Part B | 24.4
534 Part B Autonomous Ocean Vehicles, Subsystems and Control
24.4 Risk of Unavailability
System availability is defined as the probability of the
system being available given that the system is needed
at a given time. System unavailability is the complement of this figure. The deployment of an AUV consists
of several phases. A series of tasks are carried out
in each phase of the deployment. These human- or
machine-related tasks are not immune to error. Estimating the availability for the AUV being at a given phase
is only possible if we take into account the sequence
of phases that precede that phase. This problem can be
mathematically modeled using a probability approach
denoted as Markov theory.
For a brief description of Markov theory, consider
a probability problem with a set of outcomes of interest E 1 ; E 2 ; : : : ; E k . Given that there is a probability
p j associated with each event the joint probability for
a given sequence is defined by the multiplicative property, thus P.E 1 ; E 2 ; : : : ; E k / D p 1 p 2 : : : p k . The Markov
chain theory introduces an assumption that simplifies
this expression; it considers that the outcome of any
trial depends on the outcome of the preceding trial and
only on it [24.26]. Therefore, if event E 1 precedes event
E 2 and event E 2 precedes event E 3 and so on for the remaining events, then instead of associating a probability
to an event E j , it uses a transition probability p jk for every pair of events .E j ; E k /, where p jk is the probability
of E k occurring given that E k1 occurred in the previous trial. A Markov model may have more than one
sequence of events; E j may have two or more posterior
states. Therefore j and k are not necessarily adjacent.
In addition to p jk , one must also define the probability
of E j occurring at the initial trial, a j0 . Therefore for the
initial trial, P.E j ; E k / D a j0 p jk . For the general case,
considering a sequence of several transitions, given that
event E j0 precedes E j1 which precedes E j2 and so on for
the remaining events, the joint probability distribution
is computed using the expression in (24.4)
P r .E j0 ; E j1 ; : : : ; E jn /
D a j0 p j0j1 p j1j2 : : : p jn2jn1 P jn1jn :
(24.4)
It is not unusual to find problems where one state has
more than one potential preceding state. When this
is the case, the mathematical calculation cannot be
performed using the simple equation presented above.
Instead the calculation is performed using matrix operations. The transition probabilities are arranged in
a matrix denoted as transition matrix or stochastic matrix. The transition probability together with the initial
state vector completely defines the Markov chain. The
availability after n transitions can be calculated by the
product between the transition matrix to the power of n
and the initial state vector.
Brito and Griffiths [24.27] used Markov chains for
modeling and estimating the availability of Autosub3.
The Autosub3 deployment sequence was modeled as
a Markov chain in which each state corresponds to one
phase of the vehicle deployment. A key assumption in
the proposed model is that the Autosub is deployed
from a stationary vessel. The model can be altered to
capture the case where the AUV is deployed from an
ice hole or from the coast. This would consist of adding
a state for which the transition probabilities would
depend on the failure modes that can emerge in the different type of deployment. Griffiths and Brito in [24.19]
give an example of when the approach was adopted for
deployments from ice holes.
Here we briefly describe the model proposed
in [24.27] for estimating AUV availability based on surface vessel deployments. The Markov chain model for
AUV availability consists of 11 phases, these states and
the transitions are presented in Fig. 24.3. The first state
captures the phase when the vehicle is on board of the
vessel, switched on, ready for testing (Dp). A series of
communication, actuation, and navigation checks are
carried out during this phase. Having passed the onboard checks, the deployment moves to vehicle ready
for deployment phase (Dr). Here the vehicle is waiting to be deployed overboard. While overboard there
is the risk that the vehicle may run under the vessel causing severe damage, in which case the vehicle
may need to be salvaged. This scenario is captured by
the transition from phase overboard (O) to salvage (S).
Once overboard and at a safe distance, communications
checks are carried out before the vehicle starts diving.
The vehicle is then set to stop diving when it reaches
a predefined depth and enters a holding pattern (Sh).
During this period, the vehicle is still within acoustic
range, more checks are carried out before committing
to the mission. Once underway (U), there are two possible outcomes: the vehicle can be recovered (R), or it
can be lost (L). If historic operational data exist then
the probability of AUV loss can be computed using
reliability modeling techniques combined with expert
judgment – this method is described in the Sect. 24.5.
Alternatively, if there is no historic data of AUV operation then the probability of loss can be computed
using fault tree modeling that takes into account expert
judgment with regard to the consequence of each failure
mode.
