Calculation of $MTTDL$

$MTTDL$ can be obtained from the fundamental matrix $M$, which is defined by the following equation [33].

$$M =[m_{ij}] = [I-Q]^{-1}$$ (12)

where $m_{ij}$ represents the average amount of time in State $j$ before entering the data loss state, when the Markov chain starts from State $i$.

The total amount of time expected before being absorbed into the data loss state is equal to the total amount of time it expects to make to all the non-absorbing states. Since the system starts from State 1, where there are no node failures, MTTDL is the sum of the average time spent on all states $j$, ( $1 \leq j \leq S$), i.e.,

$$MTTDL = \sum_{j=1}^{S}m_{1j}$$ (13)

When $MTTD = 0$ and $MTTF_{switch} = \infty$, our model becomes the classic model for RAID-1. If $MTTD = \infty$ and $MTTF_{switch} = \infty$, it then becomes the classic model for RAID-0. When using the same $MTTF$ and $MTTR$ to calculate the $MTTDL$ of RAID-0 and RAID-1 as Ref. [28], our model shows identical results to those given in the above references.

To further validate our model, Figure 14 shows the relationship between $MTTDL$ and $MTTD$ under different workload conditions in an CEFT-PVFS where there are 8 data server nodes in either group. The $MTTDL$ in this figure is calculated based on our model built above. This figure indicates that the $MTTDL$ decreases with an increase in $MTTD$. With the same $MTTD$ but increasing $MTTW$, $MTTDL$ increases. All of these performance trends are intuitive and realistic.

Figure 14: Influence of $MTTD$ on $MTTDL$ of 8 mirroring 8 data servers under different workloads.( $MTTF = \mbox{1 year}$, $MTTF_{switch} = \mbox{3 years}$, $MTTR = \mbox{2 days}$ and $MTTW = \mbox{5 minutes}$)