Tri-Message: A Lightweight Time Synchronization Protocol for High
Latency and Resource-Constrained Networks
Tian Chen1
Hongbo Jiang1
Xue Liu2
Liu Wenyu 1
Wang Yi1
1 Dept. of Electronics and Information Engineering, Huazhong University of Science and Technology
2 School of Computer Science, McGill University, Montreal, Quebec, Canada
{tianchen, hongbojiang, liuwy, ywang}@mail.hust.edu.cn
ABSTRACT - High precision time synchronization protocols
are critical for many distributed systems. On the other hand,
for emerging resource-constrained applications such as sensor
networks, energy efficiency and computation simplicity could
be as important as precision. Existing terrestrial
synchronization protocols including RBS, FTSP, TPSN and
LTS have already achieved high precision in radio networks,
but none of them perform well in high latency networks like
acoustic sensor networks. The reason is that they either
assume instantaneous packet transmission between nodes, or
ignore clock drift during synchronization process. By
addressing these problems, the recent proposed TSHL
protocol provides precise time synchronization for high
latency networks. However, due to its high energy and
computation consumption, TSHL is still not practical in
resource-constrained applications such as sensor networks
which have stringent constraints on system resources. A new
approach, both providing high precision and having small
energy reservation, is prerequisite for those applications.
In this paper, we present Tri-Message: a lightweight time
synchronization protocol for high latency and resourceconstrained networks. As its name suggests, only three
message exchanges are required in one synchronization
process. Meanwhile, Tri-Message utilizes very simple
mathematical operations to calculate the clock skew and offset.
Specially, Tri-Message is feasible for many extremely long
latency applications such as space exploration because it has
an increasing synchronization precision with the increasement
of distance. Our theoretical analysis and simulation results
demonstrate that Tri-Message is able to achieve good
performance of synchronization and outperform existing
protocols in terms of energy and computation cost.
Keywords - Time Synchronization, Packet Delay, Clock
Synchronization, Clock Drift, Sensor Networks.
1. Introduction
Time synchronization is a critical piece for many
distributed systems. Some distributed synchronization
protocols, such as NTP [1], have been proposed and
investigated thoroughly for many years. Emerging resourceconstrained applications such as sensor networks, however,
are extreme cases where nodes often collaboratively
process time-sensitive data like target location [10, 11]. In
[email protected]
these networks, time synchronization should be mind of
energy/computation consumption due to the limited
resource capabilities, as well as precision [14].
Many time synchronization protocols have been
proposed for sensor networks recently [2, 3, 4, 5]. By
complicated approaches to reducing uncertainty and
accounting for various processing latencies in
communication, these protocols provide a high degree of
precision as well as remain energy efficiency. Designed for
RF-based networks, however, these protocols either assume
that propagation latency is negligible, or ignore clock drift
during the synchronization process. Accordingly, for many
high latency networks like acoustic networks and space
exploration [7], none of abovementioned protocols work
well: their assumptions are not practical due to the high
propagation delay in these networks.
Time Synchronization for High Latency networks
(TSHL) [6] is a dedicated synchronization protocol that
deliberately compensates for high-latency during the
synchronization communications. While it achieves a
relatively high precision, the energy inefficiency and high
computation complexity make TSHL inadequate in sensor
networks: first, a typical synchronization process between
two nodes consists of 27 packets exchanges, and second,
computational-heavy linear regression algorithm is required
in the first phase to get an accurate estimation of node
skew.
In this paper, we present Tri-Message: a lightweight time
synchronization protocol for high latency and resourceconstrained networks. The novelty of our work is that, only
three messages in one Tri-Message synchronization process
are needed. In addition, Tri-Message utilizes some very
simple mathematical operations to calculate the clock skew
and offset as well as provides a satisfactory estimation.
Another advantage of Tri-Message is that synchronization
precision of Tri-Message is increasing with the increased
distance, which makes it feasible for high latency
applications such as space exploration.
Reference Node A
Synchronizing Node B
Reference Node A
Synchronizing Node B
Global Time
A1
…
Synchronizing Node B
{
…
B1
{B
B1
{B
A2
I1
I2
A3
t3
}
I2
A3
t3+d+δ3
B3
(a)
t2+d+δ2
(b)
2
A2
Phase2
}
Phase2
Bk = Ak + Offset
Phase Offset = [(B1-A1) – (A2-B2)]/2
Propagation Delay = [(A1-B1) + (B2-A2)]/2
A2
I1
t2
2
B2
Phase1
I3
A1
I3
Phase1
{
B1
Reference Node A
A1
B3
(c)
Figure 1 (a) Sender-Receiver two-way synchronization (b) TSHL: Synchronization (c) TSHL: Synchronization with jitter
