Multi-Channel Distributed DSA Networks ...

Multi-Channel Distributed DSA Networks: Connectivity and Rendezvous Tradeoffs Osama A. H. Al Tameemi1 , Ayad Al-Rumaithi2 , Mainak Chatterjee3 , Kevin Kwiat4 and Charles Kamhoua4 1 Electrical Engineering Technical College, Middle Technical University, Baghdad, Iraq 2 Civil, Enviornmental, and Construction Engineering, University of Central Florida, Orlando, FL 3 Electrical Engineering and Computer Science, University of Central Florida, Orlando, FL 4 Information Directorate, Air Force Research Laboratory, Rome, NY {osama, mainak}@eecs.ucf.edu, {ayad.ali89}@knights.ucf.edu, {kevin.kwiat,charles.kamhoua.1}@us.af.mil Abstract—In this paper, we investigate and characterize the effects of multi-channel and rendezvous protocols on the connectivity of Dynamic Spectrum Access networks using percolation theory. In particular, we focus on the scenario where the secondary nodes have plenty of vacant channels to choose from–a phenomenon which we define as channel abundance. To cope with the existence of multi-channel, we use two types of rendezvous protocols: naive ones which do not guarantee a common channel and advanced ones which do. We show that, with more channel abundance, even with the use of either type of rendezvous protocols, it becomes difficult for two nodes to agree on a common channel, thereby potentially remaining invisible to each other. We model this invisibility as a Poisson thinning process and show that invisibility is even more pronounced with channel abundance. Following the disk graph model, we represent the multiple channels as parallel edges in a graph and build a multilayered graph (MLG) in R2 . In order to study the connectivity, we show how percolation occurs in the MLG by coupling it with a typical discrete percolation. Using a Boolean model and the MLG, we study both cases of primaries’ absence and presence. For both cases, we define and characterize connectivity of the secondary network in terms of the available number of channels, deployment densities, number of simultaneous transmissions per node, and communication range. When primary users are absent, we derive the critical number of channels which maintains supercriticality of the secondary network. When primary users are present, we characterize and analyze the connectivity for all the regions: channel abundance, optimal, and channel deprivation. For each region we show the requirement and the outcome of using either type of rendezvous techniques. Moreover, we find the trade-off between deployment-density versus rendezvous probability which results in a connected network. Our results can be used to decide on the goodness of any channel rendezvous algorithm by computing the expected resultant connectivity. They also provide a guideline for achieving connectivity using minimal resources.1

I. I NTRODUCTION Shared communication infrastructures are becoming increasingly heterogeneous, with a dynamic composition of interdependent, interactive, and hierarchical network components [27]. Dynamic spectrum access (DSA) network is such a network where secondary networks/users (non-license holders) can ‘borrow’ idle spectrum from those who hold licenses (i.e., primary networks/users), without causing harmful interference 1 A preliminary version of this work has appeared in IEEE ICNC 2016 [2]. Approved for Public Release; Distribution Unlimited: 88ABW-2016-3735, dated 28 July 2016.

to the latter [19]. Unlike traditional radios, cognitive radios that constitute the secondary DSA network constantly monitor the radio spectrum and intelligently access the radio spectrum in an opportunistic manner both in the licensed and unlicensed bands. For any secondary DSA network to provide sustained communication services, it is important that the network remains connected where radios (nodes) can communicate with each other directly or via intermediate nodes. Thus, knowing the connectivity properties is of utmost importance as they play a major role in determining the network’s expected QoS including throughput, reliability, routing, and communication mode/range. The term network connectivity can have multiple interpretations depending on how it is defined. One such interpretation is the asymptotic connectivity which is usually used for studying the connectivity of homogeneous networks where all nodes in the network are connected with high probability when the number of nodes approaches infinity [13]. However, in DSA networks the additional constraints, particularly the interference tolerance conditions set forth by the primary networks, do not necessarily mean that all the secondary nodes remain connected with high probability even in the limiting case which makes the asymptotic connectivity infeasible. Thus, in this work, we define connectivity as the existence of the giant component2 with high probability in any given time slot. Moreover, the impact of primary transmission activities propagate both spatially and temporally through a secondary network. Hence, it becomes important to investigate what effect does the presence of a primary have on the connectivity of secondaries, i.e., reachability of a node from other nodes in the secondary network via single or multiple hops. It is known that, link establishment between a pair of communicating neighboring nodes (i.e., node-to-node connectivity), requires the two nodes to be within range of each other and to have at least one available channel between them. A channel is available to two neighboring nodes Si and Sj if and only if the channel is not being used by a primary user. Regarding channel vacancy and availability, we shed the light on: 2 More

details about the giant component are provided in section IV-A

A. Channel Abundance in DSA Networks One of the most ignored issues in studying the connectivity is the total number of candidate channels– be they vacant or occupied. This is mainly due focus only on the limited number of free channels that the secondary users contend for and not on the total number of channels. For example the IEEE 802.22 networks have about 47 candidate channels of 6 MHz each but the focus is usually on the few vacant ones rather than the 47 candidate ones. Moreover, a vacant channel by the primary (say 6 MHz) could be fragmented into many sub-channels via channel fragmentation (6 channels of 1 MHz each) [3], [21]. Thus, there could be abundance of the number of channels and not necessarily abundance of bandwidth. Although this abundant number of channels (whether vacant or occupied) seems harmless at a first glance, it might cause the rendezvous protocols not to converge within the delay budget. B. Channel Abundance and Node Invisibility It is true that, the existence of N channels opens more possibilities for Si and Sj to find available channel(s) to establish a communication link especially when the node does simultaneous transmissions on a set of channels rather than one. However, if the neighboring nodes do not agree on a common channel, they will remain invisible to each other. This is why in such multi-channel environments, finding a common channel is achieved via rendezvous protocols. Having a common channel means that the two lists of channels chosen for transmission by two neighboring nodes contain at least one common element. For two nodes, the ideal case would be having a common channel in each time slot3 . However, since this is not possible in may cases due to the spatial variations in spectrum opportunities, the alternative is to have the two nodes rendezvous with some rendezvous probability (m/T ). m is the amount of time for which the two nodes have a common channel in a frame of duration T , with m ≤ T . We highlight that, rendezvous between two nodes does not guarantee the connectedness of the network. The underlying network formed in each time slot will consist of only a fraction of the nodes which corresponds to a Poisson Blinking Model [9]. This is why we argue that channel abundance adversely affects the connectivity of a DSA network. Demonstrating this counterintuitive observation is one of the objectives of this work. In our case, the rendezvous process is an attempt to establish an initial dialogue between two secondary nodes to discover at least one of the common idle channels in a multi-channel environment. If the two nodes successfully rendezvous on one common channel, they proceed to exchange their lists of idle channels (based on their individual sensing) and agree on i) which of those channels can be used and ii) during which (future) time slots that they can be used. A node could undergo the rendezvous process with multiple nodes and agree on the future time slots when it will be communicating with the other nodes and using which channels. However, since a node is assumed to have only one transceiver, it can transmit to or 3 The

assumption of slotted time is used just for illustration purposes.

