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how can one be sure Salvatorin isn't an echo of ghost packets slowly being received from the mid 90's???

 

Let us consider all paths to be partitioned into sets ?? ? ?? ? ? ? ?, with ?? containing all paths ? with ???

??? ??? ?? ? ?

????? ? ??

? ??. Our protocol works in rounds, starting with round 1. For every ? ? ?, the ?th round has the task

to route all packets with paths in ?? to their destinations. In particular, we show that

(1) round ? takes ???

? ?

?

? time steps, and

(2) any participating packet having a path in ??? with ??

? ? successfully reaches its destination with a

probability of at least ?

? ??

?? ?

??? ?

.

This means that

? the expected time to route a packet with path in ?? is bounded by

?

????

???

? ?

?

? ? ?

?

?????

???

? ?

?

?

? ??

?? ?

??? ?

? ???

? ?

?

? ? ?

?

???

?

??

? ? ?? ? ? ??

? ???? ? ??

? ?????? ? ?? ?? ?

and

? a deviation of an additive ? from this bound occurs with a probability of at most ??

?? ?

??? ??? ??

.

Thus it remains to prove items (1) and (2) above. Let us consider a fixed round ?

? ?. Let ? ?

?

??? . Every

edge is assumed to have a bandwidth ? of ? ??? ? , i. e., every edge can forward ? packets at every time step.

(Obviously, each of these time steps can be simulated in ? ??? ? time steps by edges with bandwidth 1.) Apart

from the usual packets (called real packets in the following), so-called ghost packets are generated at each node.

In fact, each node generates ? ?

? ??? ? ghost packets for each of its links. Each ghost packet is assigned a delay

Æ

? ???

?

?, chosen independently and uniformly at random (i.u.r.). Ghost packets, however, never really leave

a node. As we will see later, their only purpose is to keep down certain correlations among the real packets.

Furthermore, each real packet ? is followed by a so-called counting packet ?? . The packet ? ? counts the time

steps in which at least ? packets want to cross an edge of the path of ? (this information has to be collected by

the nodes of the network). Let us now describe how round ? works. In the following, ??? for some integer ?

??

denotes the set

??? ? ? ? ? ??, and ??? ?? for some integers ? ? ? denotes the set ??? ? ? ? ? ??.

 

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