The remainder of this paper is organized as follows. We
introduce the background and related works in Section 2.
Section 3 presents the basic idea behind Tri-Message,
followed by the mathematical framework and analysis.
Section 4 provides a performance comparison of TriMessage and TSHL. We conclude in Section 5. Appendix
constrains the proof of some theorems.
2. Background and Related Works
2.1 Background
Boot at different times, there must be some way for
distributed network nodes to determine a common time
base, which we called the process of time synchronization.
What are the obstacles of time synchronization? There are
two quantities should be deal with: clock offset and clock
skew (clock drift speed). Clock skew is caused by variations
in crystal oscillation frequency. Without skew, offset can be
determined by a single pair of messages exchange if we can
compensate for any sources of uncertain latencies in the
path. When skew exists, already synchronized nodes would
eventually drift out of synchronization sooner or later,
hence a synchronization protocol should figure out both
offset and skew.
Here we introduce the major causes of errors in time
synchronization process itself. Deterministic delays like
Encoding and Decoding can be equivalent into propagation
delays. Those versatile sources of jitters of high nondeterminism in the estimations of message delivery delay
are listed in Table 1 [6, 9].
Due to different assumptions in which sources of
variation are dominant, existing protocols have adopted
different approaches to eliminate one or several sources of
errors simultaneously (Section 2.2). Seldom have they taken
propagation delay into consideration. In high latency
networks, clock drift continues just during the
synchronization process. An accurate time synchronization
scheme must account for this source of error.
Some notational conventions are given here. We refer to
t as global clock times and node name to denote local
clock readings in this paper. The subscript indicates the
local clock reading index, for example, A1 is the first local
clock reading recorded at node A.
Table 1. Sources of non-determinism
Source
Side
Details
Send
Sender
Message traversal delay from application
layer to MAC layer.
Access
Sender
Channel contention delay in MAC layer
Interrupt
Handling
Receiver
Propagation
None
Receive
Receiver
Delay of CPU responding to an receive
interrupt, also maybe be affected by clock
granularity
Can be deterministic if the speed of
propagation is assumed constant, and if
synchronization exchange is performed
with assumption of path symmetry.
Traverse up from MAC layer to the
receiver application
2.2 Previous Protocols
There are just two fundamental schemes to synchronize
clocks: Sender-Receiver and Receiver-Receiver. As a
Sender-Receiver two-way scheme shown in Figure 1(a),
NTP works well over Internet paths with high latency and
high variability and estimates both offset and skew [1].
Because of its long-term bi-directional time information
exchange, NTP is unsuitable for many other high latency
applications like acoustic sensor networks.
Existing terrestrial sensor network synchronization
protocols achieve high precision in radio networks.
Reference Broadcast Synchronization (RBS) [2] eliminate
transmitter side uncertainties by Receiver-Receiver style
synchronization. Flooding Time Synchronization Protocol
(FTSP) [4] eliminates timestamp uncertainty by
timestamping in the MAC and PHY (radio) message layer
and also account for byte alignment jitter. Both RBS and
FTSP deal with skew, but they do not consider propagation
delay at all because RF signal travels at the speed of light.
Due to their assumptions, none of abovementioned
protocols work well in high latency networks. Taking into
account propagation latency, Timing-sync Protocol for
Sensor Networks (TPSN) [3] performs better in high
latency networks [6]. But, TPSN does not take clock skew
effect during the synchronization process into
consideration, which is also critical to achieve a reasonable
synchronization precision and stableness for high latency
networks.
Time Synchronization for High Latency networks
(TSHL) is the first protocol that takes into account both
high propagation latency and process skew effect for high
latency networks [6]. Its procedure is shown in Figure 1(b).
Without loss of generality, we assign node A as the time
base: an original anchor or a node already compensated its
skew and offset thereafter serve as an anchor. Node B is
supposed to be synchronized which has its skew a and
offset b . Let d refer to the standard propagation delay
between nodes, and let I1 and I 2 denote two transmission
intervals between successive messages. We also let I 3
represent total time span of the first phase, and Bk denote
the clock reading of Node B at time t k . We
have Bk  atk  b, tk  ( Bk  b) / a .
TSHL splits time synchronization process into two
phases. In the first phase, node A sends a group of
timestamp beacons to node B, enabling node B to estimate
its clock skew through linear regression to time base. Due to
size limitations, all deduction details are presented in
