key: cord-0188934-yz8kkqhz
authors: Kindt, Philipp H.; Chakraborty, Samarjit
title: Performance Limits of Neighbor Discovery in Wireless Networks
date: 2021-03-08
journal: nan
DOI: nan
sha: 9db4eaed71fe4a7490296768e4839d633e948c1f
doc_id: 188934
cord_uid: yz8kkqhz

Neighbor Discovery (ND) is the process employed by two wireless devices to discover each other. There are many different ND protocols, both in the scientific literature and also those employed in practice. All ND protocols involve devices sending beacons, and also listening for them. Protocols differ in terms of how the beacon transmissions and reception windows are scheduled, and the device sleeps in between consecutive transmissions and reception windows in order to save energy. A successful discovery constitutes a sending device's beacon overlapping with a receiving device's reception window. The goal of all ND protocols is to minimize the discovery latency. In spite of the ubiquity of ND protocols and active research on this topic for over two decades, the basic question"Given an energy budget, what is the minimum guaranteed ND latency?", however, still remains unanswered. Given the different kinds of protocols that exist, there has also been no standard way of comparing them and their performance. This paper, for the first time, answers the question on the best-achievable ND latency for a given energy budget. We derive discovery latencies for different scenarios, e.g., when both devices have the same energy budgets, and both devices have different energy budgets. We also show that some existing protocols can be parametrized such that they perform optimally. The fact that the parametrizations of some other protocols were optimal was not known before, and can now be established using our technique. Our results are restricted to the case when a few devices discover each other at a time, as is the case in most real-life scenarios. When many devices need to discover each other simultaneously, packet collisions play a dominant role in the discovery latency and how to analyze such scenarios need further study.

Wireless networks that operate without any fixed infrastructure are rapidly growing in importance. Since all devices in such mobile ad-hoc networks (MANETs) run on batteries or rely on intermittently available energy-harvesting sources, the energy spent for communication needs to be as low as possible. Typically, MANET radios are duty-cycled and wake up only for short durations of time for carrying out the necessary communication and then go back to a sleep mode. While such duty-cycled communication schemes are easy to realize when the clocks of all devices are synchronized and their wakeup schedules are known by all participants of the network, asynchronous communication (i.e., communication without synchronized clocks) remains a challenging problem. One of the most important asynchronous procedures is establishing the first contact between different wireless devices, which is referred to as neighbor discovery (ND).

Neighbor Discovery: ND is used by a device for detecting other devices in range. This could be for clock synchronization and establishing a connection, after which more data can be exchanged in a synchronous fashion. Efficient ND is characterized by achieving the shortest possible discovery latency for a given energy budget. Towards this, a large number of ND protocols have been proposed till date, see [2, 4, 6, 7, 9-21, 23, 25-29, 33-35, 37-59] . Among these, e.g., [2, 10, 13, 17, 18, 23, 25, 34, 35, [38] [39] [40] [41] 53] , concern deterministic discovery. Here, given the protocol parameters, an upper bound on the discovery latency can be determined. The problem of pairwise discovery between two devices is of fundamental importance, since in many scenarios, devices join the network gradually and only a master device and the newly joining one carry out the discovery procedure simultaneously. Moreover, the process of discovering multiple devices always relies on pairwise ND.

Over the years, successive ND protocols have improved their discovery latencies for given energy budgets. For example, the Griassdi [25] protocol proposed in 2017 claims to achieve by 87% lower worst-case latencies than Searchlight-Striped [2] that was proposed in 2012. However, despite the significant attention the ND problem has received over the past 15 years, the fundamental question of what is the theoretically lowest possible discovery latency that any ND protocol could guarantee for a given energy budget still remained unanswered. We next describe what a ND protocol is from a technical perspective and how the properties of such a protocol relate to an optimal performance. Protocols for ND: The performance (e.g., worst-case discovery latency, energy consumption, etc.) of the ND procedure is fully determined by the wake-up schedules for transmission and reception of two devices discovering each other, i.e., their sequences of beacons and reception windows. While only very few constraints limit the set of feasible schedules for transmission and reception for technical reasons (e.g., no transmission and reception can be scheduled at the same time), existing protocols for ND reduce this design space considerably. Any protocol for ND is essentially a "construction plan" for creating a set of schedules for transmission and reception, and only schedules allowed by this construction plan can be realized. Moreover, all known protocols provide one or multiple parameters for adjusting the resulting schedules to practical needs, e.g., to the energy budgets of the radios executing the schedules. For example, Bluetooth Low Energy (BLE) requires all beacons and reception windows to be scheduled with periodic intervals, and the lengths of these intervals can be configured when using the protocol. Clearly, the lengths of these intervals impact the energy consumption and discovery latency.

Obviously, the highest performance of a particular ND protocol is achieved for certain, protocol-specific, optimal configurations, and it is often not trivial to identify these Paretopoints. However, even when a specific ND protocol is configured optimally, this does not mean that the resulting performance cannot be superseded by a different ND protocol. In fact, the construction plan the ND protocol inheres might not result in an optimal set of wake-up schedules, leading to a non-optimal performance even when the parameters that lead to the highest performance have been chosen. Similarly, a protocol that actually results into an optimal set of wake-up schedules for some parametrizations does not necessarily perform optimally, when a different, non-optimal parametrization is used. We next discuss the difficulties in assessing the performance of ND protocols.

Performance of ND Protocols: In the absence of a protocol-agnostic bound on the discovery latency, the performance evaluations of different ND protocols have often been very subjective. The results of such evaluations relied on the choice of protocols, their parametrizations and the assumed setups. Hence, while a certain protocol might outperform others in such a comparison, it might perform differently if the parametrization or setup is changed. In addition, most known protocols, e.g., [2, 13, 40] , subdivide time into multiple slots and are hence referred to as slotted. The device sleeps in most slots, whereas some slots are active and used for communication. In each active slot, a device sends a beacon at the beginning and/or end of the slot and listens for incoming beacons in the meantime. Discovery occurs once two active slots overlap in time. Here, performance is quantified in terms of the worst-case number of slots until discovery is guaranteed. Though a certain protocol could perform better than another in terms of the number of slots, such comparisons are heavily dependent on the supported range of slot lengths. As a result, such comparisons in terms of slots and not directly in terms of time are often not meaningful. Moreover, despite slotted protocols having been studied thoroughly in the literature, many protocols that are frequently used in practice, e.g., BLE, do not rely on a slotted paradigm. They schedule reception windows and beacon transmissions with periodic intervals and offer three degrees of freedom that can be configured freely (viz., the periods for reception and transmission, and the length of the reception window). The high practical relevance of such periodic interval (PI)-based protocols is underpinned by the 4.7 billion BLE-devices that were expected to be sold in 2018 [36] . It has recently been shown that the parametrizations for ND in BLE networks proposed by official specifications [30] lead to a performance far from the optimum [23] . This has raised the interest to fully understand such slotless ND procedures. In particular, finding beneficial parametrizations for periodic interval-based protocols has been studied in the literature recently, e.g., in [17, 23, 25] . However, until today, it is neither clear whether the proposed parametrizations are actually optimal, nor how such protocols compare to the slotted ones in terms of performance. In summary, despite the large volume of available literature, it is not possible to meaningfully assess and classify the performance of ND protocols in a purely objective fashion.

This Paper: In this paper, we study the fundamental limits of pairwise, deterministic ND. In particular, we establish a relationship between the optimal discovery latency, channel utilization (and hence beacon collision rate) and duty-cycle. No pairwise ND protocol can achieve lower discovery latencies than the ones established in this paper. The resulting bounds not only give important insights into the design of ND protocols, but will serve as a baseline for more objective performance comparisons. Surprisingly, our analysis shows that some recently proposed protocols actually perform optimally and cover parts of the latency/channel utilization/duty-cycle Pareto front. We show in this paper how to modify such protocols to cover the entire Pareto-front. The optimality results of such protocols were not known until now. In particular, the coverage of the entire Pareto front implies that there is no further potential for improvement. However, there is still potential to improve the robustness against beacon collisions, which might occur frequently when many devices carry out ND simultaneously.

Principle of ND: In general, a radio can either be in a sleep state, listen to the channel or transmit a beacon. Hence, the basic building blocks of a ND protocol are given by these three operations and any ND protocol can be represented as a unique pattern of them. For a higher power-budget, the number of beacons and/or the number or lengths of reception windows can be increased and a discovery procedure is successful once a beacon overlaps with a reception window on another device. Since the design space of all possible reception and transmission patterns allows for an infinite number of possible configurations, determining the optimal pattern and its performance through any form of exhaustive search or numerical method is not possible. Further, as outlined above, most work on ND has focused on slotted protocols and therefore studied only a small part of the design space. As a result, the problem of assessing the optimal performance of ND has so far remained unsolved.

ND Scenarios: For different scenarios, the ND problem appears in different forms, and we provide bounds on the discovery latency for many of them. First, it is obvious that if two devices E and F both have the same beacon and reception patterns, their discovery properties are symmetric. This implies that device E discovers device F with the same worst-case latency for a given duty-cycle as F discovering E. Several publications, e.g., [13, 23, 59] , have studied this special case of symmetric duty-cycles, for which we present a bound on the discovery latency. If both devices run different patterns (for example, due to different duty-cycles), the discovery properties are asymmetric. For the asymmetric case, we provide a bound on the discovery latency when each device is aware of the other device's configuration. The problem of two devices being allowed to modify their patterns autonomously during operation is also relevant. It is currently not clear whether the bounds we present for the asymmetric case can also be achieved when one device does not know the patterns of its opposite one. This question needs further study.