receive from only finite number of nodes at any given time slot (assuming it is capable of simultaneous transmissions)– thus remaining invisible to the others during that time slot. C. Motivation and Contributions Before delving into the details of our work, we highlight two perspectives that are very important to understand our work. It is true that from a rendezvous point of view, the more the channels (channel abundance), the more the time to rendezvous (TTR) which is adverse; and at the same time, channel abundance is favorable from the point of view of the secondary users. However, neither of these perspectives (rendezvous and secondary users) provide insights on whether channel abundance is good or bad regarding the network connectivity as both of the aforementioned perspectives are conflicting and monotonic in nature regarding channel abundance. In fact, the effects of channel abundance on connectivity were not clear prior to our work. Thus, in this paper, we seek to answer the following questions: With channel abundance, is it possible to maintain network connectivity with/without the use of a rendezvous protocol? What type of rendezvous protocol, naive or advanced, will be sufficient or be necessary to establish connectivity? (An advanced rendezvous protocol ensures a common channel within a bounded time assuming the existence of at least one common channel while the naive does not.) What is the rendezvous probability? If the network is disconnected due to node invisibility, to what extent should the invisibility be reduced, using rendezvous, to attain connectivity? To answer these questions, we start by characterizing the effects of channel abundance on connectivity. Such characterization requires isolating the effects of rendezvous techniques on connectivity. This is why, at first we use a naive rendezvous protocol. By naive, we mean a simple rendezvous protocol where a node randomly hops through the frequencies (excluding the subset of busy channels) in every time slot. Once the connectivity–channel abundance relations are established, we consider the advanced rendezvous protocols and their effects on the connectivity of the secondary network. We isolate the effects from interference by following the Boolean model of connectivity (disk graph connectivity), as was done in [14], [23], [27]. Boolean model connectivity means that two nodes are connected if the two circles drawn around their centers overlap the centers of each other i.e., the nodes are connected if they are within some distance from each other. As for mimicking the primary and secondary users’ activity we choose to use a Poisson point process for the generation of both [8], [23], [27]. To handle the availability of N channels, each channel is considered as a dimension/layer (N channels yield N dimensions/layers) which introduces a multi-layered graph (MLG), the projection of which on R2 is called PMLG which transforms the N -channel DSA network into a one channel network embedded in R2 . Then we show how channel abundance induces invisibility (in PMLG) which we model using Poisson thinning. At this point we use percolation theory [28] to study the connectivity and

reachability of the PMLG. Although for abstract mathematical models, the connectivity increases as the number of dimensions (d) increases [30], the same behavior is not attained when dealing with real systems like DSA networks due to physical constraints as we will illustrate. Afterwards, we show percolation in the PMLG via coupling it with percolation in the discrete lattice and study connectivity in the presence and absence of primary users. For both cases, we define and characterize connectivity in terms of the available number of channels, deployment densities, number of simultaneous transmissions per node, and communication range. In the absence of the primary users, we find the upper bound for the number of channels at which phase transition occurs i.e., getting a connected secondary network from islands of disconnected networks. In the presence of the primary users, we characterize connectivity in terms of the primary density and identify two ranges that have very different effects on connectivity. In the first range, we show how the connectivity increases as the primary density increases. The two ranges are partitioned by an optimal point where the connectivity is maximized. In the second range, we show how the connectivity decreases as the primary density grows. With the insights from the naive rendezvous protocol, we analyze the effects of advanced rendezvous protocols on the connectivity. We also find the minimum deployment density needed to maintain connectivity for a given rendezvous probability. Finally, we interpret our findings in terms of achieving connectivity using the minimal deployment density, number of simultaneous transmissions per node, and scanning time. II. R ELATED W ORK To the best of our knowledge, the combined effects of channel abundance, the number of simultaneous transmissions per node, and the rendezvous probability on the connectivity of DSA networks have not been investigated. However, the literature is rich with classical results for connectivity of both homogeneous [8], [17], [25] and heterogeneous [13], [14], [23], [27] networks. In [8], percolation was used to show that even when the SINR model is considered, the resulting connectivity is similar to that of a Boolean model given that the interference from other nodes can be sufficiently reduced at each receiving node. In [17], continuum percolation was used to obtain the critical transmission range of the asymptotic connectivity of i.i.d. nodes in the limiting case. In [25], the k-connectivity of a random graph is linked with the minimum node degree for graphs with high number of nodes. In [13], the k-connectivity of CRN is studied using percolation and asymptotic-connectivity where the critical secondary density is found. In [14], the percolation degree of the secondary users and its relation with the k-disjoint paths is studied in large scale CRNs. In [23], the activity of the primary users is assumed to be dynamic; its effects on the available number of channels for the secondary network is studied using percolation theory in which the critical primary user densities were illustrated. In [27], percolation was used to

study the connectivity of heterogeneous networks via identifying connectivity regions which were characterized by a set of primary and secondary networks in which percolation occurs. All the mentioned works provide useful theoretical insights. However, none of these i) studied the impact of rendezvous on connectivity, ii) considered that a radio can communicate simultaneously on finite multiple channels, iii) accounted for the scanning limitations. Our arguments are motivated by [7], [8], [12], [23], [24]. III. S YSTEM M ODEL In this section, we present the system model and study the concepts and implications of channel abundance and rendezvous on connectivity. A. Deployment and Connectivity We consider a secondary DSA network co-existing with primary users. The secondary users are generated using a Poisson point process, X, with a mean deployment density of λs . All users are distributed uniformly randomly with a coverage radius of rs . Thus, the secondary network can be modeled as a Poisson Boolean model B(X, λs , rs ). Similarly, the primary network is also modeled as a Poisson Boolean model B(X ′ , λp , rp ), where X ′ is the Poisson point process for generating the primary users on the plane only i.e., it does not generate their traffic dynamics. λp is the primary deployment density with coverage radius of rp . The two networks are embedded in R2 with a total of N channels (vacant and nonvacant) each of which could be accessed by primary users with equal probability i.e., 1/N . Due to primary activity and spatial diversity, all secondaries do not necessarily observe the same number of vacant channels. The number of vacant channels perceived by secondary user i is ni , 0 ≤ ni ≤ N . Commonly used notations are shown in Table I. X λs λcs λp rs , rp M N ni Ch(Si ) navg P0 ψp θp Cmax

Poisson point process for generating the secondary users Density of secondary nodes Critical density of secondary nodes Density of primary nodes Coverage radius of secondary (primary) nodes Number of simultaneous transmission per secondary node Total number of channels in the system Number of vacant channels for secondary node i The channels which will be used for simultaneous transmissions by secondary node i Avg. no. of available channels for all secondary nodes Prob. of no common channel between two secondary nodes Prob. of existence of the infinite cluster Conditional prob. of existence of the infinite cluster The infinite connected component TABLE I N OTATIONS USED

1) Node’s Limitations: We assume that each secondary node can simultaneously transmit on at most M different channels due to i) M transceivers (communication modules) , ii) the maximum number of channels that the node can scan per scanning period is M , or iii) even with the use of rendezvous techniques, the node still chooses a subset of the available channels. We assume that each node randomly selects its list

of channels to be sensed/scanned so as not to bias the results in favor of any sensing method. We emphasize that although the sensing method will not change the core idea rather than increasing or decreasing the value of M . Therefore, at any point of time a secondary user Si can communicate over (at maximum) any M of its ni available channels which are determined by the sensing process. Note that, depending on the values of N , M , and on the number of channels occupied by the primary users, it can be either ni ≤ M or ni > M . 2) Node to Node Connectivity: Two secondary nodes Si and Sj are considered to be connected (communication link exists) if and only if the following two conditions are satisfied. I- Range Condition: |xi − xj | ≤ rs , where xi (xj ) is the location of node Si (Sj ). This is the disk graph model where nodes within the disk centered at xi are considered neighbors of Si . II- Common Channel(s) Condition: There must be at least one common channel i.e., Ch(Si ) ∩ Ch(Sj ) 6= ∅, where Ch(Si ) is the set of selected for transmission channels by node Si from ni and 0 ≤ |Ch(Si )| ≤ min(M, ni ). B. Rendezvous Protocols Though there are sophisticated rendezvous protocols [1], [4], [29], at first, we consider a very naive protocol/scheme where nodes randomly and independently hops through subset of channels (of maximum size M ) from the set of available channels (ni ) in every time slot. We will refer to this naive rendezvous technique as ‘the naive protocol’ which is opposed to the other advanced techniques. In our context, the term advanced refers to any rendezvous protocol that guarantees a common channel within a certain period of time (bounded time) which makes the rendezvous probability strictly greater than zero. Such rendezvous techniques are considered later in section V. Note, the naive protocol does not guarantee common channel(s) between two neighboring nodes in a bounded time interval; however it is the easiest to implement as compared to the advanced ones whose performance vary and depends on their complexity. As outlined earlier, we first consider the naive protocol to isolate the effects of channel abundance as the use of advanced rendezvous schemes lessens such effects. C. Illustrating Channel Abundance Although the concept of channel abundance is common when dealing with rendezvous techniques however, in connectivity studies that is not the case. Thus in this subsection, we will analytically define channel abundance and illustrate how it affects the nodes’ visibility and the network connectivity. Since nodes perceive different number of vacant channels, e.g., ni by Si and nj by Sj , it is the subset of selected channels that determine whether a common channel is guaranteed or not. Thus, we define ni,j as:  0 ni or nj = 0 ni,j = (1) ni + nj − L otherwise

