JKAU: Eng. Sci., Vol. 27 No. 2, pp: 15 - 31 (1437 A.H./ 2016 A.D.) Doi: 10.4197/Eng. 27-2.2
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems Mahmoud Ali. M. Rushdi1, Omar M. Ba-Rukab2 and Ali M. Rushdi3 1
Department of Electronics and Communications Engineering, Faculty of Engineering, Cairo University, Giza 12613, Egypt, 2Department of Information Technology, Faculty of Computing and Information Technology, King Abdulaziz University, P.O.Box 344, Rabigh, 21911, and 3Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, 21589, Saudi Arabia
[email protected] Abstract. The probabilities of exactly and at least k successes out of n trials represent respectively the probability mass function (pmf) and the Complementary Cumulative Distribution Function (CCDF) of the generalized Binomial distribution. These two probabilities obey the same two-dimensional recursive relations (or signal-flow-graph structure), albeit with different boundary conditions (sources) and almost identical regions of validity that cover approximately an octant of the k-n plane. Related to these is the failure frequency of a non-repairable k-out-of-n system, which equals the probability density function (pdf) for the time to failure of this system. This failure frequency is represented herein by a two-dimensional signal flow graph (SFG) whose sources emanate from one of the SFGs of the aforementioned probabilities. Therefore, the recursive relations of this failure frequency take the shape of a threedimensional loopless or acyclic graph. Closed-form symbolic expressions for this failure frequency are obtained via Mason gain formula, i.e., simply via path enumeration. Such expressions are also deduced herein via a two-dimensional strong version of mathematical induction. An offshoot contribution of this paper is to derive expressions for the k-out-of-n failure frequency via Markov-chain modeling, and to demonstrate that the resulting expressions are equivalent to those obtained via recursive relations. Keywords: Recursive relations, Signal flow graph, Mathematical induction, Multi-dimensions, Failure frequency, k-out-of-n system.
1. Introduction
where is the Cumulative Distribution Function (CDF) of the time to failure T, which turns out to be the same as the component or system unreliability . The corresponding probability density function (pdf) of the time to failure is given by[1]
The reliability R(t) of a component or a system is the probability that it will adequately perform its required function for a semi-closed period of time (0, t] under specified working conditions. The component or system is assumed to be initially good (i. e., R(0) = 1.0), i.e., R(t) is actually the conditional probability of uninterrupted success during the time interval (0, t] given success at the initial instance of time. The reliability concept is related to that of the time to failure T, which is a random variable such that[1]
,
(2a)
and it is also known as the failure frequency of the component or the system. When is normalized by , it is called the failure rate r(t) or the hazard rate h(t), namely
(1)
(2b)
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The field of system reliability deals with the relation between the reliability of a system and the reliabilities of its components. At the core of this field is the concept of the k-out-ofn:G system (k-out-of-n system, for short) which is a system of n components that functions if at least k out of its components function[1-13]. The k-out-of-n system has many attractive features, and it is a subclass of many other important systems[4, 7, 13-18]. This paper utilizes several mathematical entities to establish several forms for the relation between the failure frequency of a kout-of-n system and the failure frequencies and reliabilities of its components. The current study invokes the derivaion of appropiate recursive relations, the manipulation of pertinent signal flow graphs, the proof of symbolic closed-form expressions via strong mathematical induction, and a verification of the results obtained via discrete-state continuous-time Markov chains. The organization of the remainder of this paper is as follows: Section 2 lists the assumptions and notation used throughout the paper. Both assumptions and notation are standard ones typically used in the reliability literature (see, e. g., Rushdi[13]). Section 3 reviews the two-dimensional recursive relations governing the probabilities of exactly and at least k successes out of n trials, and visualizes these relations via two-dimensional signal flow graphs (SFGs). Section 4 derives the recursive relations governing the failure frequency of a non-repairable k-out-of-n system, and represents it by a two-dimensional signal flow graph whose sources originate from the SFGs of the aforementioned probabilities, or equivalently by a threedimensional SFG. Closed-form symbolic expressions for this failure frequency are obtained first via Mason gain formula for loopless SFGs with small values of k and n, and then generalized for any values of k and n
via a two-dimensional strong version of mathematical induction. Section 5 verifies the above results using Markov-chain modeling. Section 6 concludes the paper and proposes some future work. 2. Assumptions and Notation 2.1 Assumptions Both the system and each of its components are of two states, i.e., either good (working) or failed (defective or malfunctioning). Components identical.
are
not
necessarily
Component independent.
states
are
statistically
The system is a mission-type one, i.e., without repair. The system is a k-out-of-n:G system, referred to herein as a k-out-of-n system. Components have constant failure rates (CFR) , i.e., they have reliabilities , and failure frequencies , . System metrics such as , , , and have an implicit time dependence through the time dependence of . This dependence might be made explicit by referring to these metrics as , , and , respectively. 2.2 Notation n = number of system components, n 1. = success of component i = indicator variable for successful operation of component i = a switching random variable that takes only one of the two discrete values 0 and 1; ( =1
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
17
= 0 iff
U(p) = 1.0 – R(p). Both R(p) and U(p) take real values in the closed real interval [0.0,1.0].
