Metastability

Tau at 200 picoseconds and T0 at 100 picoseconds are of the order published for CMOS flip-flops, and both are adjustable here. They are not a datasheet for any real component. Attaching a specific vendor's measured constants to an invented circuit would look more precise and be less true, and the shape of the answer, which is the subject, does not depend on them.

The usual illustration is a waveform wobbling before it settles, and it gives the wrong idea. This entry used to say metastability is not a slow visible event you could catch on a scope, and the paper it cites disproves that in its own figures: Chaney and Molnar publish five of them, and say “Fig. 1 is a 5-min exposure of a sampling oscilloscope display of Q and Q̄; while Fig. 2, from a real time oscilloscope, shows the details of some individual output trajectories.” They caught it. The reason not to animate one here is different and better: there is no single waveform that represents the phenomenon. Every trajectory in those figures is a different shape, and the quantity being modelled is the distribution of resolution times rather than any one of them. Drawing one wobble would teach a shape that does not exist.

A mean time between failures of four years does not mean four good years and then a fault. Failures arrive at random, and the model is the ordinary one: each sampling event fails independently with the same small probability, which makes the waiting time exponential and therefore memoryless. GIVEN that it has not failed yet, the next month is exactly as likely to be the one as the first month was. That model is an assumption, not an observation: real parts age, and clocks drift together rather than independently. That condition matters and an outside review was right to catch this page dropping it. Unconditionally the fiftieth month is far less likely than the first, because to fail in the fiftieth it has to survive the other forty-nine. The number tells you a rate, and treating it as a deadline is the most common way to misread it.

The exponential in this equation passes what a double-precision number can hold at around e^710, which is reached by entirely ordinary parameters. Computing it directly returns Infinity, and Infinity displayed as a mean time between failures reads as a guarantee. The arithmetic is done on the exponent throughout and converted only for display.

The chain here is n flip-flops on one clock. Real designs have to worry about the data being sampled more than once, about signals that must arrive together staying together, and about buses where a synchroniser per bit is the wrong answer entirely. The equation is right; it is not the whole of the discipline.

Both citations were verified against Crossref for title, authors, volume, issue, pages and date, and this entry used to say neither was read, because IEEE served nothing for either DOI. Chaney and Molnar has since been read here, from a scan at ibm-1401.info: two pages, five figures of oscilloscope traces, and no failure-rate equation anywhere in it. The paper says the opposite of one, in fact — that the authors are “unaware of any commercially available integrated circuit flip-flop that is specified in a manner that makes it possible to predict its failure probability as a synchronizing element”. Reading it is what corrected the date on the equation and the claim about what a scope can catch. Veendrick 1980 is still unread. The equation on this page comes from its standard published form rather than from those papers, and the parameters are a model. That is a weaker footing than this site prefers and it is better said than hidden.