Reliability of a Large Series-parallel System in Variable Operating Conditions

Transcription

1 International Journal of Automation and Computing 2 (2006) Reliability of a Large Series-parallel System in Variable Operating Conditions Joanna Soszynska Department of Mathematics, Gdynia Maritime University, Gdynia , Poland Abstract: In this paper, a semi-markov model of system operation processes is proposed and its selected parameters are determined. A series-parallel multi-state system is considered, and its reliability and risk characteristics found. Subsequently, a joint model of system operation process and system multi-state reliability and risk is constructed. Moreover, the asymptotic approach to reliability and risk evaluation of a multi-state series-parallel system in its operation process is applied to a port grain transportation system. Keywords: Series-parallel system, limit reliability function, semi-markov processes. 1 System operation process In this paper, we assume that a system during its operation process is performing a repertory of tasks. Namely, the system at each moment t, t < 0, θ >, θ is its operation time, is performing at most w tasks. We denote the system operation process by Z(t) = [Z 1 (t), Z 2 (t),, Z w (t)] Z j (t) = 1 if the system is executing the jth task, and Z j (t) = 0 if the system is not executing the jth task for j = 1, 2,,w. Therefore, Z(t) is a process with continuous time t, t < 0, θ >, and discrete states from the set of states {0.1} w. We number the operational states of process Z(t), assuming that it has v different states from the set: and they are of the form: Z = {z 1, z 2,, z ν } z b = [z b 1, z b 2,, z b w], b = 1, 2,,v z b j {0, 1}, j = 1, 2,,w. In practice, a convenient assumption is that Z(t) is a semi-markov process [1] with conditional sojourn times θ bl at operation state z b when its next operation state is z l, b, l = 1, 2,,v, b l. In this case, this process may be described using: the vector of probabilities of the initial operational states [p b (0)] 1xν the matrix of the probabilities of its transitions between these states [p bl ] νxν Manuscript received September 22, 2005; revised January 6, address: joannas@am.gdynia.pl the matrix of the conditional distribution functions [H bl (t)] νxν of the sojourn times θ bl, b, l = 1, 2,, v, b l. If the sojourn times θ bl, b, l = 1, 2,, v, b l, have exponential distributions with transition rates between the operation states γ bl, i.e. if for b, l = 1, 2,, v, b l, H bl (t) = P(θ bl < t) = 1 exp[ γ bl t], t > 0 then their mean values are determined by E[θ bl ] = 1/γ bl, b, l = 1, 2,,v, b l. (1) The unconditional distribution functions of the process Z(t) sojourn times θ b at operation states z b, b = 1, 2,, v, are given by H b (t) = 1 p bl exp[ γ bl t], t 0, b = 1, 2,,ν l=1 and considering (1), their mean values are M b = E[θ b ] = (2) p bl /γ bl, b = 1, 2,, v. (3) l=1 Limit values of the transient probabilities p b (t) at operation states z b are given by p b = lim t p b (t) = π b M b / π l M l, b = 1, 2,, ν (4) l=1 M b are given by (3), and the probabilities π b of the vector [π b ] 1xν satisfy the system of equations: [π b ] = [π b ][p bl ] π l. = 1 l=1

2 200 International Journal of Automation and Computing 2 (2006) Multi-state series-parallel systems In multi-state reliability analysis, to define systems with degrading components, we assume that all components in a system under consideration have a reliability state set {0, 1,, z}, z 1, in which reliability states are ordered (state 0 worst, to state z best), and component and system reliability states degrade over time t, if not repaired. The above assumptions mean that the states of a system with degrading components may change over time only from better to worse. A basic multi-state reliability structure with components which degrade over time is a series-parallel system. To define such a system, we assume that E ij, i = 1, 2,, k n, j = 1, 2,,l i, k n, l 1, l 2,, l kn, n