NOLTA, IEICE
Paper

A mode-lock free decentralized timing
synchronization algorithm for intervehicle
ad-hoc networks
Hisa-Aki Tanaka 1 a) and Kenta Shinohara∗ 1
1

Department of Electronic Engineering, The University ofElectro-Communications (UEC)
1-5-1 Choufugaoka, Choufu-shi, Tokyo 182-8585, Japan∗The author is currently with NTT Network Service Systems Laboratories,
Japan

a)

htanaka@uec.ac.jp

Received September 16, 2014; Revised December 5, 2014; Published April 5, 2015
Abstract: Timing synchronization is an important integrating component in wireless distributed systems, such as mobile ad-hoc networks, M2M networks, and wireless sensor networks,
and therefore, various timing synchronization algorithms have been proposed so far. Recently,
Imai and Suzuki developed a new synchronization algorithm based on a time division multiple access (TDMA) protocol. Despite of its eﬃciency in synchronization for vehicle-to-vehiclecommunications, their algorithm sometimes suﬀers from a certain undesired synchronous pattern, i.e., a so called mode-lock state or a deadlock. Although their algorithm takes certain
precautions to avoid this mode-lock state, in around more than 10% of instances this state
is observed to persist and the desired perfect synchronization is not realized. Then, ﬁrst we
investigate the mechanism of this persisting mode-lock state for their algorithm. With thisinsight to the mode-lock state, we propose a new mode-lock free (i.e., mode-lock eliminating) distributed algorithm that always leads to a perfect synchronization. From systematic,
comparative simulations, we observe that the proposed algorithm always eliminates mode-lockstates, and eventually leads to the perfect synchronization. In addition, we observe the algorithm realizes even faster synchronization, compared with the algorithm by Imai and Suzuki,
although these observed properties are not mathematically proved in this study.
Key Words: timing synchronization, wireless ad-hoc networks, wireless sensor networks, deadlock, mode-lock state, TDMA

1. Introduction
Applications of mobile ad-hoc networks have been actively developed for vehicle-to-vehicle communications recently. Such applications are expected to support a safe driving and prevent traﬃc accidents,
and the Japanese ministry of land, infrastructure, transport, and tourism has organized a project foradvanced safety vehicles (ASV) [1]. While in Toyota central R&D laboratories, basic researches have

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Nonlinear Theory and Its Applications, IEICE, vol. 6, no. 2, pp. 285–294c
IEICE2015

DOI: 10.1588/nolta.6.285

been carried out for intervehicle ad-hoc networks, which cover a new timing synchronization [2], MAC
protocol [3], realistic urban traﬃc simulations [4], and even ﬁeld trials [5]. In [2], Imai and Suzuki
developed an eﬃcient synchronization algorithm for vehicle-to-vehicle (V2V) communications basedon the decentralized time division multiple access (D-TDMA) protocol [3]. The algorithm by Imaiand Suzuki includes new devices for synchronizing two (or even multiple) groups of vehicles, andits performance is analyzed under certain realistic environments by systematic simulations so far [4].
However, as they have already pointed out in [2], their algorithm often suﬀers from a certain undesired
synchronous pattern; a so called mode-lock state [6] or a deadlock. Although their algorithm takes
certain precautions to avoid this mode-lock state, in more than 10% of instances this state is observed
to persist and the desired perfect synchronization is not obtained.
To realize the perfect synchronization, in this study, ﬁrst we clarify how and why this mode-lockstate persists when their algorithm is used. Based on this insight, we propose a new mode-lock free
distributed synchronization algorithm which eventually leads to the perfect synchronization. Fromsystematic, comparative simulations, we verify the proposed algorithm always eliminates mode-lockstates, and we observe it realizes even faster synchronization, compared with the algorithm by Imaiand Suzuki, although these properties are not yet mathematically proved in this study. The basic
idea of this study was presented in NOLTA’08 [7]. However, the conference paper lacks a detailed
analysis and systematic simulation results of the proposed algorithm. In the present paper we not only
provide more detailed analysis but also show systematic simulation results, both of which validate theusefulness of proposed algorithm.
This paper is organized as follows. In Section 2, we explain the background of this study andintroduce the distributed timing synchronization algorithm proposed by Imai and Suzuki. In Section
3, we consider and clarify how and why the mode-lock state persists in their algorithm. Then, we
propose a new mode-lock free, synchronization algorithm. In Section 4, we compare the performanceof the algorithm by Imai and Suzuki and our new algorithm, by systematic simulations. Finally,
discussions and conclusions are given in Section 5.