Another important question we answer in this paper is the partitioning of the duty-cycle, which corresponds to the energy-budget of a device. The duty-cycle of a device is the fraction of time it is active. On the other hand, channel utilization is the fraction of time a device occupies the channel, which is between zero and its duty-cycle. Beacon collision rates are solely determined by the channel utilizations of the devices in range. For the case when the channel-utilization (and hence collision rate) is unconstrained, we derive the ratio between transmission and reception times that minimizes the discovery latency.

In the case of many devices discovering each other, the channel utilization of each device has to be constrained for limiting the collision rate. In this paper, we therefore not only derive bounds for the discovery latency that any protocol can guarantee for a given duty-cycle, but also for the case where both duty-cycle and the maximum channel-utilization are provided. To the best of our knowledge, no such protocol-agnostic bounds on discovery latency for the ND problem has been derived until now. In particular, this paper makes the following contributions.

Technical Contributions: We present the following bounds on the discovery latency of deterministic ND protocols.

(1) The lowest discovery latency any symmetric and asymmetric pairwise ND protocol can guarantee for a given duty-cycle and hence energy consumption. Recall that in symmetric ND, all devices operate using the same duty-cycle, whereas in asymmetric ND devices use different duty-cycles. (2) A discovery latency bound for the case where the channel utilization is additionally constrained. (3) Bounds for the following three cases where two devices E and F discover each other.

(a) Only E needs to discover F, whereas F does not need to discover E. (b) Either E discovers F or F discovers E, but both discovering each other is not possible. (c) Both E and F mutually discover each other. We further study the relation between our bounds and previously known ones [34, 35, 58, 59] , which are all limited to slotted protocols. These bounds are given in terms of a worst-case number of slots until discovery is guaranteed, where the discovery latency also depends on the slot length. However, how small a slot length can be is difficult to answer, while it is known that slot lengths cannot be made arbitrarily small. Therefore, the lowest possible discovery latencies of slotted protocols in terms of time have not been derived, which we address in this paper. Finally, while most previous work has focused on slotted protocols, we show that when channel utilization is unconstrained, only slotless protocols can perform optimally, whereas slotted ones cannot. This result is important because in many IoT scenarios devices join gradually and only a pair of devices participate in ND at any point in time. Here, channel utilization is therefore not of concern.

Importance of Performance Bounds for ND: In addition to the theoretical importance and the new insights of our results, they also have many practical applications. Our results help in understanding how to configure existing protocols such as BLE towards low latencies and energy consumption. This will become increasingly important for battery-powered IoT devices, and a large number of existing BLE devices can benefit from increased battery runtimes. Second, our results help in the development of practical protocols that are tailored to a certain application, while providing latencies beyond what is possible using already deployed protocols, e.g., BLE. For example, contact tracing using smartphones or custom wearables has received significant attention in the course of the COVID-19 pandemic in 2020. Here, devices for contact tracing carry out ND continuously as their main mode of operation. For this purpose, protocols need to provide low latencies, low energy consumption, and a high resilience against colliding packets. Our results will provide a baseline on what is the optimal performance when considering only two devices, and what trade-off between discovery latency for two devices and resilience against collisions for large numbers of devices need to be made for operating reliably in all possible situations, e.g., in a crowded subway [22] .

Organization of the paper: The rest of this paper is organized as follows. In Section 2, we present related work on discovery latency bounds of ND protocols. Next, in Section 3, we provide a formal description of a generic ND procedure. Based on this, in Section 4, we derive a list of properties that deterministic ND protocols need to guarantee. Recall that deterministic ND protocols are ones for which bounded discovery latencies can be guaranteed. We derive such latency bounds in Section 5. Finally, in Section 7, we relate the latency bounds of multiple existing ND protocols to the bounds obtained in this paper. We also show how to extend existing protocols, such that every point on the Pareto-front spanned by the worst-case latency, duty-cycle and channel utilization can be reached. Throughout this paper, we make a couple of simplifying assumptions. These assumptions are only for the ease of exposition and are relaxed in Section 6. A table of symbols and additional proofs are given in the appendix.

In this section, we describe existing efforts to determine bounds on the discovery latency that any ND protocol can achieve, and relate them to this paper.

Bounds for Slotted Protocols: As discussed above, the vast majority of ND protocols proposed in the literature follow a slotted paradigm, in which reception and transmission are temporally coupled into slots. A bound on the discovery latency of slotted protocols has been studied in [58, 59] . Here, it has been shown that for guaranteeing discovery within slots, every device needs to have at least = √ active slots. Therefore, if e.g., = 2 out of = 4 slots are active, then discovery can be guaranteed within four slots with a duty-cycle of 50%, whereas if = 4 and = 16, discovery can be guaranteed within 16 slots with a duty-cycle of 25%. Determining the schedule of active slots that realizes this bound relies on cyclic difference sets [58] . Since only a very limited number of such difference sets are known, slotted protocols utilizing this bound can only be realized for a few duty-cycles that correspond to these known difference sets. Subsequently proposed protocols, such as Disco [13] , Searchlight [2] and U-Connect [18] for the same discovery latency require more active slots than defined by this bound. But they are more flexible in terms of duty-cycles they can realize. Other recent work [34, 35] claims to have superseded this bound. By sending an additional beacon outside the slot boundaries in a schedule defined by difference sets, a tighter bound than described in [58, 59] can be reached. Being on slotted protocols, the bounds in [34, 35, 58, 59] are all given in terms of a worst-case number of slots within which discovery is guaranteed. The corresponding bounds in terms of time depend on the slot length . The minimum size of such a slot, among other factors, also depends on the hardware, and cannot be made arbitrarily small. Consequently, no bounds on the discovery latency in terms of time for slotted protocols have been known until now. This issue is addressed later in this paper.

Bounds for PI-based Protocols: Given a tuple of parameter values , , , a method to compute the worst-case discovery latency for PI-based protocols was provided in [24] . However, since there are infinite numbers of possible parametrizations , , , and because of the computation scheme provided in [24] , which parametrization leads to the lowest discovery latency has so far remained unknown. Recently, [23] and [25] proposed parametrization schemes that can compute parameters , , to realize any given duty-cycle. However, the optimality of such parametrizations in terms of discovery latency has not been established.

Generic Approaches: Unlike the work described above that was specific to slotted or PI-based protocols, protocol-agnostic bounds were presented in [3, 5] . In particular, they give an asymptotic latency bound in the form of Θ , where is "the discretized uncertainty period of the clock shift between the two processors" [5] . Hence, this bound depends on the degree of asynchrony between the clocks of a sender and a receiver. First, the asymptotic nature of such a bound is very different from the concrete time bounds that have been pursued within the computer communications community, e.g., [34, 35, 58, 59] , and the ones presented by us in this paper. Second, this community has also focused on bounds that depend on duty-cycle and hence energy budget, which are of direct practical relevance. For these reasons, the bounds from [3, 5] are not comparable to those that have been more commonly pursued, and also to those presented in this paper.

In this section, we formally define the ND procedure and its associated properties. 

., where each reception window starts at time and ends time-units later (see Figure 1 ). A reception window sequence = 1 , 2 , ..., could be of finite or infinite length. In this paper, for simplicity of notation, we refer to such finite length sequences by and infinite length sequences by ∞ .

For the simplicity of exposition, throughout this paper, we always assume that any ∞ is an infinite concatenation of some finite length sequence . For such ∞ , we define = | | (i.e., the number of windows contained in ). Further, we denote the time between the ends of two consecutive instances of as the reception period . It is worth mentioning that all our bounds remain valid also for sequences ∞ that are not given by concatenating the same C, as we show in Appendix A. We assign a time-axis to every instance of . For convenience, which will become clear later, the origin of time in a certain instance of will start at the end of the last reception window of the previous instance, as depicted in Figure 1 . In this figure, consists of three reception windows (i.e., 1 , 2 , 3 ), and three concatenated instances of are shown. For example, the origin of the time-axis for Instance 2 lies at the end of 3 in Instance 1. We denote infinite length beacon sequences ∞ that are given by concatenations of a finite beacon sequence as repetitive beacon sequences. In such repetitive sequences, = | | and the time between the endings of two consecutive instances of is given by . Unlike for reception window sequences, we do not restrict ourselves to repetitive infinite beacon sequences. However, we will prove that all beacon sequences that optimize the relevant metrics of a ND procedure are repetitive when the corresponding reception window sequence is also repetitive.

We indicate an arbitrary shorter sequence ′ to be a part of a longer sequence by using the notation ′ ∈ . For example, in Figure 2 , ′ = 2 , 3 , 4 , 5 , 6 ∈ . Further, the time between the beginnings of beacon and beacon 1 is called the beacon gap . It is = 1 − . Definition 3.3 (ND Protocol): A tuple of an infinite beacon and reception window sequence ∞ , ∞ is called a ND protocol. In this paper, unless explicitly stated, we assume that ∞ and ∞ stem from two different devices E and F. When it is necessary to explicitly specify the device that a sequence is scheduled on, we use the notation ,∞ or ,∞ , where E and F refer to device E or F respectively. We also apply this notation to reception windows and beacons, e.g., ,1 refers to beacon 1 on device E and ,1 refers to reception window 1 on device F.

The most important properties of a ND protocol are its worst-case latency , its duty-cycle , and its channel utilization , as defined next. is the earliest possible time after which an overlap of a beacon from E with a reception window of F is guaranteed, measured from the point in time both devices come into the range of reception. The transmission duty-cycle of a device is the fraction of time it spends for transmission, whereas the reception duty-cycle is the fraction of time spent for reception. In general, depending on the configuration of the radio (e.g., transmit power and receiver gain), transmission incurs a different power consumption than reception. Therefore, the total duty-cycle is given as a weighted sum = , where the weight is the ratio of transmission and reception powers, i.e., = . For a radio running a tuple of sequences ∞ , ∞ , it is:

The transmission duty-cycle is the same as the channel utilization. The duty-cycle directly corresponds to the power consumption of an ideal radio. Non-ideal radios are discussed in Section 6.2.

