In a simplified treatment of mechanical failures for the virtual beanstalk, we can expect the probability of failure for each individual component to generally follow some empirical geometric function over time.

R_j(t) = C_j^(t/Tfail,j) + g_j(t)

where:

R_j(t) = Mean Reliability for Component j

t = Elapsed Mission Time

C_j = Reliability Constant for Component j

T_fail,j = Mean Time Between Failure Period for Component j

g_j(t) = Reliability Function During "Breaking In" for Component j

Reliability is the probability of successful function of the component during a given use. Unfortunately, it is always less than unity. It is equal to 1-f, where f is the probability of failure.

The g_j(t) term is itself a geometric function, only with a negative exponent. It starts out relatively high and drops to negligible values shortly after the breaking in period. The C_j^(t/Tfail,j) term starts out at some negligible value during the breaking in period, but gradually rises as g_j(t) drops, producing a nearly constant value of R_j(t) over a broad range. This period of nearly constant behavior can be considered the useful lifetime of the component. Eventually, the C_j^(t/Tfail,j) term dominates the outcome, corresponding to a period during which the component begins to wear out. The C_j^(t/Tfail,j) term will eventually rise above some acceptable value, at which point the component must be considered no longer adequately functional.

The reliability of the entire system is equal to the product of the reliabilities of all of its components.

PR = R₁ * R₂ * R₃ * … * R_n

In order for PR to be very high, the average values of R_j have to be very high as well. The system is always less reliable than its least reliable component, so reliability must be high for all components.