2. Background of this study; synchronization for D-TDMA protocol
First, we introduce the D-TDMA protocol [3] as a basis of this study, which is outlined as follows. As
shown in Fig. 1, each node (i.e., node A, B, C and D in Fig. 1) is preassigned a single time interval
for packet transmission (i.e., a slot with an equal length in Fig. 1), and these slots are evenly spaced
along the time in one frame as shown in Fig. 1, in order to avoid packet collisions. The number ofslots is generally set to several hundreds, which is much larger than the number of vehicles directly
communicating with each other. This point is explained in Section 4, in detail. (Note that the
number of slots in Fig. 1 is impractically small, which is just for simplicity of illustration.) Thus,
all neighboring nodes (i.e., nodes A, B, C, and D in Fig. 1) send their packets, one after another in
one frame, and this sequence is repeated if the nodes are maintained in the group. Since the frame
length is equal in all nodes and frames are repeated regularly in each node, we can naturally introduce
the phase variable φ ∈ [0, 2π] as shown in Fig. 1, in which one frame length becomes 2π. Namely,
we identify the time t to the phase φ by φ ≡ (t mod T ) × 2π, where T is one frame length. This
phase variable is useful for understanding the mode-lock state as shown in Section 3. Here we note
one important requirement in this TDMA protocol, i.e., a perfect synchronization of local timings(i.e., timers) in neighboring nodes, has to be satisﬁed in a precision within the guard interval in
Fig. 1, otherwise packet collisions may happen. Note that this is the frame synchronization amongneighboring nodes, and if this frame synchronization is attained, slots for packet transmission are
regularly coordinated among neighboring nodes.
Second, we consider issues of local time synchronization. If we assume a certain global timing issomehow available (for instance, from GPS [8, 9]), each node acquire the global timing independently,
and the synchronization can be realized relatively easily. However, in real environments this global
timing is not always accessible and hence neighboring nodes have to communicate with each otherto realize a common local timing, i.e., a distributed timing synchronization. For distributed timing286

synchronization methods in vehicle-to-vehicle communications, there have been a few studies reported
so far [2, 10, 11]. These synchronization methods share the same idea in using an ‘averaged timing’
over neighboring mobile nodes directly communicating with each other. This averaged timing isdeﬁned by the average over the local timings of those mobile nodes, which enables each mobile node
update (i.e., synchronize) its local timing by referencing the average timing in a fully distributed way.
For instance in [2] this synchronization process is repeated in each frame as follows.
(1) Firstly, each node periodically receives the local timing of the i-th neighboring nodes (≡ ti ) frompreambles in their packets.
(2) Then, secondly, each node periodically obtains all timing errors (≡ Δti = ti − t0 ) to the ith neighboring nodes’ local timings by subtracting its local timing (≡ t0 ) from neighboring nodes’
timings. (= ti )(3) Finally, each node updates its local timing such that the average of all timing errors becomesnminimized, i.e., the local timing is set to t0 + τ with τ ≡ n−1 i=1 Δti , where n is the number ofneighboring nodes.

Fig. 1.

Time structure in the TDMA protocol in this study.