It follows from the above that the duty-cycle of a tuple of sequences ∞ , ∞ , that are concatenations of finite length sequences and respectively, can be computed as follows.

A beacon needs to be transmitted entirely within a reception window of a receiving device for being received successfully. Each beacon has a certain transmission duration , and if the beacon transmission starts after the last time-units of a reception window (cf. after the start of the hatched area in Figure 3a) ), it cannot be received successfully. Nevertheless, for simplicity of exposition, for now we assume that any overlap between a beacon and a reception window leads to a successful discovery. We further assume that all beacons have the same length and neglect the contribution of the transmission duration of the first successfully received beacon to the worst-case latency. We study the relaxation of these assumptions in Section 6.1 and 6.3.

A device F can successfully discover another device E only if E sends a beacon during one of the reception windows of F. We refer to the other direction as E discovering F. In what follows, we first consider F discovering E only, and later generalize it towards mutual discovery.

On device E, let ′ = 1 , 2 , ... be a subsequence of ∞ . From here on, we will always assume that 1 is the first beacon that is in range of a remote device F. This is because any prior beacons of ∞ , when E is not within the range of F, are not relevant for ND. Further, let F run an infinite reception window sequence ∞ . Though ∞ and hence ′ ∈ ∞ could be of infinite length, let us think of ′ as a fixed-length sequence. This assumption is valid because in the case of a successful discovery, beacons that are sent thereafter are no longer relevant for the discovery procedure. Now recall that the reception windows of ∞ are formed by concatenations of a finite sequence and every instance of has its own time origin, as defined by Definition 3.1 (cf. Figure 1 ). The first beacon 1 in ′ lies within a certain instance of and has a certain (random) offset Φ 1 from the time origin of this instance of . This is depicted in Figure 3b ), which shows an infinite reception window sequence consisting of concatenations of = 1 , 2 , 3 , of which one full instance is depicted. In addition, the figure contains the last reception window 3 of the preceding instance and the first reception window 1 of the succeeding one. Further, three beacons 0 , 1 and 2 are shown, of which only 1 and 2 are in range. Here, ′ consists of 1 , 2 and some later beacons that are not shown in the figure. Beacon 1 falls into the depicted instance of and has an offset of Φ 1 time-units from its origin.

For some valuations of Φ 1 , at least one beacon of ′ will coincide with a reception window of ∞ . For other valuations of Φ 1 , there might be no beacon in ′ that coincides with any reception window of ∞ , irrespective of the length of ′ . If an overlapping pair of a beacon and a reception window exists for all possible offsets Φ 1 , the tuple ′ , ∞ guarantees discovery within a bounded amount of time and hence realizes deterministic ND. We, in the following, formalize the properties that such a tuple ′ , ∞ needs to fulfill for guaranteeing discovery.

A tuple ( ∞ , ′ ), along with Φ 1 , is depicted in Figure 3b ). For a given ( ∞ , ′ ), it is obvious that the offset Φ 1 , which is a measure of the shift between ′ and ∞ , solely determines whether a beacon in ′ overlaps with a reception window in ∞ or not. The time-duration after which such an overlap takes place, and hence the discovery latency, is also determined by Φ 1 . For which values of Φ 1 will beacon 1 fall into one of the reception windows? Clearly, these are given by the set Figure 3b) ). In other words, if Φ 1 lies within any interval belonging to Ω 1 , then 1 is successfully received. Similarly, if Φ 2 is the offset of 2 , then for Φ 2 belonging to any interval in Ω 1 , 2 will be successfully received (see Figure 3b) ). Now, what are the offsets Φ 1 of 1 , such that beacon 2 is successfully received? These are given by the set

where 1 is the time-distance between the beacons 1 and 2 , as already defined in Section 3 (see Figure 3b) ). Therefore, Ω 2 is obtained by shifting all elements of Ω 1 by 1 time-units to the left. Similarly,

Now consider a beacon sequence ′ = 1 , ..., of length . If Φ 1 belongs to any interval in Ω 1 ∪ Ω 2 ∪ ... ∪ Ω , then one beacon from ′ will be successfully received. We now extend this result and define a coverage map, which can be used to reason about valuations of the initial beacon offset Φ 1 that lead to successful discovery.

A coverage map is a formal representation of all offsets Φ 1 for which any beacon in ′ overlaps with a reception window in ∞ . It also allows for a graphical representation, from which several properties of the tuple ′ , ∞ can be easily understood.

Recall that ∞ is a repeated concatenation of a sequence of reception windows (i.e., ∞ = ...). Now, we need to be able to specify specific instances of within ∞ . For this purpose, let us consider a simple example where has two reception windows and , and ∞ is therefore given by ∞ = ..., and in order to distinguish between different instances of these reception windows, we will denote ∞ = 0 0 1 1 2 2 ... . The reception windows and 1 , as well as and 1 , are time-units apart (see Figure 4a ) and also Figure 1 ). Figure 4a ) shows a sequence of beacons ′ = 1 , ... 7 from a transmitting device. Below, two reception windows 0 , 0 from a receiving device are depicted, together with their periodic repetitions 1 , 1 , which are time-units later. Again, 1 ∈ ′ has a certain random offset Φ 1 from the origin of . Figure 4b) shows the coverage map for the sequences in Figure 4a ). Definition 4.1 (Covered): An offset Φ 1 is covered, if at least one beacon in ′ overlaps with any reception window in ∞ for this offset.

Given the parameters of ′ , ∞ , the construction of a coverage map as in Figure 4b ), is straightforward. We believe that the notion of such a coverage map and its use go beyond deriving latency bounds as done in this paper. It would also be useful for analyzing and optimizing various kinds of different ND protocols, including already known ones.

From coverage maps, we can derive the following properties.

• Beacon-to-beacon discovery latency l * : For a given offset Φ 1 , let * Φ 1 be the latency measured from the transmission time of the first beacon that is in range, to the first time a beacon is successfully received. In Figure 4 ,

, where is the smallest row number in which Φ 1 is covered. For example, for an offset Φ 1 slightly above 0 (i.e., an offset within the highlighted frame in Figure 4b )), the beacon-to-beacon discovery latency will be * = 3 − 1 , since 3 is the earliest successful beacon for this offset.

• Determinism: By ensuring that all possible initial offsets are covered by at least one beacon, we can guarantee that ′ is deterministic with respect to ∞ (see next section for a formal definition of determinism).

• Redundancy: For certain valuations of Φ 1 , one can see in Figure 4b ) that a beacon will be received by multiple reception windows. For example, for values of Φ 1 within the shaded frame, beacons 3 and 7 will be received by the windows 1 and 2 , respectively. By integrating over the length of all reception windows, for which such duplicate receptions happen, we can quantify the degree of redundancy of a tuple ′ , ∞ .

Recall that protocols that can guarantee discovery for every possible initial offset are called deterministic. This is formalized below. In particular, we distinguish between a beacon sequence ′ and a protocol ∞ , ∞ that can result in such a sequence.

Hence, deterministic ND protocols ∞ , ∞ always guarantee a bounded discovery latency, no matter when a beacon of ∞ comes within the range of a receiving device. Proof. Let us assume that a certain range

, is covered by a beacon in conjunction with a certain reception window . Since the pattern of reception windows repeats every time-units, any Φ 1 ∈ Φ , Φ will result in being received by the reception window , which is time-units after . □ Definition 4.3 (Redundant Sequences): If any offset Φ 1 within 0, is covered by more than one beacon, then the tuple ′ , ∞ is redundant. Otherwise, ′ , ∞ is disjoint, since no intervals in the corresponding coverage map overlap.

For example, in Figure 4b ), all offsets Φ 1 are covered and hence the corresponding tuple ′ , ∞ is deterministic. Further, since some offsets, e.g., the ones slightly above offset 0 (marked by the highlighted frame in Figure 4b) ) are covered twice, it is also redundant.

For a tuple ′ , ∞ , certain values of Φ 1 might be covered by multiple beacons, other values by exactly one beacon and yet others by no beacons. The notion of coverage quantifies how different values of Φ 1 ∈ 0, are covered. To understand this, recall that Ω is a set of intervals. Let us now consider those (full or partial) intervals of Ω that lie within 0, . The sum of the lengths of all such intervals for all Ω captures a notion of coverage that we formalize below. Definition 4.4 (Coverage): Given a tuple ′ , ∞ , let a certain offset Φ 1 ∈ 0, be covered by beacons, where ∈ {0, 1, 2, ...}. Let us define an auxiliary function Λ * Φ 1 = . Then, the coverage Λ is defined as

For example, in Figure 4b ), if the lengths of and are equal to unity and therefore = 8, then Λ = 14. If Λ < , a tuple ′ , ∞ cannot be deterministic, which implies that for certain values of Φ 1 , no bounded discovery latency can be guaranteed. If Λ = , then ′ , ∞ can either be deterministic and disjoint, or else, it will be redundant and not deterministic. If Λ > , than ′ , ∞ cannot be disjoint, and may or may not be deterministic.