N, are components of the system, and T ij (u), i = 1, 2,,k n, j = 1, 2,, l i, k n, l 1, l 2,, l kn, n N, are independent random variables which represent the lifetime of components E ij in the state subset {u, u + 1,,z}, while they are in state z at moment t = 0; e ij (t) are components E ij states at moment t, t (, ), while they are in state z at moment t = 0; T(u) is a random variable representing the lifetime of a system in the reliability state subset {u, u + 1,, z}, while it is in reliability state z at moment t = 0; and s(t) is the system reliability state at moment t, t (, ), given that it is in state z at moment t = 0. Definition 1. [2,3] A vector R ij (t, ) = [1, R ij (t, 0), R ij (t, 1),, R ij t (, ) R ij (t, u) = P(e ij (t) u e ij (0) = z) = P(T ij (u) > t) for t (, ), u = 0, 1,,z, i = 1, 2,, k n, j = 1, 2,,l i, is the probability that component E ij is in the reliability state subset {u, u+1,, z} at moment t, t (, ), while it is in reliability state z at moment t = 0, and is called the multi-state reliability function of a component E ij. Definition 2. [2,3] A vector R knl n (t, ) = [R knl n (t, 0), R knl n (t, 1),, R knl n R knl n (t, u) = P(s(t) u s(0) = z) = P(T(u) > t) for t (, ), u = 0, 1,,z, is the probability that the system is in reliability state subset {u, u+1,, z} at moment t, t (, ), while it is in reliability state z at moment t = 0, and is called the multi-state reliability function of a system. It is clear that from Definition 1 and Definition 2, u = 0 that we have R ij (t, 0) = 1 and R knl n (t, 0) = 1. Definition 3. [2,3] A multi-state system is called series-parallel if its lifetime T(u) in the state subset {u, u + 1,,z} is given by T(u) = max 1 i k n { min 1 j l i {T ij (u)}}, u = 1, 2,,z. Definition 4. [2,3] A multi-state series-parallel system is called regular if l 1 = l 2 = = l kn = l n, l n N. Definition 5. [2,3] A multi-state regular seriesparallel system is called non-homogeneous, if it is composed of a, 1 a k n, k n N, different types of series of subsystem, and the fraction of the ith type series subsystem is equal to q i, q i > 0, q i = a 1. i=1 Moreover, the ith type series subsystem consists of e i, 1 e i l n, l n N, types of components with reliability functions: R (i,j) (t, ) = [1, R (i,j) (t, 1),, R (i,j) R (i,j) (t, u) = 1 F (i,j) (t, u) for t (, ), j = 1, 2,,e i, u = 1, 2,,z, and the fraction of the jth type component in this subsystem is equal to p ij, p ij > 0 and p ij = e i 1. The reliability function of a multi-state nonhomogeneous regular series-parallel system is given by (t, ) = [1, (t, 1),, (5) (t, u) = 1 a {1 [R (i) (t, u)] ln } qikn i=1 for t (, ), u = 1, 2,,z, and e i R (i) (t, u) = [R (i,j) (t, u)] pij, i = 1, 2,,a.

3 J. Soszynska/Reliability of a Large Series-parallel System in Variable Operating Conditions 201 the system reliability structure as well [4]. Therefore, we denote the conditional reliability function of the system component E ij while the system is at operational state z b, b = 1, 2,,v, by [R (i,j) (t, )] (b) = [1, [R (i,j) (t, 1)] (b),, [R (i,j) (b) ] for t < 0, ), b = 1, 2,, v, u = 1, 2,,z, Fig. 1 The scheme of a regular non-homogeneous series-parallel system Under these definitions, if R knl n (t, u) = 1 for t 0, u = 1, 2,, z, then M(u) = 0 R knl n (t, u)dt, u = 1, 2,, z (6) is the mean lifetime of a multi-state non-homogeneous regular series-parallel system in the reliability state subset {u, u + 1,, z}, and the mean lifetime M(u), u = 1, 2,, z, of this system in the state u can be determined from the following relationship: M(u) =M(u) M(u + 1), u=1, 2,, z 1 M(z)=M(z). (7) Definition 6. [2,3] A probability r(t)=p(s(t) < r s(0)=z)=p(t(r) t), t (, ) that a system is in a subset of states worse than a critical state r, r {1,,z} while it is in reliability state z at moment t = 0 is called a risk function of a multistate non-homogeneous regular series-parallel system. Considering Definition 6 and Definition 2, we have r(t) = 1 R knl n (t, r), t (, ) (8) and, if τ is the moment at which the system risk function exceeds a permitted level δ, then τ = r 1 (δ) (9) r 1 (t), if it exists, is the inverse function of risk function r(t). 