Reliability formula &; limit law of the failure time of “<Emphasis Type="Italic">m</Emphasis>-consecutive-<Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis>:<Emphasis Type="Italic">F</Emphasis> system”

An “m-consecutive-k-out-of-n:F system” consists of n components ordered on a line; the system fails if and only if there are at least m nonoverlapping runs of k consecutive failed components. In this paper, we give a recursive formula to compute the reliability of such a system. Thereafter, we state two asymptotic results concerning the failure time Z _n of the system. The first result concerns a limit theorem for Z _n when the failure times of components are not necessarily with identical failure distributions. In the second one, we prove that, for an arbitrary common failure distribution of components, the limit system failure distribution is always of the Poisson class.      

Reliability evaluation of multi-state consecutive k-out-of-r-from-n: G system

This paper proposes a model that generalizes the linear consecutive k-out-of-r-from-n: G system to multi-state case. In this model the system consists of n linearly ordered multi-state components. Both the system and its components can have different states: from complete failure up to perfect functioning. The system is in state j or above if and only if at least k_j components out of r consecutive are in state j or above. An algorithm is provided for evaluating reliability of a special case of multi-state consecutive k-out-of-r-from-n: G system. The algorithm is based on the application of the total probability theorem and on the application of a special case taken from the [Jinsheng Huang, Ming J. Zuo, Member IEEE and Yanhong Wu, Generalized multi-state k-out-of-n: G system, IEEE Trans. Reliab. 49(1) (2000) 105–111.]. Also numerical results of the formerly published test examples and new examples are given.     

Bounds for increasing multi-state consecutive k-out-of-r-from-n: F system with equal components probabilities

The consecutive k-out-of-r-from-n: F system was generalized to multi-state case. This system consists of n linearly ordered components which are at state below j if and only if at least k_j components out of any r consecutive are in state below j. In this paper we suggest bounds of increasing multi-state consecutive-k-out-of-r-from-n: F system (k₁ ? k₂ ? ? ? k_M) by applying second order Boole–Bonferroni bounds and applying Hunter–Worsley upper bound. Also numerical results are given. The programs in V.B.6 of the algorithms are available upon request from the authors.     

On the multi-phase M/G/1 queueing system with random feedback

In this paper we consider an M/G/1 queue with k phases of heterogeneous services and random feedback, where the arrival is Poisson and service times has general distribution. After the completion of the i-th phase, with probability θ _i the (i + 1)-th phase starts, with probability p _i the customer feedback to the tail of the queue and with probability 1 − θ _i − p _i  = q _i departs the system if service be successful, for i = 1, 2 , . . . , k. Finally in kth phase with probability p _k feedback to the tail of the queue and with probability 1 − p _k departs the system. We derive the steady-state equations, and PGF’s of the system is obtained. By using them the mean queue size at departure epoch is obtained.     

Consecutive k-out-of-n systems with maintenance

A consective k-out-of-n system consists of n linearly or cycliccally ordered components such that the system fails if and only if at least k consecutive components fail. In this paper we consider a maintained system where each component is repaired independently of the others according to an exponential distribution. Assuming general lifetime distributions for system's components we prove a limit theorem for the time to first failure of both linear and circular systems.     

Nonparametric inference for sequential <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> systems

The k-out-of-n model is commonly used in reliability theory. In this model the failure of any component of the system does not influence the components still at work. Sequential k-out-of-n systems have been introduced as an extension of k-out-of-n systems where the failure of some component of the system may influence the remaining ones. We consider nonparametric estimation of the cumulative hazard function, the reliability function and the quantile function of sequential k-out-of-n systems. Furthermore, nonparametric hypothesis testing for sequential k-out-of-n-systems is examined. We make use of counting processes to show strong consistency and weak convergence of the estimators and to derive the asymptotic distribution of the test statistics.     

Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations

Commonly studied models of the consecutive-k-out-of-n: F repairable systems in the existing literatures were considering the systems which had one repairman without vacation or infinite repairmen without vacations. In addition to those models, multiple repairmen without vacations are studied occasionally. However, technical personnel are very short in some fields. Some failed components cannot be repaired in time. This paper deals with the phenomenon of waiting for repair by supposing R repairmen with multiple vacations in the system. Using the pairs (i, |j|), the factor that the R repairmen taking multiple vacations was embedded into the classical C(k, n: F) system. Reliability indexes are presented. Finally, the Runge–Kutta method was used to a special case, and the experimental results demonstrate the necessity and validity of the new model.     

Poisson approximations for 2-dimensional patterns

LetX=(X _ij) _n×n be a random matrix whose elements are independent Bernoulli random variables, taking the values 0 and 1 with probabilityq _ij andp _ij (p _ij+q _ij=1) respectively. Upper and lower bounds for the probabilities ofm non-overlapping occurrences of a square submatrix with all its elements being equal to 1, are obtained. Some Poisson convergence theorems are established forn . Numerical results indicate that the proposed bounds perform very well, even for moderate and small values ofn.This work is supported in part by the Natural Science and Engineering Research Council of Canada under Grant NSERC A-9216.     