While Λ quantifies the coverage due to all beacons in ′ , we now quantify the coverage induced by individual beacons. Proof. The first beacon 1 in ′ will cover exactly those time-units for which 1 directly coincides with a reception window. The sum of such matching offsets is therefore =1 time-units. Every later beacon will cover the same offsets shifted by the sum of beacon gaps =1 to the left, which does not impact the amount of offsets covered. Since ∞ is an infinite concatenation of a finite sequence , for every covered offset that is shifted out of the considered range 0, , the same amount from a later period is shifted into that range, such that each beacon covers exactly =1 time-units within 0, . □ From the above, we are able to derive a minimum length of ′ . Theorem 4.3 (Beaconing Theorem): Given a tuple ′ , ∞ , the minimum number of beacons a beacon sequence ′ needs to consist of to guarantee deterministic discovery is:

Proof. From Theorem 4.2 it follows that every beacon induces a coverage of Λ = =1 . For deterministic discovery, the coverage Λ has to be at least . Therefore, the number of beacons needed for deterministic ND must be at least ⌈ Λ⌉. □ It is worth mentioning that Theorem 4.3 is a necessary, but not sufficient condition for deterministic ND. The positioning of the beacons, along with their number, together determine whether or not a tuple ′ , ∞ is deterministic.

In this section, we derive the lower bounds on the worst-case latency that a ND protocol could guarantee in different scenarios (e.g., symmetric or asymmetric discovery). In other words, given constraints like the duty-cycle, such a bound defines the best worst-case latency that any protocol could possibly realize. First, we consider the most simple case in which one device F runs an infinite reception window sequence ,∞ without beaconing, whereas another device E only runs an infinite beacon sequence ,∞ without ever listening to the channel. We refer to this as unidirectional beaconing.

Bound. Consider a tuple ′ , ∞ , where ′ consists of beacons and is given by Theorem 4.3. Recall Theorem 4.3 and the subsequent discussion. If ′ is disjoint and deterministic, then for every value of Φ 1 , there is exactly one beacon in ′ that overlaps with a reception window in ∞ . What are the beacon gaps using which such beacons need to be spaced for minimizing the discovery latency?

The worst-case beacon-to-beacon discovery latency * , measured from the first beacon in range to the earliest successfully received one, is given by the sum of the − 1 beacon gaps between these beacons. The first beacon in ′ is the first beacon that was sent when the transmitter came within the range of the receiver. To measure the worst-case discovery latency , time begins when the two devices come in range, which might be earlier than the time the first beacon in ′ was sent. How much earlier? At most by the beacon gap that precedes ′ . Recall that ′ belongs to an infinite sequence ∞ . Hence, the lowest worst-case latency is achieved if the sum of these beacon gaps is minimized. At the same time, all offsets in 0, need to be covered exactly once for ensuring determinism. However, the following arguments rule out such consecutive beacon gaps to be arbitrarily short. ∞ has a transmission duty-cycle , defined by the energy budget of the transmitter. Obviously, determines the average beacon gap . If the sum of a certain consecutive beacon gaps becomes smaller than · , then the sum of a different consecutive beacon gaps within ∞ needs to exceed · in order to respect the average beacon gap of defined by . Since any beacon in ∞ could be the first beacon in range, the beacons with the largest sum of beacon gaps determine the worst-case latency . Hence, in an optimal ∞ , every sum of consecutive beacon gaps must be equal to · . It is worth noting that this requirement does not necessarily require equal beacon gaps, because the above property has to hold for a specific value of given by Theorem 4.3. This is formalized in Lemma 5.2. To illustrate the above, consider the following example. Figure 5 shows a sequence ′ = 1 , ..., 7 . Here, let the minimum number of beacons for deterministic ND be equal to 4 and let the partial sequences 1 , ..., 4 , 2 , ..., 5 , 4 , ..., 7 be deterministic. Consider the sequence 1 , ..., 4 . Let us assume that 4 would be sent somewhat earlier than depicted. Then, by decreasing 3 , the beacon gap 4 would increase accordingly, and though the sequence 1 , ..., 4 would result in a shorter discovery latency for all possible offsets, the sequence 4 , ..., 7 would lead to a larger worst-case latency. The above observations are formalized below. Theorem 5.1 (Coverage Bound): The lowest worst-case latency that can be guaranteed by a tuple ∞ , ∞ is:

Proof. Consider a sequence ′ = 1 , ..., with >> . In ′ , if any sum of consecutive beacon gaps is less than · , then the sum of a different consecutive beacon gaps will exceed · and will define . Since this is true for every , it also holds for ∞ . The mean beacon gap is given by = − 1 −1 and the worst-case latency by = · . Expressing the mean beacon gap by the duty-cycle for transmission (cf. Equation 1) and expanding using Theorem 4.3 leads to Equation 6 . □ Lemma 5.2 (Repetitive Beacon Sequences). Given a repetitive ∞ , every ∞ that guarantees the lowest worst-case latency is repetitive, with a period of = beacons or = · time-units.

We know that in an optimal beacon sequence, the sum of every consecutive beacon gaps is . The corresponding reception window sequence must be such that all offsets in 0, are covered by such a beacon sequence. While there can be multiple such ∞ for a given ∞ , the ones that are optimal must fulfill the following property. Theorem 5.3 (Overlap Theorem): Consider a tuple ∞ , ∞ ), which guarantees a certain worst-case latency . Every ∞ that achieves this latency with the lowest possible reception duty-cycle fulfills the following property.

Proof. Let us assume that the length of is equal to · =1 − Δ, where is an integer and Δ ∈ 0, =1 . Theorem 5.1 implies the same worst-case latency for all values of Δ, since the ceiling function in Equation 6 does not change . With = · =1 − Δ, the reception duty-cycle is given by (cf. Equation 2):

From Equation 8 follows that the reception duty-cycle is minimized when Δ = 0, and hence = · =1 . □ The intuition behind Theorem 5.3 is that if Equation 8 is not satisfied, then can be increased and therefore, the reception duty-cycle can be reduced without requiring any additional beacons to guarantee discovery with the same . In other words, the coverage intrinsically induced if Equation 8 is not satisfied exceeds what is needed for determinism. By combining Theorem 5.1 and 5.3, we can derive a bound for unidirectional beaconing. Theorem 5.4 (Fundamental Bound for Unidirectional Beaconing): Given a device E that runs an infinite beacon sequence ,∞ with a duty-cycle of and a device F that runs an infinite reception window sequence ,∞ with a duty-cycle of , the minimum worst-case latency that can be guaranteed for F discovering E is as follows.

Clearly, optimal values of are of the form 1 , ∈ N and other values of do not lead to an improved compared to them. Proof. By combining = · =1 from Theorem 5.3 and Equation 1, we can write Equation 6 as follows.

This holds true for in the form of 1 , ∈ N. The proof for other duty-cycles follows from the above discussion. □

In this section, we extend Theorem 5.4 towards bidirectional (i.e., device E discovers device F and vice-versa), symmetric (i.e., both devices E and F use the same duty-cycle ) ND. For achieving bidirectional discovery, every device runs both a beacon and a reception window sequence, and we assume that ∞ and ∞ can be designed such that both sequences on the same device never overlap with each other. We relax this assumption in Appendix B.

We can achieve bidirectional discovery by running the optimal sequences ∞ and ∞ we have identified for unidirectional beaconing on both devices simultaneously. The latency of each partial discovery procedure (viz., the discovery of E by F and of F by E) is bounded by Theorem 5.4. As a result, the worst-case latency for both partial discoveries being successful will also be bounded by Theorem 5.4. Since both devices transmit and receive, we can optimize the share between and , which leads to the following bound. Theorem 5.5 (Symmetric Bound for Bi-Directional ND Protocols): For a given duty-cycle , no bi-directional symmetric ND protocol (i.e. every device runs the same tuple ∞ , ∞ ) can guarantee a lower worst-case latency than the following.

Proof. Because of Theorem 5.3, optimal reception duty-cycles are given by 1 = , = 1, 2, 3, .... By inserting = (cf. Definition 3.5) into Equation 9 and setting 1 = , we obtain

We now have to find the value of that minimizes . Let us for now allow non-integer values of in Equation 12 . By forming the first and second derivative of Equation 12 by , one can show that a local minimum of exists for = 2 , which is a non-integer number for most values of . By analyzing , we can further show that Equation 12 is monotonically decreasing for values of < 2 and monotonically increasing for values of > 2 . Hence, the only integer values of that potentially minimize are ⌈ 2 ⌉ and ⌊ 2 ⌋. Inserting = ⌈ 2 ⌉ or = ⌊ 2 ⌋ into Equation 10 and taking the minimum latency among both possibilities leads to Equation 11 . □ In fact, Theorem 5.5 also holds true for unidirectional beaconing, if the joint duty-cycle = · of two devices E and F is to be optimized. Further, one can easily see that for small values of , the floor-and ceiling functios in Equation 11 only marginally affect the value of , which can therefore be approximated by

Even when both devices E and F transmit as well as receive, it is possible to design unidirectional protocols in which only one of the two devices, E or F, can discover the other.

Here, the beacons on both devices contribute to a joint notion of coverage, leading to a reduced latency bound compared to the case where both devices can discover each other mutually. A bound for this possibility is given below. 

