Figure 11 shows the Markov state diagram for
Protocol 1, which can also be applied to the other protocols. In
this diagram, signifies that the state number/index is
, and there are
and
failed nodes in the primary and
backup group, respectively. All the states shown are working
states, with the exception of
, which is the data loss state.
The total number of states in the Markov state diagram is denoted
by
and is equal to
. The Markov chain begins
with State 1 (
), followed by State 2(
), and so on.
To facilitate the solution to this model, we derive
a function, given in Eqn. 2, that maps from the system state
with failed primary nodes and
failed backup nodes to
the state index
of the Markov state diagram:
Similarly, the inverse mapping function is given in Eqn. 3
and 4.
Figure 12 shows the transition rate between the
neighboring states. In the diagram, denotes the
probability that the system remains functional, also referred to
as safety probability, given that one more primary node
fails while
primary nodes and
backup nodes have already
failed. Similarly,
denotes the probability, or safety
probability, of the system remaining functional when one more
backup node fails while
primary nodes and
backup nodes
have already failed.
can be calculated as
Eqn. 5.
Similarly, we have
![]() |
(7) |
The transition probability from State to the data loss state,
denoted as
, can be calculated as Eqn. 8.
The stochastic transitional probability matrix is defined as
, where
and
is the transition probability
from State
to State
. In summary,
can be calculated as follows.
If , then
![]() |
(9) |
If , then
![]() |
(10) |
If , then
![]() |
(11) |
If , i.e., the primary node and backup node are always
kept consistent, like in RAID-1, and a fault-free network is
assumed, the model shown in Figure 11 can be
simplified to the classic RAID-1 model [28], as shown
in Figure 13. This is proven by the fact that
numerical results generated by both models with the same set of
input parameters are identical.