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Infinite Parallel Redundancy
Posted:
Jul 12, 1996 6:30 PM
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A common strategy for increasing the mean time between failure
(MTBF) of a system is to add redundant paths in parallel. If
each path has an exponential failure rate of q per hour, then its
MTBF is 1/q hours, and the MTBF of two parallel paths is 3/(2q).
In other words, the redundant path increases the MTBF by 50%.
It isn't uncommon for critical systems to have 3 or more parallel
redundant channels. In general the overall system with n channels
can be modelled as a simple Markov chain as illustrated below:
_______ _______ _______ _______
| n | | n-1 | | n-2 | | 0 |
| paths | nq | paths | (n-1)q | paths |(n-2)q 1q | paths |
|working|----->|working|------->|working|----- ... -->|working|
|_______| |_______| |_______| |_______|
The mean time from the "n paths working" state to the "0 paths
working" state is just the sum of the mean times from each state
to the next. This immediately gives the well known result that
the MTBF of a system with n parallel redundant paths (without
repairs) is proportional to the partial sum of the harmonic
series
1 / 1 1 1 1 MTBF(n) = --- ( 1 + --- + --- + --- + ... + --- )
q \ 2 3 4 n /
Since the harmonic series diverges, this implies a system consisting
of an infinite number of parallel paths, each of which has a mean time
to failure of 1 hour, would NEVER fail.
Now consider the variance of the MTBF for a highly parallel system.
The variance of an exponential distribution with mean 1/q is simply
(1/q)^2. Since each transition in the above Markov chain is
exponential with means 1/q, 1/2q, ..., 1/nq, and since the variance
of the sum of two (or more) random variables is the sum of their
variances, it follows that the variance of the MTBF for an n-path
system is given by
1 / 1 1 1 1 VARIANCE(n) = --- ( 1 + --- + --- + --- + ... + --- )
q^2 \ 2^2 3^2 4^2 n^2 /
Interestingly, although the MTBF itself goes to infinity as more
parallel paths are added, the variance on the MTBF converges on the
finite value
pi^2
VARIANCE(inf) = -----
6 q^2
I wonder if any aspect of nature can be modelled as an infinitely
parallel system?
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