part level predictions, two separate fields of investigation are common: The physics of failure approach uses an understanding of physical failure mechanisms involved, such as mechanical crack propagation or chemical corrosion degradation or failure; The parts stress modelling approach is an empirical method for prediction based on counting the number and type of components of the system, and the stress they undergo during operation. Reliability theory Reliability is defined as the probability that a device will perform its intended function during a specified period of time under stated conditions. Mathematically, this may be expressed as, , where is the failure probability density function and is the length of the period of time (which is assumed to start from time zero). There are a few key elements of this definition: Reliability is predicated on "intended function:" Generally, this is taken to mean operation without failure. However, even if no individual part of the system fails, but the system as a whole does not do what was intended, then it is still charged against the system reliability. The system requirements specification is the criterion against which reliability is measured. Reliability applies to a specified period of time. In practical terms, this means that a system has a specified chance that it will operate without failure before time . Reliability engineering ensures that components and materials will meet the requirements during the specified time. Note that units other than time may sometimes be used (e.g. "a mission", "operation cycles"). Reliability is restricted to operation under stated (or explicitly defined) conditions. This constraint is necessary because it is impossible to design a system for unlimited conditions. A Mars rover will have different specified conditions than a family car. The operating environment must be addressed during design and testing. That same rover may be required to operate in varying conditions requiring additional scrutiny. Two notable references on reliability theory and its mathematical and statistical foundations are Barlow, R. E. and Proschan, F. (1982) and Samaniego, F. J. (2007). Quantitative system reliability parameters—theory Quantitative requirements are specified using reliability parameters. The most common reliability parameter is the mean time to failure (MTTF), which can also be specified as the failure rate (this is expressed as a frequency or conditional probability density function (PDF)) or the number of failures during a given period. These parameters may be useful for higher system levels and systems that are operated