A) B) C) D)

ni=1

ChL(Si) f1 - -

N=1

nj=1

ChL(Sj) f1 - -

ni,j=1

ni=3

ChL(Si) f1 f2 f3

N=3

nj=3

ChL(Sj) f1 f2 f3

ni,j=3

ni=3 nj=5 ni=3 nj=6

ChL(Si) f1 f2 f3 ChL(Sj) f3 f4 f5 ChL(Si) f1 f2 f3 ChL(Sj) f6 f8 f9

N=5 ni,j=5

N=9 ni,j=6

Fig. 1. Worst case scenarios for two nodes choosing a common channel when M = 3 with N =1, 3, 5 and 9 for cases A, B, C and D respectively.

L is the number of common channels between Si and Sj . In general there can be only two cases– either ni,j ≤ 2M −1 or ni,j > 2M − 1. For the first case, 1 ≤ ni,j ≤ 2M − 1, there is bound to be a common channel between the two nodes Si and Sj since Ch(Si ) ∩ Ch(Sj ) 6= ∅ with probability 1. For example, with M = 3 (i.e., two nodes choosing and transmitting at maximum of 3 channels independently) there is bound to be a common channel between the two nodes for 1 ≤ N ≤ 5 since it leads to 1 ≤ ni,j ≤ 5 as illustrated in Figs. 1(A), 1(B), and 1(C), where N = 1, N = 3, and N = 5 respectively. However, for the second case, ni,j > 2M − 1 (i.e., ni,j > 5 when M = 3), there is no guarantee on a common channel. For example, consider the case when N = 9 as shown in Fig. 1(D) with the channels numbered from f1 through f9 . Suppose Si chooses f1 , f2 , and f3 . It might so happen that Sj chooses channels f6 , f8 , and f9 which results in no common channel between the two nodes. This availability of a large number of channels is referred to as ‘channel abundance’. Thus, the condition for channel abundance between two nodes Si and Sj is M + M ≤ ni,j (alternatively 2M − 1 < ni,j ). Under channel abundance, it is to be noted that though there could be a number of available channels over which communication link(s) could be established; it is the agreement on the common channel that would establish the link between two
nodes. In fact, there are ni,jM−M different ways for nodes Si ad Sj to choose M channels each without having any common channel,
assuming ni , nj ≥
M ; else the number of ways is ni,j −ni j for Si . As a consequence, two for Sj and ni,jn−n nj i nodes might remain invisible even if they satisfy the range condition. We want to emphasize that invisibility is not due to lack of channels but due to excess of them. D. Multi-Layered Graph (MLG) Following the arguments from the previous section, the possibility of multiple channels between two nodes can be represented by parallel edges in a graph. Or, in other words, two neighboring nodes can be connected in more than one channel/dimension, where each channel is a dimension.

si

: Secondary User

Pj

s2 P1

s1

: Primary User s3

P2

s4

s5

L1/f1

s2

s1

GL2(V,E2)

s3

P3 s5

L2/f2

P4

s2

s1

GL1(V,E1)

s4

P6

GL3(V,E3)

s3 P5

L3/f3

s2 P1

s1 P5P3

f1 + f2 + f3

P7

s5

G (V,E1+E2+E3) L1+L2+L3

P6 s3

P2 P7 P4

s5

s4

s4

s2

s1

G(V,E)

s3

PMLG

s5

s4

Fig. 2. MLG with N=3 and its corresponding projection (PMLG).

When N = 1, we obtain a traditional graph G(V, E) in R2 with V comprising the set of secondary nodes and E is the set of edges (communication links). The edges are established over that single channel. When N > 1, the dimensions can be visualized as independent layers as shown in Figure 2. This opens the possibility for the secondary nodes to establish edges in M of the N layers. Clearly, this gives a rise to a MultiLayered Graph (MLG). Layer i of the MLG corresponds to a graph whose edges are established over channel/frequency fi i.e., Gf1 (V, Ef1 ) is the graph whose edges are established in layer 1 (channel f1 ) only. Where:   Efx = ∪ Ch(Si ) ∩ Ch(Sj ) = fx ∀ neighboring Si , Sj ∈ V

Since connectivity can be achieved in any of the N layers, we are interested in the Projected-MLG (PMLG) [23] on R2 which is G(V, E). An edge between two nodes Si and Sj in the PMLG indicates that they have a common edge in at least one of the N layers of the MLG which is sufficient for their connectedness in the PMLG. From a connectivity perspective, it does not matter which layer(s) provide connectivity. The reduction of the N -dimensional graph to 1-layer (in frequency) PMLG embedded in R2 helps us use the concepts of percolation in R2 .

IV. C ONNECTIVITY A NALYSIS With the naive rendezvous protocol being used by the secondary nodes, we proceed to analyze the effects of changing N and M from both connectivity and rendezvous perspectives. First, we consider that there are no primary users i.e., the DSA network functions as a traditional ad hoc network. To do so, we simply use λp = 0. Then, we consider λp > 0; hence the secondary users cannot use any channel that is being used by the primary user(s). First, let us introduce some basic percolation theory definitions. A. Percolation Theory Definitions Percolation theory is the simplest model displaying a phase transition [6]. It is categorized into discrete and continuum

percolation according to the placement of the nodes whether uniformly placed or randomly scattered. We will focus on continuum percolation as it better mimics the random scattering of the nodes in our network. Continuum percolation was established by Gilbert [15] during his search for a solution to the problem of achieving long range communication (over the connected component) using multiple short range transmissions of randomly scattered nodes that are placed close enough to each other i.e., multi-hop connectivity of wireless broadcasting stations. In essence, percolation4 deals and tells under what conditions will multiple finite disconnected components transition to form one infinite component [6]. Although there is no infinite cluster for finite networks; however for sufficiently large networks a giant component emerges after a critical point [5], [10], [18], [26].

(a)

(b)

Fig. 3. a) Non-percolated network (no spanning component). b) Percolated network (spanning component).

In our context, percolation can help characterize phenomena such as impact propagation and reachability as it provides the appropriate tools to study the connectivity of heterogeneous entities in i) the continuum space and ii) the multi-dimensional lattice Ld , where d is the number of dimensions [28]. For our network, percolation will be used to choose critical parameters (primary and secondary densities, coverage radius, percolation probability) that achieve the phase transition i.e., formation of an infinite/giant connected secondary DSA network from multiple isolated smaller DSA networks. In a graph, an infinite component is usually referred to as the infinite cluster Cmax and is defined as: (|Cmax | = ∞), where |.| is the cardinality operator. With no condition on any particular node inclusion, the formation probability of the infinite cluster is given by ψp . By Kolmogorov zeroone law, ψp is either 0 or 1 [12]. Percolation probability, θp , refers to the probability of the formation of the infinite cluster conditioned on including a particular node (which could be the origin). Thus, θp also refers to the probability of any node being contained in Cmax . In this work, we will use the words ‘percolation’ and ‘connectivity’ synonymously to indicate the formation of the infinite component Cmax i.e., percolation implies θp = P (|Cmax | = ∞) > 0 and ψp = 1. λcs refers to the critical density of the secondary network which allows the secondary network to be percolated in R2 . The terms superand sub-critical are used to indicate whether the network is 4 The

non-expert readers on percolation are encouraged to read [6], [11].

percolated or not, respectively.

