ITEM TOOLKIT

Understanding Analysis Results
Chapter 9 Markov 215
The following shows how the various reliability measures are computed. X refers to the state or group of states that have
been marked as ‘Unavailable States’. In these definitions, the following notation is used:
x: a state.
X: a group of states.
Pr( x; t ) : The probability that the system is in state x at time t.
Pr( X ; t ) : The probability that the system is any of the states belonging to group X at time t.
λx
→ y : The rate at which transition takes place from state x to state y. These transition rates concern continuoustime
transitions.
P () x→
y
t : The probability of a transition from state x to state y at time t. These probabilities concern discretetime
transitions.
Unavailability Q: / Availability A
The point unavailability is computed as the probability that the system is in any of the states belonging to group X at time t.
The availability is computed as 1 minus this value.
U ( t)
= Pr( X ; t)
A(
t)
= 1−U
( t)
TDT (Total Down Time): / Total Up Time (TUT)
The Total Down Time and Total Up Time are respectively computed as the expected amount of time spent in any state
belonging to group X, and in any state not belonging to group X, between 0 and t.
t
∫
TDT = Pr( X ; τ ) ⋅dτ
τ = 0
TUT = t −TDT
Mean Unavailability: / Mean Availability
The mean unavailability is computed as the expected amount spent in X divided by the total mission time. The mean
availability is computed as 1 minus this value.
t
Uˆ
1
= X d
t
∫ Pr( ; τ ) ⋅ τ
τ = 0
Aˆ
= 1−Uˆ
Expected Failure: / Expected Repair
The expected number of repairs r is computed as the expected number of transitions from X to states outside X. The
expected number of failures f is computed as the expected number of transitions from states outside X to states inside X.