Cost Elasticities of Reliability and MTTF for k-out-of-n Systems

The application of the concept of cost elasticity of reliability was extended from the parallel system to the partially redundant or k-out-of-n:G(F) system (or k-out-of-n system, for short). An expression for the cost elasticity of reliability was derived for a general k-out-of-n system. The expression yielded acceptable results for a wide range of values for k, n, and component reliability. For systems of practical interest characterized by good components, the expression became highly susceptible to round-off errors, and catastrophic cancellations took place. These numerical problems seemed unavoidable as they were inherently associated with the definition of the cost-elasticity-ofreliability metric itself. We introduced another metric, the cost elasticity of the Mean Time To Failure (MTTF), which measures the relative change in the life expectancy that can be obtained for a given relative change in cost. We believe the cost-elasticity-of-MTTF metric is a more tangible and a more cumulative measure than the cost-elasticity-of-reliability metric. We derived a very simple expression for the cost elasticity of MTTF for a k-out-of-n system and showed that it is a function of only k and n, i.e. it is independent of component characteristics such as component failure rate or component reliability. This expression is insensitive to round-off errors since it is a purely additive formula. We provided charts for the cost elasticity of MTTF that can be used to assess the cost incurred in achieving a certain life expectancy for a k-out-of-n system. These charts can be used with any coherent system, since the MTTF for a coherent system can be approximated by that of a k-out-of-n system.


INTRODUCTION
An important goal for reliability engineering is to achieve cost minimization [1] . However, this goal has rarely been achieved, primarily because of the lack of suitable mathematical models or metrics [2] . A recently introduced metric that captures the value of reliability from a financial viewpoint is the cost elasticity of reliability [3] , defined as (1) This metric measures the relative change in reliability R that can be obtained for a given relative change in cost C. As its name indicates, this metric mimics a wellknown material constant, viz., the modulus of elasticity which relates an applied stress to the resulting strain or relative change in length [4] . However, the metric C R, ∈ is more analogous, from a cause-effect point of view, to the price elasticity of demand or supply, a concept well known in microeconomics [5] .
The new metric C R, ∈ was studied in [3] in the case of a parallel system. We extend this study by applying this new metric to a k-out-of-n:G(F) system. The k-outof-n:G(F) system is a system of n components that functions (fails) if at least k out of its n components function (fail). Situations in which this system serves as a useful model are frequently encountered in practice [6] . The k-out-of-n system plays a central role for the general class of coherent systems, as it can be used to approximate the reliability of such systems [7] . While virtually all nontrivial network reliability problems are known to be NP-hard for general networks, the regular structure of the k-out-of-n system allows the existence of efficient algorithms for its reliability analysis that are of quadratic-time linear-space complexity in the worst case [6] . The k-out-of-n:G system covers many interesting systems as special cases. These include the perfectly reliable system (k = 0), the parallel system (k = 1), the voting or N-modular redundancy (NMR) system ( , the fail-safe system (k=n-1), the series system (k=n), and the totally unreliable system (k=n+1). For 1<k<n the k-out-of-n system is sometimes called a partially-redundant system [6] , as it lies somewhere between the extreme cases of the (nonredundant) series system and the (fully-redundant) parallel system. The k-out-of-n:G system and the k-outof-n:F system are mirror images of each other; their successes are dual switching functions. The k-out-ofn:G system is exactly equivalent to the (n-k+1)-out-ofn:F system [6] .

