# Reconstruction of Temporal Order for Burst Periodic Events: A Sufficiency Condition

In a DRTS, let a set of periodic events E = *[e _{1},e*

_{2},...,

*e^*recur with a period A.

If *e _{pq} Vp* e [1,2, ... , n],

*q*e Z+,

*q >*0 denotes the

*qth instance*(or occurrence) of an event

*e*E, then the following holds:

_{p}e

Since the events are burst periodic, it is assumed further that

that is, for a particular instance, all events in E occur within a time interval n constituting a burst of events.

Now, if n ^ 5 holds, then the actual time stamps *t _{r}(e_{pq})* for the qth instance shall be same for all events

*e*E, and the corresponding time stamp generated by an arbitrary ith clock in the ensemble shall differ by ±1, that is,

_{p}e

and

If a causal relationship exists between the qth instance of an event *e _{p}* and another event

*e*(

_{s}*e*,

_{p}*e*e E), then the following must hold:

_{s}

Again, since *t _{r}(e_{pq}) = t_{r}(e_{sq}),* Equation 2.25 implies

and the sufficient condition for which is

or

By definition, the left-hand side of Equation 2.28 is A, and, thus, it can be stated that if a burst of events E occurring within a time span n recurs after a time period A, then the DRTS is termed the п/A precedent. From Equation

2.28 it is seen that the temporal order of events belonging to two consecutive instances of E can always be constructed by any arbitrary clock *i* in the ensemble if the burst repeats after a period A = *3G* and the system is said to be 0/3*G* precedent.

The significance of the п/A precedence can be understood if one considers a practical case like a networked control system (NCS) where a number of process control loops are integrated over a shared network. Each control loop consists of a sensor that senses the process variables periodically with an interval *T _{s}* and sends the sensor data to a controller that generates a controller output with the last two samples of sensor data. If all the sensors transmit at the same time, then the transmission events constitute a burst. Since the controller has to establish a precedence among two successive samples of sensor data,

*T*^ A must hold.

_{s}^{[1]}

^{[2]}

^{[3]}

^{[4]}

^{[5]}

^{[6]}

^{[7]}

initiated and the time at which it is completed, however small it might be. This is particularly important for a DRTS where the component subsystems are integrated over a network where the time taken to read other clocks may vary. Thus, there exists a finite difference between the clocks even after they are synchronized and the smallest difference by which two clocks can be synchronized is defined as *convergence* v. Now, if p_{max} is the maximum drift rate of a clock during the resynchronization interval Г, then two clocks may drift by a maximum interval

In order for a precision 5 to be maintained, the following must hold:

Equations 2.29 and 2.30 may be used to ascertain the resynchronization interval in a DRTS.

- [1] 2.4 Time Synchronization Time synchronization in a DRTS is a process by which the clocks of all component subsystems are adjusted so that the difference in time lies withina bound. The time interval at which time synchronization is repeated istermed the resynchronization interval Г The main requirements of time synchronization in a DRTS are as follows:
- [2] The synchronization algorithm must ensure that a predefined precision is always achieved at any point of time.
- [3] When temporal order has to be established using physical time, theglobal clock should be synchronized to a physical clock.
- [4] The synchronization algorithm should be fault tolerant, that is, capable of handling processor and communication faults.
- [5] The synchronization algorithm should not be drastic. That is to say,the adjustment should not be such that the time becomes negative atany instant on a timeline.
- [6] The synchronization algorithm should be scalable.
- [7] The synchronization algorithm should preferably be light, that is, itshould not degrade the performance of a DRTS. Accordingly, synchronization may either be external, where the global clockis synchronized to a physical clock, or internal, where logical clocks aresynchronized. In a DRTS, since time synchronization requires a finite time, for example,the time taken to read other clocks and the time taken to adjust a clock, thereelapses a finite time interval between the instant time synchronization is