In Section 5.1, we have studied unidirectional discovery in the sense that one device F could discover E without E discovering F. However, it is also possible to design the tuple ∞ , ∞ on each device such that either device E or F can directly discover its opposite, which we study in this section. This form of unidirectional discovery is realized using beacon sequences ∈ ∞ , in which the beacons on one device are sent with a fixed temporal relation to the reception windows on the same device. For example, let beacon ,1 on device F be sent by time-units after reception window ,1 , as depicted in Figure 6 . Further, let such a relation exist on both devices and in every period of the reception window sequence. As previously explained ,1 has a random offset of Φ ,1 time-units from the coordinate origin of device E. The temporal correlation between ∞ and ∞ on every device implies that the offset Φ ,1 beacon ,1 has from the coordinate origin of device F is fully determined by Φ ,1 (cf. Figure 6 ). It is:

By exploiting this temporal relation, a quadruple of sequences ,∞ , ,∞ , ,∞ , ,∞ can guarantee deterministic one-way discovery, even if the pair ,∞ , ,∞ only covers half of the offsets Φ ,1 ∈ 0, , by having the pair ,∞ , ,∞ covering the remaining ones. Thereby, the number of beacons that need to be sent per device for guaranteeing one-way discovery can be halved. The upper part of Figure 7 depicts the beacons (arrows) and reception windows (rectangles) of two devices E and F. On each device, the reception windows and beacons have a fixed temporal relation, whereas beacon ,1 has a random offset Φ ,1 from the coordinate origin of device E. Dashed arrows depict beacons that would need to be sent if every device would have to cover all offsets in the entire period on its own. When exploiting temporal correlations between ∞ and ∞ on the same device, these beacons can be omitted without increasing the one-way worst-case latency. The lower part of Figure 7 depicts the coverage map of the beacons ,1 , ..., ,4 of device F and ,1 , ..., ,4 of device E. This coverage map represents all offsets Φ ,1 , for which either a beacon from device F overlaps with a reception window of device E or a beacon from device E overlaps with a reception window of device F. Covered offsets of omitted beacons have been left white. As can be seen, every possible initial offset Φ ,1 is covered by either a beacon of ,∞ falling into a reception window of ,∞ , or a beacon of ,∞ falling into a reception window of ,∞ , and hence the number of beacons per device is halved compared to direct bi-directional discovery. This leads to the following latency bound, which is lower than the one given by Theorem 5.5. Since there are no further possibilities to improve pairwise discovery, this is also the tightest fundamental bound for all pairwise deterministic ND protocols. Theorem 5.6: The lowest worst-case latency a pair of devices can guarantee for mutual exclusive one-way discovery (i.e., either of both devices can discover its opposite one) is given by:

Proof. For a given set of offsets Ω covered by ∈ ,∞ on device E, Equation 14 defines a set of offsets Ω that are automatically covered by ∈ ,∞ on device , and vice-versa. If Ω and Ω are disjoint, the amount of offsets contained in Ω ∪ Ω must sum up to time-units for guaranteeing one-way discovery. The lowest worst-case latency is achieved if each device provides the same amount of coverage (since otherwise, for some offsets, the device that provides the larger amount of coverage would need to send at least one additional beacon until discovery occurs). Hence, the beacon sequence on every device needs to cover only 1 2 · time-units to guarantee one-way determinism, and Equation 6 becomes:

The rest of this proof is identical to the one for direct symmetric discovery (cf. Theorem 5.5). □ Theorem 5.6 is valid for one-way discovery (i.e., device E discovers device F or vice-versa). An indirect reverse discovery can be realized as follows. Each device transmits its next point in time at which it listens to the channel along with its beacons. The receiving device then schedules an additional beacon at the received point in time. This technique is called mutual assistance [25] , and is actually a form of synchronous connectivity. Here, the latency for two-way discovery will be increased by the maximum temporal distance between any beacon and its succeeding reception window on the same device. An upper bound for this penalty for two-way discovery is time units, which can be reduced significantly in sequences with more than one reception window per period . Mutual assistance comes with two significant drawbacks: 1) The packets lengths are increased for transmitting information on the next reception window, which increases the duty-cycle. 2) If multiple devices simultaneously receive a packet containing a temporal hint on the next reception window of the transmitting device, they will all schedule an additional beacon at the received point in time, which greatly increases the collision rate. Due to the higher practical relevance, we focus on direct protocols in the rest of this paper.

For achieving the bound given by Theorem 5.5, we have assumed that the beacons of multiple devices never collide. This assumption is reasonable for a pair of radios, in which collisions only rarely occur. However, as soon as more than two radios are carrying out the ND procedure simultaneously, collisions become inevitable and some of the discovery attempts fail. As a result, some devices might discover each other after the theoretical worst-case latency has passed, or, depending on the protocol design, might not discover each other at all. Therefore, it is often required to limit the channel utilization and hence collision rate, which leads to an increased worst-case latency bound.

In protocols with disjoint sequences (i.e., every Φ 1 is covered exactly once), every collision will lead to a failure of discovering within . The collision probability is solely determined by the channel utilization . We in this section study the worst-case latency that can be achieved if both and (and hence the collision probability) are given. We in addition discuss possibilities to reduce the number of failed discoveries for a given collision probability in Section 8.1.1.

Consider a number of senders, of which each occupies the channel by a time-fraction of . The first beacon of an additional sender that starts transmitting (or comes into range) at any random point in time will face a collision probability of (cf. [1] ):

Once a beacon has collided, the repetitiveness of infinite beacon sequences (cf. Lemma 5.2) implies that the fraction of later beacons colliding with this device is predefined. Nevertheless, since all offsets between the two sequences occur with the same probability, the collision probability of every individual beacon is given by Equation 17 . When constraining the channel utilization to a maximum value that must never be exceeded, the following latency bound applies. Theorem 5.7 (Bound for Symmetric ND with Constrained Channel Utilization): For a given upper bound on the channel utilization , no symmetric ND protocol can guarantee a lower worst-case latency than the following.

Here, A and B are given by Equation 11 and = 1 ⌈ 2 ⌉, if A ≤ B, and 1 ⌊ 2 ⌋, otherwise. Proof. Given , if the channel utilization that results from choosing the optimal value of (see proof of Theorem 5.5) does not exceed , the bound given by Equation 11 remains unchanged. Otherwise, the bound is obtained from Equation 9 by eliminating using = (cf. Definition 3.5). □

So far, we have assumed that two devices E and F run the same tuple of sequences. Often, different devices have different energy budgets, which can be due to different capacities or states-of-charge of their batteries, different energy harvesting capabilities or different required lifetimes. In such scenarios, ND protocols that allow all devices to have different duty-cycles are required, and hence the sequences on both devices differ. Next, we study the latencies of asymmetric protocols with different sequences on both devices, which allow for configurations with ≠ . We thereby assume that each device knows the tuple of sequences on its opposite device. This scenario is relevant e.g., when connecting a gadget with limited power supply to a smartphone using BLE. Here, different sequences on both devices, which account for their different power budgets, can be specified e.g., through the standardization documents of the service offered by the gadget. The case of every device being allowed to choose its duty-cycle autonomously during runtime, and hence, asymmetric discovery procedures in which devices are unaware of the sequences of remote devices, is also relevant. The possible degradation of the optimal performance for this case needs to be studied in further work.

We first consider tuples of duty-cycles , , for which 2 and 2 are integers. We then extend this towards all duty-cycles. Theorem 5.8 (Simplified Bound for Asymmetric ND): Consider two devices E and F with duty-cycles and , where 2 and 2 are integers. The lowest worst-case latency for two-way discovery is as follows.

Proof. According to Theorem 5.4, if 1 and 1 are integers, the lowest worst-case one-way discovery latency for device F discovering device E and the latency for the reverse direction are as follows.

The global worst-case latency for two-way discovery is given by = , . Because of this, every optimal asymmetric ND protocol must fulfill = , since in cases of e.g., > , one could decrease the reception duty-cycle of device E. . If e.g., 1 exceeds its next-lower integer value, we could decrease and therefore decrease . Because only certain discrete values of and are optimal, the latency functions become discontinuous and hence, = can not always be realized. Finding the tuple of integers that minimizes = min , is not straightforward, since there is an infinite number of integers and the optimal solution cannot be found using analytical methods. We in the following present an algorithm that limits the solution space to a finite number of integers. By iterating through the resulting candidate solutions, the configuration that minimizes can be identified with low computational complexity. Towards this, it is beneficial to re-write Equation 22 as follows.

Here, Δ is the deviation of from 2 and Δ from 2. For values of Δ for which 1 is an integer, in Equation 23 becomes:

Similarly, all values of Δ for which 1 is an integer lie on the curve that is given by:

Optimizing F given E : Let us first assume that the optimal value of Δ was known. Then, = * , since the optimal value of Δ leads to 1 being an integer. For such a given Δ , finding the optimal value of Δ and hence the corresponding worst-case latency works as follows.

Lemma 5.9 (Latency for a given Δ ). Given any value Δ for which 1 is an integer , the worst-case latency is given by min , , where Proof. All values of Δ for which 1 is an integer are given by the following equation Figure 8 depicts , * , and * . Recall that = * , since we consider only values of Δ for which 1 is an integer. Because * shrinks and * grows for increasing values of Δ , the lowest value of min * , * is achieved when * = * . We can solve * = * by Δ and denote the resulting value as Δ 0 . This value, in general, does not lead to 1 being an integer. However, all optimal values of Δ lie on the curve of * Δ . Further, increasing differences |Δ − Δ 0 | lead to larger latencies min , . Hence, only the pair of integer-values of 1 that are neighboring Δ can minimize the latency. All other values of Δ for which 1 is an integer will lead to a larger latency of either or (cf. Figure 8 ). When replacing Δ by Δ from Equation 27 , rounding 1 to the next higher integer results into , rounding it to the next lower integer to . □ Optimizing E : All integer values of 1 are given by Equation 27 . Which integer will minimize the worst-case latency? We in the following discuss finding the value of that minimizes . However, the same procedure also holds true for optimizing . Differentiating Equation 26 is not possible, since it contains a ceiling term. For this reason, we cannot directly identify its minimum by computing = 0. However, by exploiting the relation ≤ ⌈ ⌉ ≤ 1, we can derive a differentiable upper and a lower bound for from Equation 23 . These bounds are as follows. Figure 9 depicts and . It also shows , which lies always in-between. By analyzing the derivative of , one can easily identify the minimum of , which lies at 0 . The value 0 is not necessarily an integer. Clearly, 0 will always lie below 0 (cf. Figure 9) . We now solve = 0 , and obtain the values 1 and 2 , with 1 < 2 . Since ≤ , all integer values that are potentially optimal lie between ⌊ 1 ⌋ and ⌈ 2 ⌉ (cf. Figure 9 ). Note that 1 and 2 always exist, since there is exactly one minimum of and , and < . Further, no other values of can be optimal, since no value of can be smaller than the corresponding value of , and all values of that lie outside of ⌊ 1 ⌋, ⌈ 2 ⌉ always exceed those that lie within ⌊ 1 ⌋, ⌈ 2 ⌉.