1 M=1 M=2 M=3 M=4 M=5

B. λp = 0 (Primaries absent) 0.8

When primaries are absent (abstracted by λp = 0), all the N channels are available to the secondary users resulting in ni = ni,j = N . Two cases arise: i) 2M > N and ii) 2M ≤ N which are addressed in the following two lemmas. Lemma 1: Under the Boolean model, neither the value of M nor N will affect the values of θp or ψp , when λp = 0 and 2M > N . The network with 2M > N is equivalent to a network with M = N =1. Proof: With 2M > N , |Ch(Si ) ∩ Ch(Sj )| ≥ 1 for neighboring nodes Si and Sj . This satisfaction of the channel condition is equivalent to having an edge in the PMLG, irrespective of the N layers of MLG and the number of edges contained in each layer. Thus, we are able to reduce the channel-dimensionality from N channels/dimensions to only one channel/dimension since it results in the same set of connected nodes/vertices in the PMLG5 . This results in M = N = 1. Note that, satisfaction of the channel condition requires the two nodes to be neighbors as well. Thus two nodes that are out of range on channel k are out of range on every other channel (under the disk graph model). It follows that the values of N and M have no effect on out-of-range nodes. Since θp and ψp deal with the number of connected nodes, it follows that they are invariant with respect to M or N as long as 2M > N .  From Lemma 1, it immediately follows that for networks with 2M > N there is no need to implement advanced rendezvous techniques as the naive technique (random selection of channels) will result in a common channel on every time slot. Moreover, Lemma 1 can be used to find the smallest value of M that satisfies 2M > N i.e., using M = ⌊ N2 + 1⌋ results in the smallest integer value of M that still achieves 2M > N . Having a smaller M saves on the cost the number of transceivers and saves on the duty cycle. Lemma 2: Under the Boolean model, with any given M , there exists a critical value (N c ) for N which if exceeded leads to ψp = 0 and θp = 0 (no-percolation), when λp = 0 and 2M ≤ N . Proof: Under channel abundance (i.e., when 2M ≤ N ), satisfaction of common channel condition is not guaranteed. Rather, it is probabilistic. We define the probability of no common channel between two neighboring nodes, P0 , as:   P0 = P (M channels of Si ) ∩ (M channels of Sj ) = ∅

Of course, probability of having common channel(s) is 1−P0 . For any two neighboring nodes Si and Sj , P0 is given by:
ni,j −M M
(2) P0 = ni,j M

5 We are assuming a typical DSA setting where the carrier frequencies of the various channels are in proximity of each other hence endure similar propagation losses which results in the same communication range.

P

0

0.6

0.4

0.2

0 0

2

4

6

8

10 12 14 16 18 20 22 24 N

Fig. 4. Effect of ni,j = N on P0 for different values of M .

As per Eqn. (2), for a fixed M , as ni,j increases6 the probability for any two neighboring nodes of having a common channel decreases. An illustration is shown in Fig. 4. Thus, two nodes in range without any common channel (among their 2M selected channels) remain invisible to each other. Notably, invisibility is not due to shortage of channels, rather, it is due to abundance of channels. We go one step further and mention that: when 2M ≤ N , the density of Poisson point process (of the secondary network) used in the Boolean model B(X, λs , rs ) gets thinned (Poisson thinning theorem [22]) with probability (1 − P0 ) resulting in B(X, (1 − P0 )λs , rs ). That is, though the secondary nodes are deployed with density λs , however from connectivity (percolation) point of view, the perceived density is (1 − P0 )λs . This is equivalent to the transformation of G(V, E) into G(V ′ , E ′ ) where |V | ≥ |V ′ |. This thinning effect is adverse from a continuum percolation point of view, because percolation occurs (ψp = 1) when λcs < λs [24], where λcs is the critical deployment density. Thus any reduction in the value of λs will adversely affect connectivity and can render a percolated network to a non-percolated one. This means, the thinned secondary network remains percolated as long as: λcs < (1 − P0 )λs .

(3)

Rearranging, we get: 1−

λcs > P0 λs

Substituting for ni,j = N and P0 from Eqn. (2) yields:
N −M λcs M > 1−
N λs M

(4)

(5)

This shows that, for a connected network, with the use of the naive rendezvous technique, as the value of N increases, so does the thinning of λs (due to increase in P0 ). Eventually 6 In

this scenario having λp = 0 results in ni,j = N .

the density of the network (1 − P0 )λs goes below λcs resulting in no percolation/connectivity (i.e., ψ = 0 and θp = 0). We define the critical value for the number of channels, N c : N c = sup(N : θp = 0) N c is obtained by setting λcs = (1 − P0 )λs in Eqn. (3) . Thus:
N c −M λc M (6) 1− s =
Nc λs M

Fig. 5. Secondary and primary coverage zones with regions A, B, and C.

Which can be re-written as:

2

Γ (N c − M + 1) λc =1− s c + 1) Γ (N − 2M + 1)) λs

(Γ (N c

(7)

where Γ(·) is the Gamma function. In order for the secondary network to be percolated (for a fixed λs and rs ) the value of N has to be smaller than N c ; otherwise M has to be increased7 . Doing so prevents P0 from increasing to the extent where (1− P0 )λs ≤ λc . In other words, increase M to prevent rendering the original network from being too sparse to be percolated.  The limiting case of N follows directly from Eqn. (2), where P0 = 1 as N → ∞ and the resultant density is (0) × λs , thus ψp = θp = 0. From lemma 2, it follows that, even in networks with 2M ≤ N , connectivity can still be maintained using the naive rendezvous technique at the expense of losing a fraction of the nodes (λs ×P0 ) due to thinning. Nevertheless, when 2M ≤ N an advanced rendezvous protocol can substantially improve the resulting the connectivity especially when N c < N . In other words, to achieve the same level of connectivity, there is a trade-off between a denser deployment (to compensate for λs × P0 ) with the advantage of using a naive rendezvous technique versus using less nodes per unit area at the expense of using a more advanced rendezvous technique. When the goal is just to establish a giant component with the minimum deployment density (λmin ), Lemma 2 gives us: s λcs +ǫ 1 − P0 where ǫ is a infinitesimally small positive number. λmin = s

(8)

C. λp > 0 (Primaries present) Here, the primary users are present with density λp each of whom have radius rp . We assume each primary user can only use one of the N channels at a time8 . Each channel is chosen uniform randomly with probability 1/N . From a channel utilization perspective, this results in thinning the primary density from λp to λp /N . Similar to the previous case, two cases arise: i) 2M ≤ N and ii) 2M > N . Case 1: 2M ≤ N : Due to presence of primaries, not all the nodes will observe N vacant channels. Contrary to the popular belief that the 7 Alternatively λ can be increased to maintain connectivity, combination s of M and λs can be used as well. 8 λ with 1 channel per user is similar to λ /2 with 2 channels per user p p for non-overlapping nodes.