Appendix. In the second phase node B entered skewsynchronized state, and a skew-compensated two-way
exchange is taken. The offset of node B is
b  ( B2  B3 )  a( A2  A3 ) / 2
(1)
However, time jitters during the synchronization process
will affect the synchronization accuracy. Here we analyze
impact of those errors on TSHL. Time deviations from d
are represent by  k , where
k is correspondent to message
index. Here we use a superscript on the timestamp to
indicate an error-affected value. For example, B3 ' means
B3 value is polluted by jitter. As shown in Figure 1(c),
A2 and B3 deviate to A2 ' and B3 ' respectively.
Nevertheless, the calculated a' is different from real
skew a . We refer  a to relative skew error
and a'  a(1   a ) . Let tk  t3  d   ,  denotes time
real
passed since synchronization completed. Clock offset error
tk can be calculated according the following theorem.
Theorem 1: The long-term offset error of TSHL is
dominated by skew error  a . The instant Offset error t k
of TSHL is proportional to the propagation delay d and
can be expressed by
tk  ( 2   3 ) / 2   a (  d  I 2 / 2) (2)
The proof of the theorem is in the appendix.
We argue that TSHL is not applicable for recourseconstrained networks in three-fold. First, while it achieves
considerable high precision, the energy and computation
consumption of TSHL are significantly high: a typical
synchronization between two-nodes costs 27 packets;
computation-heavy linear regression algorithm is required
for an accurate estimation of node skew. Furthermore,  a is
affected by first phase beacon numbers and jitters. Finally,
we found  a is sensitive to I 3 , neglected by [6].
3. Tri-Message
3.1 Assumptions
Many sources of non-determinism in message exchange
can be removed by previous works like MAC/PHY layer
timestamping. Consistent with [6], here we assume these
errors had already been compensated, and all remain
uncertainties can be treated as a time jitter, which follows
Gaussian distribution, add to propagation delay d . The
second assumption is that clocks are short-term-skewstable. That is, clock skew maintains constant during the
synchronization process. Long term instability can be
countered by resynchronization.
3.2 Overview
As its name suggests, only three message exchanges are
needed for a single Tri-Message synchronization process.
We focus on two nodes’ situation, one node and one anchor,
to illustrate its operation. We refer to the skew rate and
clock offset of node B as  and    respectively. The
reason why clock offset is denoted by    is that it can
facilitate mathematical formulation later in Section 3.3.
The process is shown in Figure 2(a). First, the anchor
node A sends a message to node B, at the same time
captures the transmit timestamp A1 in MAC/PHY layer and
put the timestamp in the message; node B captures its own
receive timestamp B1 during the reception of the message
and, save the send timestamp A1 contained in the first
message. Second, two nodes swap their roles and B saves
the transmit timestamp B2 and A records the receive
A2 . Then they swap again. A sends the third
message and put the transmit timestamp A3 together with
A2 in the packet. At last, B receives the third message so
timestamp
that all 6 timestamps are known to B.
Global Time
Reference A
Synchronizing B
A1
t1
Global Time
Reference A
A1
t1
(A1)
(A1)
t1+d
I1
t2
t2+d
A2
t3
A3
}
{
t1+d+δ1
B1
I1
B2
t2
I2
A3
t3
3)
t3+d
B3
{
B1'
B2
A2'
t2+d+δ2
(A2
A
Synchronizing B
}
I2
(A2
A
3)
t3+d+δ3
(a)
B3'
(b)
Figure 2 (a) Tri-Message: Synchronization (b) Tri-Message: Synchronization with jitter
After the message-exchanges, node B has 6 timestamps
enough to figure out its clock skew and offset. The
mathematical formulation of Tri-Message is given in
Section 3.3. Once clock skew and offset is known, B can
estimate its skew-offset-compensated “real clock” form its
clock readings.
3.3 Mathematical formulation
Assume that anchor A has no skew and offset error. B has
local clock skew and offset, for global clock t k , its clock
reading can be expressed as Bk
message
exchanges,
B
  (tk   ) . After threehas
6
timestamps
A1, A2 , A3 , B1, B2 , B3. From the global clock view, we
have 6 reference equations, and we can get
  B3  B1  /  A3  A1 
  ( B1  B2）/ 2  ( A1  A2 ) / 2
(3)
For a local clock reading Bk , node B estimate its skewoffset-compensated global time
t k as
tk  Bk /   
(4)
Our algorithm draws the concept of skew modeling from
RBS [2], skew compensation during the synchronization
exchange from TSHL [6], and the mathematical formulation
of local clock readings from PinPoint location system [8].
As opposed to prior works, we synchronize a node to a time
base anchor by only three messages, which is extremely
energy efficient. The computational complexity is also
tractable: no linear regression is needed as in [6]. To our
best knowledge, this is the most efficient and practical
synchronization protocols for high latency networks.
3.4 Discuss and analysis
In this subsection, we investigate the impact of jitters to
the synchronization performance of Tri-Message. By taking
into account time deviations  k shown in Figure 2(b), the
estimated skew
 ' can be expressed by real 
B3 ' B1 '
1   3