With the above, the following scheme is guaranteed to result in the lower bound , for asymmetric discovery within a finite number of computational operations.

(1) Compute 1 and 2 by solving = 0 (2) Compute the minimum latency , by evaluating Equation 26 for all values of ∈ ⌊ ,1 ⌋, ⌈ 2 ⌉.

(3) Repeat Steps 1) and 2) for , which leads to , . (4) The worst-case latency , is given by min , , , . In practice, the required computation time is negligible, allowing for the computation of for large numbers of different duty-cycles within milliseconds on a laptop. Figure 10 exemplifies this bound for asymmetric ND. Here, has been fixed to 0.5, while we sweep through all values of . Values for which the simplified bound from Theorem 5.8 applies are highlighted using a circle. 

In Section 4, for the sake of ease of presentation, we have made multiple simplifying assumptions. In this Section, we relax all assumptions that have an impact on the discovery latency, study how the fundamental bounds are impacted by this and numerically evaluate the difference between the ideal and real bounds. In the appendix, we relax further assumptions that do not directly affect these bounds.

Throughout the paper, we have assumed that also beacons that only partially overlap with a reception window are received successfully. In this section, we relax this assumption.

To account for the fact that beacons cannot be received if their transmissions start within the last time-units of each reception window (since they must entirely overlap with the window), we have to artificially shorten the actual length of each reception window by one beacon transmission duration when computing discovery latencies, while still accounting for the full length of each reception window in computations of the duty-cycle. This leads to the following bound. Theorem 6.1 (Unidirectional Beaconing with Reduced Reception Window Length): Consider a device E that runs an infinite beacon sequence ,∞ with a duty-cycle of and a device F that runs an infinite reception window sequence ,∞ with a duty-cycle of . When accounting for the fact that beacon transmissions starting within the last time-units of each reception window cannot be received successfully, the minimum worst-case latency that can be guaranteed for F discovering E is as follows.

Proof. For accounting for the failure of transmissions starting within the last time-units of a reception window, the coverage per beacon Λ in Equation 6 from Theorem 5.1 needs to be reduced by one beacon transmission duration for each reception window. This results into the following equation.

Clearly, this overhead increases the worst-case latency that can be achieved at least for some given reception duty-cycles = =1 , whereas others remain unaffected. Equation 30 implies that the latency is minimized for = 1 (i.e., one reception window per period). Further, should become as large as possible for minimizing . Therefore, we have to identify the maximum value of . Consider a deterministic beacon sequence = , , ... Fig. 11 . Relative difference between real and ideal bound on radios without switching overheads. and one reception window per , as depicted in Figure 11 . Let beacon be the first beacon sent after the devices have come into range and let its predecessor be sent infinitesimally after the beginning of reception window 0 . Beacon cannot overlap with the same reception window as , since otherwise a beacon sequence containing and would cover the same offsets more than once and hence induce redundant coverage. Similarly, all other sequences of beacons, in which multiple beacons overlap with 0 , are infeasible. Therefore, the first beacon after that can overlap with a reception window is , which overlaps with window 1 in Figure 11 . Since = 1, the windows 0 and 1 are spaced by time-units. is the time difference between the transmission of and . Hence, since overlaps with 0 and with 1 , ≥ . Note that ≥ also holds true for sequences that are deterministic within a multiple of . If > 1, is limited by the largest time distance between the beginnings of two subsequent reception windows. Hence, can have a maximum value of · , while also · time-units of overhead per is induced. As a result, the overhead per worst-case latency remains identical (cf. Equation 

If now is increased starting from such a value, as long as the ceiling-term in Equation 31 does not wrap over, the worst-case latency remains constant. Moreover, when is increased such that the ceiling-term wraps over its the next smaller value, also is decremented by one. We can therefore write as in Equation 29 . □

considers unidirectional discovery. In symmetric ND, every device operates using a duty-cycle = , and we have to identify the optimal share between and . For this case, the following bound applies. Theorem 6.2 (Symmetric Bound for Bi-Directional ND Protocols): When considering that beacon transmissions starting within the last time-units of each reception window are not received successfully, for a given duty-cycle , no bi-directional symmetric ND protocol (i.e., every device runs the same tuple ∞ , ∞ ) can guarantee a lower worst-case latency than the following. Equation 31 can be written as follows.

As long as the ceiling-term in Equation 34 does not wrap over, we can reduce and therefore optimize the term − . The optimal values of are therefore those that fulfill 1 − = , = 1, 2, 3..., since any smallest decrease of would cause the ceiling-term to increase its value. We can write as

We can easily solve this equation by . Let us for now allow also non-integer values of . By solving = 0, we can show that = 1 · √︀ 2 minimizes , and that there are no further extrema for ≥ 0. Since the slope of is negative for < and positive for > , only the pair of integer values that are neighboring can minimize . Therefore, the only possible integers that could lead to the global minima are ⌊ ⌋ and ⌈ ⌉. This leads to Equation 33 . □

Throughout this paper, we have assumed that the radios do not require any energy to switch from sleep mode to transmission or reception, and vice-versa. We now assume an overhead to switch the radio from the sleep mode to transmission and back, and an overhead to switch from the sleep mode to reception and back. These overheads are the effective durations of additional active time, i.e., the actual durations that are needed to switch the radio's mode of operation, weighted by the quotient of the average power consumption during the switching phase over the power consumption for reception. For the sake of simplicity of exposition, we also assume the same overheads for switching directly between reception and transmission, without going to a sleep mode in between.

and . Here, the following bound applies. Theorem 6.3 (Unidirectional Discovery with Radio Overheads): Consider a device E that runs an infinite beacon sequence ,∞ with a duty-cycle of and a device F that runs an infinite reception window sequence ,∞ with a duty-cycle of . If the radio induces an overhead of time-units to switch between sleep mode and transmission, and of time-units to switch between sleep mode and reception, the minimum worst-case latency that can be guaranteed for F discovering E is as follows.

Proof. The duty-cycle for reception and for transmission of a radio that is subjected to these overheads are as follows.

Equivalently to Theorem 5.4, it is

since is minimized for = 1 and = , as in the previous section. We first consider reception duty-cycles for which − = , = 1, 2, 3, .... For these duty-cycles, the ceiling-term in Equation 38 can be omitted and hence, a closed-form term for can be derived easily. If is further increased, the latency will not decrease until the ceiling-term wraps around. Hence, = ⌈ ⌉ · , which directly leads to Equation 29 . □

We now study symmetric discovery, in which each device has the same duty-cycle . Theorem 6.4 (Bound for Symmetric Bidirectional ND with Radio Overheads): For a given duty-cycle , no bi-directional symmetric ND protocol (i.e. every device runs the same tuple ∞ , ∞ ) can guarantee a lower worst-case latency than the following, if the radio induces an effective overhead of time-units to switch between sleep mode and transmission, and an overhead of to switch between sleep mode and reception.

= min

Proof. Consider one partial discovery procedure, e.g., device E discovering device F. When inserting and from Equation 37 into = ⌈︂ =1 ⌉︂ · , we obtain the following latency.

As for the unidirectional case, is minimized, if = 1 and = . Further, only values of for which − = , = 1, 2, 3... can be optimal (cf. Theorem 5.3). By inserting these values into Equation 41 and by differentiating the resulting by , we can identify a local minimum of at . Since > 0 for > and < 0 for < , only the pair of integers ⌊ ⌋, ⌈ ⌉, which is closest to , minimizes . Inserting = ⌊ ⌋ and = ⌈ ⌉ into Equation 41 leads to Equation 39 . □

Throughout the paper, we have neglected the transmission duration of the successfully received beacon. We can account for this by adding time-units to Equation 30 , from which the bounds for unidirectional and for symmetric discovery are derived. By forming the first and second derivative, we can show that the optimal share between transmission and reception for symmetric discovery is not influenced by this. When accounting for this beacon transmission, all our presented bounds become by time-units longer (e.g., Equation 13 becomes = 4 2 ), if no other assumptions are relaxed simultaneously. Besides from this, there are no changes, since finding the optimal beaconing duty-cycle is the only step that is potentially sensitive to adding to . The approaches for jointly relaxing other assumptions, e.g., those described in Section 6.1 or 6.2, remain unchanged, but the resulting equations become more complex.

In this section, we numerically evaluate the impact of the simplifying assumptions described above on the latency bound for unidirectional discovery. We assume a transmission duration of 32 µs, which corresponds to a 4-byte beacon on a 1 MBits -radio used for e.g, BLE. We consider a range of duty-cycles of the sender and of the receiver between 0.055 % and 5.55 %. This range of duty-cycles leads to a practically relevant range of discovery latencies from 0.1 s to 100 s for optimal protocols on ideal hardware platforms (cf. Equation 9). We assume = 1. Let denote the ideal latency bound (i.e., Equation 9 ) and the latency bound with relaxed assumptions. As can be seen from Figure 12 , in the considered range of duty-cycles, the relative deviation − ranges between nearly 0 % to nearly 8 %. While Figure 12 provides a platform-independent comparison for any ideal 1 MBits radio, what performance can be achieved on existing hardware platforms? For a Nordic nRF51822 SOC [32] , the switching overheads are approximately given by = = 140 µs. Within the considered range of duty-cycles, the relative deviation from the ideal bound ranges between 438 % and 481 %.