With this synchronization algorithm using averaged timings, groups of vehicles realize a temporally
and spatially smooth synchronization of their local timings with each other, and therefore this method
is expected to be suitable for the D-TDMA protocol in intervehicle communications [3, 4]. Note that
other synchronization methods including the IEEE 802.11 TSF (timing synchronization function; forinstance, see [8, 9]) often show an abrupt change of local timings which violates the above ‘smooth’
synchronization. We also note that distributed timing synchronization methods in this study arenatural and reasonable in a sense that the local timing informations can be easily shared throughpacket communications among neighboring nodes; packets include the local time information in theirpreambles.
The synchronization algorithm by Imai and Suzuki [2] shares the same idea to the conventional
synchronization methods [10, 11], since it is based on the averaged timing. However, compared withthese conventional methods the algorithm by Imai and Suzuki [2, 4] includes two new additional
devices as below, and hence it can be advantageous over previous methods when applied to larger,
distributed intervehicle communication networks. The new devices are as follows.
(i) This algorithm takes into account of a relative degree of synchronization between two (or more)groups of vehicles, and tries to accelerate synchronization over two groups of vehicles [4], and(ii) this algorithm tries to avoid a certain undesired asynchronous state; a so called mode-lock state ora deadlock, knowing a degree of synchronization for each node to the neighboring nodes by computing
the variance of all timing errors in the above (2) [2].
Note that the device (i) is not essentially involved in the situation of this study as well as in [2],because this device is eﬀective only when more than two independent groups of vehicles are going tobe synchronized, in which each group of vehicles are already perfectly synchronized. While in thisstudy, neighboring two nodes are generally not perfectly synchronized. We also note that even if this(i) is included in the simulations, we have obtained the same results. Hence, in contrast to (i), thedevice (ii) becomes essential in this study, and we focus this (ii) and consider its eﬀect in detail, in
Section 3.

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3. A mode-lock eliminating timing synchronization algorithm
As observed in [2], undesired synchronous states; mode-lock states often emerge and persist if no prevention is given in a decentralized synchronization algorithm based on the averaged timing. Althoughmode-lock states have some variations in their dynamics and complexity ([6, 13, 14]) the basic structure is given as in Fig. 2; all nodes are placed on a two dimensional regular array (just for simplicity)and the central zone of the mode-lock state (i.e., a so called ‘core’) is set to the center of the array(also just for simplicity). Then, the precise deﬁnition of mode-lock states in this particular setting isgiven as follows. Using the phase variable φ introduced in Section 2 (see Fig. 1), the local timings
of nodes are characterized as in Fig. 2; thicker marked nodes (•) correspond to φ  2π. In contrast,thinner marked nodes (◦) correspond to φ  0. Nodes with the same phase symmetrically radiatefrom the central zone of the mode-lock state. Hereafter, we call central zone of the mode-lock as core,for simplicity. Around the core, the phases of nodes are evenly ordered, which form a two dimensionalrotating wave as φ(t) increases as time. (A particular example in Fig. 2 exhibits a counter-clockwise
rotating wave. The direction of rotation is determined by the choice of initial local timings (i.e., phases
φ(0)) of nodes). Note that such a mode-locked state emerges in non-regular networks (including an
example shown later in Fig. 4(b)), and even multiple cores can coexist in one mode-locked state [6,14]. We also note that this rotating, mode-lock state is notorious for its diﬃculty in elimination;
examples are known in VLSI clocking [13], millimeter wave generation [6], and heart diseases [14].

Fig. 2.

Basic structure of mode-lock state.

3.1 Imai and Suzuki’s algorithm
The D-TDMA protocol introduced in Section 2 requires one strict constraint; each node is always
synchronized with neighboring nodes in the precision within one guard interval. However, in themode-lock state as shown in Fig. 2, this requirement is clearly violated because nodes around the core
have a widely distributed local timings φ(t) ∈ [0, 2π]. (Note that this is nothing to do with evenly
spaced packet transmission timings in Fig. 1). Imai and Suzuki observed that this undesired statepersists and they include the following precautions for it in their algorithm [2];(i) once in the frame period T, each node judges if it is in the core of the mode-lock state by checking
nthe variance of timing errors to neighboring nodes n−1 i=1 Δt2i (cf. the device (ii) in Section 2); if
the variance exceeds a certain threshold, the node assumes it belongs to the core of the mode-lock.
Otherwise, the node assumes it is out of the core, and then(ii) if a node is assumed to be in the core, this node immediately updates its local timing (≡ t0 ) to themost recently received local timing (≡ t∗ ) from neighbouring nodes (namely, t0 is set to t∗ , insteadof updating its timing to the averaged timing given by (3) in Section 2), and this node transmits its
updated local timing information to neighboring nodes once in the frame period.
At ﬁrst glance, these precautions (i) and (ii) seem eﬀective to perfectly synchronize the nodes in
the core, and eventually lead to a perfect synchronization over all nodes. However, in reality, what288

