Optimal timer based selection schemes

Timer-based mechanisms are often used to help a given (sink) node select the best helper node from among many available nodes. In these, a node transmits a packet when its timer expires. The timer value is a monotone non-increasing function of its local suitability metric, which ensures that the best node is the first to transmit and is selected successfully if no other node´s timer expires within a ´vulnerability´ window after its timer expiry and so long as the sink can hear the available nodes. In this paper, we show that the optimal metric-to-timer mapping that (i) maximizes the probability of successful selection or (ii) minimizes the average selection time subject to a minimum constraint on the probability of success, maps the metric into a set of discrete timer values. We specify, in closed-form, the optimal scheme as a function of the maximum selection duration, the vulnerability window, and the number of nodes. An asymptotic characterization of the optimal scheme turns out to be elegant and insightful. For any probability distribution function of the metric, the optimal scheme is scalable, distributed, and performs much better than the popular inverse metric timer mapping. It even compares favorably with splitting-based selection, when the latter´s feedback overhead is accounted for.