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Availability is a performance criterion for repairable systems that accounts for both the reliability and maintainability properties of a component or system. It is defined as the probability that the system is operating properly when it is requested for use

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  1. 1. Availability Page 1 of 6 Availability If one considers both reliability (probability that the item will not fail) and maintainability (the probability that the item is successfully restored after failure), then an additional metric is needed for the probability that the component/system is operational at a given time, t (i.e. has not failed or it has been restored after failure). This metric is availability. Availability is a performance criterion for repairable systems that accounts for both the reliability and maintainability properties of a component or system. It is defined as the probability that the system is operating properly when it is requested for use. That is, availability is the probability that a system is not failed or undergoing a repair action when it needs to be used. For example, if a lamp has a 99.9% availability, there will be one time out of a thousand that someone needs to use the lamp and finds out that the lamp is not operational either because the lamp is burned out or the lamp is in the process of being replaced. (Note: Availability is always associated with time, much lik reliability and maintainability. As we will see in later sections, there are different availability classifications and for some of which, the definition depends on the time under consideration. Since no discussion about these classifications has been made yet, the time variable has been left out of this 99.9% availability statement.) This metric alone tells us nothing about how many times the lamp has been replaced. For all we know, the lamp may be replaced every day or it could have never been replaced at all. Other metrics are still important and needed, such as the lamp's reliability. The next table illustrates the relationship between reliability, maintainability and availability. A Brief Introduction to Renewal Theory For a repairable system, the time of operation is not continuous. In other words, its life cycle can be described by a sequence of up and down states. The system operates until it fails, then it is repaired and returned to its original operating state. It will fail again after some random time of operation, get repaired again, and this process of failure and repair will repeat. This is called a renewal process and is defined as a sequence of independent and non-negative random variables. In this case, the random variables are the times-to-failure and the times-to-repair/ restore. Each time a unit fails and is restored to working order, a renewal is said to have occurred. This of renewal process is known as an alternating renewal process because the state of the component alternates between a functioning state and a repair state, as illustrated in the following graphic. A system's renewal process is determined by the renewal processes of its components. For example, consider a series system of three statistically independent components. Each component has a failure distribution and a repair distribution. Since the components are in series, when one component fails, the entire system fails. The system is then down for as long as the failed component is under repair. Figure 7.1 illustrates this. 4/22/2004
  2. 2. Availability Page 2 of 6 Figure 7.1: System downtime as a function of three component downtimes. Components A, B and C are in series. One of the main assumptions in renewal theory is that the failed components are replaced with new ones or are repaired so they are "as good as new," hence the name renewal. One can make the argument that this is the case for every repair, if you define the system in a high enough detail. In other words, if the repair of a single circuit board in the system involves the replacement of a single transistor in the offending circuit board, then if the analysis (or RBD) is performed down to the transistor level, then the transistor itself gets renewed. In cases where the analysis is done at a higher level, or if the offending component is replaced with a used component, additional steps are required. We will discuss this in later chapters using a restoration factor in the analysis. For more details on renewal theory, interested readers can refer to Elsayed [7] and Leemis [15]. Availability Classifications The definition of availability is somewhat flexible and is largely based on what types of downtimes one chooses to consider in the analysis. As a result, there are a number of different classifications of availability, such as: z Instantaneous (or Point) Availability. z Average Up-Time Availability (or Mean Availability). z Steady State Availability. z Inherent Availability. z Achieved Availability. 4/22/2004
  3. 3. Availability Page 3 of 6 z Operational Availability. Instantaneous or Point Availability, A(t) Instantaneous (or point) availability is the probability that a system (or component) will be operational (up and running) at any random time, t. This is very similar to the reliability function in that it gives a probability that a system will function at the given time, t. Unlike reliability, the instantaneous availability measure incorporates maintainability information. At any given time, t, the system will be operational if the following conditions are met [7]: The item functioned properly from 0 to t with probability R(t) or it functioned properly since the last repair at time u, 0 < u < t, with probability: With m(u) being the renewal density function of the system. Then the point availability is the summation of these two probabilities, or: Average Uptime Availability (or Mean Availability), The mean availability is the proportion of time during a mission or time period that the system is available for use. It represents the mean value of the instantaneous availability function over the period (0, T] and is given by: (2) Steady State Availability, The steady state availability of the system is the limit of the instantaneous availability function as time approaches infinity or: (3) (Note: For practical considerations, the instantaneous availability function will start approaching the steady state availability value after a time period of approximately four times the average time-to-failure.) Figure 7.2 also illustrates this graphically. 4/22/2004
