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arXiv:1507.06239v1 [cs.SY] 22 Jul 2015
1
Fast Desynchronization For Decentralized
Multichannel Medium Access Control
Nikos Deligiannis, Jo˜ao F. C. Mota, George Smart, and Yiannis Andreopoulos
Abstract—Distributed desynchronization algorithms are key to
wireless sensor networks as they allow for medium access control
in a decentralized manner. In this paper, we view desynchro-
nization primitives as iterative methods that solve optimization
problems. In particular, by formalizing a well established desyn-
chronization algorithm as a gradient descent method, we establish
novel upper bounds on the number of iterations required to reach
convergence. Moreover, by using Nesterov’s accelerated gradient
method, we propose a novel desynchronization primitive that
provides for faster convergence to the steady state. Importantly,
we propose a novel algorithm that leads to decentralized time-
synchronous multichannel TDMA coordination by formulating
this task as an optimization problem. Our simulations and
experiments on a densely-connected IEEE 802.15.4-based wireless
sensor network demonstrate that our scheme provides for faster
convergence to the steady state, robustness to hidden nodes,
higher network throughput and comparable power dissipation
with respect to the recently standardized IEEE 802.15.4e-2012
time-synchronized channel hopping (TSCH) scheme.
Index Terms—Medium access control, desynchronization, gra-
dient methods, decentralized multichannel coordination.
I. INTRODUCTION
IN WIRELESS sensor networks (WSNs), achieving and
maintaining (de)synchronization among the nodes supports
various functionalities, including data aggregation, duty cy-
cling, and cooperative communications. In particular, devising
protocols that perform desynchronization at the medium access
control (MAC) layer is keyin achieving fair TDMA scheduling
among the nodes in a channel [2]–[7].
In order to extend fair TDMA scheduling to large-scale
networks, protocols that achieve (de)synchronization across
multiple channels [4], [5] are required. Typical approaches are
infrastructure-based (i.e., centralized), as they use a coordi-
nation channel and/or node and a global clock (e.g., via a
GPS system) [5]. Channel hopping is been accepted as a good
solution for MAC-layer coordination for dense WSN topolo-
gies. According to channel hopping, nodes hop between the
available channels of the physical layer such that they are not
constantly using a channel with excessive interference. Form-
ing the state-of-the-art, the time-synchronized channel hopping
(TSCH) [5] protocol is now part of the IEEE 802.15.4e-
2012 standard [8]. In TSCH, each node reserves timeslots
This work has been presented in part at the 14th International Conference
on Information Processing in Sensor Networks (IPSN ’15) [1].
N. Deligiannis is with the Department of Electronics and Informatics,
Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium, and also with
iMinds, Ghent 9050, Belgium (email: ndeligia@etro.vub.ac.be).
J. F. C. Mota, G. Smart, and Y. Andreopoulos are with the Electronic
and Electrical Engineering Department, University College London, Roberts
Building, Torrington Place, London, WC1E 7JE, UK (e-mail: {j.mota,
george.smart, i.andreopoulos}@ucl.ac.uk).
within the predefined slotframe interval and within the 16
channels of IEEE 802.15.4. However, filling up the avail-
able slots follows an advertising request-and-acknowledgment
(RQ/ACK) process on a coordination channel. This channel is
prone to interference and self-inflicted collisions when nodes
advertise slots aggressively. Moreover, when nodes leave the
network, their slots may remain unoccupied for long periods
until another advertisement process reassigns them to other
nodes. This limits the bandwidth usage per channel and does
not allow for fast convergence to the steady state1. It is also
important to note that TSCH requires a coordinator to maintain
global time synchronization [3], [5].
To achieve infrastructure-less (i.e., decentralized) WSN
MAC-layer coordination, distributed (de)synchro-nization al-
gorithms have attracted a lot of interest [2], [5], [6], [9]–[17].
These algorithms are inspired by biological agents modeled
as pulse-coupled oscillators (PCOs) [6], [14], [18], namely, as
timing mechanisms following a periodic pulsing (i.e., beacon
packet transmission at the MAC) that is updated via the
timings of pulses heard from other nodes.
Most work on distributed (de)synchronization is based on
the PCO dynamics model introduced by Mirollo and Strogatz
[18], and derives several algorithms with properties of practical
relevance to WSN deployments, namely: (i) limited listening
[2], [19], [20], a property that is imperative for low energy
consumption in wireless transceivers; (ii) solutions amenable
to multi-hop network topologies and the existence of hidden
nodes [2], [11], [17]; (iii) solutions scalable to large groups
of nodes [6], [15]; and (iv) modifications that lead to fast
convergence to steady state [12]–[14], [21]. PCO-based syn-
chronization methods have also been interpreted as consensus
algorithms for multi-agent systems [22]–[24]. The work in
[22] studied synchronization of networked oscillators under
heterogeneous time-delays and varying topologies. In [24], the
synchronization of networked oscillators was modeled using
coupled discrete-time phase locked loops.
Regarding the study of the convergencespeed of desynchro-
nization algorithms, mostly estimates based on simulations
or empirical measurements have been derived. In effect, only
lower bounds [14], [19], order-of-convergence estimates [2],
[6], [19] and operational estimates [25] have been established.
However, no upper bounds are currently known for the conver-
gence speed of desynchronization algorithms, despite the fact
that such bounds provide for worst-case guarantees of time and
1Both high network throughput and quick convergence are important for
WSNs that operate with a periodic wake-up cycle (or are event-triggered) and
must quickly converge to a steady operational state and transmit high data
volumes before being re-suspended.
2
0
θi+1(ti−1)
θi−1(ti−1)θi(ti−1)
θ′
i(ti−1)
Fig. 1. Phase update of node iaccording to the DESYNC algorithm: node i−
1fires at time ti−1, and node iupdates its phase from θi(ti−1)to θ′
i(ti−1),
towards the average of the phases of nodes i−1and i+1, its phase neighbors.
energy consumption to achieve the state of desynchrony. Fur-
thermore, despite the plethora of works on PCOs, the problem
of extending distributed (de)synchronization algorithms to the
multichannel case (which is key in today’s wireless networks)
has received limited attention. A preliminary attempt was done
in [7], where desynchronization was independently applied per
channel. The limitation of the scheme in [7] is that, since the
nodes in different channels are not synchronized, when a node
switches channels convergence needs to be established anew.
In this work, we view the problem of desynchronization
as an optimization problem. In particular, we show that a
minor modification of the well established DESYNC algorithm
[2], [10] is the gradient descent method applied to a specific
optimization problem. Although desynchronization can also
be viewed from a consensus perspective [23], the optimiza-
tion approach is more powerful as it allows deriving faster
algorithms [26], [27]. Our contributions are as follows:
•We establish novel upper bounds on the convergence rate
of the DESYNC process. Such bounds can yield reliable
estimates of worst-case energy consumption and time re-
quired for convergence, which are important for systems
that operate under delay and/or energy constraints.
•We propose a novel desynchronization algorithm based
on Nesterov’s accelerated gradient method [28], [29].
We show, both theoretically and experimentally, that the
proposed algorithm leads to faster convergence to steady
state than the conventional DESYNC algorithm [2], [10].
•We propose a novel distributed multichannel method
that jointly performs synchronization across channels
and desynchronization within each channel. Contrary to
[7], the proposed algorithm leads to time-synchronous
multichannel TDMA coordination (where nodes allocated
the same timeslot in adjacent channels are synchronized).
In this way, nodes can swap channels (thus, avoiding
persistent interference in certain channels and achieving
higher connectivity) without the network exiting the
steady state.
•Finally, via simulations and experiments using a real
WSN deployment abiding by the IEEE802.15.4 stan-
dard, we show that our approach leads to decentralized
time-synchronous multichannel MAC-layer coordination
that achieves higher network throughput compared to
the state-of-the-art TSCH [5] protocol, while incurring
comparable power consumption.
The paper continuous as follows: Section II presents the
background on PCO methods, while Section III derives our
upper bound for the desynchronization process and proposes
our novel accelerated desynchronization algorithm. Section IV
presents our novel formulation of multichannel coordination.
