Numerical Problems in the Solution of Markov Models for RAID Systems

Markov models are used in a wide variety of reliability problems because it is easy to model complex devices and systems using nodes to represent combinations of failed elements and arcs between the nodes to represent failures and repairs. For example, Figure 1 represents a system in which there are two groups of elements. Node (State) 00 represents no failures in either group of elements, State 10 represents one failure in Group 1, State 01 represents one failure in Group 2 and State 11 represents one failure in each group. All of these four states are successful. If there are two failures in either group, the system enters the failed state X. Since there are N disks in each group, each with a failure rate $\lambda$, the failure rate of the transition between states is $N\lambda$. The repair rates for each group of disks is $\mu_{1}$ or $\mu_{2}$. Thus, a Markov model for a complex system is fairly easy to construct. Each Markov state leads to a first-order differential equation. Solution of this set of equations can be difficult. This paper examines some of the solution difficulties for Markov models in general, and specifically for several fairly complex RAID disk storage system models.