3 Multi-state series-parallel system in its operation process We assume that changes in process Z(t) states have an influence on system components E ij reliability and [R (i,j) (t, u)] (b) = P(T (b) ij (u) > t Z(t) = zb ) and the conditional reliability function of a system while it is in operational state z b, b = 1, 2,,v, by: R (b) (t, ) = [1, R (b) (t, 1),, R (b) for t < 0, ), b = 1, 2,, ν, u = 1, 2,,z. R (b) (t, u) = P(T (b) (u) > t Z(t) = z b ). The reliability function [R (i,j) (t, u)] (b) is a conditional probability that component E ij lifetime T (b) ij (u) in the state subset {u, u + 1,,z} is not less than t, while process Z(t) is in operation state t, t 0. Similarly, reliability function R (b) (t, u) is a conditional probability that system lifetime T (b) (u) in the state subset {u, u + 1,,z} is not less than t, while process Z(t) is in operation state z b. In the case that system operation time θ is large enough, the unconditional reliability function of system is (t, ) = [1, (t, 1),, (t, u) = P(T(u) > t) for u = 1, 2,, z, and T(u), is the unconditional lifetime of the system in the reliability state subset {u, u + 1,,z}, as given by (t, u) = p b R (b) (t, u) (10) for t 0, and the mean values of system lifetimes in the reliability state subset {u, u + 1,,z}: µ(u) = for u = 1, 2,,z, M b (u) = p b M b (u) (11) 0 R (b) (t, u)dt (12) for b = 1, 2,,ν, t 0, and p b are given using (4). The mean values of system lifetimes in particular reliability states u, by (7), are µ(u) = µ(u) µ(u + 1), u = 1, 2,,z 1 µ(z) = µ(z). (13)

4 202 International Journal of Automation and Computing 2 (2006) Large Multi-state series-parallel system In an asymptotic approach to multi-state system reliability analysis, we are interested in the limit distributions of a standardised random variable (T(u) b n (u))/a n (u), u = 1, 2,, z, T(u) is the lifetime of a multi-state non-homogeneous regular seriesparallel system in the state subset {u, u+1,, z}, and a n (u) > 0, b n (u) (, ), u = 1, 2,, z are suitably chosen numbers, called normalising constants. Since P((T(u) b n (u))/a n (u) > t) = P(T(u) > a n (u)t + b n (u)) = (a n (u)t + b n (u), u), u = 1, 2,, z, we assume the following definition. Definition 7. [2,3]. A vector R(t, ) = [1, R(t, 1),,R, t (, ) is called the limit multi-state reliability function of a multi-state non-homogeneous regular series-parallel system with reliability function (t, ), if there exist normalising constants a n (u) > 0, b n (u) (, ) such that lim R (a n (u)t + b n (u), u) = R(t, u) n for t C R(u), u = 1, 2,, z, C R(u) is a set of continuity points of R(t, u). Knowledge in the system limit reliability function allows us, for sufficiently large n and t (, ), to apply the following approximate formula: (t, ) = [1, (t, 1),, = [1, R( t b n(1) a n (1), 1),, R( t b n(z), z)]. a n (z) (14) Proposition 1. If the components of a nonhomogeneous regular multi-state series-parallel system have exponential reliability functions: R (i,j) (t, ) = [1, R (i,j) (t, 1),,R (i,j) for u = 1, 2,, z, i = 1, 2,,a, j = 1, 2,,e i, then R (i,j) (t, u) = exp[ λ ij (u)t], t 0, λ ij (u) > 0 k n k, l n a n (u) = 1/[λ(u)l n ], b n (u) = 0 λ i (u) = min 1 j e i {λ ij (u)}, λ(u) = max 1 i a {λ i(u)} R(t, ) = [1, R(t, 1),,R R(t, u) = 1 (i:λ i(u)=λ(u)} {1 exp[ t]} qik, t 0 for u = 1, 2,,z, as its limit reliability function. From Proposition 1, and after applying (7) with u = 1, 2,, z, we have the following approximate formula: (t, u) =1 {1 exp[ λ(u)l n t]} qik, t 0. (i:λ i(u)=λ(u)} (15) 5 Large multi-state series-parallel system in its operation process On the basis of previous considerations, it is possible to evaluate approximately the reliability functions of a large multi-state non-homogeneous regular seriesparallel system in its operation process. Definition 8. [2,3] A reliability