Simulations and experiments using a WSN deployment are
given in Section V, while Section VI concludes the paper.
II. BACKGROUND ON PULSE-COUPLED OSCILLATORS
Consider a fully-connected WSN comprising nnodes, each
acting as a pulse-coupled oscillator [18]. When a node does
not interact with others, it broadcasts a fire message or
pulse periodically. This is modeled by assigning to node i
aphase θi(t), whose value at time tis given by [2], [19]
θi(t) = t
T+φimod 1 ,(1)
where φi∈[0,1] is the phase offset of node iand mod 1
denotes the modulo operation with respect to unity. Fig. 1
illustrates (1) graphically: the phase θi(t)of node ican be
seen as a bead moving clockwise on a circle, whose origin
coincides both with 0and 1[6], [18], [19], [30]. If φiis
constant, which happens when the nodes do not interact, node i
broadcasts a fire message every Ttime units, when θi(t) = 1,
and then sets its phase to zero. When the nodes interact,
e.g., by listening to each others’ messages, they modify
their phases (specifically, their phase offsets), according to an
update equation that expresses the PCO dynamics [18]. One of
the most prominent PCO algorithms for desynchronization at
the MAC layer of WSNs is the DESYNC algorithm [2], [10].
In DESYNC, the nodes are ordered according to their initial
phases: 0≤θ1(0) < θ2(0) <··· < θn(0) <1. Assuming
perfect beacon transmission and reception, the order of the
firings in DESYNC will remain the same [2], [10]. The phase
θiof each node iis updated based on the phases θi−1and θi+1
of its phase neighbors, nodes i−1and i+ 1, respectively. This
is illustrated in Fig. 1: immediately after node i−1transmits
a fire message, node imodifies its phase according to
θ′
i(ti−1) = (1−α)θi(ti−1) + αθi−1(ti−1) + θi+1(ti−1)
2,(2)
where ti−1is the time instant in which node i−1fires, i.e.,
θi−1(ti−1) = 1, and i= 1,2,...,n, with periodic extension at
the boundaries. The jump-phase parameter α∈(0,1) controls
the phase increment [2], [10].
When node iupdates its phase, it has stale knowledge of the
phase of node i+ 1, namely, it only knows the previous value
of θi+1 and not the current one. This is because node i+ 1
modified its phase when node ifired, but the value of the
new phase has not been “announced” yet [10]. In DESYNC,
each node: (i) updates its phase once in each firing round (we
say that a firing round is completed when each node in the
network has fired exactly once); (ii) does not need to know
the total number of nodes, n, in the network; (iii) requires
limited listening, as only the messages from the two phase
neighbors are required. These features make DESYNC quite
popular [2], [10]. For a fully-connected network, it has been
shown that (2) converges to the state of desynchrony at time t,
after which the interval between consecutive firings is T /n up
to a small threshold ǫ. Under partial connectivity or hidden
3
nodes, convergence is still achieved under a wide variety of
topologies, but the node firings may not be equidistant [2]. It
has been conjectured via simulations [10], [30] that DESYNC
converges to desynchrony (i.e., perfect TDMA scheduling) in
rDESYN C =O1
αn2ln 1
ǫ(3)
firing rounds. Recently, under the assumption of uniformly
distributed initial firing phases, an operational estimate for
the number of firing rounds for the DESYNC algorithm’s
convergence was derived [25]. However, no upper bounds are
known for the desynchronization process.
III. DESYNC AS A GRADIENT METHOD
We start by showing that, considering a fully-connected
network, a minor modification of DESYNC [2], [10] can be
viewed as a gradient descent method solving an optimization
problem. Then, we establish novel convergence properties of
the resulting method and derive a new accelerated desynchro-
nization primitive.
Staleness of DESYNC:Fig. 2 shows five consecutive con-
figurations of the phases of the nodes of a network with four
nodes. The purpose is to illustrate how the phases of the nodes
are updated in the first iteration of DESYNC [2], [10] and to
highlight our minor modification. For simplicity, we omit the
time dependence of the phases, but use a superscript to indicate
how many times they have been updated. In Fig. 2(a), no
firing has yet occurred. The first update occurs when node 2
fires, whereby node 3updates its phase from θ(0)
3to θ(1)
3[see
Fig. 2(b)]. According to (2), this update requires knowing θ2
(which is equal to 1because node 2is firing) and θ4(which
is known because node 4was the first to fire). The second
phase update occurs in Fig. 2(c): node 1fires, and node 2
updates its phase from θ(0)
2to θ(1)
2. According to (2), this
update requires the value of θ1(known because node 1is
firing) and θ3. The current value of θ3(actually, θ(1)
3) is not
known because node 3has not fired since it updated its phase.
Therefore, node 2will use θ(0)
3rather than θ(1)
3. This is why
we say that DESYNC is stale: each update uses stale versions
of the phases. In step (d), node 1updates its phase and also
uses a stale version of the phase of node 2. Finally, in step (e),
node 4updates its phase using a stale version of the phase of
node 1. We assume, however, that in contrast with the other
nodes, this update uses the value θ(0)
3(in gray) and not θ(1)
3.
Assumption 1. In DESYNC, node nupdates its phase at
iteration kusing θ(k−1)
n−1in place of θ(k)
n−1.
Via Assumption 1, all updates in Fig. 2 use the initial
values θ(0)
1,θ(0)
2,θ(0)
3, and θ(0)
4. In practice, this assumption
does not lead to a discernible difference in the performance
of DESYNC.
Vector notation: Suppose we are in the k-th firing round,
i.e., all nodes have updated their phases k−1times. We have
already mentioned how the firing of a node i, say at time ti,
enables other nodes to determine the current value of φ(k−1)
i
in (1): φ(k−1)
i= 1 −ti/T . Knowing this, each node can
determine the value of θ(k−1)
i(t)for any time instant. We will
now see how the update rule (2) translates into the updates
of the phase offsets. Replacing (1) into (2) at firing round
(iteration) k, we obtain
θ′
i(ti−1) = ti−1
T+φ(k)
i
= (1 −α)hti−1
T+φ(k−1)
ii
+α
2hti−1
T+φ(k−1)
i−1+ti−1
T+φ(k−1)
i+1 i
=ti−1
T+ (1 −α)φ(k−1)
i+αφ(k−1)
i−1+φ(k−1)
i+1
2.
Eliminating the term ti−1/T , we get: φ(k)
i= (1 −α)φ(k−1)
i+
αφ(k−1)
i−1+φ(k−1)
i+1
2.In a strict sense, this expression is only valid
for i= 2,...,n−1as the updates for nodes 1and nrequire
a correcting term to compensate the fact that each θwraps
around 1. Therefore, the updates for all nodes are
φ(k)
1= (1 −α)φ(k−1)
1+α
2φ(k−1)
2+φ(k−1)
n−1(4)
φ(k)
i= (1 −α)φ(k−1)
i+α
2φ(k−1)
i−1+φ(k−1)
i+1 ,2≤i≤n−1
(5)
φ(k)
n= (1 −α)φ(k−1)
n+α
2φ(k−1)
n−1+φ(k−1)
1+ 1.(6)
Without Assumption 1, φ(k−1)
1in (6) would be replaced
with φ(k)
1. It is, however, this assumption that enables us to
write (4)-(6) in vector form:
φ(k)=
1−αα
20··· 0α
2
α
21−αα
2··· 0 0
.
.
.....
.
..
.
.
α
20 0 ··· α
21−α
φ(k−1)−α
2d,
(7)
where φ(k)= (φ(k)
1, φ(k)
2,...,φ(k)
n)∈Rnis a vector
containing the phases of all the nodes at iteration k, and
d:= (1,0,...,0,−1) ∈Rn. Equation (7) has the format
of the updates usually found in the discrete-time consensus
literature [23], [31], [32]. In particular, the matrix in (7)
can be seen as the Perron matrix of a network with a ring
topology and the vector dcan be seen as an input bias [23].
