You would like to build a synchronizer that can receive asynchronous inputs with

Question

You would like to build a synchronizer that can receive asynchronous inputs with an MTBF of 50 years. Your system is running at 1 GHz, and you use sampling flip-flops with = 100 ps, T0 = 110 ps, and tsetup = 70 ps. The synchronizer receives a new asynchronous input on average 0.5 times per second (i.e., once every 2 seconds).

(a) What is the required probability of failure per second to satisfy this MTBF?

(b) How many clock cycles would you have to wait before reading the sampled input signal to give that probability of error?

Explanation / Answer

MTTF: Mean time to failure describes the expected time to failure for a non-repairable system.

For example, assume you tested 3 identical systems starting from time 0 until all of them failed. The first system failed at 10 hours, the second failed at 12 hours and the third failed at 13 hours. The MTTF is the average of the three failure times, which is 11.6667 hours.

If these three failures are random samples from a population and the failure times of this population follow a distribution with a probability density function (pdf) of , then the population MTTF can be mathematically calculated by:

(1)

Assuming the failure times follow a Weibull distribution, we can use Weibull++ to estimate the parameters for the distribution and calculate the population MTTF. The analysis settings and estimated parameters are:

Table 1: Results from Weibull++

Distribution Weibull-2P

Analysis RRX

CB-Method FM

Ranking MED

Beta 7.2393

Eta 12.3559

Rho 0.9904

LK-Value -5.2592

Fail/Susp 3/0

The Mean Life (MTTF) can be calculated in the Quick Calculation Pad (QCP):

MTBDE: Mean Time between Downing Event, describes the expected time between two consecutive downing events for a repairable system.

For example, assume you are testing a system that can be repaired when there is a failure. The failures causes the system to go down. The first failure happens at 10 hours and it takes 5 hours to fix. The second failure is at 27 hours and the repair duration is 3 hours. Then after working for 13 hours, the system fails at 43 hours. The repair lasts for 7 hours and the system is restored at 50 hours. This failure and repair process can be illustrated using the following graph.

The MTBDE = x (T1 + T2) = 16.5 hours, if you use only the observations of complete cycles. You can add one more cycle by combining x0 and y3. Then the MTBDE = x (T1 + T2 + x0 + y3) hours.

If all the uptime durations xi are independent and identically distributed (i.i.d) and all the repair durations yi are i.i.d, then:

MTBDE = MTBF + MTTR (Mean Time to Repair)

(2)

Eqn. (2) shows that the MTBDE is the sum of the average uptime and the average downtime (MTTR). The definition of MTBF is given next.

MTBF: Mean Time between Failures. This average time excludes the time spent waiting for repair, being repaired, being re-qualified, and other downing events such as inspections and preventive maintenance and so on; it is intended to measure only the time a system is available and operating.

For the above example, it will be:

MTBF = (x0 + x1 + x2) = 11.6667

The above equation assumes that all the downing events are caused by failures. The duration of the downing events are the duration of repairs.

Again, this calculation assumes the uptime durations xi are i.i.d. However, for a repairable system, the i.i.d assumption for xi is rarely true unless the system can be treated as brand new after each repair or the distribution of xi is exponential. When the i.i.d assumption is not true

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