  4. 4. Availability Page 4 of 6 Figure 7.2: Illustration of point availability approaching steady state. In other words, one can think of the steady state availability as a stabilizing point where the system's availability is a constant value. However, one has to be very careful in using the steady state availability as the sole metric for some systems, especially systems that do not need regular maintenance. A large scale system with repeated repairs, such as a car, will reach a point where it is almost certain that something will break and need repair once a month. However, this state may not be reached until, say, 500,000 miles. Obviously, if I am an operator of rental vehicles and I only keep the vehicles until they reach 50,000 miles, then this value would not be of any use to me. Similarly, if I am an auto maker and only warrant the vehicles to X miles, is knowing the steady state value useful? Inherent Availability, Inherent availability is the steady state availability when considering only the corrective downtime of the system. For a single component, this can be computed by: This gets slightly more complicated for a system. To do this, one needs to look at the mean time between failures, or MTBF, and compute this as follows: This may look simple. However, one should keep in mind that until steady state is reached, the MTBF may be a function of time (e.g. a degrading system), thus the above formulation should be used cautiously. Furthermore, it is important to note that the MTBF defined here is different from the MTTF (or more precisely for a repairable 4/22/2004
  5. 5. Availability Page 5 of 6 system, MTTFF, mean time to first failure). Achieved Availability, Achieved availability is very similar to inherent availability with the exception that preventive maintenance (PM) downtimes are also included. Specifically, it is the steady state availability when considering corrective and preventive downtime of the system. It can be computed by looking at the mean time between maintenance actions, MTBM and the mean maintenance downtime, or: Operational Availability, Operational availability is a measure of the average availability over a period of time and it includes all experienced sources of downtime, such as administrative downtime, logistic downtime, etc. Operational availability is the ratio of the system uptime and total time. Mathematically, it is given by: (4) Where the operating cycle is the overall time period of operation being investigated and uptime is the total time the system was functioning during the operating cycle. (Note: The operational availability is a function of time, t, or operating cycle.) When there is no specified logistic downtime or preventive maintenance, Eqn. (4) returns the Mean Availability of the system. The operational availability is the availability that the customer actually experiences. It is essentially the a posteriori availability based on actual events that happened to the system. The previous availability definitions are a priori estimations based on models of the system failure and downtime distributions. In many cases, operational availability cannot be controlled by the manufacturer due to variation in location, resources and other factors that are the sole province of the end user of the product. Introduction to Repairable Systems Example 1 As an example, consider the following scenario. A diesel power generator is supplying electricity at a research in Antarctica. The personnel are not satisfied with the generator. They estimated that in the past six months, they were without electricity due to generator failure for an accumulated time of 1.5 months. Therefore, the operational availability of the diesel generator experienced by the personnel of the station is: 4/22/2004
  6. 6. Availability Page 6 of 6 Obviously, this is not satisfactory performance for an electrical generator in such a climate so alternatives to this source of electricity are investigated. One alternative under consideration is a wind-powered electrical turbine, which the manufacturer claims to have a 99.71% availability. This is much higher than the availability experienced by the crew of the Antarctic research station for the diesel generator. Upon investigation, it was found that the wind-turbine manufacturer estimated the availability based on the following information: Failure Distribution Repair Distribution Exponential, MTTF = 2400 Exponential, MTTR = 7 Based on the above information, one can estimate the mean availability for the wind turbine over a period of six months to be: This availability, however, was obtained solely by considering the claimed failure and repair properties of the wind-turbine. Waiting downtime was not considered in the above calculation. Therefore, this availability measure cannot be compared to the operational availability for the diesel generator since the two availability measurements have different inputs. This form of availability measure is also known as inherent availability. In order to make a meaningful comparison, the inherent availability of the diesel generator needs to be estimated. The diesel generator has an MTTF = 50 days (or 1200 hours) and an MTTR = 3 hours. Thus, an estimate of the mean availability is: Note that the inherent availability of the diesel generator is actually a little bit better than the inherent availability of the wind-turbine! Even though the diesel generator has a higher failure rate, its mean-time-to-repair is much smaller than that of the wind turbine, resulting in a slightly higher inherent availability value. This example illustrates the potentially large differences in the types of availability measurements, as well as their misuse. In this example, the operational availability is much lower than the inherent availability. This is because the inherent availability does not account for downtime due to administrative time, logistic time, the time required to obtain spare parts or the time it takes for the repair personnel to arrive at the site. See Also: Introduction to Repairable Systems Go to Go to ©1999-2003. ReliaSoft Corporation. ALL RIGHTS RESERVED. 4/22/2004