This observation can be used to provide upper bounds on
the convergence rate of (7). However, one can view (7) as
an algorithm solving an optimization problem since, besides
also providing upper bounds, this interpretation enables the
derivation of an accelerated version of desynchronization. This
interpretation is formalized next.
Proposition 1. Let φ(k)= (φ(k)
1, φ(k)
2,...,φ(k)
n)denote the
phases of all nodes at firing round k. If Assumption 1 holds,
then DESYNC (2) and (7) is the steepest descent method
applied to
minimize
φg(φ) := 1
2
Dφ −v1n+en
2
2(8)
where v= 1/n,1n∈Rnis the vector of ones, en=
4
0
θ(0)
4
θ(0)
3
θ(0)
2
θ(0)
1
(a)
0
θ(0)
4
θ(0)
2
θ(0)
1
θ(0)
3
θ(1)
3
(b)
0
θ(0)
4
θ(1)
3
θ(0)
1
θ(0)
2
θ(1)
2
(c)
0
θ(0)
4
θ(1)
3
θ(1)
2
θ(0)
1
θ(1)
1
(d)
0
θ(0)
4
θ(1)
3
θ(0)
3
θ(1)
2
θ(1)
1
θ(0)
4
θ(1)
4
(e)
Fig. 2. Updates during the first iteration of DES YNC in a 4-node network: (a) initial phases; no firing has occurred yet; (b) the first update occurs when
node 2fires and after nodes 4and 3have fired. The firing of node 2causes node 3to update θ(0)
3to θ(1)
3. In the remaining steps, node ifires and node j
updates its phase, where (i, j)is (1,2) in (c), (4,1) in (d), and (3,4) in (e). All phases are updated as a function of the initial values, i.e., although some
of the phases have already changed, the updates use always θ(0)
1,θ(0)
2,θ(0)
3, or θ(0)
4, and not the new values.
(0,0,...,0,1) ∈Rn, and
D=
−1 1 0 0 ... 0
0−1 1 0 ... 0
.
.
........
.
.
0... 0 0 −1 1
1... 0 0 0 −1
∈Rn×n.(9)
Specifically, the updates in (7) can be written as
φ(k)=φ(k−1) −α
2∇g(φ(k−1)).(10)
Proof: Since DT1n=0n, we have
∇g(φ) = DT(Dφ −v1n+en) = DTDφ +d,(11)
where d=DTenis the vector that appears in (7). Therefore,
the steepest descent applied to (8) yields
φ(k)=φ(k−1) −β∇g(φ(k−1))
=φ(k−1) −βDTDφ(k−1) −βd
= (In−βDTD)φ(k−1) −βd,(12)
where Inis the identity matrix in Rn. Replacing β=α/2,
we obtain
φ(k)= (In−α
2DTD)φ(k−1) −α
2d,(13)
The last equation is exactly (7).
We set v=1
nin (8) to emphasize that the goal of DESYNC
is to disperse the nphases throughout [0,1]. However, any
other value for vwould lead to the same update rule, since
the gradient of the objective function does not depend on v;
see (11) in the proof. This confirms the fact that DESYNC does
not require the knowledge of the number of nodes, n, in the
network [10]. Notice also that Dis not full rank; therefore, the
objective of (8) is not strictly convex. Indeed, the nullspace
of Dis {z1n:z∈R} ∪ {0n}. Consequently, if φis a
solution of (8), so is φ+z1nfor any z∈R. We notice
that the interpretation of Proposition 1 is akin to the one that
views consensus algorithms as gradient descent methods for
minimizing Pn
i=1(φi−θi)2, where θiis the observation of
agent i[26], [33].
This interpretation of DESYNC provides for: (i) an alterna-
tive way to establish the values of αfor which convergence
holds, and (ii) an upper bound on the number of the firing
rounds until convergence.
Corollary 1. Every limit point of the sequence produced by
the DESYNC algorithm (7) with α∈(0,1) is a stationary
point of (8).
Proof: The proof is given in Appendix A.
Corollary 2. Let φ(0) represent the vector of initial phases,
and let φ⋆be any solution of (8). Suppose 0n≤φ(0) ≤1n.
Then, the number of firing rounds, rD, that DESYNC (4)–(6)
requires in order to generate a point φthat has accuracy
ǫ:= gφis upper bounded as
rD≤kφ(0) −φ⋆k2
2
2α(1 −α)1
ǫ−1
gφ(0)(14)
≤1
6nα(1 −α)7
2n2+ 3n+ 41
ǫ−1
gφ(0).(15)
Proof: The proof is given in Appendix A.
Corollary 1 confirms Theorem 1 in [10] regarding the sta-
bility and convergence of DESYNC, albeit using different tools
and without requiring simulations to illustrate the avoidance
of limit cycles. Corollary 2 complements the existing order-
of-convergence estimate of (3) and the operational estimates
derived by Buranapanichkit et al. [25] by deriving an upper
bound for the firing rounds to achieve convergence. Such an
upper bound allows for reliable estimates of worst-case energy
consumption and time, expressed in number of firing rounds or
iterations, required to reach convergence. These estimates are
important for systems that operate under delay and/or energy
constraints. Notice that the bound in (15) is a function of
known system parameters, namely, the number of nodes n,
the jump-phase parameter α, the tolerance parameter ǫ, and
the evaluation of g(·)on the initial phase vector (the latter
can be ignored yielding a looser bound).
The FAST-DESYNC algorithm based on Nesterov: A key
advantage of viewing desynchronization as an optimization
problem is that we can create new primitives that converge to
desynchrony much faster. Particularly, we can use Nesterov’s
fast gradient algorithm [28], [29] (here we use the adaptation
in [34]):
φ(k)=µ(k−1) −β∇g(µ(k−1))(16a)
µ(k)=φ(k)+k−1
k+ 2 φ(k)−φ(k−1) ,(16b)
5
where µ(k)∈Rnis an auxiliary vector. Nesterov’s method is
applicable under the same assumptions as the steepest descent,
i.e., when gis continuously differentiable and its gradient is
Lipschitz continuous with constant L. However, it requires
0< β ≤1/L rather than 0< β < 2/L. At the expense
of small extra memory and computation, Nesterov’s method
takes O(1/√ǫ)iterations to produce a point φthat satisfies
g(φ)−g(φ⋆)≤ǫ, where φ⋆minimizes g. Recall that the
steepest descent takes O(1/ǫ)to produce such a point [cf.
(15)]. We shall show that this improved performance in terms
of bounds is also observed experimentally. Note that µ(k),φ(k)
converge to the same point, i.e., kφ(k)−µ(k)k → 0as
k→ ∞. More importantly, Nesterov showed in [29] that (16)
has optimal convergence rate among first-order methods, i.e.,
methods that use information about first-order derivatives only,
possibly from all past iterations.
We propose applying Nesterov’s algorithm (16a)-(16b) to
solve (8). This yields a primitive that we call FAST-DESYNC.
Node i= 1,...,n holds two variables φiand µi, which are
updated at iteration kas
φ(k)
i= (1 −α)µ(k−1)
i+α
2µ(k−1)
i−1+µ(k−1)
i+1 −di(17a)
µ(k)
i=φ(k)
i+k−1
k+ 2φ(k)
i−φ(k−1)
i,(17b)
where d1= 1,dn=−1, and di= 0 for i= 2,...,n −1.
Note that (17a) is identical to the DESYNC updates (4)–(6).
The only detriment is that each node needs an extra memory
register to store φ(k−1)
i, which is used in (17b), and perform
the extra computations in (17b). Under this modification, the
following holds:
Corollary 3. Let α∈(0,1/2] and let 0n≤φ(0) =µ(0) ≤1n
represent the vectors of initial phases. Let also φ⋆be any
solution of (8). Then, the number of firing rounds that FAST-
DESYNC (17a)-(17b) requires to generate a point φthat has
accuracy ǫ:= g(φ)is upper bounded as
rFD ≤2
√α ǫ
φ(0) −φ⋆
2(18)
≤2s1
3nαǫ 7
2n2+ 3n+ 4.(19)
Proof: The proof is given in Appendix A.
