Imposing a Reference Timing onto Firefly Synchronization in Wireless ...

Imposing a Reference Timing onto Firefly Synchronization in Wireless Networks Alexander Tyrrell

Gunther Auer

DoCoMo Euro-Labs 80687 Munich, Germany Email: [email protected]

Abstract— In this paper1 a reference timing is imposed onto a network that synchronizes in a completely distributed manner. Synchronization is not always obtained because normal nodes are deaf while transmitting and cannot always receive synchronization messages from reference nodes. To counter this unwanted effect the transmission scheme of reference nodes is modified so that synchrony can be ensured and all network members agree on the imposed reference timing.

for the convergence of synchrony was published in 1990 by Mirollo and Strogatz [7]. In this original model communication through pulses of negligible duration is assumed, which is generally not realistic in a wireless environment. In [8, 9] Firefly synchronization was applied to wireless networks, where communication through long synchronization messages, such as a Pseudo-Noise (PN) sequence, is considered.

I. I NTRODUCTION Time synchronization is a classical and important building block in networked systems. Besides the obvious advantage of having an accurate clock, synchronization offers various benefits for communication protocols. For example, medium access control can be designed in a more efficient manner, if all network nodes are accurately synchronized. A well-known example, which illustrates this benefit, is the comparison of unslotted and slotted ALOHA: the slotted version offers twice the maximum throughout, since it reduces the probability of message collisions [1]. Furthermore, cooperative diversity schemes (i.e., virtual antenna arrays) typically require the nodes to be synchronized [2]. Other classical means to exploit synchrony are implicit messaging and coding in time [3]. Imposing a reference timing is typically done in a centralized manner. The Internet, for instance, applies the Network Time Protocol (NTP) [4] to maintain the clocks of hosts and servers well synchronized. In wireless networks a global time reference may be provided by satellites, through the Global Positioning System (GPS) [5]. However, in an indoor environment, it may not always be possible to receive the required signals. Furthermore, for sensor networks the implementation of a GPS receiver may be prohibitive, due to constraints in cost, power consumption and/or size. However, time synchronization for distributed clocks, preferably using a distributed algorithm to avoid central entities, is a non-trivial task. The issue of time synchronization for wireless sensor networks is more clearly detailed in [6]. Turning to this problem, we can learn from nature, where synchrony is not uncommon. For example, in South-East Asia alongside riverbanks, fireflies gather on trees at dawn and synchronize their blinking. It seems as though the whole tree is flashing in perfect synchrony. A theoretical framework 1 This work has been performed in the framework of the IST project IST4-027756 WINNER (World Wireless Initiative New Radio), which is partly funded by the European Union.

1550-2252/$25.00 ©2007 IEEE

% $ Fig. 1.

Merging of two synchronized groups through one link

The present paper extends the work on Firefly synchronization to address an inherent problem of distributed synchronization procedures: nodes agree on a relative time reference, as opposed to a global time reference, e.g. provided by GPS. What happens if two groups that have synchronized separately merge? Fig. 1 shows two disjoint groups A and B that cannot communicate as they are too far from one another. Suppose that initially A and B have synchronized separately. An individual from B moving in the direction of A and creates a bridge between the two groups. As these groups only have a relative time reference, this bridge forces all members to resynchronize and agree on a new reference, which is costly if networks A and B are large. To avoid this difficulty, both groups need to synchronize to the same reference before merging. Hence, we consider a scenario where only a few nodes have access to an absolute clock reference. The objective is to let these reference nodes impose a global time reference to the entire network. For instance, in sensor networks the existence of fusion centers with superior capabilities is a common assumption. The TimeDiffusion Synchronization Protocol (TDP) [10] addressed this issue by distributing a time reference among a sensor network. In this paper we impose a reference timing to Firefly synchronization, with the following key objectives: • The number of reference nodes is significantly smaller

222

Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on January 20, 2009 at 20:10 from IEEE Xplore. Restrictions apply.

than the number of normal nodes The behavior of normal nodes is not modified at all, i.e. the simple rules of [8, 9] that govern normal nodes remain unchanged. Unlike TDP [10], the exchange of timestamps is avoided. The remainder of this paper is structured as follows. In Section II the synchronization scheme of [8] is presented, and reference nodes are directly applied to a fully meshed network. In Section III the issue of deafness among reference and normal nodes is presented, and a modification to the reference nodes is made in order to regain synchrony. •

