Scalable wake-up of multi-channel single-hop radio networks

We consider single-hop radio networks with multiple channels as a model of wireless networks. There are n stations connected to b radio channels that do not provide collision detection. A station uses all the channels concurrently and independently. Some k stations may become active spontaneously at arbitrary times. The goal is to wake up the network, which occurs when all the stations hear a successful transmission on some channel. Duration of a waking-up execution is measured starting from the first spontaneous activation. We present a deterministic algorithm that wakes up a network in O(klog1/b klog n) time, where k is unknown. We give a deterministic scalable algorithm for the special case when b> dlog log n, for some constant d>. 1, which wakes up a network in O(kblog nlog (blog n)) time, with k unknown. This algorithm misses time optimality by at most a factor of O(log n(log b+log log n)), because any deterministic algorithm requires Ω(k/b log n/k) time. We give a randomized algorithm that wakes up a network within O(k^1/b ln 1/ε) rounds with a probability that is at least 1 - ε, for any 0 < ε < 1, where k is known. We also consider a model of jamming, in which each channel in any round may be jammed to prevent a successful transmission, which happens with some known parameter probability p, independently across all channels and rounds. For this model, we give two deterministic algorithms for unknown k: one wakes up a network in time O(log^-1 (1/p)k log n log ^1/b k), and the other in time O(log^-1 (1/p)k/b log n log (blog n)) when the inequality b> log (128 blog n) holds, both with probabilities that are at least 1 - 1/poly( n).