Contrasting (15) and (19) we notice that FAST-DESYNC
allows for significant reduction in the order-of-iterations for
convergence compared to DESYNC, particularly, O(pn/ǫ)
versus O(n/ǫ), respectively.
IV. EXTENSION TODECENTRALIZED MULTICHANNEL
COORDINATION
We now describe our algorithm that jointly applies syn-
chronization across channels and desynchronization in each
channel. We assume that all nodes can receive all fire message
broadcasts in their channel. We will show experimentally,
however, that our proposal works even for densely-connected
WSNs (when some nodes cannot be reached by others), as
DESYNC still converges in such cases [2]. We first describe
our protocol.
(a) (b)
Fig. 3. (a) Initial random state of n= 14 nodes in C= 4 channels;
(b) steady state of the proposed protocol with nc= 3 nodes for channels
c= 1 and c= 2, and nc= 4 nodes in channels c= 3 and c= 4. The
DESYN C nodes (in white) allow for intra-channel desynchronization, while the
SYNC nodes (in grey) provide for cross-channel synchronization. Nodes that
belong to balanced channel and that fire synchronously can swap channels.
The horizontal position of a node indicates the firing moment.
A. Proposed Decentralized Multichannel MAC-layer Coordi-
nation
Let a WSN comprise nnodes that are initially randomly
distributed in Cchannels [see Fig. 3(a)]—for example, the
C= 16 channels of the IEEE 802.15.4 standard [35], [36].
The maximum achievable throughput per node is obtained
when the nodes are uniformly distributed across the available
channels and a perfect TDMA scheduling is reached in each
channel. When the total number of nodes in the network, n, is
divisible by Cour protocol will lead to nc=n
Cnodes being
present in each channel, alternatively, nc=n
C,n
C
nodes will be present in each channel, as shown in Fig. 3(b).
Existing mechanisms, such as the one in [7], can take
place during convergence to balance the number of nodes.
Specifically, a node lying in channel cmay switch to channel
c+1 (with cyclic extension at the border), if it detects that less
nodes are present there. Detection of the number of nodes in
a channel is possible by integrating this information in the fire
messages transmitted by the nodes. In [7], in order to detect
the number of nodes in channel c+ 1, nodes within channel
cproactively switched channels for short time intervals [7].
Here, however, we follow a different approach, which is akin
to the proposed algorithm. In particular, a single node (which
we later call SYNC) lying in channel cis elected to listen for
fire messages in channel c+1. This specific node may jump to
the next channel if it detects that less nodes are present there.
When a SYNC node jumps from one channel to the next, both
channels are set to elect their SYNC nodes anew. In order
to avoid a race condition, where nodes continuously jump
channels, the following conditions are defined for channel
switching:
(nc−nc+1 ≥1,if c∈[1, C)
nc−nc+1 ≥2,if c=C
where ncdenotes the number of nodes present in channel
c, with n=
C
P
c=1
nc. The switching rule and conditions ensure
that, after a few firing periods, there will be nc∈ {n
C,n
C}
nodes in each channel c.
When the channels have been balanced, the proposed it-
erative joint synchronization-desynchronization algorithm is
6
applied. By considering that each node acts as a pulse-coupled
oscillator with a period of Tseconds, our novel algorithm
(see Section IV-B) leads to decentralized multichannel round-
robin scheduling. The nodes in each channel are divided in
two classes. Specifically, all but one node in each channel
apply desynchronization so as to achieve TDMA within the
channel (these nodes are denoted as “DESYNC”). DESYNC
nodes operate only within their channel, firing and listening to
messages from the other nodes in their channel. In addition,
one “SYNC” node per channel performs cross-channel syn-
chronization to achieve a time-synchronous slot structure [Fig.
3(b)]. The SYNC node of each channel listens for the SYNC
fire message in the next channel2. A node can be designated as
the SYNC node in a channel based on a pre-established rule,
e.g., the node with the smallest node ID, or the node with the
highest battery level (all nodes can be made to report their
node ID and battery status in their beacon messages).
We highlight that the existence of a SYNC node in each
channel calls for an iterative algorithm performed jointly
across the available channels (see Section IV-B). This is
fundamentally different from prior schemes, e.g., [7], which
applied desynchronization in each channel independently. In
contrast, cross-channel synchronization allows for a channel
swapping mechanism to be applied in the converged state.
Specifically, nodes (both of SYNC and DESYNC type) that
fire synchronously in adjacent channels can swap channels and
time-slots in pairs using a simple RQ/ACK scheme3[see Fig.
3(b)]. Channel swapping allows for communication between
nodes initially present in different channels without leaving the
steady network state, thereby achieving increased connectivity.
Conversely, in [7], when a node changes channels, conver-
gence to TDMA in the channel needs to be established anew.
According to our protocol, starting from any random state,
the network reaches a steady state, where: (i) the same number
of nodes is present in adjacent channels, (ii) the nodes in each
channel have converged to a TDMA scheduling and (iii) the
nodes in channels with the same number of nodes have a
parallel TDMA scheduling, where nodes allocated with the
same time-slot order transmit synchronously [see Fig. 3(b)].
B. Proposed Joint SYNC-DESYNC Algorithm
We now describe the proposed joint algorithm that al-
lows for synchronization of SYNC nodes across channels and
desynchronization of DESYNC nodes in each channel. Let θc,i
(resp. φc,i) denote the phase (resp. phase offset) of node i=
1,...,ncin channel c= 1,...,C. Without loss of generality
and to simplify notation, let the node i= 1 be the SYNC node
in each channel4. DESYNC nodes i= 2,...,ncin channel c
are coupled with phase neighboring nodes (both DESYNC and
SYNC) in the same channel. Namely, any DESYNC node iin
channel cupdates its phase offset φc,i when node i−1in the
2We consider a cyclic behavior between channels 1 and 16 of IEEE 802.15.4
[35], [36]. Namely, the SYNC node at channel 16 listens for the fire message
from the SYNC node in channel 1.
3Swap RQ/ACK packets are transmitted at another channel during a short
interval after and before a node’s fire message transmission.
4As explained in Section IV-A, any node in a channel can be the SYNC
node. This convention is only used to simplify our notation.
Fig. 4. Example of the phase updates performed by the proposed multichan-
nel MAC algorithm: In channel 1, the DES YNC (white) node 4 undergoes
a phase update receiving coupling from nodes 1 and 3, present in the same
channel. In channel 2, the SY NC (grey) node 2 does not receive coupling from
the DESYNC node 4 that fires. In channel 3, the phase of the SYNC node 1
is updated due to the firing of the SYNC node in channel 4. The firing of the
latter node also triggers a phase update of the DES YNC node 2 in channel 4.
same channel transmits a fire message, i.e., when θc,i−1= 1.
The SYNC node in channel c, in turn, receives coupling only
from the SYNC node in channel c+ 1 (channel 1 for c=C).
Specifically, it updates its phase offset φc,1when the SYNC
node in the next channel fires, that is, when θc+1,1= 1. An
illustrative example of the phase updates performed by the
proposed algorithm is given in Fig. 4.
Problem formulation: Inspired by the interpretation given
in Proposition 1, we address the multichannel coordination
problem by solving
minimize
φ1,...,φC
h(φ1,...,φC) :=
C
X
c=1
1
2
Dcφc−1
nc
1nc+ec
2
2
+
C
X
c=1
1
2wc+1Tφc+1 −wcTφc2,(20)
where φc= (φc,1, φc,2,...,φc,nc)∈Rncis the vector
containing the phase offsets of all nodes of channel c,
Dc∈Rnc×ncis the matrix of (9) with dimensions nc×nc,
ec= (0,0,...,1) ∈Rncand wc= (1,0,...,0) ∈Rnc.
While the first term of henforces desynchronization among
the nodes of the same channel [note that each summand
has the same format as gin (8)], the second term enforces
synchronization among the first nodes of each channel. We
remark that the second term of (20) is commonly found in the
design of optimization-based consensus algorithms [26], [27],
[33].
Intuition: We show that the direct application of the gra-
dient descent method to solve (20) leads to updates (for the
SYNC nodes) that cannot be implemented in a practical WSN.
