Calculations for Functional Safety
Quantities, Formulas and Methods

E Glossary and list of abbreviations

Table 8: Terms and abbreviations


\(\lessapprox \)

About the same, but certainly smaller. Means here: The formula is slightly conservative, but for correctly designed systems (unavailabilities only slightly larger than zero) practically well usable.

Failure density

\(f(t)\), derivative of \(\rightarrow \)unreliability.

Failure rate

\(\rightarrow \)occurrence rate.

Basic event

An event of a \(\rightarrow \)element.


Condition Event. A basic event described only by a probability (typ. unavailability), not by an occurrence rate.

\(\beta \)

The common cause factor relating to the occurrence rate or unavailability of multiple events that are not completely independent.

Occurrence rate

The rate of occurrence of an event conditioned with respect to availability or reliability. When it refers to an event describing a failure, also called failure rate.


Any \(\rightarrow \)component, human behavior, or environmental condition, that affects the behavior of the system.


A situation or state that an element or system can enter.


Equipment under Control, term from [IEC 61508], here always denoted by "process".


\(\rightarrow \)failure density


\(\rightarrow \)unreliability

Failure rate

\(\rightarrow \)Occurrence rate


Fault Tree, Fault Tree


Fault tree analysis


\(\rightarrow \)occurrence rate


Hazard Rate, actual or calculated (i. e., estimated) occurrence rate of a hazard.


The representation of a basic event in a Markov model.


A technical unit that can usually fail with different failure modes.


Mean repair time, mean time between detection of a failure and repair, if the process (in [IEC 61508]: the "EUC") is still operated in case of a detected failure.


Mean time to detect, mean time to detect failure.


Mean time to failure and also mean time of fault free operation between two failures.


Mean time to restoration. Recovery time. Includes the \(\rightarrow \)MTTD and the \(\rightarrow \)MRT.


probability \(Q(t)\) that a \(\rightarrow \)element will not work, if it is requested at time \(t\).


Probability of Failure on Demand, denoting the mean \(\rightarrow \)unavailability \(\overline {Q}\) in [IEC 61508].


Probability of Failure per Hour, designation of the mean conditional failure frequency (failure rate) \(\rightarrow {overline{h}}\) of a system in [IEC 61508].


Process Fault Tolerance Time, also Process Safety Time, time a process can be operated with incorrect manipulated variables without entering an unsafe status.


Prime Implicant, Prime Implicant. Equivalent of a minimum cut in the case of incoherent fault trees. In the case of coherent fault trees, prime implicants are identical to minimal cuts.


\(\rightarrow \)unavailability


A partial fault tree referenced by transfer gate as \(\rightarrow \) branch in a higher fault tree.

System lifetime

The planned deployment time of the system under consideration. Needed when the quantity of interest is not constant and not periodic, i. e., in particular to determine \(\rightarrow \) unreliability, or when basic events of the model "non-restorable" are included in the fault tree or Markov model.


Tolerable Hazard Rate, target for safety functions with continuous or frequent demand.


Tolerable Probability of Failure on Demand, acceptable unavailability, target value for safety functions with rare (low) demand.


Tolerable Probability of Failure per Hour, acceptable failure rate, target for safety functions with continuous or frequent demand. Mathematically and at the top level also logically identical to \(\rightarrow \)THR.


probability \(F(t_1,t_2)\) that a \(\rightarrow \)element fails during the time period \(t_1 \dots t_2\).


occurrence frequency in the case of testable and/or repairable events. Unlike \(h(t)\), \(w(t)\) is conditional only with respect to the last test or replacement, not with respect to the availability of the system. Thus, \(w(t)\) is often referred to as unconditional occurrence frequency. Unlike \(f(t)\), \(w(t)\) is conditional with respect to the last test or swap, so its integral over time can become larger than 1.


The part of a fault tree, which is below a certain gate including the gate itself. Special case: a base event is also a branch.