= failure of component i = indicator variable for unsuccessful operation of component i; ( = 0 iff component i is good, while = 1 iff component i is failed). The success and the failure are complementary variables
R(k, n, p) = reliability of a k-out-of-n:G system of component reliabilities p, 0 k (n+1).
iff component i is good, while component i is failed).
= a vector of n elements representing the component successes =[
T
...
].
Pr (…) = probability of the event (…). E (...) = expectation of the random variable (...). pi, qi = reliability and unreliability of component i; Both pi and qi are real values in the closed real interval [0.0,1.0].
R(k, n, p) = the value of R(k, n, p) when component reliabilities are all equal to a common value p, f(k, n, p) = failure frequency of a k-out-of-n:G system of component reliabilities p, 0 k (n+1). r(k, n, p) = failure rate (hazard rate) of a k-out-of-n:G system of component reliabilities p = f(k, n, p) /R(k, n, p), 0 k (n+1), i.e., it is the failure frequency normalized by reliability. c(k, n) =
the binomial (combinatorial)
pi = Pr ( Xi = 1 ) = E ( Xi ) = 1.0 – qi = 1.0 – E( ).
coefficient = the number of ways of choosing k objects from a set of n objects, when repetition is not allowed and order does not matter. Binomial coefficients satisfy al-Karkhi's (Pascal's) identity
p = a vector of n elements representing the component reliabilities
c(k, n) = c(k, n–1) + c(k–1, n–1), 0 < k < n, (3a)
= [p1 p2... pm-1 pm pm+1... pn]T. q = a vector of n elements representing the component unreliabilities = 1 – p, where 1 is an n-tuple of 1’s. p/pm = a vector of (n-1) elements obtained by omitting the mth element of vector p = [p1 p2... pm-1 pm+1... pn]T. = a vector of elements representing the first component reliabilities = [p1 p2...
]T.
R(p), U(p) = reliability and unreliability of the system. Both R(p) and U(p) are real values in the closed real interval [0.0,1.0]. R(p) = Pr ( S(X) = 1 ) = E (S(X )).
Together with the boundary conditions c(k, n) = 1, (k = 0 or k = n) and n 0. (3b) N = the set {1, 2, 3, …, n}, n 1. K = a subset of N that has cardinality k . There are c(k, n) such sets starting at {1, 2, 3,…, k} and ending at{(n–k +1), (n–k +2),…, n}. We use K also to denote a Markov-chain state in which only components with indexes in K are good. This is called a critical state of a k-out-of-n:G system. |Y| = cardinality of the finite set Y = the number of elements in the set Y. 3. Two-Dimensional Recursion Relations This section reviews the twodimensional recursive-relations governing the two related probabilities and
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[4, 5, 7, 8, 13]
. The reliability of a k-out-of- n:G system is the probability of at least k sucesses in n trials. It is governed by the two-dimensional recursive relation (4a) together with the boundary conditions ,
(4b) (4c)
while the probability of exactly k sucesses in n trials is governed by the two-dimensional recursive relation , (5a) together with the boundary conditions (5b) (5c) (5d) The two probabilities are related by the relations
and
(6a) (6b) This means that is obtained from via a summation operation, while is obtained from via a differencing operation, in conformity with the fact that is complementary to the Cumulative Distribution Function (CDF) of the generalized binomial distribution, while equals the probability mass function (pmf) of the generalized binomial distribution [13, 19, 20] . The probabilities and are usually mistaken to be the same,
because they obey the similar sets of recursive relations (4a) and (5a), but these relations extend over non-identical regions of validity and have definitely different boundary conditions. The set of relations (4) and (5) were first derived by Rushdi[4] by utilizing properties of symmetric switching (two-valued Boolean) functions. Rushdi and Rushdi[21] provided a much simpler (albeit admittedly less intuitionistic) novel proof of (4) via the Total Probability Theorem[1, 13]. In this proof (and in an analogous proof for (5)), the underlying mutually-exclusive and exhaustive events are the two complementary events: {component n is not working} and {component n is working}, with respective probabilities and . A k-out-of-n event becomes a k-out-of(n ) event under the condition that component n is not working, and becomes a (k )-out-of-(n ) event under the condition that component n is working. This is true whether the concerned events are described by the adverbs "at least" or "exactly". Figures 1 and 2 show regular Mason signal flow graphs (SFGs)[22-27] that illustrate the computation of and , [4, 7, 8, 11, 13, 16-20] respectively . Note that in column i each diagonal arrow has a transmittance equal to pi, while each horizontal arrow carries a transmittance equal to its complement qi = (1.0 – pi). There are two types of nodes: (a) Source nodes of known values which are either black or white. A black node has a value of 1.0, while a white node has a value of 0.0, and (b) Non-source nodes drawn as shaded ones, which include (at least) one sink node whose value is the final result sought. Figure 1 first appeared in Rushdi[4], and can be viewed in the Boolean domain as an SFG for a symmetric switching function representing the success function of a k-out-of-n system. In such a graph, algebraic multiplication and addition are