However, the proposed solution will be a modification of those
updates.
Taking into account that DcT1nc=0ncfor any c, the
gradient of hwith respect to φcis given by
7
∇φch(φ1,...,φC) = DcTDcφc+dc
+2wcTφc−wc−1Tφc−1−wc+1Tφc+1 wc,(21)
where dc:= (1,0,...,0,−1) ∈Rnc. Therefore, the partial
derivative of hwith respect to φc,i is
∂
∂ φc,i
h(φ1,...,φC)
=4φc,i −φc,i−1−φc,i+1 −φc−1,i −φc+1,i , i = 1
2φc,i −φc,i−1−φc,i+1 + (dc)i, i 6= 1 ,
where (dc)idenotes the i-th component of dc. The gradient
descent with stepsize βapplied to (20) yields for node iof
channel c:φ(k)
c,i =φ(k−1)
c,i −β∂
∂ φc,i h(φ1(k−1),...,φC(k−1) ).
Replacing βwith α/2, we obtain
φ(k)
c,i = (1−2α)φ(k−1)
c,i +α
2(φ(k−1)
c,i−1+φ(k−1)
c,i+1 +φ(k−1)
c−1,i +φ(k−1)
c+1,i ),
(22)
for i= 1, and
φ(k)
c,i = (1 −α)φ(k−1)
c,i +α
2(φ(k−1)
c,i−1+φ(k−1)
c,i+1 −(dc)i),(23)
for i6= 1. The update of (23) is similar to the DESYNC
algorithm phase update in (4)–(6). However, the derived update
for the SYNC node, given in (22), does not abide by the
coupling rules mentioned in Section IV-A. Specifically, to
implement (22) in a wireless transceiver, each SYNC node
has to listen for fire messages in its own channel, as well
as in the previous and the next channel. This is impractical
with the half-duplex transceiver hardware in IEEE 802.15.4-
based WSNs. This issue stems from the symmetry of the
matrix In−βDTD[cf. (21) and (12)]. To alleviate this
issue, we propose modifying directly the matrix associated
with the iterations (22) and (23). Our modification is based
on the insight that there is one degree of freedom in each
channel. Therefore, we can fix the phase of one of the nodes
at an arbitrary value. Our approach is to modify (22) and (23)
to have the first nodes of each channel performing a simple
consensus algorithm [31] (while the remaining nodes perform
a DESYNC algorithm).
Multichannel SYNC-DESYNC (MUCH-SYNC-DESYNC):
For simplicity and without loss of generality, we assume that
all channels have the same number of nodes: n:= n1=n2=
···=nC. The iteration we propose is
φ1(k)
φ2(k)
.
.
.
φC(k)
=
Q1Q20··· 0
0Q1Q2··· 0
.
.
.....
.
.
0 0 ··· Q1Q2
Q20 0 ··· Q1
|{z }
=:M
φ1(k−1)
φ2(k−1)
.
.
.
φC(k−1)
+β
en
en
.
.
.
en
|{z }
=:b
,(24)
where 0is the n×nzero matrix, en:= (0,0. . . , 0,1) ∈Rn,
Q2:= Diag(γ, 0,...,0) ∈Rn×n,0< γ < 1, and Q1is the
n×nmatrix defined as
Q1:=
1−γ0 0 0 ··· 0 0
β1−2β β 0··· 0 0
0β1−2β β ··· 0 0
.
.
.....
.
..
.
.
β0 0 0 ··· β1−2β
.
In other words, in each channel c, node i6= 1 performs the
update (23), while node 1performs
φ(k)
c,1= (1 −γ)φ(k−1)
c,1+γφ(k−1)
c+1,1.(25)
Recall that the phase update of the SYNC node in channel c
is performed when the SYNC node in channel c+ 1 fires, i.e.,
when θc+1,i(tc+1,i ) = 1. Adding tc+1,i
Tin both sides of (25)
as well as replacing φc+1,i = 1 −tc+1,i
Tand using (1) leads to
the following phase update for the SYNC node in channel c:
θ′
c,1(tc+1,1) = (1 −γ)θc,1(tc+1,1) + γmod 1.(26)
Since 0≤θc,1(t)≤1, it is straightforward to show that, for
0< γ < 1, (26) provides for inhibitory coupling5between
the SYNC nodes in subsequent channels, thereby leading to
synchronization of their phases. In the following proposition
we establish that the update (24) converges to a solution of the
optimization problem (20). In this case, however, we cannot
obtain an explicit convergence rate. Note that the matrix M
is not symmetric, which complicates the convergence analysis.
Note also that, when the number of nodes per channel varies,
the sizes of vectors φ,e, and matrices Q1and Q2in (24)
vary per channel c= 1,...,C, but their format is the same.
Moreover, the update equations, described in (23) and (25)
remain the same.
Proposition 2. Let 0< γ < 1and 0< β < 1
2. Then, the
sequence produced by (24) converges to a solution of (20).
Proof: The proof is given in Appendix B.
In MUCH-SYNC-DESYNC—formed by (23) and (25)—the
DESYNC and SYNC nodes per channel update their phases
only once during a firing round in the channel. Similarly
to existing (de)synchronization algorithms, the role of the
parameters αand γin the updates of (23) and (25) is to
compensate for missed fire messages and to not allow their
propagation throughout all nodes and channels in the network.
Since the update of the DESYNC nodes in each channel fol-
lows the phase update in (4)–(6), the corresponding Nesterov
modification can be applied to speed-up desynchronization in
each channel. This approach leads to the FAST-MUCH-SYNC-
DESYNC version of our algorithm, of which the convergence
speed is assessed in the next section.
V. EXPERIMENTAL EVALUATIONS
A. Simulation Results
All simulations were performed in MATLAB, by extending
the event-driven simulator in [2]. Initially, we examine the
5Similar to other synchronization algorithms [15], every time the SYNC
node in channel c+ 1 fires the SYNC node in the previous channel will
increase its phase towards 1 according to (26).
8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
20
40
60
80
100
120
140
160
180
α
F ir i n g Ro un d s
DE S Y NC , ǫ= 10 −4
FA ST - D ES Y N C , ǫ= 10−4
DE S Y NC , ǫ= 10 −3
FA ST - D ES Y N C , ǫ= 10−3
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
50
100
150
200
250
300
350
400
α
F ir i n g Ro un d s
DE S Y NC , ǫ= 10 −4
FA ST - D ES Y N C , ǫ= 10−4
DE S Y NC , ǫ= 10 −3
FA ST - D ES Y N C , ǫ= 10−3
(b)
Fig. 5. Average number of firing rounds for convergence to TDMA
scheduling for DESYNC and the proposed FAST-DESYNC with the Nesterov
modification: (a) n= 4 and (b) n= 8.
performance of DESYNC versus its fast counterpart based on
Nesterov’s algorithm. Then, we assess the performance of
the proposed MUCH-SYNC-DESYNC algorithm and its fast
version. We use two convergence thresholds, i.e., ǫ= 10−3
and ǫ= 10−4. Convergence is reported at the firing round
where the phases φof the nodes minimize the objective
function in (8) with accuracy gφ≤ǫ. Following existing
desynchronization schemes [2], [10], our algorithms’ updates
are performed on the nodes’ phases θi, as Assumption 1
does not need to be followed in practice. This simplifies the
implementation, as we do not need to know the order of firings.
All simulations were repeated 400 times and average results
are reported.
The results of applying desynchronization at a given channel
using either DESYNC [2], [10] or the proposed FAST-DESYNC
algorithm are presented in Fig. 5(a) and (b) for n= 4 and n=
8nodes, respectively. Although our analysis proves that FAST-
DESYNC converges for α∈(0,0.5], convergence is actually
achieved for α∈(0,1). In fact, FAST-DESYNC systematically
reduces the required number of iterations to convergence (i.e.,
irrespective of the value of the parameter α), leading to a
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
102
104
106
108
α
F ir i n g Ro un d s
D E SY NC , ǫ= 10 −4
FAS T -D E S YN C , ǫ= 10−4
D E SY NC , ǫ= 10 −3
FAS T -D E S YN C , ǫ= 10−3
Bo u n d FAST - D E S Y N C , ǫ= 10−4
Bo u n d FAST - D E S Y N C , ǫ= 10−3
Bo u n d DE S Y N C , ǫ= 10−3
Bo u n d DE S Y N C , ǫ= 10−4
Fig. 6. Maximum required firing rounds to convergence for DESYNC and
FAST-DESYNC versus the corresponding upper bounds, n= 8.