is often observed (through numerical simulations in Section 4) under these precautions (i) and (ii),is shown as in Fig. 3(a). Here we arrive at the following insight. As indicated by the bidirectional
arrows (↔), the local time information ﬂow is always bidirectional, which imply nodes in the core,and other (majority of) nodes are coupled with each other. However, majority of nodes outside thecore are not so much aﬀected by nodes in the core for the following reasons; (a) the mode-lock state
itself is a quite robust pattern, (b) the range of interaction between two nodes is relatively short due
to V2V communications in a wide area, and hence, (c) majority of nodes far from the core virtuallyshow no change in their local timings.
Therefore, as shown in snapshots 1, 2, and 3 of Fig. 3(a), the position of the core is gradually shifteddue to ‘random perturbations’ from adjacent nodes to the core; the core keeps slowly meandering. If
the core fortunately reaches to the outside boundary of the network, it disappears and this leads tothe perfect synchronization. However, as already observed in [2], there is no guarantee for this lucky
event within a time of hundreds ﬂames.

3.2 Our proposed algorithm
In contrast to such a random perturbing approach, here we propose a simple, but quite eﬀective
synchronization algorithm that somehow destroys the mode-lock state itself. The proposed algorithm
is basically on the same line of previous methods [2, 10, 11], in a sense that it is based on the averaged timing from neighbouring nodes. However, as opposed to these previous methods, this method
destroys the structure of the mode-lock itself in a distributed manner as follows;(i) each node judges if it is in the core of the mode-lock state; this step is exactly the same to theabove (i). However, the following step is quite diﬀerent from the above (ii),(ii) if a node is aware of being inside or near the core (by using (ii) in Section 2), this node temporally
quits sending its (erroneous) local timing information, only receives the local timing information (i.e.,
ti in Section 2) from other nodes as indicated by the unidirectional arrow (→) in Fig. 3(b), and it
updates its local timing to the averaged timing. (Note, if such a node does not receive the local timing
information from the outside of the core, its local timing becomes quite diﬀerent to the one in theoutside of the core, resulting in violation of ‘smooth’ synchronization assumed in Section 2.)

Fig. 3. Essential mechanisms of the both two algorithms. (a) Imai and
Suzuki’s algorithm [2]. (b) Proposed algorithm. Arrows (↔, →) indicate theﬂow of local time information.

The algorithm is thus, quite simple in itself. However, when all nodes start synchronizing with eachother by this algorithm, the eﬀect appears drastically; a typical process to the perfect synchronization
is illustrated as in Fig. 3(b). As indicated by the unidirectional arrows (→) in snapshots 1 and 2289

in Fig. 3(b), the time information ﬂow initially becomes inward to the core. This implies that theadjacent nodes placed outside the core no more receive the time information form nodes in the core,
which results in a slight change of the averaged timing at each adjacent node (to the core). This slightchange is due to the randomness of the preassigned (i.e., randomly coordinated among the nodes [3])slots, and this gradually breaks the symmetry of the phases around the core (cf. Fig. 5(b)). This breakof symmetry eventually lead to a near synchronization among nodes outside the core (cf. snapshot 2
in Fig. 3(b) and Fig. 5(d)), because perfect synchrony is now the other unique stable state. Once theseoutside nodes are mutually synchronized, the nodes in the core are synchronized to them, resulting in
the perfect synchronization over all nodes (cf. snapshot 3 in Fig. 3(b) and Fig. 5(e)). Note that aftersnapshot 2 in Fig. 3(b) the core is no more detected (by the above (i) ) and the time information ﬂow
becomes bidirectional as in snapshot 3 in Fig. 3(b). A detailed analysis of the above observation is
given in Section 4.1, and a systematic performance evaluation for both algorithms is given in Section
4.2.