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
replaced by their logical counterparts (ANDing and ORing), and the graph can be identified to be an early precursor of a Reduced Ordered Binary Decision Diagram (ROBDD), which is currently known to be the state-of-the art data structure for encoding and manipulating switching functions[13, 21, 28-32]. Moreover, Fig. 1 has certain similarities and minor dissimilarities with al-Karkhi's Triangle (Pascal's Triangle) that reflect the similarities and dissimilarities of the recursive equations (4a) and the boundary conditions (4c&d) with their counterparts (3a) and (3b), respectively[13]. 4. Three–Dimensional Recursive Relations According to (2a) and (4), the negative of the time-derivative of system reliability is
,
19
Equations (7) constitute a set of recursive relations and boundary coditions for the failure frequency of a non-repairable k-outof-n system. They first appeared in Amari, et al. [12], albeit with one erroneous non-zero boundary condition for therein. Figure 3 shows a Mason signal flow graph (SFG) that illustrates the computation of via (7a)-(7c), together with (4). The failure frequency is represented herein by a two-dimensional SFG that does not have sources of its own (All boundary nodes in (7b) and (7c) are zeroes). However, has external sources originating from the SFG for . In a sense, the overall SFG in Fig. 3 can be viewed as a threedimensional one. Its nodes are depicted as squares in a lower plane, while its nodes are drawn as circles in an upper plane. Equation (7a) can be rewritten as,
or, equivalently (Using ,(7d) or, equivalently, with an aid from (6b) as
and
(7e)
,(7a) ,
(7b) (7c)
Figure 4 shows a Mason signal flow graph (SFG) that illustrates the computation of via (7e), (7b), and (7c), together with (5). Again, the failure frequency is represented herein by a two-dimensional SFG that does not have sources of its own, but has an external source originating from the SFG
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for . In a sense, The overall SFG in Fig. 4 is simpler than the one in Fig. 3, since it has a single source node at and it has fewer arrows from its upper plane to its lower one. Again, Fig. 4 can be viewed as a three-dimensional SFG. Its nodes are still depicted as squares in a lower plane, while its nodes are drawn as triangles in an upper plane. All SFGs in Fig. 1-4 are loopless. Closed-form symbolic expressions for pertinent quantities could be obtained via Mason gain formula, simply via path enumeration, and evaluation of the gain or transmittance of each path, which is the product of transmittances on the edges traversed by the path. The following equations represent computation of via (7a)(7c), together with (4) for . Identical results are obtained if (5) is used instead of (4).
,
,
The above results have beautiful symmetries and suggest that the failure frequency of a k-out-of-n:G system is a weighted sum of the failure frequencies of the n components, where the failure frequency of a particular component is weighted by the probability that exactly (k 1) out of the other (n 1) components are successful. On the other hand, equation (7e) indicates that is a straight-line relation in . Hence, by symmetry, it is a multi-affine relation in . Since it is a homogenous relation in these variables (it is 0 if each of them is 0), then we guess it must be a weighted sum of them, with the weight of in this weighted sum being . Once we have made this conjecture, and hoping that it is indeed true, we formalize it as the follwing theorem, which we prove via mathematical induction. Despite the existence of many forms of mathematical induction and its intimate relation to both recursion and to recursive relations[33-50], we employ herein the strong version of mathematical induction as it suits the present two-dimensional situation. Theorem 1 (
) ,
(7b)
(8) (7c) In a rigorous proof of this two-dimensional theorem { } by the
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
strong version of mathematical induction, we prove The
base or boundary cases { } and { }, which are obviously true, thanks to the original bounday conditions (7b) and (7c).
