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• Flows f(k,i),j are all nonnegative and positive flows are not assigned to nonexistent links.

• Packets that have reached the destination are not injected back into the network.

We can show the following lemma.

Lemma 7.3

If λΛ1 then there exists a stationary randomized algorithm to choose μ(i,k),j(t) independent of the history of the arrivals and departures up to timeslot t, such that

$E\left(\sum _{k\in {\mathit{N}}_{\text{out}\left(i\right)}}{\mu }_{\left(i,k\right),j}\left(t\right)-\sum _{k\in {\mathit{N}}_{\text{in}\left(i\right)}}{\mu }_{\left(k,i\right),j}\left(t\right)\right)>{\lambda }_{i,j}$ (7.20)

(7.20)

for all 1 ≤ iN, and 1 ≤ jJ.

The following is an informal proof. Recall that any link capacity vector in $\text{Co}\left(\mathit{S}\right)$ can be achieved by a stationary randomized schedule that chooses the link ...

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