arrival of the primary users always degrades the connectivity of the secondary users, we present a counter-intuitive observation in Lemma 3. Lemma 3: Under the Boolean model when λp > 0 with 2M ≤ N , connectivity of the secondary network (θp ) increases with λp as long as λp < λopt p , and the connectivity of the secondary network is maximized at λp = λopt p . The secondary network L U U remains percolated as long as λL p ≤ λp ≤ λp , where λp (λp ) opt is the upper (lower) bound of the primary density, λp is the opt U optimal primary density with λL p ≤ λp ≤ λp . Proof: Before the arrival of the first primary user, the number of available channels for each secondary user is N . Since 2M ≤ N , the secondary network is thinned due to excess of channels. As the first primary user arrives it randomly selects one of the N available channels making navg < N , where navg is the average number of available channels for all secondary users in the network. As more primary users become active (arrive), the number of available channels for each secondary user decreases. Decrease in the number of available channels for the secondary users (given 2M < navg ) lowers P0 as shown in Eqn. (2). Basically, the primaries’ arrivals minimize the thinning of λs . This allows more secondary nodes to participate in the percolation process which in turn increases the size of the connected component. This proves the first part of the Lemma. As λp increases, more channels are used by the primaries. We are interested in the following values of λp : i) λL p , the primary density for which the secondary network percolates for the first time, i.e., the smallest λp value at which the secondary network percolates, ii) λopt p , which maximizes the connectivity of the secondary network, and iii) λU p which is the highest λp value for which the secondary network stays U percolated. The interval [λL p λp ] marks λp values for which the secondary network percolates. In other words, we are interested in: 1) λL p = min(λp : θp >  0)  opt 2) λp = arg maxλp θp (λp ) . 3) λU p = max(λp : θp > 0) Consider two secondary nodes Si and Sj which are in range (i.e, within rs ) and using some channel k as shown in Fig. 5. Any primary user using channel k that comes within rp of Si or Sj will disrupt communications between Si and Sj . Thus, the shaded region in Fig. 5 must be void of primary users

using channel k. It is to be noted that the exact location of the primary user in the shaded region will have varied impact on the communications between Si and Sj . Let us define ⊚Si as the circle of radius rp with Si as the center. Similarly, we define ⊚Sj as the circle of radius rp with Sj as the center. We define three non-overlapping regions (slightly abusing the set theoretic notations): i) region A as ⊚Si − ⊚Sj , ii) region B as ⊚Sj − ⊚Si , and iii) region C as ⊚Si ∩ ⊚Sj as shown in Fig. 5. A primary user in region C forces Si and Sj to abandon the primary’s channel, which increases their probability of selecting a common channel from the set of remaining/available ones (i.e., increases rendezvous probability). However, a primary user in region A affects only Si and a primary in region B affects only Sj which may lead to no common channel. Let the number of primary users in regions A, B, and C be lA , lB , and lC which are Poisson distributed with means µA = λp × A, µB = λp × B, and µC = λp × C, respectively. With N available channels, there will be a finite countable number of combinations for the lA +lB +lC users to randomly choose one channel each (with or without spatial reuse). The randomly selected channels by the primaries might result in no common channel between Si and Sj in-spite of channels being available in regions A and B. With lA + lC users affecting Si and lB +lC users affecting Sj , there are multiple combinations that lead to no common channel. Now, we enumerate all the possible channel selection combinations of Si , Sj , and the primaries such that there are no common channel between Si and Sj . Each such combination occurs with some probability, the sum of which is equal to P0 . Theoretically, the upper bound of lA , lB , and lC is ∞ as they are Poisson distributed. For better tractability, we bound their maximum value to 10µA , 10µB , and 10µC respectively. This reasonable assumption ensures that 99.36% of the cases P10µ k e−µ ≥ 0.9936 for µ ≥ 5. With as accounted for as k=0 µ k! larger µ, the sum tends to 1. We represent the set of channels selected by all lA primary users in A by A = [A1 , · · · , AlA ], where Ai is the channel selected by primary user i. Similarly, we define B = [B1 , · · · , BlB ] and C = [C1 , · · · , ClC ]. Considering and summing all possibilities of no common channel, we get

P0 (λp ) =

10µ XA 10µ XB 10µ XC

lA =0 lB =0 lC =0

 1 lA +lB +lC × N

lC X lB X lA X N N Y N Y Y

P (no common channel|A, B, C) =

N X

M1,1 =1 6=Aj ,Bk N X

···×

M1,p =1 6=Aj ,Bk ,∀z

N , increase in λp decreases θp until percolation disappears at λp > λU p. Proof: When 2M > N and before the arrival of the first primary user, P0 = 0. With the arrival of the primary users, secondary user(s) will start losing channels making P0 > 0 and driving the secondary network into the channel deprivation opt region thus the concept of λL does not apply in this p and λp U case. However λp still exists and shows the same behavior that was shown in Lemma 3. That is, when λp > λU p percolation disappears. Analytical value for λU p for this case is obtained by solving Eqn. (13) for λp as well.  V. A DVANCED R ENDEZVOUS P ROTOCOLS AND C ONNECTIVITY In this section, we analyze the effects of advanced rendezvous protocols on connectivity. A common misconception that is associated with the use of rendezvous protocols (naive and advanced) is that their use eliminates the invisibility among the nodes. However, we have to keep in mind that, rendezvous among the nodes will occur over different time instances (even in the advanced techniques). Such phenomena has a major impact on the network connectivity when viewed from connectivity perspective. More specifically, the advanced protocols will have neighboring nodes ni , nj converging to at least one common channel by the end of duration T (assuming ni ∩ nj 6= ∅). The channel sharing will occur over different time instances9 . In other words, utilizing such a rendezvous protocol for a network with 3 channels (1, 2, and 3) and three nodes Si , Sj , and Sk , can result into nodes Si and Sj having a common channel (channel 1) on time t1 , Si and Sk can have a common channel (channel 2) at t2 , and Sj and Sk having a common channel (channel 3) at t3 . It is true that by t3 all the nodes would have shared a common channel; however, at any time instance ti only a subset of them is connected which defines the connectivity at ti . For example, Si in the example above Si was visible to either Sj or Sk but not both. Such phenomena introduces the Poisson Blinking Model where a set of nodes is partitioned into smaller subsets that emerge at different times. As mentioned earlier, m refers to the amount of time for which two nodes are visible to each other during the period T . Thus, at any point of time, two nodes Si and Sj have a common channel with probability m/T . In other words: m P0 = 1 − T 9 Assume

duration T is divided into t1 , t2 , and t3 .

This means, with rendezvous probability m/T , the network percolates as long as: m (14) T We emphasize that the value of the rendezvous probability m/T varies between the advanced techniques as it entirely depends on the inner workings of the advanced rendezvous scheme itself. Usually a higher m/T value corresponds to a more complex rendezvous scheme as well as to the hardware requirements 10 . Eqn. 14 enables us to choose a rendezvous technique that is compliant with the hardware limitations. For example, if a specific m1 /T value is not attainable by the nodes and instead the nodes can only attain m2 /T with m1 > m2 , then Eqn. (14) provides the compensation level in terms of λs such that the resultant network will be connected when m2 /T is used. Finally, the information provided by Eqn. (14) in conjunction with Lemmas 1, 2, 3 and 4 provides knowledge to decide between the naive or the advanced technique while accounting for the invisibility, density, and hardware limitations. λcs < λs ×

crossed from top to bottom by a sequence of neighboring secondary users as shown in Figure 7. 3) For each pair of consecutive/neighboring secondary nodes in the component mentioned above, the range and channel conditions are met i.e., a communication link is established. Note that the open vertical edges of L can be defined by simply rotating the rectangles by 90 degrees. We shift L by (d/2, d/2) resulting in the dual lattice L′ . At this point, the center of a rectangle in L lies on the side of another rectangle in L′ . Thus bond-percolation in L′ results in infinite open path in L.