2.6%–28.6% speed-up with respect to DESYNC. Furthermore,
the convergence speed-up increases when a strict threshold
(ǫ= 10−4) is used. The improvement is more significant at
low and medium values of α, which are typically used in
practice to attenuate the impact of missed fire messages.
Fig. 6 depicts the maximum number of required firing
rounds for convergence of DESYNC and FAST-DESYNC versus
the bounds in Corollaries 2 and 3. The difference between
DESYNC and FAST-DESYNC is not visible now due to the
logarithmic scale. Because of the low ǫvalue in the denomi-
nator of (15) the DESYNC upper bound appears to be loose.
However, the FAST-DESYNC bound in (19) offers a tighter
characterization of the simulation-based convergence iterations
and follows a trend very similar to the simulation results.
We now evaluate the convergence properties of the proposed
MUCH-SYNC-DESYNC and its fast version. The results are
given in Fig. 7(a) and (b) for nc= 4 nodes per channel
in C= 6 and C= 16 channels, respectively. Contrasting
these results with the ones in Fig. 5, we observe that the
proposed multichannel algorithm requires approximately only
10–20% more firing rounds to reach convergence than the
single-channel DESYNC algorithm. It is also worth noticing
that the proposed FAST-MUCH-SYNC-DESYNC version offers
a notable convergence speed-up (i.e., 6.01%–42.54%) with
respect to the simple MUCH-SYNC-DESYNC algorithm, ir-
respective of the number of channels.
B. Experiments with TelosB Motes
Experimental setup: We implemented the proposed
MUCH-SYNC-DESYNC and its FAST version as applications
in the Contiki 2.7 operating system running on TelosB motes.
By utilizing the NullMAC and NullRDC network stack
options in Contiki, we control all node interactions at the MAC
layer via our code. By utilizing the TelosB high-resolution
timer (rtimer library), we can achieve the scheduling of
transmission and listening events with sub-millisecond accu-
racy, and set T= 100 ms. The phase-jump parameters are
set as α=γ= 0.6. All nodes first listen constantly until
convergence is achieved in their channel, at which point data
9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
50
100
150
200
250
α
F ir i n g Ro un d s
MU C H - SY N C -D E S YN C , ǫ= 10 −4
FA ST - M UC H - S YN C - D ES Y N C , ǫ= 1 0 −4
MU C H - SY N C -D E S YN C , ǫ= 10 −3
FA ST - M UC H - S YN C - D ES Y N C , ǫ= 1 0 −3
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
50
100
150
200
250
α
F ir i n g Ro un d s
MU C H - SY N C -D E S YN C , ǫ= 10 −4
FA ST - M UC H - S YN C - D ES Y N C , ǫ= 1 0 −4
MU C H - SY N C -D E S YN C , ǫ= 10 −3
FA ST - M UC H - S YN C - D ES Y N C , ǫ= 1 0 −3
(b)
Fig. 7. Average number of firing rounds for convergence to decentralized
multichannel TDMA scheduling for the proposed MUCH-SYNC-DESYNC
algorithm and its fast counterpart; nc= 4 nodes per channel are considered
with: (a) C= 6 and (b) C= 16 channels.
transmission starts and nodes switch to sparse listening to save
energy. Due to interference in the 2.4 GHz band of IEEE
802.15.4 and timing uncertainties in the fire message broadcast
and reception, we apply three practical modifications to ensure
that, once the network reaches the steady state, it remains there
until the entire network operation is suspended, or nodes join
or leave the network:
1) Each node can transmit data in-between its own fire mes-
sage and the subsequent fire message from another node,
albeit allowing for guard time of 6 ms before and after
the anticipated beacon broadcast times; this ensures no
collisions occur between data and fire message packets.
2) In the steady state, each node turns its transceiver on
solely for the 12 ms guard time corresponding to each
beacon message. Moreover, all nodes switch to “sparse
listening”, i.e., they listen for beacons only once every
eight periods, unless high interference noise is detected6.
3) To remain in sparse listening and avoid interrupting data
transmission due to transient interference, all nodes are
set to switch to full listening only if Nc= 10 consecutive
fire messages are missed. Our choice of Ncprovides
stable operation under interference at the cost of slower
reaction time.
As mentioned in Section IV-A, once all nodes are activated,
they are first balanced across the available channels. Note also
that, although our time-synchronized slot structure provides
channel swapping between synchronous nodes, this is not
considered in the experiments.
We select TSCH as benchmark for our comparisons, since
it is a state-of-the-art centralized MAC protocol for densely-
connected WSNs [3], [4]. Our implementation follows the
6tisch simulator and TSCH standard [4], [8], [36], namely:
channel 11 of IEEE 802.15.4 was used for advertisements,
the RQ/ACK ratio was set to 1
9, the slotframe comprised 101
slots of 15 ms each, and one node was set to broadcast the
slotframe beacon for global time synchronization. Finally, the
WSN under TSCH is deemed as converged to the steady state
when 5% or less of the timeslots changed within the last 10
slotframes.
Adhering to scenarios involving dense network topologies
and data-intensive communications (e.g., visual sensor net-
works [37]), we deployed n= 64 nodes in the C= 16
channels of IEEE 802.15.4. This leads to nc= 4 nodes per
channel after balancing. The 64 TelosB motes were placed in
four neighboring rooms on the same floor of an office building,
with each room containing 16 nodes.
Power dissipation results: We assessed the average power
dissipation of our scheme against TSCH by placing selected
TelosB motes in series with a high-tolerance 1-Ohm resistor
and by utilizing a high-frequency oscilloscope to capture the
current flow through the resistor in real time. During this
experiment, no other devices (or interference signal genera-
tors) operating in the 2.4 GHz band were present in the area.
Average results over 5min of operation are reported. The
average power dissipation of MUCH-SYNC-DESYNC without
transmitting or receiving data payload was measured to be 1.58
mW. The average power dissipation of a TSCH node under
minimal payload (128 bytes per 4 s) was found to be 1.64 mW,
which is very close to the value that has been independently
reported by Vilajosana et al. [4]. Therefore, under the same
setup, our proposal and TSCH were found to incur comparable
power dissipation for their operation.
Convergence speed results: We investigate the conver-
gence time of MUCH-SYNC-DESYNC, FAST-MUCH-SYNC-
DESYNC and TSCH under varying interference levels. Rapid
convergence to the steady state is very important when the
WSN is initiated from a suspended state, or when sudden
changes happen in the network (e.g., nodes join or leave). We
carried out 100 independent tests, with each room containing
an interference generator for 25 tests. To generate interference,
6In the converged state, each node determines the interference noise floor
in-between transmissions by reading the CC2420 RSSI register. If high
interference is detected, the node switches to regular listening. Thus, sparse
listening does not affect the stability of MUCH-SYNC-DE SYNC.
10
−20 −15 −10 −5 0 5
0
5
10
15
20
25
30
35
Jamming Signal Power (dBm)
Time (s)
MUCH−SYNC−DESYNC
FAST−MUCH−SYNC−DESYNC
TSCH
Fig. 8. Average time required for MUCH-SYNC-DESYNC, its FAST version,
and TSCH to converge under various interference levels.
TABLE I
AVERAGE CONV ERGENCE TIME (IN SECONDS) UNDER HIDDEN NODES .
NUMBERS IN PARENTHE SIS SHOW THE CONVERGENCE TIME OF THE
FAST (NESTEROV-BASED)VERSION OF OUR PROPOSAL.
