4 Reliability Theory
137
Whatever we build will eventually fail. Nevertheless, we would like to access
the service that a device is providing for us for any time period we choose. We can
(and do) maximise the lifetime of a device by improving its design, but in order to
enable the service to be prolonged up to an arbitrary amount of time, we sometimes
have to replace the failed device or its parts with new, functioning ones or perform
a repair to make them functional again. Maintenance theory is generally concerned
with the overall system operation, including its economical aspects, like the costs
and logistics of enabling the repair at all. In this section, we will introduce some
basic aspects of the maintenance in the current subsection.
In renewal theory, we are no longer focusing on reliability of a mission—
meaning mission is successful if system does not fail until the mission’s end. We
now admit that a failure may occur and shift our focus to the performance measures
instead—i.e. what proportion of time is the system operational (and possibly how
that influences other performance measures). In this scenario, we assume that once a
system (or its components) fails, a repair process is initiated with a random time to
renewal denoting its duration, a time span after which the system will usually regain
its functionalities. Variations exist when failures are not directly observed, and we
need to plan also for inspections of the system or when the repair does not renew
the system but only make it minimally functional again. The state of the system is
again a random process in time with states {0, 1} as before, but now we pose no
restrictions on monotonicity.
The renewal function N(t) describes the number of repairs in interval [0, t].
It depends on the design of the system and the policies for its maintenance and
directly influences the economical aspects of the system (i.e. how much will it cost
to maintain the operation of the system, or can we supply enough spare parts?).
Because the times to failure are random variables, the renewal function will be a
random process, and mostly, for the sake of simplicity, we focus on its mean value.
Another important performance measure is the function describing the availability of a system—the probability that system is functioning at a specific moment in
time. Even though we can replace a failed component, the replacement may not be
immediate so the provided service may be unavailable (not functioning) during the
time of maintenance. The availability is, again, a time dependent function, which
we will denote as A(t) : T → [0, 1].
Specific form of N(t), A(t) and E{N(t)} (the mean value of N(t)) depends on
the qualitative properties of the system, namely on the models of the processes of
maintenance and failure inspection. Many real world systems will fall into one of
the following categories.
• Systems with immediate repair—the maintenance length is negligible so a
renewed unit is considered to start working immediately after the last unit failure.
The system itself is considered operational at all times (A ≡ 1), and we are only
interested in N(t).
• Systems with significant maintenance time—the maintenance length cannot
be neglected, but maintenance still commences right after the failure. The system
may not be operational at any time, since it may be undergoing maintenance.
137
Whatever we build will eventually fail. Nevertheless, we would like to access
the service that a device is providing for us for any time period we choose. We can
(and do) maximise the lifetime of a device by improving its design, but in order to
enable the service to be prolonged up to an arbitrary amount of time, we sometimes
have to replace the failed device or its parts with new, functioning ones or perform
a repair to make them functional again. Maintenance theory is generally concerned
with the overall system operation, including its economical aspects, like the costs
and logistics of enabling the repair at all. In this section, we will introduce some
basic aspects of the maintenance in the current subsection.
In renewal theory, we are no longer focusing on reliability of a mission—
meaning mission is successful if system does not fail until the mission’s end. We
now admit that a failure may occur and shift our focus to the performance measures
instead—i.e. what proportion of time is the system operational (and possibly how
that influences other performance measures). In this scenario, we assume that once a
system (or its components) fails, a repair process is initiated with a random time to
renewal denoting its duration, a time span after which the system will usually regain
its functionalities. Variations exist when failures are not directly observed, and we
need to plan also for inspections of the system or when the repair does not renew
the system but only make it minimally functional again. The state of the system is
again a random process in time with states {0, 1} as before, but now we pose no
restrictions on monotonicity.
The renewal function N(t) describes the number of repairs in interval [0, t].
It depends on the design of the system and the policies for its maintenance and
directly influences the economical aspects of the system (i.e. how much will it cost
to maintain the operation of the system, or can we supply enough spare parts?).
Because the times to failure are random variables, the renewal function will be a
random process, and mostly, for the sake of simplicity, we focus on its mean value.
Another important performance measure is the function describing the availability of a system—the probability that system is functioning at a specific moment in
time. Even though we can replace a failed component, the replacement may not be
immediate so the provided service may be unavailable (not functioning) during the
time of maintenance. The availability is, again, a time dependent function, which
we will denote as A(t) : T → [0, 1].
Specific form of N(t), A(t) and E{N(t)} (the mean value of N(t)) depends on
the qualitative properties of the system, namely on the models of the processes of
maintenance and failure inspection. Many real world systems will fall into one of
the following categories.
• Systems with immediate repair—the maintenance length is negligible so a
renewed unit is considered to start working immediately after the last unit failure.
The system itself is considered operational at all times (A ≡ 1), and we are only
interested in N(t).
• Systems with significant maintenance time—the maintenance length cannot
be neglected, but maintenance still commences right after the failure. The system
may not be operational at any time, since it may be undergoing maintenance.