VI. P ERCOLATION IN THE S ECONDARY N ETWORK The analyses in the previous section were derived based on concepts of percolation in the Boolean model. However our model (PMLG with its range and channel conditions) is not a typical Boolean model where two points are said to be connected if they are in the range of each other. In our model, two points are connected if and only if the range and channel conditions are satisfied. In this section, we show that despite the range and the channel conditions of the PMLG, the concepts of percolation still hold true and apply to it. The domain of our secondary network (represented by the PMLG) is the continuous space R2 . Proving percolation in the continuum model is difficult given the additional constraints of the DSA network. However, if we could somehow map the continuum model to a discrete model, we will be able to use the well-known percolation results of the discrete model. Our objective is to couple the continuum percolation model with a two-dimensional discrete lattice L and show percolation in the lattice implies percolation in R2 . Let us first formally define the discrete lattice then show the coupling between the two. A. Discrete Grid (L) We construct a discrete square lattice L, with distance between neighboring vertices d > 0. The center of each edge q ∈ L will be denoted xq , yq . Let Eq be the event that the edge q is open. Thus Eq occurs if and only if: 1) The rectangle [xq −3d/4, xq +3d/4]×[yq −d/4, yq +d/4] is crossed from left to right by a sequence of neighboring secondary users as shown in Figure 7. 2) The squares [xq − 3d/4, xq − d/4] × [yq − d/4, yq + d/4] and [xq + d/4, xq + 3d/4] × [yq − d/4, yq + d/4] are 10 These include the hopping speed, oscillator settling time, receive filter settings, sensing threshold.

Fig. 7. Illustration of crossing of the connected component (sequence of neighboring nodes) from left to right and from top to bottom.

B. Coupling Percolation of L and R2 We use the results from Lemma 1 in [23] which couples percolation in the continuous and discrete models. When percolation occurs in L an infinite open path appears. The edges in this path correspond to the centers of the rectangles in L′ ; thus, there exists an infinite open path of rectangles. Connectivity of rectangles means satisfaction of conditions 1, 2, and 3 in Section VI-A. We emphasize: condition 3 indicates range and channel satisfaction for secondary users that are covered by two adjacent squares along the open edges. For any two adjacent edges, their associated rectangles intersect in the same square of L′ . Since both the edges are open (by assumption of the infiniteness of the open path), there exists a connected component crossing the two rectangles of the edges. The squares of all open edges on L′ are also infinite forming an infinite connected component in G(λs , rs , (1 − P0 )). Thus formation of the infinite open path on L, implies percolation (infinite path) in G(λs , rs , (1 − P0 )) (i.e., PMLG). VII. S IMULATION M ODEL AND R ESULTS In this section, we show the relationship between N , λp , λs , θp and the rendezvous protocols. We use |C| to denote the size of the biggest connected component in the secondary network while S denotes the total number of deployed secondary nodes. The ratio |C|/S denotes the relative size of the biggest connected component to the total number of deployed nodes11 . Obviously C alone does not imply whether percolation has occurred or not, this is why we use Cmax to refer to C when percolation occurs. The network is percolated once the biggest 11 Plots

of |C|/S and θp are obtained strictly from the simulation.

0.9 Sub-Critical Region

0.85

Super-Critical Region

0.8

θp= |Cmax| / S

0.75 0.7 0.65 0.6 0.55 0.5 0.45 0

40

80

120 λs

160

200

230

260

Fig. 8. θp versus λs with N = M = 1

Scenario 2: λp = 0, 2M ≤ N Here, we illustrate the effects of N and M on θp . We also show the value of N c as defined in Lemma 2. We use λs = 280 (for entire deployment area) and rs = 25. The value of N is incremented in steps of 1. The number of the transceivers per node is varied from M = 2 to M = 4. Other parameters remain the same as in Scenario 1. The resultant connectivity is shown in Fig. 10.

0.9

0.9 0.85 0.8

max

|/S

0.75 0.7

0.7

p

θ = |C

max

|/S

0.8

0.65

p

θ = |C

0.6

0.6

0.55 0.5 0.45 0

20

40

60

80 100 120 140 160 180 200 220 240 260

0.5 0

40

80

λs

(a)

120 λs

160

200

230

260

(b)

Fig. 9. a) θp versus λs with N = 5 and M = 4. b) θp versus λs with N = 5 and M = 3.

0.9 M=2 M=3 M=4

0.8 0.7 0.6 |C| / S

component (C) contains at least half of the secondary nodes i.e., S/2 ≤ |Cmax | ≤ S. Thus when percolation occurs, we get 0.5 < (θp = |Cmax |/S) ≤ 1. We present 4 scenarios and discuss the results accordingly. Scenario 1: λp = 0, 2M > N In this scenario we show the effects of λs on |C|/S and then we illustrate the results of Lemma 1. First, secondary nodes are deployed on a square region of area 300 × 300 following a Poisson distribution with density λs for the entire deployment area (i.e., λs /90000). λp is set to zero, N = 1 and each secondary user is equipped with one transceiver module (i.e., M=1) with rs = 25. The resultant connectivity is shown in Fig. 8. The figure shows a typical behavior of the Boolean model where the network’s connectivity transits from the sub-critical region to the supercritical region (i.e., network percolates) as λs > λcs . We see that phase transition occurs at λcs = 189. Moreover, in the super-critical region, |C|/S increase as λs increases. Next we illustrate the results of Lemma 1 by changing the values of N and M while maintaining 2M > N i.e., P0 = 0 with λp = 0. In our simulation, all the sets of M and N values with 2M > N produced similar connectivity to the ones obtained with M = N = 1. We have chosen to compare θp from N = M = 1 with θp from N = 5, M = 4 where the former is shown in Fig. 8 while the latter is shown in Fig. 9(a). Similarity between the two figures support our findings in Lemma 1. From Lemma 1, it also follows that with N = 5, the lowest theoretical value for M that still achieves 2M > N is M = 3. The corresponding connectivity (simulation) is shown in Fig. 9(b). Notably with M = 3, the network’s connectivity is similar to that with M = 4 (Fig. 9(a)) i.e., Lemma 1 provides a guideline to achieve the same results with minimal resources.

Super-Critical Region

0.5 0.4

Sup-Critical Region

0.3 0.2 0.1 0 0

2

4

6 8 10 12 14 16 18 Availailable Number of Channels N

20

22 23

Fig. 10. Relation between the number of channels and θp when M = 2, 3, 4.

From Fig. 10 it can be noted that for all the values of N when 2M > N , θp has similar values. As the value of N grows such that 2M ≤ N , channel abundance occurs and thinning takes place which reduces the value of θp . Decrease in θp continues as N increases until it reaches the critical value of N = N c . At that point, |C|/S becomes less than 0.5 and hence the network will no longer be percolated. From Fig. 10 it can also be seen that the rate of decrease of θp is inversely proportional to M . This is because as M increases, P0 decreases which reduces the thinning of the network. For example, in Fig. 10 when M = 2 and N = 8, the network is not percolated because |C|/S = 0.2963 while the same network is percolated at |Cmax |/S = 0.8484 with M = 4 and N = 8. Note that, although the two networks have the same number of nodes and channels, it is the value of M which changed the connectivity of the network from non-percolated to percolated. This can be used as a basis for trade-off between the cost of having more transceiver modules (M ) or number of channels to be scanned (in the given time budget) and the connectivity of the network (θp ). We find the critical value, N c , from Eqn. (6). We also find the same from Fig. 10 through simulations. The values obtained are shown in Table II. Percolation-wise, it can be noted that, the connectivity of the thinned network corresponds to the connectivity of a network whose density is (1−P0 )λs under no thinning (i.e., 2M > N ). For example, in Fig. 10 when M = 2, N = 10 and λs = 280, the network’s connectivity is in the sub-critical region. As per