MUCH-SYNC-DESYNC TSCH
Without Hidden Nodes 1.1356 (0.7351) 15.5845
With Hidden Nodes 1.8514 (1.2896) 15.2957
an RF signal generator was used to create an unmodulated car-
rier in the center of each WSN channel. The carrier amplitude
was adjusted to alter the signal-to-noise-ratio (SNR) at each
receiver [38]. The nodes were set to maximum transmit power
(+0 dBm) in order to operate under the best SNR possible.
Fig 8 shows the time required for MUCH-SYNC-DESYNC,
FAST-MUCH-SYNC-DESYNC and TSCH to converge under
varying interfering signal power levels. The results corroborate
that our proposal reduces the convergence time by an order
of magnitude in comparison to TSCH and that the Nesterov-
based algorithm offers 36.48%-41.07% increased convergence
speed under a realistic setup. Moreover, the difference in
convergence time between the proposed mechanism and TSCH
increases with the interference level because TSCH nodes miss
most of the RQ/ACK messages in the advertisement (control)
channel. This result demonstrates the key advantages of our
decentralized MAC mechanism with respect to TSCH, namely:
(i) it is fully decentralized and (ii) it does not depend on an
advertisement and acknowledgement scheme.
Results under hidden nodes: We now investigate the
robustness and convergence speed of our scheme when some
nodes in the WSN are hidden from other nodes. We measure
the time to achieve convergence to steady state when a random
subset of 20 nodes in our WSN setup was programmed to
ignore transmissions from 4randomly chosen nodes. The
results in Table I show that, irrespective of the presence
of hidden nodes, the convergence of MUCH-SYNC-DESYNC
and its FAST version is an order-of-magnitude faster than
that of TSCH. When hidden nodes are present, the required
convergence time of MUCH-SYNC-DESYNC (resp. its FAST
version) increases by 63.03% (resp. 75.43%), while that of
TSCH is actually sightly decreased by 2.13%. This is to be
−20 −15 −10 −5 0 5 10 15
0
10
20
30
40
50
60
70
80
90
100
Jamming Signal Power (dBm)
Throughput (kbps)
MUCH−SYNC−DESYNC
TSCH
Fig. 9. Total network throughput between MUCH-SYNC-DES YNC and TSCH
under varying signal power levels.
expected, as TSCH nodes simply ignore RQ packets from
hidden nodes. Conversely, due to the DESYNC (resp. FAST-
DESYNC) process within each channel, applied by MUCH-
SYNC-DESYNC (resp. its FAST version), prolonged beaconing
will take place until all hidden nodes are placed amongst non-
hidden DESYNC phase neighbors. This spontaneous robustness
of MUCH-SYNC-DESYNC (and its FAST version) to hidden
nodes is an interesting property that deserves further study7.
Bandwidth results: We measure the total network through-
put (i.e., total payload bits transmitted by all nodes per second)
achieved with MUCH-SYNC-DESYNC and TSCH under var-
ious interference levels. Since the measurement is performed
after the network is converged, the throughput of MUCH-
SYNC-DESYNC coincides with its fast version. The results
in Fig 9 show that MUCH-SYNC-DESYNC systematically
achieves substantially higher network throughput (more than
40% increase w.r.t. TSCH), irrespective of the interference
level. Both protocols suffer a significant throughput loss
of under high interference (i.e., above 10 dBm), which is,
however, substantially more severe for TSCH. In effect, when
interference is above 12 dBm, the bandwidth obtained with
TSCH drops to zero because of the inability to recover
lost slots through advertising. Conversely, even under high
interference levels, MUCH-SYNC-DESYNC recuperates band-
width utilization due to the elasticity of SYNC and DESYNC
mechanisms and the high value used for Nc.
VI. CONCLUSION
We have shown that DESYNC, which is a well established
desynchronization method for MAC layer coordination in
WSNs, can be viewed as a gradient method for solving
an optimization problem. This interpretation led to a novel,
faster desynchronization algorithm (based on Nesterov’s mod-
ification of the gradient method) and resulted in the deriva-
tion of upper bounds for the convergence of desynchroniza-
tion. Importantly, casting the problem of time-synchronous
desynchronization across channels as a convex optimization
7For instance, one can try to determine conditions that guarantee that no
configuration of hidden nodes can lead to instability.
11
problem, led to the derivation of novel multichannel MAC
algorithms. Our proposed MUCH-SYNC-DESYNC algorithm
and its fast counterpart were benchmarked against the IEEE
802.15.4e-2012 TSCH and were shown to provide for: (i)
an order-of-magnitude decrease in the convergence time to
the network steady state, (ii) more than 40% increase in
the total network throughput, and (iii) significantly-increased
robustness to interference and hidden nodes in the network,
while requiring comparable power dissipation.
APPENDIX A
Proof of Corollary 1: It is known that every limit point
of the steepest descent method with a constant stepsize β, i.e.,
φ(k)=φ(k−1) −β∇g(φ(k−1)),is a stationary point of g(φ)
whenever ∇gis Lipschitz continuous, i.e., there is an L > 0
such that k∇g(y)− ∇g(x)k ≤ Lky−xkfor all x,y∈Rn,
and β∈(0,2/L); see [39, Prop.1.2.3]. In problem (8), gis
twice differentiable, and ∇2g(φ) = DTD, for all φ. We can
then set L≥λmax(DTD), where λmax(·)is the maximum
eigenvalue of a matrix. Notice that, for Din (9), DTD
coincides with the Laplacian matrix of the ring graph, whose
eigenvalues are given by 2−2 cos(2πk/n),k= 1,...,n [40,
Lemma 2.4.4]. We then have
λmax(∇2g(φ)) = arg max
k
2−2 cos 2πk/n≤4.(27)
Setting L= 4, and taking into account that α= 2β, we
obtain that DESYNC converges whenever α∈(0,1). Notice
that when nis even, the maximum is achieved in (27), i.e.,
λmax(∇2g(φ)) = 4.
Proof of Corollary 2: Let g:Rn−→ Rbe a convex,
continuously differentiable function whose gradient is Lips-
chitz continuous with constant L. It is known that the se-
quence generated by the steepest descent method with constant
stepsize β∈(0,2/L), i.e., φ(k)=φ(k−1) −β∇g(φ(k−1)),
satisfies [29, Thm.2.1.14]
g(φ(k))−g(φ⋆)
≤2(g(φ(0))−g(φ⋆))kφ(0) −φ⋆k2
2
2kφ(0) −φ⋆k2
2+k β(2 −Lβ)(g(φ(0))−g(φ⋆)) ,(28)
where φ⋆is any minimizer of g. As shown in the proof of
Corollary 1, L= 4 in our case. Furthermore, g(φ⋆) = 0.
Taking this into account in (28), using α= 2β, and after
some manipulations, we get (14).
To obtain (15), we note that (14) holds for any solution φ⋆
of (8). That is,
rD≤min
φ⋆∈S⋆kφ(0) −φ⋆k2
2·1
2α(1 −α)1
ǫ−1
gφ(0),
(29)
where S⋆is the set of all solutions of (8). We have S⋆={φ+
z1n:z∈R}, where φis any solution of (8). Henceforth,
we will take φ= (0,1/n, 2/n, . . . , (n−1)/n). Then, the
minimization problem in (29) is equivalent to the minimization
of kφ+z1n−φ(0)k2
2over z, which yields z⋆=1
n1T
n(φ−
φ(0)). Hence,
min
φ⋆∈S⋆kφ(0) −φ⋆k2
2=
φ+1
n1T
n(φ−φ(0))1n−φ(0)
2
2
=
In+1
n1n1T
nφ−φ(0)
2
2.
(30)
To find a worst case scenario, we maximize (30) with respect
to φ(0), subject to the constraints 0n≤φ(0) ≤1n. This is a
non-convex problem, but the solution can be found in closed-
form with the following observation. Since In+ (1/n)1n1T
n
is a circulant matrix and its entries are all positive, maximiz-
ing (30) subject to 0n≤φ(0) ≤1nis equivalent to
maximize
φ(0)
φ−φ(0)
2
2
subject to 0n≤φ(0) ≤1n.