II. F IREFLY S YNCHRONIZATION This section describes decentralized time synchronization by means of pulse-coupled oscillators [7]. Then, a reference timing is imposed, and a novel scheme is presented that improves accuracy and speeds up convergence. A. Synchronization of Pulse-Coupled Oscillators Pulse-coupled oscillators refer to systems that oscillate periodically in time and interact each time they complete an oscillation. This interaction takes the form of a pulse that is perceived by neighboring oscillators. As a simple mathematical representation, a pulse-coupled oscillator i is completely described by its phase function φi (t). This function evolves linearly over time until it reaches the threshold value φth . When this happens, the oscillator is said to fire, meaning that it will transmit a pulse and reset its phase. If not coupled to any other oscillator, it naturally oscillates and fires with a period equal to T . Fig. 2(a) plots the evolution of the phase function during one period when oscillator i is isolated.

IIW I

ILUH

TH

ILUH

A Fig. 2.

4

TJ

B

Trefr > 2 · T0[max] where T0

[max]

4

T

Time evolution of the phase function

When coupled to others, an oscillator is receptive to the pulses of its neighbors. When node j fires at t = τj , it transmits a pulse that will instantly increment phases of its neighbors by an amount that depends on the current value: φi (τj ) → φi (τj ) + ∆φ(φi (τj )) when receiving a pulse Fig. 2(b) plots the time evolution of the phase when receiving a pulse. The received pulse causes the oscillator to fire early. ∆φ is determined by the Phase Response Curve (PRC), which was chosen to be linear in [7]: φi (τj ) + ∆φ(φi (τj )) = min (α · φi (τj ) + β, 1)

(1)

(2)

is the maximum propagation delay in the network.

B. Imposing a Reference Timing A reference node is characterized by an oscillator that emits pulses every T seconds without being influenced by other oscillators. The objective is to impose the global timing of reference nodes to the entire network. As normal nodes do not increment their phase while in refractory period, a reference pulse received during refractory will not be acknowledged. This blind spot of width Trefr limits the attainable accuracy. In order to unlock the situation where firing instants of reference and normal oscillators are stuck within Trefr , we propose a slightly higher phase frequency for reference oscillators compared to normal nodes. This frequency offset is denoted fsw and it is equivalent to reference oscillators transmitting pulses periodically every Tsw . The sweeping period is necessarily shorter than the period of normal oscillators: Tsw < T

'IIITJ



where α and β determine the coupling between oscillators. The threshold φth is normalized to 1. It was shown in [7] that if the network is fully meshed, the system always converges, i.e. all oscillator will fire as one, for α > 1 and β > 0. The time to synchrony is inversely proportional to α. (i,j) When a propagation delay T0 occurs between two oscillators, the system can become unstable [11]. The pulse of one oscillator causes the other oscillator to fire and transmit after (i,j) T0 . This echo then causes the first oscillator to increment (i,j) its phase again after T0 , and so on. To avoid this avalanche effect and regain stability, a refractory period of duration Trefr is introduced after firing [12]. When a node is in this state, no phase increment is acknowledged. Stability is maintained if echos are not received. This translates to a condition on Trefr :

(3)

Thanks to this novel strategy normal nodes receive reference pulses earlier, causing them to fire earlier and eventually follow reference nodes. Therefore the accuracy bounded by T0[max] is restored, without modifying the internal behavior of normal nodes. It is referred as sweeping because reference nodes sweep for normal oscillators that were stuck within the deaf spot. Fig. 3 plots the mean time to synchrony, denoted T¯, required by 30 nodes after 1000 trials when one oscillator imposes a reference. Simulations are done by decomposing each period T into 1000 steps, and calculating at each step the state of each node and interactions between nodes. The coupling settings are α = 1.25 and β = 0.01. Successful synchronization is declared when the firing instants are spread within T0[max] when sweeping is used and within Trefr when no sweeping is used. For Trefr = 0.01 · T , T¯ is comparable for both methods, but in the case of sweeping, T¯ rapidly drops to an average of 3.2 · T when Trefr increases. When no sweeping is used T¯ averages around 4.2 · T . Hence, both accuracy and time to synchrony are improved by sweeping reference nodes.

