Reliability Analysis of $M$-for- $N$ Shared Protection Systems With General Repair-Time Distributions

In this paper, we investigate the user perceived reliability of
$M$
-for-
$N$
shared protection systems. The
$M$
-for-
$N$
shared protection system is a general type of the fault-tolerant configuration with shared backup units. We assume that there are
$N$
working units each serving a single group of users and
$M$
shared protection (spare) units in the system. We also assume that the time to failure of the unit is subject to an exponential distribution, and the time to repair it is subject to a general distribution. Under these assumptions, we derive the Laplace transform of the reliability function (The cdf of the failure time, i.e. the cumulative distribution function of the time beyond which the system will survive) for each user, as well as the user-perceived Mean Time to First Failure (MTTFF) by combining the state transition analysis and the supplementary variable method. We also show the effect of the repair time distribution of the units through the case study of the cloud computing systems in which
$N$
service-independent servers share
$M$
spares for backup purpose. The analysis reveals that there exist optimal values for
$M$
and
$N$
in terms of the reliability and the cost for the given conditions of failures and repairs.