(31)
Since φ= (0,1/n, 2/n, . . . , (n−1)/n), the solution of (31)
is φ(0) = (1,1,...,1,0,0,...,0), where the transition from 1
to 0occurs at the first index mwhere m≥n/2. Denoting
B:= max
φ(0) min
φ⋆∈S⋆kφ(0) −φ⋆k2
2
s.t. 0n≤φ(0) ≤1n
we have
B≤2m−1
X
i=0 1−i
n2+
n−1
X
i=mi
n2(32)
=2
n2m−1
X
i=0
(n−i)2+
n−1
X
i=m
i2(33)
=2
n2mn2−nm(m−1) + n(n−1)(2n−1)
6(34)
=1
3n6nm −6m2+ 6m+ 2n2−3n+ 1(35)
≤1
3n7
2n2+ 3n+ 4.(36)
The bound in (32) is due to replacing φ(0) =
(1,1,...,1,0,0,...,0) in (30) and using
(In+1
n1n1T
n)(φ−
φ(0))
2
2≤
In
2
2+ (1/n2)
1n1T
n
2
2
φ−φ(0)
2
2=
2
φ−φ(0)
2
2. From (33) to (34), we developed the
square in the first summand and used the identities
Pm−1
i=1 i=m(m−1)/2and Pn−1
i=1 i2=n(n−1)(2n−1)/6.
From (35) to (36), we used the bound n/2≤m≤(n+ 1)/2.
Using (36) in (29) we get (15).
Proof of Corollary 3: Equations (17a)-(17b) are applying
Nesterov’s method (16a)-(16b) to problem (8) with α= 2β. It
is known that the number of iterations that (16a)-(16b) requires
to generate a point φthat has accuracy ǫ=g(φ)is bounded
as [34]
rFD ≤p2/β
pǫ−g(φ⋆)
φ(0) −φ⋆
2,(37)
where φ⋆minimizes g. This expression is valid for β∈
(0,1/L], where Lis the Lipschitz constant of ∇g. We saw
in the proof of Corollary 1 that L= 4 is a valid choice.
Since g(φ⋆) = 0 for any optimal φ⋆, and using α= 2β
in (37), we get (18). To obtain (19) from (18), we use (36)
from the proof of Corollary 2.
12
APPENDIX B
Proof of Proposition 2: If {φ(k)}converges, its limit will
be a fixed point of (24). Before showing that {φ(k)}converges,
we show that any fixed point of (24) solves (20). Let φ⋆=
(φ⋆
1,φ⋆
2,...,φ⋆
C)be a fixed point of (24). For each c, we have
φ⋆
c,i = (1 −γ)φ⋆
c,i +γ φ⋆
c+1,i , i = 1 (38)
φ⋆
c,i =β φ⋆
c,i−1+ (1 −2β)φ⋆
c,i +β φ⋆
c,i+1 +β(en)i, i 6= 1 .
(39)
From (38), and since γ > 0, we have φ⋆
c,1=φ∗
c+1,1, for all c
modulo C. This makes the second summation term in (20)
equal to zero, that is, wc+1 Tφ⋆
c+1 =wcTφ⋆
c, for all c.
From (39), and since β > 0, we have
φ⋆
c,i =φ⋆
c,i−1+φ⋆
c,i+1
2, i = 2,3,...,n−1(40)
φ⋆
c,n =φ⋆
c,n−1+φ⋆
c,1+ 1
2.(41)
These equations are equivalent to φ⋆
c,i+1 −φ⋆
c,i = 1/n, for i=
1,...,n −1, and φ⋆
c,1+ 1 −φ⋆
c,n = 1/n, and this makes
the first term of the objective of (20) equal to zero. To see
why the above equivalence holds, note that (40)-(41) imposes
that all n−1phases φ⋆
c,2,φ⋆
c,3,...,φ⋆
c,n be placed in the
interval [φ⋆
c,1, φ⋆
c,1+ 1]. Furthermore, each phase has to equal
the average of the previous phase with the next phase, where
the phase previous to φ⋆
c,2is φ⋆
c,1and the phase next to φc,n
is φ⋆
c,1+ 1. The only possibility is all phases, including the
extreme points, being equispaced.
We now prove that {φ(k)}converges. Writing (24) in a
more compact form,
φ(k)=Mφ(k−1) +b.(42)
It is known that the sequence {φ(k)}produced by (42)
converges to (I−M)−1bwhenever the spectral radius of M,
denoted as ρ(M), is strictly smaller than 1[41, §1.2]. In our
case, however, 1is an eigenvalue of M, so ρ(M)≥1.
By computing all the eigenvalues of M, we will see that
actually ρ(M) = 1. Before proceeding, note that the vector
of ones, 1nC , is a right eigenvector of Massociated to the
eigenvalue 1, and u:= (e1,e1,...,e1)∈(Rn)Cis a left
eigenvector of Malso associated to the eigenvalue 1.8To
compute the eigenvalues of M, first decompose Q1as
Q1=1−γ0T
n−1
r T ,
where r= (β, 0,...,0, β )∈Rn−1, and
T=
1−2β β 0··· 0 0
β1−2β β ··· 0 0
.
.
..
.
....··· ···
0 0 ··· β1−2β
.
8If the Perron-Frobenuis theory [42], [43] were applicable, we would
conclude that ρ(M) = 1, and that the eigenvalue 1would have algebraic
multiplicity 1. This would enable us to skip the computation of all eigenvalues
of Mand jump to the next paragraph. However, the Perron-Frobenuis theory
is not applicable, since M, although being positive, is not irreducible.
There is exists a permutation matrix Psuch that
PTMP =
T r
T r
......
T r
0C×(n−1)CR
,
where
R=
1−γ γ 0··· 0
0 1 −γ γ ··· 0
.
.
.....
.
.
γ0 0 ··· 1−γ
∈RC×C.
Such permutation matrix corresponds to a reordering of the
nodes such that φis mapped onto
(φ1,2, φ1,3,...,φ1,n, φ2,2, φ2,3,...,φ2,n,...,
φ1,1, φ2,1,...,φC,1),
that is, the first nodes of each channel are in the end of the
vector in the new coordinate system. The matrices Mand
PTM P have the same eigenvalues. The upper triangular
structure of PTM P reveals that its eigenvalues are the
roots of det(T−λI)Cdet(R−λI) = 0 ,where Iis an
identity matrix with appropriate dimensions. In other words,
the eigenvalues of Mare the union of the eigenvalues of T,
each with multiplicity C, with the eigenvalues of R, each with
multiplicity 1. Since Tis tridiagonal Toeplitz, its eigenvalues
are λj(T) = 1 −2β+ 2βcos(π
nj), for j= 1,...,n−1[43,
p.514]. The matrix R, on the other hand, is a circulant
matrix and hence its eigenvalues are the Fourier transform
of the vector that generates the matrix. In this case, they
are λj(R) = 1 −γ+γexp( 2πi
Cj), for j= 1,...,C,
where i:= √−1. Since 0< γ < 1,Rhas one eigenvalue
equal to 1(multiplicity 1) and the remaining ones have mag-
nitude smaller than 1. As 0< β < 1/2, all eigenvalues of T
have magnitude smaller than 1. We conclude that ρ(M) = 1,
and that its algebraic (and geometric) multiplicity is 1.
Define M:= M−1nC uT. Then, ρ(M)<1[42, Lemma
8.2.7], and (42) can be written as
φ(k)=Mφ(k−1) +1nC uTφ(k−1) +b.(43)
Since uTM=uand uTb= 0, (42) tells us that uTφ(k)=
uTMφ(k−1) +uTb=uTφ(k−1). In particular,
uTφ(k)=uTφ(k−1) =···=uTφ(1) =uTφ(0) .
Defining b=b+1nC uTφ(0), (43) can then be written as
φ(k)=Mφ(k−1) +b,(44)
where ρ(M)<1. Thus, according to [41, §1.2], the se-
quence {φ(k)}produced by (44), and thus by (24), converges
to (I−M)−1b= (I−M)−1b+ (I−M)−11nC uTφ(0),
which is well-defined and unique (note that I−Mis invertible
because ρ(M)<1). This shows that the sequence {φ(k)}
converges.
13
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