METHODOLOGY
The methodology adopted combines analysis and simulation. We derive an expression for C R, ∈ for a parallel system based on the "continuous" limit ∆n → 0. We also derive an expression for C R, ∈ for a general k-out-of-n:G system, but since it cannot be based on the continuous limit, we base it on what we call the best discrete increment ∆n = 1, since this is the nearest possible increment to the continuous limit. We also introduce another metric, viz., C T , ∈ or the cost elasticity of the MTTF, which we believe is a more tangible and a more accumulative measure than the C R, ∈ metric. We derive a very simple expression for  Cost-reliability characterization for a parallel system: The unreliability of a parallel system with n identical but independent components of component reliability R 0 is given by: and hence, the reliability is given by: Following Saleh et al. [3] , we let the cost of a single component be C 0 , and assume that the cost of n components, C, scales linearly with the number of components, i.e., The change ∆R in reliability due to a change ∆n in the number of components is: The change in cost ∆C due to a change ∆n in the number of components is Finally, the cost elasticity of reliability is given by ∈ is based on the use of i.e., it is obtained for the "continuous" limit ∆n → 0. Figure 1 shows a cost-reliability characterization of a parallel structure with n redundant components of a relatively good component reliability R 0 = 0.9. Figure  1(a) is a plot of unreliability (1-R) versus cost (expressed in units of C 0 ), which is essentially (1-R), versus n, while Fig. 1(b) depicts cost elasticity of the structure's reliability versus n. Note that the system unreliability curve becomes quickly indistinguishable from the horizontal axis of value 0. Cost-reliability characterization for a k-out-of-n system: The reliability of a k-out-of-n:G system (with independent components of identical reliabilities R 0 ) is given by [6] :   Formula (14) for the cost elasticity of reliability C R, ∈ gives satisfactory results up to R 0 = 0.8 ( Fig.   2(b)) and then starts to exhibit some unacceptable negative values (values of -0 rather than +0), i.e. it exhibits erratic behavior for very small or negligible values of C R, ∈ ( Fig. 3(b)).
We must stress that the erratic behavior obtained is solely due to aggravated cumulative round-off error and is definitely not a result of some error in formulation or programming. Formula (14) gives acceptable and verifiable results for a wide range of values of k, n, and R 0 . However, it fails to assess C R, ∈ properly for systems having good components (i.e., for systems of practical interest). Anyhow, for such systems C R, ∈ diminishes and becomes indistinguishable from zero.
Equation (14) for C R, ∈ is obtained by using the smallest possible nonzero discrete increment ∆n = 1. This increment is the best discrete increment since it is the nearest one to the "continuous" limit ∆n → 0. For a parallel system (k = 1), we have two estimates for C R, ∈ , one based on the continuous limit ∆n → 0 in equation (7) and another based on the best discrete increment ∆n = 1 in equation (14). Our computational experience reveals that there is no significant difference between these two estimates for large n, and small R 0 , i.e. when (14) is not spoiled by accumulated round-off errors.

Cost elasticity of MTTF for a k-out-of-n system:
From cost considerations, MTTF seems to be a more tangible and cumulative measure than reliability itself. Therefore, we introduce the concept of the cost elasticity of the MTTF, which we define as where For a k-out-of-n:G system having components subject to a common constant failure rate (CFR) λ, the component reliability is ). , A simpler and a purely additive expression for T, however, can be obtained form the state diagram for a k-out-of-n:G system [8] . When the system is in state m ≥ k, it has m working components and it behaves exactly as a series system of a failure rate mλ. The system resides in this state for an average time (1/ (mλ)), and hence the MTTF of the system is  1 The cost elasticity we obtain the following tight bounds on (25) Table 1 lists

RESULTS
∈ values in proper-fraction form (exact integer arithmetic) for small k and n values. Figures 5 and 6 represent the cost elasticity of MTTF for a parallel system and for a 50-out-of-n:G system ∈ as a function of k and n for 1 ≤ k ≤ n ≤ 6.   versus the number of components n. Figure 7 represents the cost elasticity of MTTF for a k-out-of-100:G system versus the number of components k required for system success. Table 1 and Figure 7 explain why the fail-safe ((n-1)-out-of-n:G system or 2out-of-n:F system) is so popular. Among redundancy systems it has the best cost for added redundancy since it has the highest C T , ∈ . Of course, the series system (nout-of-n:G system or 1-out-of-n:F system) has an C T , ∈ = (n 2 /(n+1)) that is higher than that of the fail-safe system, but the series system has no redundancy at all and cannot tolerate even a single failure.