 '  A  A   (1  t  t )   (1    )
3
1
3
1

   (   ) /(t  t )
1
3
3
1
 
where
(5)
  is the relative skew error. We now have
Theorem 2: Tri-Message causes decreasing offset error
with the increasement of propagation delay and this error is
proportional to jitters, given by

4d  I1  2 I 2

)(1   3 )
2d  I1  I 2 4d  2 I1  2 I 2
(6)
 ( 2  1 ) / 2
tk  (
The proof of the theorem is in the appendix. This
characteristic makes Tri-Message feasible for high latency
applications like acoustic sensor networks and space
exploration networks, as we mentioned in Section 1.
4. Performance Evaluation
In this section we present some simulation results of TriMessage and the comparison with TSHL [6], which is the
closest one to our work, considering precision in high
latency networks. We compare two protocols under varying
configurations over parameters such as propagation delay
and message intervals, under varying conditions such as
receive jitter and node clock skew.
(a)
(b)
2.5
10
Mean
Variance
2.5
1.5
1
0.5
TSHL
Tri-Message
9
8
2
Instant error (usec)
Skew error (ppm)
2
Skew error (ppm)
(c)
3
Mean
Variance
1.5
1
7
6
5
4
3
2
0.5
1
0
0.5
1
1.5
2
2.5
3
Beacon total interval (seconds)
0
3.5
0
5
10
(d)
30
0
35
TSHL
Tri-Message
TSHL
Tri-Message
7
160
140
120
50
60
50
6
5
4
3
100
30
40
Jitter (usec)
60
Offset error (usec)
180
Instant error (usec)
8
20
70
9
200
10
(f)
10
220
0
(e)
240
Offset error (usec)
15
20
25
Beacon number
40
30
20
2
80
10
TSHL
Tri-Message
1
60
0
10
20
30
40
Jitter (usec)
50
60
0
0
2
4
6
Propagation delay (seconds)
8
0
0
2
4
6
Propagation delay (seconds)
8
Figure 3 (a) TSHL: Effect of beacon interval (b) TSHL: Effect of beacon number (c) instant error on varying jitters
(d) offset error on varying jitters (e) instant error on varying propagation delays (f) offset error on varying propagation delays
4.1 Simulation setup
The Tri-Message and TSHL protocols are both simulated
in a custom event driven, packet level simulator designed
for an acoustic networks with high latency. There are two
nodes in the simulation scenario: node A is an anchor with
no skew and zero offset; Node B's clock has some skew and
offset relative to the global time. We modeled all
uncertainties in one message delivery process to a single
 k by introducing a Gaussian receive jitter, similar to that
in FTSP [4] and TSHL [6]. In our simulations, granularity
is fixed because we consider error caused by granularity
can be combined with error caused by interrupt handling.
The granularity of the clocks is set to 1μs, which is common
in sensor networks. We allow the following adjustable
parameters in our simulations:
• Initial Node clock skew and offset.
• Jitter distribution.
• Propagation delay
• Message intervals
Each data point shown in a graph is the mean value of 100
simulation runs. Error bars show standard deviations.
Unless specifically mentioned, the following parameters are
used in all experiments: Skew = 40 parts per million,
Offset = 10μs, Propagation Delay = 1s, Receive Jitter =
5μs.
Consistent with [3, 4, 6], three evaluation metrics are
used in our work:
1) Skew Error: the absolute value of skew estimation
error is the most important metric. A small skew error
implies small future deviation after a particular delay from
the time the final synchronization completed.
2) Instant Error: here we measure the absolute difference
between the global time and the skew-offset-compensated
local time of the node at the instant time immediately after
the final message exchange. These results represent the best
case performance.
3) Offset Error: here we measure the offset error of
calculated time of each scheme after a particular delay from
the time the final synchronization exchange occurred. This
is the most direct metric of algorithms. Usually we set this