function R(t, ) = [1, R(t, 1),,R, t (, ) R(t, u) = v p b R (b) (t, u) is called a limit reliability function of a multi-state series-parallel system in its operation process with reliability function: (t, ) = [1, (t, 1),, (t, u), u = 1, 2,, z, are given by (10). If there exist normalising constants: a (b) n (u) > 0, b (b) n (u) (, ) b = 1, 2,,v, u = 1, 2,, z such that for t C R (b) (u), u = 1, 2,,z, b = 1, 2,, v, lim n R(b) (a (b) n (u)t + b(b) n (u), u) = R(b) (t, u). then the following approximate formula is valid: (t, u) v = p b R (b) ( t b(b) n, u), u = 1, 2,,z. a (b) n (16) Finally, linking results (15)-(16), we have an expression for the multi-state reliability function of a considered system in its operation process: (t, ) = [1, (t, 1),, (t, u) = v p b R (b) (t, u)

5 v = p b R (b) ( t b(b) n (u) ) a (b) n (u) v = 1 p b J. Soszynska/Reliability of a Large Series-parallel System in Variable Operating Conditions 203 (i:λ (b) i (u)=λ (b) (u)} {1 exp[ λ (b) (u)l n t]} qik (17) and, after linking (6), (11), and (12), for the mean values of system lifetimes in the reliability state subsets: µ(u) = 0 (t, u)dt, u = 1, 2,,z (18) (t, u) is given by (17). 6 Application As an example, we will analyse the reliability of a port grain elevator in its operation process [5]. This system composed of four multi-state non-homogeneous series-parallel transportation subsystems, is the basic structure in the Baltic Grain Terminal at the Port of Gdynia, and is assigned to handle the clearing of exported and imported grain. One basic elevator function is loading railway trucks with grain. In loading railway trucks with grain, the following elevator transportation subsystems take part: S 1 horizontal conveyors of a first type, S 2 vertical bucket elevators, S 3 horizontal conveyors of a second type, S 4 worm conveyors. Taking into account the quality of work of the considered transportation system, we distinguish the following three reliability states (z = 2) of its components: state 2 a state ensuring the largest quality of conveyor work, state 1 a state ensuring less quality of conveyor work caused by grain being thrown off the belt, and state 0 a state involving a failure of the conveyor. The structure of the port grain transportation system at operation state 1 is given in Fig. 2. Further, we will analyze the reliability of subsystem S 1. Subsystem S 1 consists of k n = 2 identical belt conveyors of the first type, each composed of l n = 129 components. In each conveyor, there is one ribbon belt with reliability functions: R (1,1) (t, 1) = exp[ t], R (1,1) (t, 2) = exp[ t] two drums with reliability functions: R (1,2) (t, 1) = exp[ 0.044t], R (1,2) (t, 2) = exp[ 0.048t] 117 channelled rollers with reliability functions: R (1,3) (t, 1) = exp[ t], R (1,3) (t, 2) = exp[ t] and nine supporting rollers with reliability functions: R (1,4) (t, 1) = exp[ t], R (1,4) (t, 2) = exp[ t]. Fig. 2 A scheme for a grain transportation system in operation state 1 Taking into account the operation process of the considered transportation system, we distinguish the following as its three tasks: task 1 system operation with greatest efficiency, when all components of subsystems S 1, S 2, S 3, and S 4 are used task 2 system operation with less efficiency, when the first conveyor of subsystem S 1, the first and second elevators of subsystem S 2, the first conveyor of subsystem S 3, and the first and second conveyors of subsystem S 4 are used task 3 system operation with least efficiency, when only the first conveyor of subsystem S 1, the first elevator of subsystem S 2, the first conveyor of subsystem S 3, and the first conveyor of subsystem S 4 are used. Since the system tasks are disjoint, its operation states belong to the set: Z = {z 1, z 2, z 3 } z 1 = [1, 0, 0], z 2 = [0, 1, 0], z 3 = [0, 0, 1]. Moreover, we arbitrarily assume the following matrix of the conditional distribution functions of system sojourn times θ bl, b, l = 1, 2, e 2.78t 1 e 5t [H bl (t)] = 1 e 20t 0 1 e 5t. 1 e 12.5t 1 e 20t 0 On the basis of data coming from experts, the probabilities of transitions between the states are given by [p bl ] =