In this section, we relate the worst-case performance of popular protocols and previously known bounds to the fundamental limits described in the previous section. We thereby consider symmetric bi-directional discovery. Due to their relevance in practice, we consider only small duty-cycles . For such duty-cycles, the numerical difference between the simplified bound for symmetric protocols given by Equation 13 and the exact bound given by Equation 11 is negligible, allowing for a simplified presentation. 

As already described in Section 2, a worst-case number of slots within which discovery can be guaranteed is known for slotted protocols [58, 59] . The corresponding worst-case latency in terms of time is proportional to the slot length , for which there is no known lower limit. In this section, we for the first time transform this worst-case number of slots into a latency bound and establish the relations to the fundamental bounds on ND presented in this paper. We will also address the bound presented in [34, 35] , which has been claimed to be tighter than the bound in [58, 59] .

According to [58, 59] , no symmetric slotted protocol can guarantee discovery within slots by using less than ≥ √ active slots per . The associated worst-case latency is · time-units, which is directly proportional to the slot length . We in the following derive a theoretical lower limit for and hence for .

Slotted protocols can only function properly if the beacon length is "at least one order of magnitude smaller than " [59] . If this requirement is not fulfilled, often a beacon might not overlap with a reception window even though the active slots of two devices overlap, as illustrated in Figure 13 . Here, the slot length in a slot design as proposed in [59] has been set to 2 · . As can be seen, practically none of the offsets for which two active slots overlap would lead to a successful reception, since every beacon would only partially overlap with a reception window. If would be increased, the fraction of successful offsets would gradually become larger. For achieving zero collisions independently of the slot length, let us assume a full duplex radio, which can both transmit and receive during the same points in time. Then, the theoretical limit on the slot length becomes as low as one beacon transmission duration , which leads to the following duty-cycle:

Since the limit from [58, 59] requires that ≥ √ = √ , with a slot length of = , Equation 42 leads to the following latency limit:

For = 1, this bound becomes 4 2 and hence identical to the fundamental bound for symmetric protocols given by Theorem 5.5 . For all other values of , this bound exceeds the one given by Theorem 5.5.

However, the assumption of full-duplex radios is not fulfilled by most wireless devices. Further, every wireless radio requires a turnaround time to switch from transmission to reception, during which the radio is unable to receive any beacons. Even for recent radios, this time is large against the beacon transmission duration (e.g, for the nRF51822 radio [32] , it lies around 140 µs, whereas beacons can be as short as 32 µs). Therefore, will be orders of magnitude larger than , which linearly increases the worst-case latency slotted protocols can guarantee in practice. It is worth mentioning that this increase occurs in addition to the duty-cycle overhead induced by the turnaround times of the radio.

We now study the bound presented in [34, 35] , which has been claimed to be lower in terms of slots than the one presented in [58, 59] . It is achieved by assuming two beacon transmissions per active slot ( [58, 59] assumes only one), of which one beacon is sent slightly outside of the slot boundaries. By accounting for the two beacons per active slot, Equation 42 becomes = · 2 , which leads to the following bound for the protocols proposed in [34, 35] :

This bound becomes minimal for = 1 2, for which it is identical to the bound in Theorem 5.5. Hence, the bound in [34, 35] is lower in terms of slots than the bound in [58, 59] , but identical or larger in terms of time.

All previously known bounds for slotted protocols are in the form of relations between the worst-case number of slots and the dutycycle. The channel utilization, which is directly related to the beacon collision rate, has not been considered before. However, in slotted protocols, the channel utilization depends both on the number of active slots per period and on the slot length. For sufficiently large slot lengths, the turnaround times of the radio only play a negligible role. Further, the time for reception in each slot approaches nearly the whole slot length . Hence, for >> , we can compute the duty-cycle of slotted protocols as follows. With the requirement of ≥ √ from [58, 59] , one can express the slot length by the desired channel utilization in Equation 45 , which results in the following bound.

From comparing Theorem 5.7 (cf. Equation 18) to Equation 46 , it follows that if lies below 2 , the worst-case latency a slotted protocol can guarantee with a channel-utilization of = is identical to the corresponding fundamental bound (recall that we only consider optimal duty-cycles). For > 2 , slotted protocols cannot reach the fundamental bound from Theorem 5.7. Figure 14 visualizes both this fundamental bound and the bound for slotted protocols from Equation 46 . As can be seen, they coincide for low channel utilizations , but the worst-case latency of slotted protocols is increased for higher channel utilizations. In practice, this means that slotted protocols can potentially perform optimally in busy networks with many devices discovering each other simultaneously, but cannot offer optimal performance in networks in which new devices join gradually and hence only a master node and the joining device need to carry out ND at the same time.

We in the following evaluate the popular protocols Disco [13] , Searchlight-Striped [2] , U-Connect [18] and diffcode-based protocols [58] and compare them to the performance bound given by Theorem 5.7. In Disco, active slots are repeated every 1 and 2 slots, where 1 and 2 are coprimal numbers. The Chinese Remainder Theorem implies that there is a pair of overlapping slots among two devices every 1 · 2 time-units. U-Connect also relies on coprimal numbers for achieving determinism. In contrast, Seachlight defines a period of and a hyper-period of 2 slots. The first slot of each period is active, whereas a second active slot per period systematically changes its position, until all possible positions have been probed. Diffcode-based solutions are built on the theory of block designs and hence guarantee a pair of overlapping slots among two devices with the minimum possible number of active slots per worst-case latency. More details on these protocols can be found in [8] .

, Table 1 . Worst-case latencies of slotted protocols.

Slot length-dependent equations on the worst-case latency and duty-cycle of these protocols are available from the literature. When assuming sufficiently large slots and by expressing the slot length by the channel utilization similarly to Equation 45 , one can derive the equations that relate the worst-case latency, duty-cycle and channel utilization given in Table 1 . Clearly, only Diffcode-based schedules reach the optimal performance in this metric, whereas all other ones perform below the optimum.

In summary, slotted protocols can perform optimally in the latency/duty-cycle/channel utilization metric, if the channel utilization remains low. In the latency/duty-cycle metric, however, higher required channel utilizations prevent slotted protocols from performing optimally.

In slotted protocols, the number of beacons is always coupled to the number of reception phases. As a result, such protocols lack optimality in the latency/duty-cycle metric. Slotless protocols are not subjected to this constraint. Can they reach optimal latency/duty-cycle relations?

In [23] , two parametrization schemes for slotted protocols, called SingleInt and MultiInt, have been proposed, which have been claimed to provide the best latency/duty-cycle performance among all known slotless protocols. We therefore in the following relate their performance to the bounds presented in Section 5.

In such slotless protocols, beacons are sent periodically with a period . Similarly, the device listens to the channel for time-units once per period . The SingleInt scheme specifies the following configuration:

= − , = 1 · , = 1, 2, 3, .... One can easily verify that such parametrizations lead to disjoint coverage. Since the distance between two consecutive beacons does not exceed the length of the effective reception window (i.e., the length of the reception window minus one beacon transmission duration, as already described), the discovery procedure will be successful within time-units. Therefore, the worst-case latency is as follows (cf. [23] for details).

For our bounds, we have assumed that 1) beacons that are sent within the last time-units of each reception window are received successfully and 2) the transmission duration of the successfully received beacon is neglected. When applying these assumptions to the protocol described above, we can set = 0 in Equation 47 and obtain = 1 . The length of the reception window, , is determined by the duty-cycle the protocol should realize. It is

which can be solved by easily. This leads to a worst-case latency of 1 2 1−1 time-units. By forming the first and second derivative of , one can find that = 2 − 1 minimizes . Since needs to be an integer number, we consider the pair of neighboring integers, i.e., 1 = ⌊ 2 − 1⌋ and 2 = ⌈ 2 − 1⌉, which leads to the following latencies:

We parametrize the protocol using 1 , if 1 < 2 , and using 2 , otherwise. With this scheme, we obtain the following latency.

This is identical to Theorem 5.5. Hence, under the assumptions described above, a slotless protocol parametrized using the SingleInt scheme is optimal in the latency/duty-cycle metric. Which degradation of the latency bound of the SingleInt scheme do these assumptions imply in practice? When assuming a beacon transmission duration of = 32 µs and a range of duty-cycles between 0.1 % and 100 % in steps of 0.1 %, the normalized root mean square error between Equation 11 and the equations presented in [23] for SingleInt is 1.24 %.

Metric. Slotless protocols parametrized as described in the previous section always use the channel utilization that minimizes the worst-case latency. They cannot obey a given limit on the channel utilization. Hence, they cover only a small part of the Pareto-front formed by the duty-cycle, the channel utilization and the worst-case latency. Next, we for the first time propose a parametrization scheme for PI protocols that can account for a given limit on the channel utilization and show that the resulting latencies are optimal.

Let us again assume = and = 1 , assuming that beacons being sent within the last time-units of a reception window are successfully received. Here, the channel utilization is given by = . Hence, can be controlled by the length of the reception window . In particular, for enforcing ≤ , we have to ensure that ≥ . By rearranging = and expanding and , we obtain the following value for .

Clearly, the larger becomes, the larger also becomes. The smallest for which ≥ (and hence, ≤ ) is therefore as follows.

With = 1 (cf. Section 7.2.1), we obtain the following worst-case latency.

Section 7.2.1 describes the value of that mimizes the worst-case latency if the channel utilization is unconstrained. From Equation 48 follows that a certain value of leads to a channel utilization of = 1 · − 1 1. If the channel utilization obtained for the optimal from Section 7.2.1 lies below the limit , we can safely use this value and obtain the worst-case latency given by Equation 51 . Otherwise, we have to use the value for given by Equation 53 , leading to the latency given by Equation 54 . For all of these cases, the latencies achieved are equal to those given by Theorem 5.7. Hence, a periodic interval protocol parametrized as described above is the first one to cover the entire Pareto-front given by the duty-cycle, the channel utilization and the worst-case latency. Given a tuple , , the resulting worst-case discovery latencies are always optimal.

