1. Proof of liveness: an example
Clock phase synchronization
System of n clocks ticking at the same rate.
Each clock is 3-valued, i,e it ticks as 0, 1, 2, 0, 1, 2…
A failure may arbitrarily alter the clock phases.
The clocks need to return to the same phase. . 1 2 3
n-1

2. Proof of liveness: an example
k: c[k]  {0.1.2}
Clock phase synchronization
{Program for each clock}
(c[k] = phase of clock k, initially arbitrary)
do  j: j  N(i) :: c[j] = c[i] +1 mod 3  c[i] := c[i] + 2 mod 3
[]  j: j N(i) :: c[j] ≠ c[i] +1 mod 3  c[i] := c[i] + 1 mod 3 od
Show that eventually all clocks will return
to the same phase (convergence), and
continue to be in the same phase (closure) 1 2 3
n-1

3. Proof of convergence
Let D = d[0] + d[1] + d[2] + … + d[n-1]
d[i] = if no arrow points towards clock i;
= i + 1 if a pointing towards clock i;
n - i if a  pointing towards clock i;
= 1 if both  and point towards
clock i.
By definition, D ≥ 0.
Also, D decreases after every step in the system. So the number of arrows must reduce to 0. 1 2
n-1 2 2 2 1 1 1 1 2 2 2 2 2
Understand the game of arrows

4. Time and Clock Primary standard = rotation of earth
De facto primary standard = atomic clock
(1 atomic second = 9,192,631,770 orbital transitions of Cesium 133 atom.
86400 atomic sec = 1 solar day – 3 ms (requires leap second
correction each year)
Coordinated Universal Time (UTC) = GMT ± number of hours in your time zone

5. Global positioning system: GPS
Location and precise time
computed by triangulation
Right now GPS time is nearly
14 seconds ahead of UTC, since
It does not use leap sec. correction
Per the theory of relativity, an
additional correction is needed.
Locally compensate by the
Receivers.
A system of 32 satellites broadcast accurate spatial
corordinates and time maintained by atomic clocks

6. What does “concurrent” mean?
Simultaneous? Happening at the same time? NO.
There is nothing called simultaneous in the physical world.
Alice
Explosion 2
Explosion 1
Bob

7. Physical clock synchronization
Question 1.
Why is physical clock synchronization important?
Question 2.
With the price of atomic clocks or GPS coming down,
should we care about physical clock synchronization?

8. Classification Types of Synchronization External Synchronization
Internal Synchronization
Phase Synchronization
Types of clocks
Unbounded 0, 1, 2, 3, . . .
Bounded 0,1, 2, M-1, 0, 1, . . .
Unbounded clocks are not realistic, but are easier to
deal with in the design of algorithms. Real clocks are
always bounded.

9. Terminologies What are these? Drift rate  Clock skew 
Resynchronization interval R
Max drift rate  implies:
(1- ) ≤ dC/dt < (1+ )
Challenges
(Drift is unavoidable)
Accounting for propagation delay
Accounting for processing delay
Faulty clocks

10. Internal synchronization
Step 1. Read every clock in the system.
Step 2. Discard outliers and substitute them by the value of the local clock.
Step 3. Update the clock using the average of these values.
Resynchronization interval will depend on the drift rate.
Berkeley Algorithm
A simple averaging algorithm
that guarantees mutual
consistency |c(i) - c(j)| < 

11. Internal synchronization
Lamport and Melliar-Smith’s
averaging algorithm handles
byzantine clocks too
Assume n clocks, at most t are faulty
Step 1. Read every clock in the system.
Step 2. Discard outliers and substitute them by the value of the local clock.
Step 3. Update the clock using the average of these values.
Synchronization is maintained if n > 3t
Why?
Bad clock
A faulty clocks exhibits 2-faced or byzantine behavior

12. Internal synchronization
Lamport & Melliar-Smith’s algorithm (continued)
The maximum difference between
the averages computed by two
non-faulty nodes is (3td / n)
To keep the clocks synchronized,
3td / n < d
So, 3t < n k
B a d c l o c k s

13. Cristian’s method External Synchronization
Client pulls data from a time server
every R unit of time, where R <  / 2.
(why?)
For accuracy, clients must compute
the round trip time (RTT), and
compensate for this delay
while adjusting their own clocks.
(Too large RTT’s are rejected)
Time
server

14. Network Time Protocol (NTP)
Tiered architecture
Broadcast mode
- least accurate
Procedure call
- medium accuracy
Peer-to-peer mode
- upper level servers use this for max accuracy
Time
server
Level 0
Level 1
Level 1
Level 1
Level 2
Level 2
Level 2
The tree can reconfigure itself if some node fails.

15. P2P mode of NTP  = (T2 -T4 -T1 +T3) / 2 - (TPQ - TQP) / 2
Let Q’s time be ahead of P’s time by . Then
T2 = T1 + TPQ + 
T4 = T3 + TQP - 
y = TPQ + TQP = T2 +T4 -T1 -T3 (RTT)
 = (T2 -T4 -T1 +T3) / 2 - (TPQ - TQP) / 2 T2 T3 Q P T1 T4 x
Between y/2 and -y/2
So, x- y/2 ≤  ≤ x+ y/2
Ping several times, and obtain the smallest value of y. Use it to calculate 

16. Problems with Clock adjustment
1. What problems can occur when a clock value is
advanced from 171 to 174?
2. What problems can occur when a clock value is
moved back from 180 to 175?