to 100 seconds.
We distinguish the presentation of message intervals by
label them with protocol names for clarity. As shown in
Figure 2, Tri-Message has two intervals: Tshl _ I1 and
Tshl _ I 2 . With an additional Beacon interval, intervals of
TSHL are denoted as Tshl _ I1 , Tshl _ I 2 and Tshl _ I 3
in Figure 1(b) and (c). Note that Tshl _ I 3 is the total span
of beacons, and should be divided by beacon numbers to
get per-message interval between successive beacons. The
total synchronization time of Tri-Message can be presented
as
,
and
TSHL
3d  Tri _ I1  Tri _ I 2
is 3d
 Tshl _ I1  Tshl _ I 2  Tshl _ I 3 .
Simulations of [6] have shown that TSHL accuracy is
directly proportional to beacon message number, receive
jitters or the granularity the clocks used. However, we
found that TSHL is also related to beacon interval. Before
comparative evaluation, parameters are investigated to
figure out a proper parameter settings for TSHL.
4.2 TSHL Parameters investigation
First, we vary beacon interval Tshl _ I 3 to investigate
its effect in terms of skew error. Beacon numbers are fixed
to 25. As shown in Figure 3(a), skew error decreases when
beacon interval increases. Although not mentioned in [6],
we believe this is reasonable because the longer delay
between successive data point, the better linear regression
solution converge to real solution. For the rest simulations
in this paper, we fix Tshl _ I 3 to 2 seconds to achieve a
be 1 seconds. The jitter is addictively incremented by 10μs.
The result is shown in Figure 3(c) and (d). It is noted that,
only mean value are presented in Figure 3(c)-(f) to be clear.
Similar to our theoretical analysis, Both Tri-Message and
TSHL offset error is directly proportional to the receive
jitter in Figure 3(d). We also show the effect of receive
jitter on instant error in Figure 3(c). We conclude that TriMessage is as sensitive as TSHL with respect to jitters. TriMessage looks inferior to TSHL in this simulation because
propagation delay is 1second only. If delay is longer, TriMessage can outperform TSHL, as we will demonstrate in
next simulation.
Next, we measure instant error and offset error as a
function of propagation delay. Here jitter is set to be 5μs
constant. We expect that the increase in propagations will
reduce the skew error of Tri-Message, as discussed in
Section 3. The delay is incremented by 0.5 second step.
Figure 3(e) shows that, TSHL instant error is increased
along with propagation delay, consistent with our
theoretical analysis. Figure 3(f) demonstrates that offset
error using Tri-Message decreases along with propagation
increases. On the contrary, TSHL is insensitive to
propagation delay as proved in [6]. This characteristic
makes Tri-Message more applicable for extremely high
latency networks.
Finally, we vary the node skew with respect to the global
clock. Since both protocols model the skew, they should be
adapted to any skew. Table 2 validates our expectation: the
skew error of Tri-Message is considerably comparable to
TSHL.
Table 2. Skew error
Skew
TSHL
Tri-Message
relative precise skew.
10ppm error mean
0.65153847
0.65152399
Next, we vary the number of beacons in the first phase.
Shown in Figure 3(b), TSHL skew is sensitive to beacon
numbers, which is also verified by [6]. For the rest
simulations, we use a constant 25 of beacon number for
TSHL.
10ppm error variance
0.46749385
0.46169087
100ppm error mean
0.65688778
0.65544851
100ppm error variance
0.46903175
0.46087955
4.3 Comparison of errors
We now compare Tri-Message with TSHL in terms of
three kinds of errors we mentioned before. Because
message intervals could affect both algorithms, we set their
total process time to be equal. Since Tshl _ I1 and
