Energy efficient adversarial routing in shared channels

We investigate routing on networks modeled as multiple access channels, when packets are injected continually. An energy cap is a component of the system, understood as a bound on the number of stations that can be switched on simultaneously. Each packet is injected into some station and needs to be delivered to its destination station via the channel. A station has to be switched on in order to receive a packet when it is heard on the channel. Each station manages when it is switched on and off by way of a programmable wake-up mechanism, which is scheduled by a routing algorithm. Packet injection is governed by adversarial models that determine upper bounds on injection rates and burstiness. We develop deterministic distributed routing algorithms and assess their performance in the worst-case sense. An algorithm knows the number of stations but does not know the adversary. One of the algorithms maintains bounded queues for the maximum injection rate 1 subject only to the energy cap 3. This energy cap is provably optimal, in that obtaining the same throughput with the energy cap 2 is impossible. We give algorithms subject to the minimum energy cap 2 that have latency polynomial in the total number of stations n for each fixed adversary of injection rate less than 1. An algorithm is k-energy-oblivious if at most k stations are switched on in a round and for each station the rounds when it will be switched on are determined in advance. We give a k-energy-oblivious algorithm that has packet delay O(n) for adversaries of injection rates less than _n^k−₋¹₁, and show that there is no k-energy-oblivious stable algorithm against adversaries with injection rates greater than _n^k . An algorithm routes directly when each packet makes only one hop from the station into which it is injected straight to its destination. We give a k-energy-oblivious algorithm routing directly, which has latency Oⁿ_k² for adversaries of sufficiently small injection rates that are O _n^k2₂ . We develop a k-energy-oblivious algorithm routing directly, which is stable for injection rate _n^k(_(n^k−₋¹₁⁾₎, and show that no k-energy-oblivious algorithm routing directly can be stable against adversaries with injection rates greater than _n^k(₍^k_n⁻₋¹₁⁾₎.