223 Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on January 20, 2009 at 20:10 from IEEE Xplore. Restrictions apply.

Mean Time to Synchrony / T

5.5 no sweep: fsw = 0 sweep: fsw = 0.1

5

4.5

4

network [8]. If long transmission bursts are considered, transmission delays are critical. The scheme needs to be modified to counter this unwanted effect. A strategy was introduced in [8] to regain acceptable accuracy: after firing, a node waits and delays its transmission for a time equal to: Twait = T − (TTx + Tdec )

3.5

(4)

(i,j) T0

3

2.5

0

0.05

0.1

0.15

0.2

0.25

Trefr / T Fig. 3.

Mean Time to Synchrony

III. A PPLICATION TO W IRELESS S YSTEMS In a wireless environment different delays need to be taken account to correctly apply the model presented in the previous section. These delays further compromise the achievable accuracy. The time advance strategy of [8] is presented to help regaining acceptable accuracy. Then the sweeping scheme is applied to impose a global time reference. A. Delays in Wireless Systems In the synchronization model of pulse-coupled oscillators described previously, it is assumed that communication is done through pulses and that a pulse is instantly received and decoded by other oscillators. In a wireless environment solitary pulses are hardly used alone as they are virtually impossible to detect. More realistically a sequence of pulses or a burst of length TTx is to be considered. During this time a transmitter is not able to receive. Once transmitted, the synchronization message will not be instantly received as some propagation delay, (i,j) denoted T0 , occurs between nodes i and j. After reception, some processing time is required to correctly declare that a synchronization message has been detected, denoted by the decoding delay Tdec . Altogether, four delays need to be taken into account to properly model a wireless network: (i,j) • T0 : Propagation delay — time for a burst to propagate from emitting node i to the receiving node j. • TTx : Transmitting delay — length of the burst. A node cannot receive during this time. • Tdec : Decoding delay — time taken by the receiver to decode a burst. • Trefr : Refractory delay — time necessary after transmitting to maintain stability. These delays are the most significant difference from the Mirollo and Strogatz model, which assumes no propagation and decoding delays, and an infinitely short transmission time [7]. B. Time Advance Strategy The maximum total delay determines the attainable accuracy when directly applying the original scheme to a wireless

is neglected, receiving nodes With this approach, if will increment their phases exactly T seconds after the transmitter fired. During wait, an oscillator is shut-off and therefore cannot receive. With this new transmission scheme oscillators have a natural oscillatory period of 2 · T . The time during which the phase function increments is reduced by the waiting, transmitting and refractory delays. It is now equal to: TRx = 2 · T − (Twait + TTx + Trefr )

(5)

For a system of N oscillators, firing instants are initially randomly distributed over a period of 2 · T , and each oscillator follows the same rule of waiting before transmitting. Over time oscillators split into two groups, each group firing T seconds apart and helping each other to synchronize [8]. Therefore T is still used as a reference. Fig. 4 plots the new phase function and the four states of a node for two synchronized groups.

IW I

ILUH

TH

4DEC :$,7

7;

IW  I

5()5

T

/,67(1

4

ILUH

TH

4DEC /,67(1



:$,7

4

7;

5()5

T

4

Fig. 4. Time evolution of two synchronized groups without a reference node

The separation into two groups shown on Fig. 4 is the only stable state that does not cause any phase increment. Therefore if a third node enters the network without being synchronized, all nodes would continue following the simple rules of incrementing the phase after detecting a synchronization word, and reform two groups and synchronize again. C. Reference Oscillators A reference node corresponds to an oscillator that fires periodically every T seconds without ever modifying its phase.

224 Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on January 20, 2009 at 20:10 from IEEE Xplore. Restrictions apply.

Therefore it simply transmits a synchronization burst every period following the time advance strategy (after firing, wait during Twait before transmitting). Fig. 5(a) plots the transmission scheme of a reference during one period as it is not sweeping.

4DEC

ILUH

A

:$,7

IREFW I

7;

4

LETTHE SYSTEM STABILIZE

ILUH

T

:$,7

.SW

.STAB

SWEEPFOR ISOLATED NODES

LETTHE SYSTEM STABILIZE

Fig. 6.