M =2 M =3 M =4

Analytical Simulation 6 6 12 13 20 22 TABLE II c VALUES OF N

Eqn. 2 the corresponding P0 value is 0.63 and the resultant thinned density is (1 − 0.63) × 280 = 103.6. From Fig. 8, it can be noted that for λs = 103.6 the network’s connectivity is also in the sub-critical region, thus the two networks are in the same connectivity regions. Likewise, from Fig. 10 when M = 3 and N = 8, the connectivity is 0.763 and P0 = 0.18. Thus the thinned density is (1 − 0.18) × 280 = 230. It can also be verified from Fig. 8 that for λs = 230 the connectivity is 0.759, which is very close to 0.763. Lemma 2 also provides a guideline to establish connectivity12 using the minimum deployment density (λmin ). Two s for a network to cases arise. The first is to determine λmin s be deployed such that it percolates. The second is to find the minimal number of nodes that needs to be activated for an already deployed network with density λs1 to percolate. For the first case, our solution is attained via Eqn. (8), where we find the minimal deployed density. Thus for N = 4, and = 226.8 and the simulation M = 2, Eqn. (8) results in λmin s result in Fig. 11(a) shows λmin = 221. s For the second case, with λs1 > λmin , our solution is to s have λmin nodes active out of the λs1 deployed nodes. That s is, have each node randomly active with probability τ=

The secondary users are deployed according to a Poisson distribution with mean of 280. We consider rs = 25, N = 10, and M = 2. The primary user density, λp , is incremented from 0 to 200 primary nodes for the entire area while rp is set to opt 50. The resultant connectivity and P0 along with λL p , λp , U and λp are shown in Fig. 12. P0 in Fig. 12 was found as the ratio of all pairs with no common channel to the total pairs, while the corresponding analytical P0 was shown in Fig. 6. θp with the sub- and super-critical regions are shown in Fig. 13. 0.7

0.7

|C|/S P

opt p

λ

0

0.6

0.6

0.5

0.5

U

λp

L p

λ 0.4

0.4

0.3

0.3

0.2 0

10

20

30

40

50

60

70

80

0.2 90 100 110 120 130 140 150 160 170 180 190 200

λ

p

Fig. 12. Relation between λp , |C|/S, and P0 , for N = 10, M = 2, and λs = 280

0.7

λmin /λs1 s

λopt

0.65

p

0.6

Super-Critical Region

0.55

p

0.5

θ

which results in λmin nodes participating in percolation. Or, s in other words, 1 − λmin /λs1 nodes are put to sleep. s With λs1 = 280, N = 4, and M = 2, the equation above results in τ = 0.81. Fig. 11(b) shows the resultant connectivity for different values of τ , notably connectivity is first achieved at τ = 0.8222 which is within 1.5% of our theoretical finding.

L p

Sub-Critical Region

0.45

λ

U p

λ

0.4 0.35 0.3 0.25

0.8

0.8

0.75

0.75

0.7

0.7

0.2

θp

θp

0

0.65

0.6

0.55

0.55

0.5 0

50

100

150 λs

(a)

200

250

280

0.5 0

20

30

40

50

60

70

80

90 100 110 120 130 140 150 160 170 180 190 200

λp

Fig. 13. Relation between λp and θp for N = 10, M = 2, and λs = 280

0.65

0.6

10

0.2

0.4

0.6

0.8

1

τ

(b)

Fig. 11. θp versus λs for N = 4 and M = 2. b) θp versus τ for N = 4, M = 2 and λs1 = 280.

Scenario 3: λp > 0, 2M ≤ N Now, we illustrate the effects of λp on the connectivity of the secondary network (θp and |C|/S) and illustrate the two connectivity regions which were mentioned in Lemma 3. 12 When N >= 2M , λc < λ cannot be used directly as the connectivity s s condition due to thinning effects.

Notably, in Fig. 12 increase in primary density from 0 to λopt = 130 is accompanied by an increase in connectivity p of the secondary network (|C|/S) and decrease in P0 . This is because the original secondary network is in the channel abundance region and is suffering from thinning which is degrading its connectivity. The arrival of the primary users reduces the abundance of available channels for the secondary users thereby reducing P0 . It can be seen that percolation first occurs at λL p = 91. As λp increases, the set of available channels for the secondaries decreases which in turn increases the probability of the secondary nodes choosing the same channel. The

Fig. 14. Percolation of the secondary network with λs = 280, M = 4 and N = 10. a) λp = 5 leads to θp = 0.48. b) λp = 45 leads to θp = 0.93. c) λp = 180 leads to θp = 0.15.

0.95

1 0.9

0.9 0.8

0.8

0.7

0.75

0.6

p

0.85

P

0

θ

maximum connectivity is attained in the optimal point with λopt = 130 where the value of P0 is minimized. Further p increase in λp pushes the secondary nodes into the channeldeprivation region. In this region, the primary users are causing channel shortage/deprivation for the secondary nodes. This increases the secondary nodes that are in range and do not share a common channel i.e., increasing P0 . This is why P0 as was shown in Fig. 12. This starts to increase for λp > λopt p increase in P0 is accompanied by decrease in |C|/S as well as θp ; this continues until the secondary network is too thinned to be percolated which occurs at λU p = 191. From Figs. 12 and 13 we see that the secondary network percolates (θp > 0) opt in the interval [91 191]. The theoretical values for λL p , λp , U and λp are 96, 142, and 179 respectively. They are found by solving Eqns. (9) and (13) for λcs = 189. Fig. 14 illustrates the corresponding connectivity regions for a network with λs = 280, N = 10, M = 4 and 3 different values for λp . Scenario 4: Advanced Rendezvous Protocols Now, we illustrate the effects of using an advanced rendezvous protocol on the secondaries’ connectivity. We do not consider a specific technique, rather we consider a generic technique that yields rendezvous with m/T . We start with λs = 250, theoretically attaining connectivity requires λs m/T > 180 has to be greater than 189, thus we use m/T = 0.77. Next we increase λs and observe the resultant connectivity while keeping m/T = 0.77. The resulting connectivity is shown in Fig. 15(a). Next, we increase the number of channels (N ) and compare the resulting thinning probabilities for both the naive and the advanced protocols as shown in Fig. 15(b). It can be seen that the thinning probability of the naive technique is a function of N , however the advanced technique provides constant thinning probability 1 − m/T . Scenario 5: Lattice Deployment and Channel Abundance Though our results hold true for random deployment, we would like to emphasize that they hold true for other deployments as well. We point out that any other deployment does not affect the thinning probability P0 in Eqn. (2) because

0.7

M=1 M=2 M=3 1-m/T

0.5 0.4

0.65

0.3 0.6

0.2 0.55 0.5 240

0.1 260

280

300

320

0

340

λs

5

10

15

20

N

(a)

(b)

Fig. 15. a) Connectivity versus λs for m/T = 0.77. b) Comparison between the naive and an advanced rendezvous protocols with m/T = 0.77.

it depends on the number of channels and the number of transceivers only. However, the deployment type (random, grid, in-homogeneous) affects the percolation condition and the deployment density; hence all the equations that depend on them have to be re-computed based on the connectivity and density conditions of the new given deployment. For example, if a grid deployment is used (square lattice), then Eqn. (3), which shows the connectivity condition in the continuum domain, will no longer be λcs < (1−P0 )λs . Instead we will use the percolation condition of the square lattice which is P c < P [16]. P is the occupancy probability of the lattice site, while P c is the critical site occupancy probability i.e., the minimum occupancy probability required for the lattice sites to percolate. For square lattices, P c is found to be 0.591 [20]. Following the same approach from section IV, the percolation condition for a network deployed on the grid (with site occupancy of P ) under channel abundance becomes: 0.591 < (1 − P0 )P In terms of N c , the above equation can be re-written as:
! N c −M 0.591 =

1−

M
Nc M

P

(15)