Successions in integer partitions

A partition of an integer n is a representation n=a ₁+a ₂+⋅⋅⋅+a _k , with integer parts 1≤a ₁≤a ₂≤…≤a _k . For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a _i+1−a _i =p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n. This material is based upon work supported by the National Research Foundation under grant number 2053740.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.     

Two Paths Joining Given Vertices in 2k-Edge-Connected Graphs

Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {s_i, t_i}⊆X_i⊆V(G) (i=1,2) and X₁∩X₂=∅. We here prove that if k is even and e(X_i)≤2k−1 (i=1,2), then there exist paths P₁ and P₂ such that P_i joins s_i and t_i, V(P_i)⊆X_i (i=1,2) and G − E(P₁∪P₂) is (k−2)-edge-connected (for odd k, if e(X₁)≤2k−2 and e(X₂)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing given edges.     

Reliability and covariance estimation of weighted k-out-of-n multi-state systems

In the literature of reliability engineering, reliability of the weighted k-out-of-n system can be calculated using component reliability based on the structure function. The calculation usually assumes that the true component reliability is completely known. However, this is not the case in practical applications. Instead, component reliability has to be estimated using empirical sample data. Uncertainty arises during this estimation process and propagates to the system level. This paper studies the propagation mechanism of estimation uncertainty through the universal generating function method. Equations of the complete solution including the unbiased system reliability estimator and the corresponding unbiased covariance estimator are derived. This is a unified approach. It can be applied to weighted k-out-of-n systems with multi-state components, to weighted k-out-of-n systems with binary components, and to simple series and parallel systems. It may also serve as building blocks to derive estimators of system reliability and uncertainty measures for more complicated systems.     

Algorithm for a general discrete k-out-of-n: G system subject to several types of failure with an indefinite number of repairpersons

A discrete k-out-of-n: G system with multi-state components is modelled by means of block-structured Markov chains. An indefinite number of repairpersons are assumed and PH distributions for the lifetime of the units and for the repair time are considered. The units can undergo two types of failures, repairable or non-repairable. The repairability of the failure can depend on the time elapsed up to failure. The system is modelled and the stationary distribution is built by using matrix analytic methods. Several performance measures of interest, such as the conditional probability of failure for the units and for the system, are built into the transient and stationary regimes. Rewards are included in the model. All results are shown in a matrix algorithmic form and are implemented computationally with Matlab. A numerical example of an optimization problem shows the versatility of the model.     

Separation properties forMW-fractals

Corresponding to the irreducible 0–1 matrix (a _ij ) _n×n , take similitude contraction mappingsϕ _ij for eacha _ij =1, ina _ij =1, in R ^d with ratio 0<r _ij <1. There are unique nonempty compact setsF ₁,…,F _n satisfying for each1≤i≤n, F _i. We prove that open set condition holds if and only ifF _i is ans-set for some1≤i≤n, wheres is such that the spectral radius of matrix (r _ij ³ ) _{n x n} is 1. Partly supported by Natural Science Foundation of China, and partly by Natural Science Foundation of Hubei Province     

On a conjecture concerningk-Hamilton-nice sequences

In this paper, we prove that a non-negative rational number sequence (a ₁,a ₂, ...,a _k+1) isk-Hamilton-nice, if (1)a _k+12, and (2) _j ^=1/h (i _j –1)k–1 implies for arbitraryi ₁,i ₂,...i _h {1,2,... ,k}. This result was conjectured by Guantao Chen and R.H. Schelp, and it generalizes several well-known sufficient conditions for graphs to be Hamiltonian.This project is supported by the National Natural Science Foundation of China.     

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

Reliability Approximation for Markov Chain Imbeddable Systems

In the present article, a simple method is developed for approximating the reliability of Markov chain imbeddable systems. The approximating formula reduces the problem to the reliability assessment of smaller systems with structure similar to the original systems. Two specific reliability structures which have attracted considerable research interest recently (r-within-consecutive-k-out-of-n system and two dimensional r-within-k ₁ × k ₂-out-of-n ₁ × n ₂ system) are studied by the new approach and numerical calculations are carried out, which reveal the high quality of our approximations. Several possible extensions and generalizations are also presented in brief.     

Graphic Sequences with a Realization Containing a Union of Cliques

An integer sequence π is said to be graphic if it is the degree sequence of some simple graph G. In this case we say that G is a realization of π. Given a graph H, and a graphic sequence π we say that π is potentially H-graphic if there is some realization of π that contains H as a subgraph. We define σ(H,n) to be the minimum even integer such that every graphic sequence with sum at least σ(H,n) is potentially H-graphic. In this paper, we determine σ(H,n) for the graph H = K_m1∪ K_m2∪...∪ K_mk when n is a sufficiently large integer. This is accomplished by determining σ(K_j + kK₂,n) where j and k are arbitrary positive integers, and considering the case where j = m − 2k and m = ∑ m_i.     

Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions

A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed components. Some examples related to some lifetime distribution functions are given. We present a numerical example for evaluating the relationship between the mean residual life of the k-out-of-n: G system and that of its components.     

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000