightforward: apply Markov’s inequality to the non-negative random variable . So in particular , where , and Cramer’s Theorem asserts that after taking a limit in n on the LHS, equality holds, provided .

We assume that the traffic has the form , where these summands are iid, interpreted as one of the loads used on source j. We have

so

so we want this to hold for large .

We might then choose to restrict attention to

So, when operating near capacity, say with call profile n* on (ie near) the boundary of A, with s* the argmin of the above. Then the tangent plane is , and since A’s complement is convex, it suffices to stay on the ‘correct’ side (ie halfspace) of this tangent plane.

We can rewrite as . Note that this is reasonable since s* is fixed, and we call , the *effective bandwidth*. It is with respect to this average that we are bounding probabilities, hence ‘effective’.

Observe that is increasing by Jensen as for t>1 implies that for t>s, .

In particular,

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