For a network with P = 1 i.e., fully occupied lattice and M = 2, the resultant connectivity13 is shown in Fig. 16(a). It 13 The

connectivity result is obtained via simulation

can be seen that for N < 4 (no thinning), the corresponding connectivity is 100%. However as N grows to more than 4 channels, P0 becomes greater than zero (thinning occurs) and some nodes become invisible to the giant component which reduces the connectivity. As N increases further, more nodes are rendered invisible until the giant component disappears at 8 channels i.e., N c = 8. This corresponds to the theoretical value for N c = 8 obtained from solving Eqn. (15) for P = 1 and M = 2. Next, we keep M = 2 and set P = 0.8 i.e., 80% occupied grid. The resultant connectivity is shown in Fig. 16(b). With increase of N , the figure shows that the that network loses connectivity at N c = 6 which coincides with the theoretical value of N c = 6 obtained from solving Eqn. (15) for P = 0.8 and M = 2. For this network, connectivity is lost at N = 6 as compared to N = 8 for the previous one. The reason is that with P = 0.8 and no thinning, the network is 80% connected which is the best possible hence it takes less thinning (smaller N ) to disconnect the network. It is these kinds of scenarios where the effects of multi-channel become more relevant and the selection of the rendezvous protocol becomes important. 1

0.8

0.8

0.6

0.6 θp

θp

1

0.4

0.4

0.2

0

0.2

2

4

6 N

(a)

8

10

0 0

2

4

N

6

8

10

(b)

Fig. 16. a) Connectivity of full lattice for various values of N and M = 2. b) Connectivity of 80% occupied lattice for various values of N and M = 2.

VIII. C ONCLUSIONS Though connectivity in secondary DSA networks is a wellstudied problem, the physical constraints are usually not considered neither the effects of rendezvous protocols. In this research, we consider that the DSA nodes can simultaneously transmit over a finite number of channels. We argue that the probability of nodes being invisible increases with channel abundance when the naive technique is used. On the contrary, we show that advanced rendezvous protocols can maintain network connectivity as long as the number of channels is within a specified range. the underlying network connectivity does not change. Using concepts from percolation theory, we show how and under what conditions the secondary network percolates for both protocols. Using the projection of a multi-layered graph, we show how dimensionality can lead to thinning. We define connectivity of the secondary network and characterize it in terms of the available number of channels, node densities. number of simultaneous channel transmissions per node, and communication range. We find the critical values at which phase transition occurs and identify

the parameters when the secondary network has plenty of channels, just enough channels, and lack of them. We also show the regions where the naive technique can result into network connectedness as well as regions where it fails and it becomes necessary to use advanced protocol. R EFERENCES [1] M. J. Abdel-Rahman, H. Rahbari, M. Krunz, “Multicast Rendezvous in Fastvarying DSA Networks”, IEEE Transaction on Mobile Computing, Issue: 99, 2014, pp. 1-14. [2] O. A. H. Al Tameemi, A. Al-Rumaithi, M. Chatterjee, K. Kwiat, C. Kamhoua, “ Connectivity and Rendezvous in Distributed DSA Networks”, IEEE ICNC, 2016, pp. 1-7. [3] S. Anand, S. Sengupta, R. Chandramouli, “Is Channel Fragmentation/bonding in IEEE 802.22 Networks Secure?” IEEE ICC, 2011, pp. 1-5. [4] K. Bian, J.M. Park, “Asynchronous channel hopping for establishing rendezvous in cognitive radio networks”, in Proceedings of IEEE INFOCOM, 2011, pp. 236-240. [5] P. Balister, B. Bollobas, and M. Walters, “Continuum percolation with steps in an annulus”, Annals of Applied Probability, Vol. 14, 2004, pp. 1869-1879. [6] K. Christensen, “Percolation Theory,” Imperial College London, 2002, pp. 1-2. [7] O. Dousse, F. Baccelli, and P. Thiran, “Impact of interference on connectivityin ad hoc networks”, IEEE/ACM Transaction on Networking, Vol. 13, No. 2, April 2005, pp. 425- 436. [8] O. Dousse, F. Baccelli, N. Macris, R. Meester and P. Thiran, “Percolation in the Signal to Interference Ratio Graph”, J. Appl. Prob, Vol. 43, 2006, pp. 552-562. [9] O. Dousse, P. Mannersalo, P. Thiran, “Latency of Wireless Sensor Networks with Uncoordinated Power Saving Mechanisms”, ACM MobiHoc, 2004, pp. 109-120. [10] M. Franceschetti and R. Meester, “Critical node lifetimes in random networks via the Chen-Stein method,” IEEE Trans. Inform. Theory, Vol. 52, 2006, pp. 28312837. [11] M. Franceschetti, L. Booth, M. Cook, R. Meester, and J. Bruck, “Percolation of Multi-Hop Wireless Networks,” University of California, Berkeley, EECS Dept., UCB/ERL M03/18, 2003, pp. 115. [12] M. Franceschetti, R. Meester, “Random Networks for Communication”, New York: Cambridge Univerisity Press, 2007. [13] L. Fu, Z. Liu, D. Nie, X. Wang, “K-connectivity of cognitive radio networks”, IEEE International Conference on Communications (ICC), June 2012, pp. 83-87. [14] L. Fu, L. Qian, X. Tian, H. Tang, N. Liu, G. Zhang, X. Wang, “Percolation Degree of Secondary Users in Cognitive Networks”, IEEE J. on Sel. Areas in Communications, Vol. 30, No.10, November 2012, pp. 1994-2005. [15] E. N. Gilbert, “Random plane networks”, Journal of SIAM, 1961, pp. 533-543. [16] G. Grimmet, “Percolation, 2nd ed”, Springer-Verlag, 1999. pp. 1-439. [17] P. Gupta and P. R. Kumar, “Critical Power for Asymptotic Connectivity in Wireless Networks”, in Proceedings of IEEE Conference on Decision and Control, Dec. 1998, pp. 1106-1110. [18] M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, “Stochastic Geometry and Random Graphs for the Analysis and Design of Wireless Networks”, IEEE Journal on Se. Areas in Communications, Vol 27, 2009, pp 10291046. [19] S. Haykin, “Cognitive radio: brain-empowered wireless communications”, IEEE J. on Sel. Areas in Communications, 23 (2005) (2), pp. 201-220. [20] J. L. Jacobsen, “High-precision percolation thresholds and Potts-model critical manifolds from graph polynomials”, Journal of Physics A Mathematical and Theoretical, 2014, pp. 1-16. [21] L. Jiao, et. al., “On the performance of Channel Assembling and Fragmentation in Cognitive Radio Network”, IEEE Tran. on Wireless Communications, Issue 99, 2014, pp1-14. [22] J.F.C. Kingman, “Poisson Processes”, Clarendon Press, 1992. [23] D. Lu, X. Huang, P. Li, J. Fan, “Connectivity of large-scale Cognitive Radio Ad Hoc Networks”, IEEE INFOCOM 2012, March 2012, pp. 1260–1268. [24] R. Meester and R. Roy, “Continuum Percolation”, New York: Cambridge Univerisity Press, 1996. [25] M. D. Penrose, “On k-connectivity for a geometric random graph”, Random Structures and Algorithms, Vol. 15, No. 2, Sept. 1999, pp. 145-164. [26] M. Penrose and A. Pisztora, “Large deviations for discrete and contin-uous percolation,” Advances in Applied Probability, Vol. 28, 1996, pp. 29-52. [27] W. Ren, Q. Zhao, A. Swami, “Connectivity of Heterogeneous Wireless Networks”, IEEE Transactions on Information Theory, vol. 57, no.7, July 2011, pp. 4315-4332. [28] D. Stauffer, “Introduction to Percolation Theory”, Taylor and Francis, 1994. [29] N. C. Theis, R.W. Thomas, L. A. DaSilva, “Rendezvous for Cognitive Radios”, IEEE Tran. on Mobile Computing, Vol. 10, No.2, February 2011, pp. 216-227. [30] S. Torquato, Y. Jiao, “Effect of dimensionality on the percolation thresholds of various d-dimensional lattices”, American Physical Society, vol.87, Issue 3, 2013.