Lamport s conditions on Physical Clocks

1. Lamport�s conditions on Physical Clocks Armin R. Mikler

2. On anomalous behavior The coordination of distributed processes based on Lamport�s clocks resulted in a total ordering of processes (?). Now let�s consider what can happen if events are causally related outside our system S : Suppose we had a nationwide system of computers Alice generates an event a on node A and then calls Bob in another state to generate an event b on node B. It is then possible that b is assigned a lower timestamp than a.

3. anomaly.. Let S be the set of all system events (i.e., internal to the system. Let�s introduce a superset S of events that include S as well as relevant external events (such as the call from Alice to Bob). Let ? denote the �happen before� relation over S Then in our example, a ? b but a -/-> b

4. Avoiding anomalies It is obvious that no algorithm entirely based on events in S that does not relate to other events in S can guarantee proper ordering. There are two possible ways to avoid anomalies: Introduce into the system explicit information about the ordering ?. e.g. B may request the assignment of a timestamp tb > ta , placing the responsibility on the user. Construct a system of clocks that satisfies a stronger clock condition

5. Strong Clock Condition The strong clock condition is stated as follows: For any event a, b in S: if a ? b then C(a) < C(b) here ? is a stronger condition as its ? counterpart which is not in general satisfied by logical clocks. It is possible to construct a system of physical clocks which are running (quite) independently that will satisfy the Strong Clock Condition � thereby eliminating anomalies! (but how)

6. From logical to physical Lamport introduces physical time into the space-time context. Let Ci(t) denote the reading of clock Ci at t. Further assume that dCi(t)/dt is the rate at which time progresses. In order for Ci to be a true physical clock, we must have dCi(t)/dt � 1 for all t.

7. (physical) clock conditions PC1 & 2 more precisely, we demand that the following clock condition holds: PC1:: There exists a constant ? << 1 such that for all clocks i: | dCi(t)/dt - 1| < ? As discussed earlier, for a crystal controlled clock we would expect ? = 10-6 While this controls the permissible drift, we need to establish permissible bounds on skew!

8. more PC1 and PC2 ? we require that Ci(t)�Cj(t) for at i, j, and t. Formally, we require the following condition to hold: PC2:: for all i, j: | Ci(t)-Cj(t) | < e where e is a small constant. The question is: how small do we need to make ? and e to prevent anomalous behavior??

9. now what? let � be a number such that if event a occurs in physical time t and event b in another process satisfies a ? b, then C(b) > t + �. � is less than the shortest transmission time time for any IPC. We could always pick �= dmin /c where (c = speed of light). Depending on how messages are transmitted in S, � could be much larger.

10. avoiding anomalies to avoid anomalies we must ensure that: for any i, j, and t: Ci(t + �) � Cj(t) > 0. PC1 and PC2 relate ? and e to � as follows: PC1 implies that Ci(t + �) � Ci(t) > (1 � ?) � From PC2 we deduce that Ci(t + �) � Cj(t) > 0 if the following holds: e/(1 � ?) = �