Issue 74, April 2007

Using Duty Cycles in System Reliability Analysis

Components of a system may not operate continuously during a systems mission, or may be subjected to loads greater or lesser than the rated loads during system operation. To model this, a factor called the "duty cycle" is used. The duty cycle can also be used to account for changes in environmental stress, such as temperature changes, that may affect the operation of a component. The use of duty cycles in reliability modeling, especially in system reliability modeling and analysis, provides additional flexibility for the analyst. This article explains how to perform analysis using duty cycles in BlockSim 7.

Analysis with Duty Cycles
The duty cycle is a positive value, with a default value of 1 representing continuous operation at rated load. Any values other than 1 represent other load values with respect to the rated load value (or total operating time). The value of the duty cycle depends on the Life-Stress Relationship (LSR) of the component. If a linear LSR can be assumed, the duty cycle is simply the ratio of the load on the component (V₂) to the rated load (V₁), i.e. d_c = (V₂/V₁). For an inverse power LSR, d_c = (V₂/V₁)ⁿ. A duty cycle value higher than 1 indicates a load in excess of the rated value. A duty cycle value lower than 1 indicates that the component is operating at a load lower than the rated load or not operating continuously during the systems mission. For instance, a duty cycle of 0.5 may be used for a component that operates only half of the time during the systems mission.

The reliability metrics for a component with a duty cycle are calculated as follows. Let d_c represent the duty cycle during a particular mission of the component, t represent the mission time and t' represent the accumulated age. Then:

t' = d_c t

The reliability equation for the component is:

R(t') = R(d_c t)

The component pdf is:

Example 1: Components with Intermittent Usage
The use of duty cycles in BlockSim allows you to specify the blocks usage as a percentage of the system usage. A component with intermittent usage, such as the CD drive in a computer, may accumulate only 18 minutes of usage for every hour the computer operates, and would have a duty cycle (usage rate) of 0.30.

Consider a computer system with three components in series: a processor, a hard drive and a CD drive, as shown in the next reliability block diagram (RBD).

Assume that all three components follow a Weibull failure distribution and that under normal operating conditions the parameters are β₁ = 1.5 and η₁ = 5000 days for the processor, β₂ = 2.5 and η₂ = 3000 days for the hard drive and β₃ = 2 and η₃ = 4000 days for the CD drive. The CD drive is used only 30% of the time.

The failure distribution for the CD drive is entered as follows:

The other components' distributions are entered in the same fashion.

The duty cycle for the CD drive is specified in the Block Properties window as follows:

The reliability of the computer system after one year (365 days) of operation, assuming that the CD drive is used only 30% of the time, is estimated to be 0.9747, as shown next.

Example 2: Stressed Components
Duty cycles can also be used to model components that are over-stressed (d_c > 1) or under-stressed (d_c < 1). Assume that under certain elevated stress conditions of temperature and humidity, the components of the above RBD are stressed by a d_c factor of 1.5. The d_c properties of the blocks would be entered as follows:

Notice that the CD drive's d_c factor reflects the combination of the stressed environment and the usage rate (i.e. d_c = 0.3 1.5 = 0.45).

The reliability of the computer system after one year (365 days) of operation under the elevated conditions and assuming that the CD drive is used only 30% of the time is estimated to be 0.9492, as shown next.

Example 3: Changing Usage Rates/Stress Factors over Different Phases of an Operation
All components of a system may be collectively subjected to different operating conditions as the system goes through different phases of its mission. Such a scenario may be modeled by assigning a duty cycle value for each phase of the system. For example, the load on an aircrafts components may be different during taxiing (e.g. d_c = 1), take-off (e.g. d_c = 1.5), cruising (e.g. d_c = 1.1) and landing (e.g. d_c = 1.3).

Phase diagrams will be discussed in future Reliability Edge and Reliability HotWire articles. For more information about reliability phase diagrams, see ReliaSoft's online System Analysis Reference.

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