ILUH

TH

B 4$

T



4SW Fig. 5. Evolution of a reference node during one period: (a) without sweeping, (b) when sweeping

The phase function with the adjustable offset fsw is defined as:
0 when t ∈ [0, T∆ ] φref (t) = (6) t−T∆ (1 + fsw ) · TRx when t ∈ [T∆ , Tsw ] where T∆ = Tref − Tdec = T − TRx , and TRx is given by (5). The corresponding sweeping period Tsw yields:   TRx 1 Tsw = T∆ + (7) = T − TRx · 1 − 1 + fsw 1 + fsw



SWEEPFOR ISOLATED  NODES

T

Global sweep and stabilize strategy

D. The Deafness Problem The deafness problem among normal oscillators is caused by the fact that a node cannot receive and transmit simultaneously [9]. In the case of reference nodes, the deafness problem is due to the fact that a reference node does not receive any message, and it simply transmits synchronization bursts periodically. After running simulations where a reference node is present and does not sweep, the system does stabilize, but half the time, four groups form. Two groups correspond to the 1 and b. 1 Two other reference node and are noted Groups a 2 and , 3 are synchronized among themselves groups, noted  (each firing T seconds apart), but have fallen into the deaf spot of the reference group. To understand this situation, Fig. 7 1 and plots the state evolution of a reference node noted  2 during two periods when the firing a normal node noted  instants are spaced by Tdeaf and the reference group is late compared to the other.

4

ILUH



Fig. 5(b) plots this new phase function. It should be noticed from Fig. 5 that if a reference node fires before it finishes transmitting, the synchronization burst is not fully transmitted. This imposes a constraint on fsw such that Tsw ≥ Twait +TTx . The frequency offset limit can be calculated as: Tdec [max] fsw = (8) T − Trefr The variation boundaries for fsw are: 0 ≤ fsw ≤ fsw . a) Sweep and Stabilize: The sweeping scheme of Section II-B presented high instability when directly applied to meshed networks, and synchrony was not always reached. Therefore sweeping should be applied more seldom during Nsw periods, and then fsw should be set back to zero to let the system stabilize during Nstab periods. Fig. 6 represents the global sweep and stabilize scheme. b) Retaining a global reference timing: As a particular case of the presented scheme, it can be interesting for the reference node to come back exactly in the same time slot as it was before sweeping. This is the case if Nsw ·(T −Tsw ) = T . This translates to the following conditions: 1 (9) fsw = Nsw · TTRx − 1

.SW

REFERENCE

7;

:$,7



ILUH

7;

4

ILUH

:$,7

7;

ILUH

:$,7 ILUH

NORMAL

:$,7

/,67(1

T

7;

5()5

/,67(1

T

[max]

4

4

4DEAF Fig. 7.

1 and normal nodes
2 Deafness between reference nodes


2 is never able During the two periods plotted on Fig. 7,  to fully receive and decode the two synchronization bursts transmitted by the reference oscillator. During the first period, transmitting periods overlap. As an oscillator cannot receive 2 is deaf to this message. During the while transmitting,  second period, it is able to fully receive the message, but it is not able to fully decode it, because it fires while decoding it. We note that the width of the deaf spot is equal to the transmitting delay TTx . To understand this suppose Tdeaf had

225 Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on January 20, 2009 at 20:10 from IEEE Xplore. Restrictions apply.

2 would have received the first been larger than TTx . Then  message, and would have progressively shifted its firing instant closer to the reference instant, until it does not exceed TTx . To cover the nodes that are within the deafness window, fsw and Nsw need to be chosen such that Nsw · (T − Tsw ) ≥ TTx . This translates to the following equation:   TTx 1 ≥ (10) Nsw · 1 − 1 + fsw TRx The deafness problem could be solved within one period if the reference node simply shortens its period by TTx during one  period. This corresponds to Nsw = 1 and fsw =  TTx TRx −TTx . This way, the nodes that were deaf to reference messages would hear them again, and be able to synchronize within one period. However this is relatively unstable and sweeping with a lower frequency increment during more periods is preferable. E. Multiple Reference Nodes To verify that the sweeping strategy resolves the deafness problem simulations are conducted. We consider multiple reference nodes within a meshed network. This reflects a multihop network topology, where relay stations or mobile terminals forward signals from/to base stations that have access to a global time reference. Simulations are conducted using the network topology shown on Fig. 8. The network contains 30 nodes including four reference nodes that are represented as large squares. 18

5

22

4

12

7 28

3 17

21 26

30

1 6

10

25 11

2 9 23 13

16 24

8 14

19 15

27

29

20

Fig. 8.

