Analysis of Practical Backoff Protocols for Contention Resolution with Multiple Servers

摘要

Backoff protocols are probably the most widely used protocols for contention resolution in multiple access channels. In this paper, we analyze the stochastic behavior of backoff protocols for contention resolution among a set of clients and servers. each server being a multiple access channel that deals with contention like an ethernet channel. We use the standard model in which each client generates requests for a given server according to a Bernoulli distribution with a specified mean. The client-server request rate of a system is the maxi- mum over all client--server pairs (i, j) of the sum of all request rates associated with either client i or server j. (Having a subunit client-server request rate is a necessary condition for stability for single-server systems. ) Our main result is that any superlinear polynomial backoff protocol is stable for any multiple--server system with a subunit client--server request rate. Our result is the first proof of stability for any backoff protocol for contention resolution with multiple servers. (The multiple-server problem does not reduce to the single-server problem, because each client can only send a single message at any step. ) Our result is also the first proof that any weakly acknowledgment based protocol is stable for contention resolution with multiple servers and such high request rates. Two special cases of our result are of interest. Hastad, Leighton, and Rogoff have shown that for a single--server system with a subunit client--server request rate any modified super- linear polynomial backoff protocol is stable. These modified backoff protocols are similar to standard backoff protocols but require more random bits to implement. The special case of our result in which there is only one server extends the result of Hastad, Leighton, and Rogoff to standard (practical ) backoff protocols. Finally, our result applies to dynamic routing in optical networks. Specifically, a special case of our result demonstrates that superlinear polynomial backoff protocols are stable for dynamic routing in optical networks.