21
according to (7e), (11a), (10b), and (5a), we have for , LHS of (12) =
The inductive case, which is suggested by the nature of the recursive relation, pictorially revealed by the SFG of Fig. 4, namely (9) where
is given by ,
(10a) + , (10b)
while
is given by . (11a) (11b)
and
is given by
,
(12)
Note that does not cover the case { } and hence does not need a boundary condition similar to (7c). Likewise, does not entail the case { } and therefore lacks a boundary condition analogous to (7b). Similarlry, neither (7b) nor (7c) is included in which invokes neither { } nor { }. Now,
The proof is the same for , but and all take zero values. It is also the same for , though and all take zero values. As a corrolary of Theorem 1 above concernining failure frequencies , the summation of such frequencies over k is given by: Theorem 2 (
) , n 1.
(13)
In fact, thanks to (8), the LHS of (13) for n 1 is given by
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5. Markov-Chain Interpretation
which equals the RHS of (13), thanks to the fact that the probabilities , , are those of a set of mutually exclusive and exaustive events and hence add to one[19, 20]. Recall that is a vector of elements. Alternatively, we prove Theorem 2 via the weak version of mathematical induction by proving The base or boundary case which states that and is obviously true,
The inductive case, namely ,
1,
(14) ,
1, (15)
Failure frequency could be assessed by drawing a continuos-time discrete-state Markov chain[1, 51-56] that models the system behaviour. One has to identify critical states, i.e., border states in which the system is successul but about to make an immediate direct transition to a failure state. The failure frequency is a summation over all critical states of the probability of a critical state multiplied by the sum of transition rates emanating from this state to a failure state. For a k-out-of-n system, there are critical states . A critical state is one in which (n k) components are already failed, and the system behaves as a series system of the k remaining working components, and hence it fails when any of these k components fails. Therefore, terms, since each of the
, 1, (16) Now, making use of (15), (7b), (7c), and the fact that the probabilities , , add to 1, we obtain
consists of k critical states has
k transition rates to failure states. The number of terms given by Theorem 1 is n which is precisely the same nubmer. failure frequency is given by:
The
(17)
…
of (16).
As a concrete example, Fig. 5 shows a Markov chain for a system of three components. The chain has eight states each labelled by the set of components still working. The chain is drawn over a torus-like Karnaugh-map layout[57-60]. The map region labelled Xm, constitues half of the map in which component m is working, and consists exactly of four states. The transition rate across the boundary of this region is the failure rate of component m, since it indicates a transition to the other half of the
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
map in which component m is not working. Figure 6 reproduces three copies of the Markov chain in Fig. 5 to identify the critical and failure states for three systems, namely, kout-of-3 systems . Realistically, failure states in each case should be combined into a single state. Table 1 reports the failure frequencies of these systems, computed via (17), as could be visualized from Fig. 6. For comparison puposes, Table 1 contains also the equivalent failure frequencies obtained via Theorem 1 (Equation 8). 6. Conclusions and Future Work Most of the literature on failure freuqency pertains to repairable systems, and uses availability (rather than reliability) as a starting point[61-71]. The current paper delas with a prominent type of non-repairable coherent systems, viz, the k-out-of-n system. It is based on two related probabilities, namely the probability of exactly k successes out of n trials, and that of at least k successes out of n trials, which is the reliability of a k-out-of-n system. The paper utilizes a variety of mathematical tools including recurssive relations, mathematical induction, and signal flow graphs. The paper offers five equivalent methods for evaluating the failure frequency of a k-out-of-n system. These are: Solution of a set of two-dimensional recurssive relations supplemented with an appropriate set of boundary conditions. Application of Mason gain formula to a signal flow graph representing failure frequency nodes, with a multitude of unit source reliability nodes. These nodes are infinite in number, but only a finite number (at most n) of them are needed.
23
Application of Mason gain formula to a signal flow grpah depicting failure frequency nodes with a single unit source node representing the probability of exactly 0 successes out of 0 trials among nodes representing the probability of exactly k successes out of n trials. Closed-form expressions obitaned via multi-dimensional mathematical induction. Alternative (albeit equivalent) closedform expression obtained via Markove-chain modelling. The concepts of recursive relations and mathematical induction have been seen to be intimately related; A recursive relation expresses a super-version of a certain entity in terms of some sub-versions of the same entity, while in a proof by mathematical induction, one or more sub-versions of a specific theorem is used to derive another version, which might be called a super-version. Both concepts can benefit from the insight and utility provided by signal flow graphs. A future immediate extension of the present work is to cover systems other than the k-out-of-n system. These include the much wider classes of coherent systems[6, 10] and threshold systems[16, 17], as well as systems closely related to the k-out-of-n system, such as the k-to- -out-of-n system[14, 15] and the consecutive-k-out-of-n system[21]. A more ambitious work entails an exploration of the general relation between failure frequency, probabilities of critical or border states, and the Fréchet derivative of the system reliability w. r. t. the component reliabilities
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Fig. 1. A Mason signal flow graph that illustrates the computation of R(k,n,p).
