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4 IEEE TRANSACTIONS ON RELIABILITY
TABLE II
M
AINTAIN ACTIONS AND THEIR CORRESPONDING TIME DETERMINED
BY THE
MONITOREDSTATE(t)
NO. MonitoredState(t) Action at Action at Time of
AS level OS level the action
0 rr no action no action -
4 Fr Rea. Rest. no action T
A 4
5 FD Proa. Rest. Proa. Rebo. T
O 3
6 FF Rea. Rest. Rea. Rebo. T
O 4
7 rD Proa. Rest. Proa. Rebo. T
O 3
8 Dr Proa. Rest. no action T
A 3
9 DD Proa. Rest. Proa. Rebo. T
O 3
1) The time to inspection trigger is a constant, T
in
, which is
actually the constant variable in this paper.
2) There is no delay in carrying out the inspection for the
M&A, which means that the system makes instantaneous
diagnoses after it is triggered.
3) When the AS fails, the M&A will be triggered immedi-
ately to detect the state of the OS.
4) After each rejuvenation of the AS or the OS, the timer of
the M&A will be reset.
5) The holding time T
A1
(or T
O 1
) of AS (or OS) from robust
state to degradation state has the exponential distribution
with parameter λ
A1
(or λ
O 1
). The holding time T
A2
(or
T
O 2
) of the AS (or OS) from degradation state to failed
state also has the exponential distribution with parameter
λ
A2
(or λ
O 2
).
6) The rejuvenation time T
A3
(or T
O 3
)oftheAS(orOS)
from degradation state to robust state has a general distri-
bution F
A3
(t) (or F
O 3
(t)). The reactive restart (or reboot)
time T
A4
(or T
O 4
) of the AS (or OS) also has a general
distribution F
A4
(t) (or F
O 4
(t)).
Let TrueState(t) be the true state of the system at time t.
Assume that the M&A does not always make correct diagnoses.
MonitoredState(t) = TrueState(t) means that the M&A makes
a misdiagnosis. Recall that the AS and the OS have their sepa-
rated memory usage. Suppose that the M&A makes diagnoses
independently for the AS and the OS. Let MonitoredState
AS
(t)
and MonitoredState
OS
(t) be the diagnosis results of AS and OS,
respectively. (t)=‘rD’ means that MonitoredState
AS
(t)=‘r’
and MonitoredState
OS
(t)=‘D.’ Suppose that if AS and OS are
at one of the states in {failed, rejuvenation}, they can be rightly
diagnosed by the M&A. Let
p
r
= Pr{MonitoredState
AS
(t)=r |TrueState
AS
(t)=r}
p
D
= Pr{MonitoredState
AS
(t)=D |TrueState
AS
(t)=D}
q
r
= Pr{MonitoredState
OS
(t)=r |TrueState
OS
(t)=r}
q
D
= Pr{MonitoredState
OS
(t)=D |TrueState
OS
(t)=D}.
(1)
When and which rejuvenation technique should be executed
depends on the diagnosis results from the M&A. For example,
if TrueState(t) is “rD,” there are four possible actions according
Fig. 2. True state diagram for the two-granularity software aging and rejuve-
nation model.
to the inspection results MonitoredState(t):
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
no action → rD, if (t)=rr
Reju. OS → rr, if (t)=rD
Reju. AS → rD, if (t)=Dr
Reju. OS → rr, if (t)=DD
by noting our assumption that if the AS and the OS are at one
of the states in {failed, rejuvenation}, they can be rightly di-
agnosed by the M&A. The meaning of the above brace can
be explained as follows. Suppose that TrueState(t) is “rD.” If
MonitoredState(t) is “rr,” which means that the M&A makes
a right report regarding the AS level and makes a wrong re-
port regarding the OS level, then no maintenance action will
be taken. The true state of the system will still be “rD.” If
MonitoredState(t) is “Dr,” which means that the M&A makes
wrong reports regarding both the AS level and the OS level,
then the action of AS rejuvenation will be taken. The true state
of the system will still be “rD” after AS rejuvenation. The true
state transition path in this case is rD → RD → rD.
Based on the rejuvenation scheduling defined above, the true
state diagram is determined. Fig. 2 shows the state diagram of
the ten states for our two-granularity software aging and rejuve-
nation model. In the figure, F
(·)
(t) are the distribution functions
of the corresponding sojourn times. According to our supposi-
tion, the holding time T
A1
of the AS transition from robust state
to degradation state follows exponential distribution F
A1
(t).
Also the holding times T
A2
, T
O 1
, and T
O 2
follow exponential
distribution with the distribution functions F
A2
(t), F
O 1
(t), and
F
O 2
(t), respectively. As shown in Fig. 2, because the time to
trigger inspection is a deterministic value T
in
, the stochastic
process Z(t) determined by the model is not a Markov process