In this section, we first describe open problems left for future research and then summarize the main results of this paper. For increasing numbers of devices discovering each other simultaneously, it is inevitable that their beacons will collide and hence, an increasing number of discovery attempts will fail. Therefore, generalized performance bounds for multi-device scenarios need to be derived. Such bounds are in the form of a function , , , , which needs to be interpreted as follows. For a given number of devices with duty-cycles and , in no ND protocol, a fraction of at least 1 − of all discovery attempts will terminate successfully within less than time-units. Clearly, for → 1 and → 0, this bound converges to from Equation 9. The following two mechanisms determine the performance in multi-device scenarios. 1) Lowering the Channel Utilization: The rate of collisions directly correlates to the channel utilization , as described by Equation 17 . Hence, devices can reduce the failure probability by reducing , which will, however, negatively affect the discovery latencies achieved in the two-device case (cf. Equation 9). 2) Redundant Coverage: Optimality in the , -metric for two devices implies that every initial offset is covered exactly once (cf. Theorems 4.3 and 5.3) and hence, every collision leads to a failed discovery. However, an ND protocol might cover multiple or all initial offsets more than once. Hence, for such offsets, more than one beacon would overlap with a reception window, and as long as one of them is not subjected to collisions, the discovery procedure will succeed. Moreover, it seems feasible to construct protocols that first cover every offset exactly once by a beacon sequence ′ 1 of length . In addition, the same offsets are then covered again by concatenations of multiple sequences ′ , = 1, 2, 3, .... In other words, such protocols would guarantee short latencies in the two-device case, while performing potentially optimally also in multi-device scenarios.

The collision of a pair of beacons from two devices often induces an increased collision probability of subsequent pairs of beacons. For example, consider protocols in which beacons are sent with periodic intervals. Since all devices in a symmetric scenario transmit with the same interval, a collision implies that all later beacons will also collide. To make protocols robust against failures due to collisions, a beacon schedule needs to fulfill the following property. Given any two beacons that both overlap with a reception window for the same offset Φ 1 , their individual collision probabilities should exhibit the lowest possible correlation. It is currently not clear which degree of such a decorrelation can be actually achieved. Further, measures for decorrelating collision probabilities might reduce the latency performance, because they could prevent beacons from being sent at their optimal points in time. Hence, not all initial offsets can be covered with the fewest possible number of beacons, making additional beacon transmissions necessary. Besides open questions on decorrelating collisions, for protocols being optimal in the multiple-device case, how many times should every initial offset be covered? These questions need to be studied further in order to derive agnostic bounds in the form of , , , .

Our results also outline an important direction for the development of future ND protocols. Protocols that contain decorrelation mechanisms to make the collision of each beacon independent from the occurrence of previous collisions have not received significant attention by the community. Though BLE applies some random delay for scheduling its beacons [31] , the optimal randomization technique to obtain the best trade-off between robustness and worst-case latency remains an open question.

We have presented and proven the correctness of multiple fundamental bounds on the performance of deterministic ND protocols. In particular, we have presented bounds for unidirectional beaconing, for symmetric and for asymmetric bi-directional ND. Further, we have shown that in the latency/duty-cycle metric, only slotless protocols can reach optimal performance. However, if the channel utilization is constrained, slotted protocols can cover large parts of the Pareto-Front, while we have presented a slotless protocol that can cover the entire one. We have also revealed new important open problems to be addressed by future research.

Let us consider an arbitrary pattern of reception windows of infinite length ∞ . Such a ∞ is characterized by its reception duty-cycle . As in Section 4, we consider a sequence ′ that consists of those beacons that are sent after both devices have come into range. Obviously, the first beacon 1 ∈ ′ is received successfully if it directly overlaps with one of the reception windows. The fraction of time-units at which a transmission of 1 leads to a reception is therefore identical to . Another beacon that is sent by 1 time units later leads to additional points in time at which 1 can be sent, such that one beacon out of 1 , 2 is received successfully. These additional points in time lie 1 time-units earlier. Hence, like in Section 4, such points in time for later beacons are given by translating those of earlier ones to the left. If every point in time is covered by exactly one such translation, the tuple ( ′ , ∞ ) is disjoint and deterministic, and hence potentially optimal. This holds also true for cases in which ∞ is not an infinite concatenation of the same . The number of beacons that need to be sent for guaranteeing deterministic discovery is therefore identical to the number of translations of the reception pattern ∞ , such that every point in time overlaps with exactly one such translation. It is:

This is identical to Theorem 4.3, and hence all bounds remain unchanged.

Throughout the paper, we have assumed that ∞ does not impose any constraints on scheduling the beacons in ∞ on the same device. In this section, we study the relaxation of this assumption.

We first study the case in which both devices E and F run the same tuple of sequences ( ∞ , ∞ ). Here, ∞ is designed such that a beacon overlap with ∞ is guaranteed for all initial offsets. Hence, not only an overlap of a beacon of with ,∞ is guaranteed, but also an overlap of a beacon of with ,∞ . Such an overlap implies that the affected reception window needs to be interrupted for a certain amount of time.

For an ideal radio (i.e., a radio that does not require any time to switch from reception to transmission and vice-versa, see Section 6.2), this amount of time is identical to one beacon transmission duration . A beacon sent by another device within this period of time would collide and therefore would not be received successfully, even if the radio was able to receive and transmit simultaneously.

However, a real-world radio needs a certain amount of time to switch from transmission to reception and an overhead to switch from reception to transmission, during which no communication can be carried out. We in the following analyze the impact of this. Towards this, we next compute the time-fraction of all reception windows in ∞ , during which the radio is unable to receive.

Since an optimal tuple of sequences ( ∞ , ′ is designed such that every initial offset is covered exactly once, exactly one beacon of ′ will overlap with a reception window for every possible initial offset. For every such overlap, the radio is unable to receive incoming beacons for time-units within the affected reception windows. In a tuple ∞ , ∞ , how frequent do such overlaps occur and which fraction of the total reception time is "blocked" by them? In optimal protocols, exactly one beacon overlaps with a reception window per worst-case latency (cf. Section 5.1). From Theorem 10 follows that for optimal values of , = · 1 =1 · , and hence is always divisible by . In every instance of , there are =1 time-units during which the radio is scanning, and therefore, the radio spends · =1 = time-units per worst-case latency for scanning. The probability of failed discoveries is identical to the fraction of "blocked" time per , which leads to the following equation.

In this equation, we assume that the amount of time during which the radio is "blocked" per beacon that overlaps with a reception window of the same device is always identical to time-units. We in the following prove this assumption. Recall from Section 4.1 that every beacon of a deterministic sequence ′ , in conjunction with a reception window from ∞ of a remote device, leads to a certain contiguous range of covered offsets, which we in the following call a coverage image. If the initial offset Φ 1 lies within one of these coverage images, ′ is received successfully. Figure 15 exemplifies a coverage map of a non-redundant and deterministic (and hence potentially optimal) ND protocol. Here, ∈ ∞ consists of only one reception window and hence, there is one coverage image per beacon. Recall that if a remote device sends a beacon during the last time-units of every scan window, it is not received successfully (cf. Section 6.1). We can therefore subdivide every coverage image of an optimal protocol into the following three parts , ℬ and (cf. Figure 15 ).

• Part has a length of time-units, and a beacon of the remote device that falls into this part will not be received successfully. Therefore, such Parts do not contribute to the overall coverage. • To nevertheless ensure discovery if a beacon falls into such a Part of a coverage image, each Part is also covered by the Part of another coverage image, which also has a length of time-units.

• The remaining part B is disjoint, i.e., no part of any other coverage image overlaps with it.

On a device E, we know that exactly one beacon of ,∞ will overlap with at least one reception window of ,∞ per , which effectively interrupts or shortens the affected scan window. Such an overlap could happen in one of the following three ways.

(1) The overlapping beacon falls into Part ℬ, such that a contiguous duration of time-units is blocked (e.g., it falls into the center of Part ℬ). (2) The overlapping beacon falls into the beginning (e.g., Part ) of the scan window. Therefore, the "blocked" amount of time would also overlap with the neighboring Part ℬ (cf. Figure 15 ). Hence, the amount of occupied scanning time is equal to is also for this situation. (3) The same holds true for a beacon falling into the end of the scan window (e.g., into Part ), where parts of the "blocked" amount of time overlap with a Part and possibly also ℬ of another scan window.

Hence, in all three cases, the amount of "blocked" time is .

For asymmetric discovery (i.e., both devices have different duty-cycles), a quadruple of beacon-and reception window sequences can be designed such that ∞ and ∞ on the same device never overlap, while allowing for optimal (i.e., disjoint coverage) and deterministic Fig. 15 . Coverage map of a deterministic beacon sequence ′ in conjunction with a certain ,∞ . The offsets covered by any reception window are composed by a Part that overlaps with the last time-units of another reception window, a Part ℬ that is disjoint and a Part , during which an incoming beacon is not successfully received. two-way discovery between the two devices. Figure 16 depicts a pair of tuples ,∞ , ,∞ and ,∞ , ,∞ along with the corresponding coverage maps. As can be seen, ,∞ , ,∞ and ,∞ , ,∞ realize disjoint and deterministic discovery, while the sequences on the same device never overlap.

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This work was partially supported by the German Research Foundation (DFG) under grant number CH918/5-1 -"Slotless Neighbor Discovery".

Throughout this paper, we have restricted our considerations to infinite length reception window sequences ∞ that are given by concatenations of some finite sequence . Though all currently known deterministic ND protocols are constructed accordingly, reception window sequences that continuously alter over time are also feasible. In what follows, we study such sequences and establish that all our presented bounds remain valid for them.