4. Performance analysis for both algorithms
To analyze and evaluate the performance of two algorithms in Section 3, we carry out systematic
simulations as follows. Firstly, the simulation setup here faithfully follows the setup in [2], which issummarized as follows.
(i) Each node periodically sends the local timing message to neighbouring nodes in the beginning of
its assigned slot (Fig. 1). In this simulation, one frame length is set to 0.1 [s] and the number of slots
is set to 400, according to the speciﬁcation of ASV [1].
(ii) Each node has a given, small frequency mismatch in its timing clock, which ranges in ±0.01%
accuracy, according to the speciﬁcation of the clock oscillator used for ASV [15].
(iii) Each node is uniformly placed on a regular two-dimensional lattice (i.e., array), and each nodecan communicate with neighbouring nodes in (1) the two-hop distance (i.e., 12 nodes in this radius)or (2) a bit larger distance (i.e., 20 nodes in this radius), as shown in Fig. 4(a) 1 . In this simulation,
we assume all events take place in a relatively short time scale (up to several hundreds of ﬂames = a
few minutes) in widely distributed, urban traﬃc environments. Therefore, mobility of each node canbe practically ignored here.
(iv) According to the setup in [2], the total number of nodes is set to 1024, and each node with acertain preassigned slot is introduced on the two-dimensional lattice, one by one in the initial 100frame time at the beginning of the simulation.
(v) As a ﬁrst approximation, a collision between timing messages is safely ignored. The reason for thisapproximation is explained as follows. Firstly, a chance of collision is quite rare, since the numberof slots is much larger than the number of vehicles directly communicating with each other, andthe preassigned slots are coordinated to avoid collisions, which is the advantage of D-TDMA overconventional CSMA/CA (cf. Fig. 18 and Fig. 19 in [4]). Secondly, even if a collision of packets
emerges, the timing information is located at the beginning of the packet; the timing information canbe safely received although the long data part is lost due to collisions.

4.1 Eliminating process of mode-lock states
Next, to correctly visualize the synchronization process and evaluate the synchronizability, we introduce a ‘phase coordinate’ as in Fig. 5. On this phase coordinate, the local timing of each node (inFig. 1) is identiﬁed to a certain point on the [0, 2π] interval. Thus, synchronization process for a given
algorithm in this study is clearly visualized as the dynamics of ‘phase points’ on the phase coordinate,
and hence an easy comparison of the mechanisms among diﬀerent synchronization algorithms becomes
possible.
A typical process of mode-lock elimination by the proposed algorithm in Section 3 is visualized as inFig. 5. Figure 5(a) shows the initial mode-lock state, where seven nodes (◦) in the core are detected by1
The same simulations in Section 4 are also carried out for non-regular distribution of vehicles as shown in Fig. 4(b),which result in a similar eliminating process of mode-lock states and synchronizability, respectively shown in Section 4.1
and Section 4.2. Hence, the assumption of ‘uniform’ distribution (i.e., (iii) in Section 4) is not essential.

290

Fig. 4. Simulation setup. (a) Case of regular distribution of vehicles. (b)Case of non-regular distribution of vehicles showing the mode-lock state.

Fig. 5.

