How does Lamport imply anomalous behavior is impossible given these constraints?

Question

Lamport's paper: Time, Clocks, and the ordering of Events in a Distributed System

Anomalous behavior occurs because Lamport's logical clocks are not enough to guarantee that the total ordering of events derived from the unique partial ordering and a non-unique total ordering is not enough to guarantee that the ordering is correct with respect to physical time.

He gives two conditions PC1 and PC2:

PC1: There exists a constant $ \kappa $ << 1 such that for all i:

$ |\frac{dC_{i}}{dt} - 1| < \kappa $

PC2: For all i, j:

$ |C_{i}(t) - C_{j}(t)| < \epsilon $

PC1 says that individual clock drift in the system must be less than some constant $ \kappa $. PC2 says that clocks must be synchronized so that the difference in time between any 2 clocks does not exceed $ \epsilon $.

This is what I do not understand. He says that anomalous behavior is impossible if this proposition holds, deduced from PC2.

proposition: PC2 implies $|C_{i}(t + \mu) - C_{j}(t)| > 0$ if $\frac{\epsilon}{1 - \kappa} \leq \mu$.

I have no idea where he comes up with this, maybe my math isn't too great.

Note: $\mu$ is a constant that must be less than the shortest transmission time for interprocess messages.

PC1 implies $|C_{i}(t + \mu) - C_{i}(t)| > (1 - \kappa)\mu$. This is apparent because (1 - $\kappa$) is less than 1. But apparently he combines this with PC2 to deduce the proposition.

How does he arrive at this??

Answer 1

The constraint as written by Lamport is $C_i(t+\mu)-C_j(t)\gt0$ or $C_i(t+\mu)\gt C_j(t)$ which can be re-written $$C_i(t+\mu) - C_i(t)\gt C_j(t) - C_i(t) \tag{*}\label{*}$$

Given Lamport's previous deductions, we recognise the boundary conditions each side must satisfy namely: $$(1-\kappa)\mu \lt C_i(t+\mu) - C_i(t)$$ $$ \epsilon \gt C_j(t) - C_i(t)$$

Now, Lamport wants to find out the range of values for 'variables' $\epsilon,\kappa$ ($\mu$ is fixed) that will ensure the strict inequality $\eqref{*}$. Obviously, for real numbers (Lamport's starting assumption) if $x \lt A$ and $y \gt B$, then as long as $y \leq x$ we know $A\gt B$.

In other words, as long as $\epsilon \leq (1-\kappa)\mu$ then $\eqref{*}$ is true and so $$C_i(t+\mu)-C_j(t)\gt0$$