Fig. 2. A Mason signal flow graph that illustrates the computation of E(k,n,p).
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
Fig. 3. A Mason signal flow graph that illustrates the computation of f(k,n,p), with the aid of computations of R(k,n,p).
Fig. 4. A Mason signal flow graph that illustrates the computation of f(k,n,p), with the aid of computations of E(k,n,p).
25
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Fig. 5. A Markov chain for a 3-component system of non-identical components. The chain is drawn on a torus-like Karnaughmap layout. A State is labeled by the set of working components.
Fig. 6. Identification of critical and failure states for three k-out-of-3 systems, 1≤k≤3, deduced from Fig. 5.
Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
Table 1. The failure frequency
27
by two methods.
Recurssive-Relation Solution
Markov-Chain Modelling
=
=
=
Methodology Oriented Treatment, Elsevier Science Publishers, Amsterdam, The Netherlands (1992).
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Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
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Multi-Dimensional Recursion Relations and Mathematical Induction Techniques: The Case of Failure Frequency of k-out-of-n Systems
عالقات المعاودة وأساليب االستقراء الرياضي متعددة األبعاد :حالة تردد الفشل لنظم ك-من-بين-ن
3
محمود علي محمد رشدي ،1و عمر محمد باركب ،2و علي محمد رشدي
1قسم هندسة اإللكترونيات واالتصاالت الكهربائية ،كلية الهندسة ،جامعة القاهرة ،الجيزة ،12613 ،جمهورية مصر العربية،
و 2قسم
تقنية المعلومات ،كلية الحاسبات وتقنية المعلومات ،جامعة الملك عبد العزيز ،ص.ب ،344.رابغ ،21911و 3قسم الهندسة الكهربائية وهندسة الحاسبات ،كلية الهندسة ،جامعة الملك عبد العزيز ،ص.ب ،80204.جدة ،21589المملكة العربية السعودية
[email protected]
المستخلص .يمثل احتمال (ك) بالضبط من النجاحات خالل (ن) من المحاوالت دالة الكتلة االحتمالية للتوزيع ذي الحدين المعمم ،أما
احتمال ك على األقل من هذه النجاحات فهو مكمل دالة التوزيع التراكمي لهذا التوزيع .يحقق هذان االحتماالن نفس العالقات المعاودة ذات البعدين (أو نفس بنية رسم سريان اإلشارة) ،وان كان ذلك يتم بشروط حدية (أو منابع) مختلفة ولمنطقتي صالحية شبه متطابقتين تغطي كل منهما تقريبا ثمن مستوى اإلحداثيين (ك-ن) .يرتبط باالحتمالين المذكورين تردد الفشل لنظام (ك-من-بين-ن) غير قابل لإلصالح ،وهذا التردد يساوي دالة الكثافة االحتمالية للزمن حتى الفشل لهذا النظامُ .يمثَل تردد الفشل هذا هنا برسم سريان لإلشارة (ر س ش) ذي بعدين تنبعث منابعه من أحد رسمي سريان اإلشارة لالحتمالين سالفي الذكر .ولذلك تأخذ العالقات المعاودة لتردد الفشل هذا شكل رسم عديم الحلقات ثالثي األبعاد .يتم الحصول على تعبيرات رمزية منغلقة الصيغة بواسطة قانون ماسون للكسب ،أي ببساطة عن طريق سرد المسارات .كما يجري استنباط هذه التعبيرات أيضا بواسطة صورة قوية لالستقراء الرياضي ذات بعدين .وثمة إسهام إضافي لورقة البحث هذه هو اشتقاق تعبيرات لتردد الفشل لنظام (ك-من-بين-ن) بواسطة النمذجة بسالسل ماركوف ،ثم إظهار أن التعبيرات الناتجة مكافئة لتلك المستنبطة بواسطة العالقات المعاود. كلمات مفتاحية :العالقات المعاودة ،رسم سريان اإلشارة ،االستقراء الرياضي ،األبعاد المتعددة ،تردد الفشل ،نظام ك-من-بين-ن.
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Mahmoud Ali. M. Rushdi et al.
