A Formula for the Reliability of a $d$-dimensional Consecutive-$k$-out-of-$n$:F System

We derive a formula for the reliability of a $d$-dimensional consecutive-$k$-out-of-$n$:F system. That is, a formula for the probability that an $n_1 \times \ldots \times n_d$ array whose entries are (independently of each other) $0$ with probability $p$ and $1$ with probability $q = 1 - p$ does not include a contiguous $s_1 \times \ldots \times s_d$ subarray whose every entry is $1$.


Introduction
Consider a pipeline punctuated by several pumping stations. Suppose the pumping stations are redundant in that, if one of them fails, then its predecessor will be strong enough to pump the fluid past it to the next pumping station. Perhaps each pump will even be able to compensate for the failure of its successor and its successor's successor, but perhaps not for the failure of the first 3 pumping stations in the chain of its succesors. In this case, the system fails if and only if among the sequence of pumping stations there exists a consecutive run of 3 or more failed pumping stations. This is an example of what is known as a consecutive-k-out-n:F system.
Suppose that such a system having n nodes fails if and only if k consecutive nodes fail, and suppose that any given node in the system works correctly with probability p ∈ [0, 1], independently of the other nodes. Then the probability that any given node fails is q = 1 − p. The Reliability of the system is the probability R(k, n; q) that the system does not fail. Then R(k, n; q) = 1 − P (k, n; q), where P (k, n; q) is the probability that the sequence of n nodes includes a contiguous interval of k or more failed nodes. Here we are using notation as in [DB214].
As noted in [DB214], the concept of the reliability of a consecutive system was introduced to Engineering by Kontoleon in 1980 [K80]. In the following year Chiang and Niu [CN81] discussed some applications of consecutive systems. The concept has been generalised in several directions, for instance to systems deemed to have failed if and only if they include: k consecutive failed components or f failed components [T82]; k consecutive components of which at least r have failed [G86]; at least m non-overlapping runs of k consecutive failed components [P90]. In 2001 Chang, Cui and Hwang published a book on the subject [CCH01]. Four reviews are also available [CFK95], [CH03], [E10] and [T15].
An exact formula for R(k, n; q) first appeared in [D1738]. De Moivre's approach depends on deriving the generating function (see [W2005]) for P (k, n; q). This work as the author knows, no such formulae were known before for d > 1, even in the case d = 2. Indeed, in [KPP93] the authors write "It is very difficult (probably impossible) to derive simple explicit formulas for the reliability of a general 2Dconsecutive-k-out-n:F system." be as in the introduction, and let

Results
(2) P (s 1 , . . . , s d , n 1 , . . . , n d ; where P(A) denotes the power set of the set A, that is, the set of subsets of A, and ∅ denotes the empty set.
Proof. In case p = q = 1 2 , all possible systems occur with equal probability, so the probability measure on the power set of the set of all systems becomes the counting measure, divided by a factor of 2 d r=1 nr .
Setting d = 1, n 1 = n and s 1 = 2, 3 and 4 in Corollary 2.2 yields the sequences [SKW99], [W199] and [W299], respectively. Setting d = 2, n 1 = n 2 = n and s 1 = s 2 = 2 in Corollary 2.2 yields the sequence [C15], whose complementary sequence is [H08]. where by the union of several n 1 × · · · × n d systems we mean the system whose generic element is 1 if and only if the corresponding element in at least one of those systems is also 1. Thus   P (s 1 , . . . , s d , n 1 The next task is to compute the number k e∈J e of 1s in the system e∈J e, where J is some nonempty subset of the set E of elementary failure cases e. We use the Inclusion-Exclusion Principle again, this time for the counting measure on the power set of the set of elements of a system, to obtain Each elementary failure case consists entirely of 0s except for a contiguous s 1 × . . . × s d subarray of 1s. Therefore e∈J ′ e also consists entirely of 0s except for a (possibly empty) contiguous subarray of dimensions t 1 × · · · × t d , where for each r ∈ {1, . . . d}, The volume of that contiguous t 1 × · · · × t d subarray is equal to k e∈J ′ e , that is, Considering (4), (5) and (6), we have which proves equation (2). Equation (1) follows immediately.
See also Figure 1.

Conclusion
We have provided a novel formula for the reliability of a general d-dimensional consecutive-k-out-of-n:F system, as an exact polynomial in q. We believe ours to be the first general exact closed-form formula published for the reliability in d dimensions.
This answers an open question in Engineering.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this manuscript.