Tshl _ I 2 has little effect compared with Tshl _ I 3 , we let
them be close to zero and set Tshl _ I 3 to maximum hence
optimize the performance of TSHL
Tri _ I1  Tri _ I 2  Tshl _ I 3  2 sec
(7)
First, we investigate the receive jitter effect on the
accuracy of both algorithms. We fix propagation delay to
4.4 Energy and Computation evaluation
In this subsection we compare the computation demand
of two protocols. TSHL linear regression can be
implemented by a mean-least-square (MLS) numerical
solution introduced in [12]. Consider Multiplication as the
dominant factor, we use packet number and Multiplication
number respectively to represent energy and computation
demand for two protocols. Let beacon number to be 25 in
TSHL, the calculation is given in the appendix. Figure 4
shows a per-synchronization-process comparison.
The significant difference shown in Figure 4
demonstrates that Tri-Message is a lightweight
synchronization protocols compared with TSHL. It can
significantly reduce the energy consumption more that 85%
in terms of number of packets, as well as computational
complexity more that 90% in terms of multiplication
operations. Overall, Tri-Message shows a similar
performance in shorter latency networks compared with
TSHL, and outperforms TSHL in extremely long latency
networks.
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5. Conclusion
Time synchronization is critical for many distributed
systems. None of the existing terrestrial synchronization
protocols are applicable for high latency networks. While
TSHL provides high precision time synchronization, it is
still not practical in some applications which have stringent
constraints on system resources such as sensor networks
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synchronization protocol for high latency and resourceconstrained networks which achieves high precision time
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Future work includes further evaluating the multi-hop
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6. ACKNOWLEDGMENTS
The project is supported by The National Natural
Science Foundation of China (Granted No.60572063) and
Specialized Research Fund for the Doctoral Program of
Higher Education (Granted No. 20040487009).
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[9] M. Horauer. K. Schossmaier, U. Schmid, and T. Vienna,
PSynUTC—evaluation of a high precision time
synchronization prototype system for ethernet lans, in
Proceedings of 34th Annual Precise Time and Time Interval
Meeting (PTTI), Reston, Virginia, USA, December 2002.
[10] S. Capkun, M. Hamdi, and J. P. Hubaux. GPS-free
positioning in mobile ad-hoc networks. Cluster Computing
Journal, 5(2):118–124, April 2002.
[11] J. Liu, Y. Zhang and F. Zhao. Robust Distributed Node
Localization with Error Management. MobiHoc 2006, May
22–25, 2006, Florence, Italy.
[12] H. William, B. P. Flannery, S. A. Teukolsky, and W. T.
Vetterling. Numerical Recipes in C: The Art of Scientific
Computing. New York, NY: Cambridge University Press,
1992
[13] R. J. Urick. Principles of Underwater Sound. McGraw-Hill,
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Sensor Networks. IPDPS Workshop PDC, 2001.
Therefore, it is reasonable to conclude that the instant Offset
error of TSHL is proportional to the propagation delay.
Appendix
Proof of Theorem 1:
Skew a of TSHL can be deduced through linear
regression. We can implement linear regression by meanleast-square (MLS) solutions. Let n denotes received TSHL
beacon numbers, we get a n -order super definition
equations group. In MLS, we set L the left (n,2) matrix,
Proof of Theorem 2:
After the Tri-Message message-exchanges, node B has 6
timestamps. From the global clock view, we have 6
reference equations:
 A1  t1

