I'm looking at the chapter on reliability theory (chapter 9) in the book on Introduction to probability models by Sheldon Ross: https://fac.ksu.edu.sa/sites/default/files/introduction-to-probability-model-s.ross-math-cs.blog_.ir_.pdf. Looking at example 9.32, there are expressions for the mean time to failure and mean time to recovery for a series and parallel system, given the failure and recovery rates of the individual components ($\lambda_i$ and $\mu_i$ respectively for the $i$th component). What is striking is the symmetry in the two expressions. For a series system, MTBF ($\bar{U}$) and MTTR ($\bar{D}$) are given by:
$$\bar{U} = \frac{1}{\sum \lambda_i}$$
$$\bar{D} = \frac{1-\prod_i \frac{\mu_i}{\mu_i+\lambda_i}}{\prod_i \frac{\mu_i}{\mu_i+\lambda_i}}\times \frac{1}{\sum \lambda_i}$$
While for a parallel system: $$\bar{D} = \frac{1}{\sum \mu_i}$$
$$\bar{U} = \frac{1-\prod_i \frac{\lambda_i}{\mu_i+\lambda_i}}{\prod_i \frac{\lambda_i}{\mu_i+\lambda_i}}\times \frac{1}{\sum \mu_i}$$
We see that if we re-interpret up as down, hence replacing $\lambda_i$ by $\mu_i$ and $\bar{D}$ with $\bar{U}$, the two expressions interchange.
Any reason for this symmetry?