probability of successful recovery (with depth m) of a task when a latent malfunction with latency l occurs in segment k is given by Equation
P k m ðlÞ ¼ a 1 À a
ð
Þ
lÀi P k m ðlÞ
ð 10:6Þ
Coefficient a is the mean probability of detecting the malfunction during the
segment execution.
a ¼
1
N
X N
i¼1
a i
The probability of event H km (L), i.e., the successful recovery of a program
affected by a latent malfunction in segment k after m recovery steps, is
P k m ðlÞ ¼
X mÀ1
i¼1
að1 À aÞ
lÀi P k m ðlÞ
ð 10:7Þ
In case of latent malfunction, the mean total recovery time for depth m is
T P
m ðlÞ ¼
X mÀ1
i¼1
að1 À aÞ
lÀi T P
m ðlÞ
The probability of system restart and task reload (event H kpf (L)) is
P k pf ðLÞ ¼ 1 À a k
ð
Þð1 À e
kT Þ
X k þ l
m¼2
P k m ðLÞ
ð 10:8Þ
where the mean length of the latency period is
l ¼
X
NÀKÀ1
l¼1
að1 À aÞ
lÀi l
If we combine all these equations into one, we can derive the mean time T of the
successful execution of segment k:
T ¼ P c T þ
X k
m¼1
P km T P
m þ
X k þ l
m¼2
P km ðLÞT P
m ðLÞ
ð 10:9Þ
We will use T throughout the analysis and comparison of the three recovery
algorithms [2–7].
158
10 Recovery Algorithms: An Analysis