Reliability Allocation
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B. Different venders may offer their own implementations of SSi at different costs. C. It may be possible to use multiple copies of SSi (for example double wheels of a truck) to achieve higher
II. Problem Formulation
We assume that a system has been designed at a higher level as an assembly of appropriated connected subsystems. In general the functionality of each subsystems can be unique, however there can be several choices for many of the subsystems providing the same functionality, but differently reliability levels.
Reliability Allocation
Yashwant K. Malaiya Computer Science Dept. Colorado State University Fort Collins CO 80523
USA
malaiya@ Phone: 970-491-7031, 970-491-2466
The ith subsystem SSi can may have several implementation choices with different reliability values:
A. By extending a continuous attribute (for example diameter of a column in building or time spent for software testing) the subsystem can be made more reliable.
Here we consider the problem formulation for a common and widely applicable case. Let there be n subsystems SSi, i = 1,..,n, each with reliability Ri and cost Ci. Let the cost Ci be a function of the reliability given by fi( Ri). Let Cs and Rs represent the total system cost and the overall reliability and RST be the specified target reliability. If all
reliability. Often the number of copies is constrained between a minimum (often one) and a maximum number because of implementation issues.
Note that in the first case, both cost and the reliability can be varied continuously, where as in the other two cases, the choices are discreet. In the first case, we can define a continuous cost function. In the second case too, the market forces may impose a cost function. In the third case, the subsystem may be modeled as a parallel or k-out-ofn system for reliability evaluation, provided the failures are statistically independent. A number of publications on reliability allocation consider only the third case, where the optimization problem becomes an integer optimization problem. It becomes a 0-1 optimization problem when choices are discreet and a component from a given list of candidates is either used or not used [majety99].
Abstract---A system is generally designed as an assembly of subsystems, each with its own reliability attributes. The overall system reliability is a function of the subsystem reliability metrics. The cost of the system is the sum of the costs for all the subsystems. This article examines possible approaches to allocate the reliability values such that the total cost is minimized. The problem is very general and is applicable for mechanical systems, electrical systems, computer hardware and software. The problem involves expressing the system reliability in terms of the subsystem reliability values and specifying the cost function for each subsystem which allows setting up an optimization problem. Software reliability allocation is examined as a detailed example. The article also examines some preliminary apportionment approaches that have been proposed. In some cases, it is possible to use exact optimization methods. In general, a complex case will require use of iterative approaches.
n
n
Minimize
Cs = Ci = fi (Ri )
(1)
i=1
i=1
Subject to
RST ≤ Rs
n
≤ ∏ Ration (1) assumes that the cost of interconnecting the subsystems is negligible. An alternative problem would be to maximize Rs while keeping Cs less than the allocated cost budget.
I. INTRODUCTION
Many systems are implemented by using a set of interconnected subsystems. While the architecture of the overall system can often be fixed, individual subsystems may be implemented differently. A designer needs to either achieve the target reliability while minimizing the total cost, or maximize the reliability while using only the available budget. Intuitively, some of the lowest reliability components may need special attention to raise the overall reliability level. Such an optimization problem may arise while designing a complex software or a computer system. Such problems also arise in mechanical or electrical systems. A number of studies since 1960 have examined such problems [kuo00].
In a non-redundant system, all the subsystems are essential, however often an individual subsystem can be made more reliable by using a more costly implementation. This additional cost may represent wider columns in a building or more thorough testing of software. In redundant implementations, higher reliability can sometimes be achieved by using several copies of a subsystems, such that form a parallel or k-out-of-n configuration.
1
the subsystems are essential to the system and if their failures are statistically independent, the system can be modeled as a series system. The cost minimization problem can be stated as:
The next section consider the problem formulation, followed by approaches used for setting up an optimization problem. As an example, software reliability allocation is examined in detail with two numerical illustrations. The last section considers reliability allocation in complex systems.
II. Problem Formulation
We assume that a system has been designed at a higher level as an assembly of appropriated connected subsystems. In general the functionality of each subsystems can be unique, however there can be several choices for many of the subsystems providing the same functionality, but differently reliability levels.
Reliability Allocation
Yashwant K. Malaiya Computer Science Dept. Colorado State University Fort Collins CO 80523
USA
malaiya@ Phone: 970-491-7031, 970-491-2466
The ith subsystem SSi can may have several implementation choices with different reliability values:
A. By extending a continuous attribute (for example diameter of a column in building or time spent for software testing) the subsystem can be made more reliable.
Here we consider the problem formulation for a common and widely applicable case. Let there be n subsystems SSi, i = 1,..,n, each with reliability Ri and cost Ci. Let the cost Ci be a function of the reliability given by fi( Ri). Let Cs and Rs represent the total system cost and the overall reliability and RST be the specified target reliability. If all
reliability. Often the number of copies is constrained between a minimum (often one) and a maximum number because of implementation issues.
Note that in the first case, both cost and the reliability can be varied continuously, where as in the other two cases, the choices are discreet. In the first case, we can define a continuous cost function. In the second case too, the market forces may impose a cost function. In the third case, the subsystem may be modeled as a parallel or k-out-ofn system for reliability evaluation, provided the failures are statistically independent. A number of publications on reliability allocation consider only the third case, where the optimization problem becomes an integer optimization problem. It becomes a 0-1 optimization problem when choices are discreet and a component from a given list of candidates is either used or not used [majety99].
Abstract---A system is generally designed as an assembly of subsystems, each with its own reliability attributes. The overall system reliability is a function of the subsystem reliability metrics. The cost of the system is the sum of the costs for all the subsystems. This article examines possible approaches to allocate the reliability values such that the total cost is minimized. The problem is very general and is applicable for mechanical systems, electrical systems, computer hardware and software. The problem involves expressing the system reliability in terms of the subsystem reliability values and specifying the cost function for each subsystem which allows setting up an optimization problem. Software reliability allocation is examined as a detailed example. The article also examines some preliminary apportionment approaches that have been proposed. In some cases, it is possible to use exact optimization methods. In general, a complex case will require use of iterative approaches.
n
n
Minimize
Cs = Ci = fi (Ri )
(1)
i=1
i=1
Subject to
RST ≤ Rs
n
≤ ∏ Ration (1) assumes that the cost of interconnecting the subsystems is negligible. An alternative problem would be to maximize Rs while keeping Cs less than the allocated cost budget.
I. INTRODUCTION
Many systems are implemented by using a set of interconnected subsystems. While the architecture of the overall system can often be fixed, individual subsystems may be implemented differently. A designer needs to either achieve the target reliability while minimizing the total cost, or maximize the reliability while using only the available budget. Intuitively, some of the lowest reliability components may need special attention to raise the overall reliability level. Such an optimization problem may arise while designing a complex software or a computer system. Such problems also arise in mechanical or electrical systems. A number of studies since 1960 have examined such problems [kuo00].
In a non-redundant system, all the subsystems are essential, however often an individual subsystem can be made more reliable by using a more costly implementation. This additional cost may represent wider columns in a building or more thorough testing of software. In redundant implementations, higher reliability can sometimes be achieved by using several copies of a subsystems, such that form a parallel or k-out-of-n configuration.
1
the subsystems are essential to the system and if their failures are statistically independent, the system can be modeled as a series system. The cost minimization problem can be stated as:
The next section consider the problem formulation, followed by approaches used for setting up an optimization problem. As an example, software reliability allocation is examined in detail with two numerical illustrations. The last section considers reliability allocation in complex systems.