The AUV can be recovered while it is in the overboard phase, during diving, in holding pattern or underway. The vehicle can also be lost from any of these
534 Part B Autonomous Ocean Vehicles, Subsystems and Control
24.4 Risk of Unavailability
System availability is defined as the probability of the
system being available given that the system is needed
at a given time. System unavailability is the complement of this figure. The deployment of an AUV consists
of several phases. A series of tasks are carried out
in each phase of the deployment. These human- or
machine-related tasks are not immune to error. Estimating the availability for the AUV being at a given phase
is only possible if we take into account the sequence
of phases that precede that phase. This problem can be
mathematically modeled using a probability approach
denoted as Markov theory.
For a brief description of Markov theory, consider
a probability problem with a set of outcomes of interest E 1 ; E 2 ; : : : ; E k . Given that there is a probability
p j associated with each event the joint probability for
a given sequence is defined by the multiplicative property, thus P.E 1 ; E 2 ; : : : ; E k / D p 1 p 2 : : : p k . The Markov
chain theory introduces an assumption that simplifies
this expression; it considers that the outcome of any
trial depends on the outcome of the preceding trial and
only on it [24.26]. Therefore, if event E 1 precedes event
E 2 and event E 2 precedes event E 3 and so on for the remaining events, then instead of associating a probability
to an event E j , it uses a transition probability p jk for every pair of events .E j ; E k /, where p jk is the probability
of E k occurring given that E k1 occurred in the previous trial. A Markov model may have more than one
sequence of events; E j may have two or more posterior
states. Therefore j and k are not necessarily adjacent.
In addition to p jk , one must also define the probability
of E j occurring at the initial trial, a j0 . Therefore for the
initial trial, P.E j ; E k / D a j0 p jk . For the general case,
considering a sequence of several transitions, given that
event E j0 precedes E j1 which precedes E j2 and so on for
the remaining events, the joint probability distribution
is computed using the expression in (24.4)
P r .E j0 ; E j1 ; : : : ; E jn /
D a j0 p j0j1 p j1j2 : : : p jn2jn1 P jn1jn :
(24.4)
It is not unusual to find problems where one state has
more than one potential preceding state. When this
is the case, the mathematical calculation cannot be
performed using the simple equation presented above.
Instead the calculation is performed using matrix operations. The transition probabilities are arranged in
a matrix denoted as transition matrix or stochastic matrix. The transition probability together with the initial
state vector completely defines the Markov chain. The
availability after n transitions can be calculated by the
product between the transition matrix to the power of n
and the initial state vector.
Brito and Griffiths [24.27] used Markov chains for
modeling and estimating the availability of Autosub3.
The Autosub3 deployment sequence was modeled as
a Markov chain in which each state corresponds to one
phase of the vehicle deployment. A key assumption in
the proposed model is that the Autosub is deployed
from a stationary vessel. The model can be altered to
capture the case where the AUV is deployed from an
ice hole or from the coast. This would consist of adding
a state for which the transition probabilities would
depend on the failure modes that can emerge in the different type of deployment. Griffiths and Brito in [24.19]
give an example of when the approach was adopted for
deployments from ice holes.
Here we briefly describe the model proposed
in [24.27] for estimating AUV availability based on surface vessel deployments. The Markov chain model for
AUV availability consists of 11 phases, these states and
the transitions are presented in Fig. 24.3. The first state
captures the phase when the vehicle is on board of the
vessel, switched on, ready for testing (Dp). A series of
communication, actuation, and navigation checks are
carried out during this phase. Having passed the onboard checks, the deployment moves to vehicle ready
for deployment phase (Dr). Here the vehicle is waiting to be deployed overboard. While overboard there
is the risk that the vehicle may run under the vessel causing severe damage, in which case the vehicle
may need to be salvaged. This scenario is captured by
the transition from phase overboard (O) to salvage (S).
Once overboard and at a safe distance, communications
checks are carried out before the vehicle starts diving.
The vehicle is then set to stop diving when it reaches
a predefined depth and enters a holding pattern (Sh).
During this period, the vehicle is still within acoustic
range, more checks are carried out before committing
to the mission. Once underway (U), there are two possible outcomes: the vehicle can be recovered (R), or it
can be lost (L). If historic operational data exist then
the probability of AUV loss can be computed using
reliability modeling techniques combined with expert
judgment – this method is described in the Sect. 24.5.
Alternatively, if there is no historic data of AUV operation then the probability of loss can be computed
using fault tree modeling that takes into account expert
judgment with regard to the consequence of each failure
mode.
The AUV can be recovered while it is in the overboard phase, during diving, in holding pattern or underway. The vehicle can also be lost from any of these