Reference Nodes on a meshed network

Fig. 9 plots the simulation results after 500 sets of initial conditions. The sweeping parameters Nsw and fsw are chosen to retain a global reference timing according to (9). The time settings are TTx = 0.20·T , Tdec = 0.25·T and Trefr = 0.45·T . The coupling settings are α = 1.3 and β = 0.01. For Nsw = 4, the synchrony rate is below 60%, but rapidly improves as Nsw increases. When Nsw ≥ 8 the synchrony rate approaches 100%. This is very appealing, and suggests that with a sufficiently low sweeping frequency a global reference time can be successfully imposed to wireless multihop networks.

Synchrony Rate (%)

100

90

80

70

60

nodes 1, 3, 19, 22 reference 4

5

Fig. 9.

6

7 Nsweep

8

9

10

Synchrony Rate with several reference nodes

IV. C ONCLUSION With the proposed modification of reference nodes a common time scale can be imposed onto a set of distributed oscillators by means of Firefly synchronization. This modification does not concern normal nodes, which only follow the self-organized synchronization strategy of [8]. Furthermore the sweep and stabilize scheme can also be used as an implicit mean of communicating to normal nodes that a global reference timing is available in the network. R EFERENCES [1] L. Roberts, “Aloha packet system with and without slots and capture,” ACM Comput. Commun. Rev., vol. 5, pp. 28–42, Apr. 1975. [2] R. Pabst, B. H. Walke, D. C. Schultz, P. Herhold, H. Yanikomeroglu, S. Mukherjee, H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Aghvami, D. D. Falconer, and G. P. Fettweis, “Relay-based deployment concepts for wireless and mobile broadband radio,” IEEE Communications Mag., vol. 42, no. 9, pp. 80–89, Sept. 2004. [3] G. Tel, Distributed Algorithms, 2nd ed. Cambridge University Press, 2000. [4] D. Mills, “Internet time synchronization: the network time protocol,” IEEE Transactions on Communications, vol. 39, pp. 1482–1493, Oct. 1991. [5] W. Lewandowski and C. Thomas, “GPS time transfer,” IEEE Proceedings, vol. 79, pp. 991–1000, July 1991. [6] J. Elson and K. R¨omer, “Wireless sensor networks: A new regime for time synchronization,” ACM SIGCOMM Computer Communication Review, vol. 33, no. 1, pp. 149 – 154, Jan. 2003. [7] R. Mirollo and S. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. APPL. MATH, vol. 50, no. 6, pp. 1645–1662, Dec. 1990. [8] A. Tyrrell, G. Auer, and C. Bettstetter, “Fireflies as Role Models for Synchronization in Ad-Hoc Networks,” in Proc. Int. Conf. BioInsp. Models of Network, Information and Comp. Sys. (BIONETICS 2006), Cavalese, Italy, Dec. 2006. [9] ——, “Synchronization Inspired from Nature for Meshed Networks,” in Proc. IEEE Int. Conference on Wireless Communications, Networking and Mobile Computing (WiCOM 2006), Wuhan, China, Sept. 2006. [10] W. Su and I. Akyildiz, “Time-diffusion synchronization protocol for sensor networks,” IEEE/ACM Transactions on Networking, vol. 13, no. 2, pp. 384–397, Apr. 2005. [11] U. Ernst, K. Pawelzik, and T. Geisel, “Synchronization induced by temporal delays in pulse-coupled oscillators,” Physical Review Letters, vol. 74, no. 9, pp. 1570–1573, Feb. 1995. [12] Y.-W. Hong and A. Scaglione, “A scalable synchronization protocol for large scale sensor networks and its applications,” IEEE Journal on Selected Areas in Communications, pp. 1085–1099, May 2005.

226 Authorized licensed use limited to: UNIVERSITY OF DELAWARE LIBRARY. Downloaded on January 20, 2009 at 20:10 from IEEE Xplore. Restrictions apply.