Neighbor discovery is the procedure in which two wireless devices initiate a first contact. In low power ad-hoc networks, radios are duty-cycled and the latency until a packet meets a reception phase of another device is determined by a random process. Most research considers slotted protocols, in which the points in time for reception are temporally coupled to beacon transmissions. In contrast, many recent protocols, such as ANT ANT+ and Bluetooth Low Energy (BLE) use a slotless, periodic-interval based scheme for neighbor discovery. Here, one device periodically broadcasts packets, whereas the other device periodically listens to the channel. Both periods are independent from each other and drawn over continuous time. Such protocols provide 3 degrees of freedom (viz., the intervals for advertising and scanning and the duration of each scan phase). Though billions of existing BLE devices rely on these protocols, neither their expected latencies nor beneficial configurations with good latency-duty-cycle relations are known. Parametrizations for the participating devices are usually determined based on a “good guess”. In this paper, we, for the first time, present a mathematical theory which can compute the neighbor discovery latencies for all possible parametrizations. Further, our theory shows that upper bounds on the latency can be guaranteed for all parametrizations, except for a finite number of singularities. Therefore, slotless, periodic interval-based protocols can be used in applications with deterministic latency demands, which have been reserved for slotted protocols until now. Our proposed theory can be used for analyzing the neighbor discovery latencies, for tweaking protocol parameters and for developing new protocols.
In this section, we provide an overview on existing models for periodic interval-based, slotless protocols, such as BLE. As already mentioned, exact models only exist for the trivial case of @math . This implies that the duration of one scan window must be larger than the time between two advertising packets. Therefore, every scan attempt is successful and the discovery latency is limited to roughly @math . The first known model (for @math ) has been presented along with the introduction thew STEM-B protocol @cite_17 , which is one of the first PI-based protocols that have been proposed. In @cite_26 @cite_18 and @cite_32 , this solution has been adopted to the BLE protocol to account for multiple channels, but the limitation to @math remains.
@cite_18 @cite_26 @cite_32 @cite_17
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We study the problem of waking up a collection of @math processors connected by a multihop ad hoc ratio network with unknown topology, no access to a global clock, and no collision detection mechanism available. Each node in the network either wakes up spontaneously or gets activated by receiving a wake-up signal from another node. All active nodes transmit the wake-up signals according to a given protocol @math . The running time of @math is the number of steps counted from the first spontaneous wake-up until all nodes become activated. We provide two protocols for this problem. The first one is a deterministic protocol with running time @math . Our protocol is based on a novel concept of a shift-tolerant selector to which we refer as a (radio) synchronizer. The second protocol is randomized, and its expected running time is @math , where @math is the diameter of the network. Subsequently we show how to employ our wake-up protocols to solve two other communication primitives: leader election and clock synchronization. We study a noiseless broadcast link serving @math users whose requests arise from a library of @math files. Every user is equipped with a cache of size @math files each. It has been shown that by splitting all the files into packets and placing individual packets in a random independent manner across all the caches prior to any transmission, at most @math file transmissions are required for any set of demands from the library. The achievable delivery scheme involves linearly combining packets of different files following a greedy clique cover solution to the underlying index coding problem. This remarkable multiplicative gain of random placement and coded delivery has been established in the asymptotic regime when the number of packets per file @math scales to infinity. The asymptotic coding gain obtained is roughly @math . In this paper, we initiate the finite-length analysis of random caching schemes when the number of packets @math is a function of the system parameters @math , and @math . Specifically, we show that the existing random placement and clique cover delivery schemes that achieve optimality in the asymptotic regime can have at most a multiplicative gain of 2 even if the number of packets is exponential in the asymptotic gain @math . Furthermore, for any clique cover-based coded delivery and a large class of random placement schemes that include the existing ones, we show that the number of packets required to get a multiplicative gain of @math is at least @math . We design a new random placement and an efficient clique cover-based delivery scheme that achieves this lower bound approximately. We also provide tight concentration results that show that the average (over the random placement involved) number of transmissions concentrates very well requiring only a polynomial number of packets in the rest of the system parameters. We consider distributed plurality consensus in a complete graph of size @math with @math initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter @math . Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when @math . If @math for some small @math , we show that it converges to the initial plurality opinion within @math rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is @math . On the other side, we show that there are cases in which @math rounds are needed, w.h.p. One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least @math times that of the second-most frequent one and @math , then our protocol achieves the best possible run time of @math , w.h.p. We relax full synchronicity by allowing @math nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol. We consider a single cell wireless uplink in which randomly arriving devices transmit their payload to a receiver. Given SNR per user, payload size per device, a fixed latency constraint T, total available bandwidth W, i.e., total symbol resources is given by N = TW. The total bandwidth W is evenly partitioned into B bins. Each time slot of duration T is split into a maximum number of retransmission attempts M. Hence, the N resources are partitioned into N MB resources each bin per retransmission. We characterize the maximum average rate or number of Poisson arrivals that can successfully complete the random access procedure such that the probability of outage is sufficiently small. We analyze the proposed setting for i) noise-limited regime and ii) interference-limited regime. We show that in the noise-limited regime the devices share the resources, and in the interference-limited regime, the resources split such that devices do not experience any interference. We then incorporate Rayleigh fading to model the channel power gain distribution. Although the variability of the channel causes a drop in the number of arrivals that can successfully complete the random access phase, similar scaling results extend to the Rayleigh fading case.