Eliminating process of mode-lock state.

themselves from the algorithm (i.e., (i) in Section 3) and other nodes (×) exhibit uniformly distributed
timings, which implies a stable rotating wave emerges around the core. However, as shown in Fig. 5(b)a ‘cut’ (i.e., a void on the [0, 2π] interval) appears from the initial uniformly distributed timings (inFig. 5(a)) as this synchronization algorithm operates. And once this cut appears, it grows rapidly

291

(Fig. 5(c) and 5(d)), and ﬁnally the mode-lock is totally eliminated and the perfect synchronizationis achieved (Fig. 5(e)).
As opposed to the proposed algorithm, the original synchronization algorithm by Imai and Suzuki
does not have any such eliminating eﬀect to the mode-lock state. Instead, their method introduces
a kind of randomly perturbing eﬀect on the position of the core. Therefore, this does not guarantee
the elimination of mode-lock in a practical time scale, as mentioned in Section 3.

4.2 Comparison of synchronizability for both algorithms
Finally, we provide a systematic performance evaluation for both algorithms. To precisely measure
the degree of synchronization among all nodes, the following order parameter σ [6] is convenient,which is deﬁned asN
1  jφi (t) σ(t) =
e
(1)

,

N
i=1

where φi (t) represents the phase of the i-th node at time t and N is the total number of nodes. By
this deﬁnition, σ becomes 1 when all nodes are perfectly synchronized, and becomes 0 when theirphases (i.e., local timings) are uniformly distributed over [0, 2π]. It is noted that if the mode-lockpersists σ becomes less than 1.
Following the simulation setup (i)–(v) in Section 4, systematic simulations are carried out to evaluate
the performance for both algorithms, and using the above σ, 1, 000 random trials for both methods
for cases of two-hop transmission range and a bit larger transmission range, are shown in Figs. 6(b)and (a) respectively. In each trial, simulation is carried out for 500-frame periods, and the histogramof the ﬁnal degree of synchronization σ is made from 1,000 trials. In addition to the results for
Imai and Suzuki’s algorithm and our algorithm, the result from a ‘simple average’ algorithm is also
obtained in this simulation. This simple average algorithm means a synchronization algorithm basedon the averaged timing without any precautions to mode-lock. Therefore, compared with this simple
average algorithm, synchronizability becomes much improved both for the two algorithms (i.e., Imaiand Suzuki’s and our algorithm), as observed in the both insets in Figs. 6(a) and (b) respectively.

Fig. 6. Comparison of synchronizability. The horizontal axis and the verticalaxis respectively show the degree of synchronization σ and the corresponding
percentage of realization. Each inset shows a magniﬁcation of the lower part of
the histogram, respectively. (a) Case of the two-hop transmission range (i.e.,
12 nodes in this radius). (b) Case of a larger transmission range (i.e., 20 nodes
in this radius).

As shown in Fig. 6(a), the proposed algorithm realizes a 100% synchronization for 1,000 trials,
while Imai and Suzuki’s algorithm realizes less than 80% synchronization; more than 20% instances
remains unsynchronized and this shows the mode-lock is not yet eliminated for these instances. Also,in Fig. 6(b), near 100% synchronization is realized by the presented method. Here we note that,in these three instances out of 1,000 trials, the mode-lock state is not completely eliminated at the292

500-frame periods. However, we have veriﬁed these mode-lock states are eliminated by simply taking
more time i.e., 1000-frame periods.

5. Conclusions
We have presented a simple and eﬀective decentralized synchronization algorithm for intervehicle adhoc networks. This algorithm is based on the insight to the nonlinear nature of mode-lock states, andits eﬀectiveness is veriﬁed through a series of comparative performance evaluations to the previous
synchronization algorithm.
We expect faster and more reliable decentralized synchronization algorithms are obtained by combining the proposed algorithm to the utilization of a certain global timing information in the mobile
environments; (i) an algorithm with an eﬃcient utilization of a reliable time information, for instance,from GPS, and (ii) a frequency synchronization algorithm for multiple groups of vehicles with frequent
unions or separations with each other. These issues will be reported elsewhere.

Acknowledgments
We appreciate Dr. J. Imai (Toyota central R&D labs.) for his informative comments. We also
appreciate kind helps from Prof. N. Wakamiya (Osaka Univ.) for advanced simulation tools.

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