 A2  t2  d
A  t
 3 3
R the right (n,1) matrix, and M the solution (2,2) matrix
L*M  R
Without jitter, offset
M  ( LT L)1 * ( LT R)
(8)
b can be given by
 A2  t2  d

 A3  t3
 ( B2  B3 )  a( A2  A3 )/ 2
question is how to get  . The deduction is similar. We can
get
(9)
a'  a(1   a ),1/ a'  1/ a(1   a ) ,
we get
b'  ( B2  B3 ' )  a' ( A2 ' A3 )/ 2
 (t1  d  t2 )  2

 (t1  d  t2 )
Next we analyze the effect of jitters. By taking deviations
 k into consideration, we redefine reference equations set
 [b  at2  a(t3  d   3 )  b
(10)
 a(1   a )(t2   2  t3  d )] / 2
 b  a ( 3   2 )  a a (t2  t3  d )/ 2
Let  denotes time passed since synchronization, we can
evaluate clock offset error t k of TSHL.
tk '  ( Bk  b' ) / a'  (atk  b  b' ) / a'  (1   a )tk
 ( 2   3 ) / 2   a (t2  t3  d ) / 2
 ( 2   3 ) / 2   a [t k  (t 2  t3  d ) / 2]
 ( 2   3 ) / 2   a [t3  d    (t 2  t3  d ) / 2]
(13) as
 A1  t1

 A2 '  t 2  d   2  A2   2
A  t
 3 3
(11)
 B1 '   (t1  d  1 )    B1  1

 B2   (t 2 )  
 B '   (t  d   )    B   
3
3
3
3
 3
(12)
t1  t1

t 2  t1  d  I1
t  t  d  I  d  I  t  2d  I  I
1
2
1
1
2
3 1
t k  t k 't k
 ( 2   3 ) / 2   a (  d  I 2 / 2)
To prove Theorem 1, we fixed other parameters and
deduce differential coefficients of t k to  and
d respectively
tk /    a
The long-term offset error of TSHL is dominated by skew error
tk / d   a ,   0

2
2
[  (t1  d )   ]  [ t2   ]  [t1  (t2  d )]
(14)


2
2
 ( B1  B2 ) / 2  ( A1  A2 )  / 2
Next we analyze the effect of jitters. As stated above
 a . And, we set   0 to evaluate effect of d
(13)
We show how B could use these values to determine the
skew  . Note that   B3  B1  /  A3  A1  , the next
B2  at2  b

B3  a(t3  d )  b
b  at2  b  a(t3  d )  b  a(t2  t3  d )/ 2
B1   (t1  d )  

B2   (t2 )  
B   (t  d )  
3
 3
to instant error.
The estimated skew
(15)
 ' can be expressed by real  as
B3 ' B1 '
1   3

 '  A  A   (1  t  t )   (1    )

3
1
3
1

   (   ) /(t  t )  1   3
1
3
3
1
 
2d  I1  I 2
By Maclaurin expansion, we can get
(16)
1
'

1
(1    )
In MLS, Let n denotes received TSHL beacon numbers,
we set L the left (n,2) matrix, R the right (n,1) matrix,
 (1    ) /  ,    1
and M the solution ( 2,2) matrix.
hence
Table 3. TSHL linear regression complexity
 '  ( B1 ' B2 ) / 2 （A1  A2 '） ' / 2
( B1  1  B2 ) ( A1  A2   2 )
Step
 (1    )
2
( B     B2 ) ( A1  A2   2 )
 1 1


2
2
( A  A2   2 )
 1
 
2
   (1   2 )  / 2  (t1  t2  d )   / 2


2
we can evaluate clock offset error
tk '  ( Bk   ' ) /  ' 
 '
 tk (1    ) 
(1    )

 '
(1    )

 (t3  d   )   (1   2 ) / 2
tk  tk 'tk  tk  
 (t1  t2  d )  / 2
 (t1  2d  I1  I 2  d  ) 
 (t1  t1  d  I1  d )  / 2  ( 2  1 ) / 2
(19)
 [  (4d  I1  2 I 2 ) / 2)]   ( 2  1 ) / 2
(

4d  I1  2 I 2

)(1   3 )
2d  I1  I 2 4d  2 I1  2 I 2
 ( 2  1 ) / 2
We use a new distribution
1 3
to represents
(1   3 ) .To
prove Theorem 2, we fixed other parameters and deduce
differential coefficients of t k to 
tk /  
1 3
2d  I1  I 2
The long-tern offset error of Tri-Message is proportional to
jitters
1 3 .
With
d
in denominator, we conclude that Tri-
Message causes decreasing offset error with the increasement of
propagation delay.
Calculation of TSHL computational complexity:
4n-4
2n
2n-2
14
8
( L L) * ( L R)
4
2
total
6n+18
6n+4
T
(18)
Addition
4n
LL
LT R
( LT L)1
(17)
tk of Tri-Message
tk     '
(1    )

Multiplication
T
1
T