6 204 International Journal of Automation and Computing 2 (2006) and further according to (2), the unconditional distribution functions of process Z(t) sojourn times θ b, in the states z b, b = 1, 2, 3, are given by H 1 (t) = exp[ 2.78t] exp[ 5t] H 2 (t) = exp[ 20t] 0.2 exp[ 5t] H 3 (t) = exp[ 12.5t] exp[ 20t] and their mean values from (3): M 1 = E[θ 1 ] = = M 2 = E[θ 2 ] = = 0.08 M 3 = E[θ 3 ] = = From the system of equations: [π 1, π 2, π 3 ] = [π 1, π 2, π 3 ] π 1 + π 2 + π 3 = 1 we have π 1 = , π 2 = 0.321, π 3 = thus the limit values of the transient probabilities p b (t) at operational states z b, according to (4), are given by p 1 = 0.684, p 2 = , p 3 = (19) At system operational state 1, subsystem S 1 becomes a multi-state non-homogeneous regular series-parallel system, with parameters: k n = k = 2, l n = 129, a = 1, q 1 = 1, e 1 = 4 p 11 = 1/129, p 12 = 2/129, p 13 = 117/129, p 14 = 9/129 λ 11 (1) = , λ 12 (1) = 0.044, λ 13 (1) = λ 14 (1) = λ 11 (2) = , λ 12 (2) = 0.048, λ 13 (2) = λ 14 (2) = Since: λ 1 (1) = p 1j λ 1j (1) = (1/129) (2/129) (117/129) (9/129) = λ(1) = min{0.0790} = λ 1 (2) = p 1j λ 1j (2) = (1/129) (2/129) (117/129) (9/129) = λ(2) = min{0.0963} = applying Proposition 1 with normalising constants: a n (1) = 1/ = 1/10.191, b n (1) = 0 a n (2) = 1/ = 1/ , b n (1) = 0 we conclude that R (1) (t, ) = [1, R (1) (t, 1), R (1) (t, 2)] = [1, 1 [1 exp[ t]] 2, 1 [1 exp[ t]] 2 ], t 0 is the subsystem S 1 limit reliability function, and from (15), we have: R (1) 2,129 (t, 1) = R (1) (10.191t, 1) = 1 [1 exp[ t]] 2, t 0 (20) R (1) 2,129 (t, 2) = R (1) ( t, 2) = 1 [1 exp[ t]] 2, t 0 (21) and, according to (12), system lifetime mean values in the state subsets are: M (1) (1) = 0.147, M (1) (2) = (22) At system operational state 2, subsystem S 1 becomes a non-homogeneous regular series-parallel system with parameters: k n = k = 1, l n = 129, a = 1, q 1 = 1, e 1 = 4 p 11 = 1/129, p 12 = 2/129, p 13 = 117/129, p 14 = 9/129 λ 11 (1) = , λ 12 (1) = 0.044, λ 13 (1) = λ 14 (1) = λ 11 (2) = , λ 12 (2) = 0.048, λ 13 (2) = λ 14 (2) = Since: λ 1 (1) = p 1j λ 1j (1) = (1/129) (2/129) (117/129) (9/129) = λ(1) = min{0.0790} = λ 1 (2) = p 1j λ 1j (2) = (1/129) (2/129) (117/129) (9/129) = λ(2) = min{0.0963} = applying Proposition 1 with normalising constants: a n (1) = 1/ = 1/10.191, b n (1) = 0

7 J. Soszynska/Reliability of a Large Series-parallel System in Variable Operating Conditions 205 a 2 (2) = 1/ = 1/ , b n (1) = 0 we conclude that R (2) (t, ) = [1, R (2) (t, 1), R (2) (t, 2)] = [1, exp[ t], exp[ t]], t 0 is the subsystem S 1 limit reliability function, and from (15) we have: R (2) 1,129 (t, 1) = R (2) (10.191t, 1) = exp[ t], t 0 (23) R (2) 1,129 (t, 2) = R (2) ( t, 2) = exp[ t], t 0 (24) and, according to (12), system lifetime mean values in the state subsets are M (2) (1) = 0.098, M (2) (2) = (25) At system operational state 3, subsystem S 1 becomes a non-homogeneous regular series-parallel system with the parameters: k n = k = 1, l n = 129, a = 1, q 1 = 1, e 1 = 4 p 11 = 1/129, p 12 = 2/129, p 13 = 117/129, p 14 = 9/129 λ 11 (1) = , λ 12 (1) = 0.044, λ 13 (1) = λ 14 (1) = λ 11 (2) = , λ 12 (2) = 0.048, λ 13 (2) = λ 14 (2) = Since: λ 1 (1) = p 1j λ 1j (1) = (1/129) (2/129) (117/129) (9/129) = λ(1) = min{0.0790} = λ 1 (2) = p 1j λ 1j (2) = (1/129) (2/129) (117/129) (9/129) = λ(2) = min{0.0963} = applying Proposition 1 with normalising constants: a n (1) = 1/ = 1/10.191, b n (1) = 0 a n (2) = 1/ = 1/ , b n (1) = 0 we conclude that R (3) (t, ) = [1, R (3) (t, 1), R (3) (t, 2)], = [1, exp[ t], exp[ t]], t 0 is the subsystem S 1 limit reliability function, and from (15) we obtain R (3) 1,129 (t, 1) = R (3) (10.191t, 1) = exp[ t], t 0 (26) R (3) 1,129 (t, 2) = R (3) ( t, 2) = exp[ t], t 0 (27) and, according to (12), the system lifetime mean values in the state subsets are M (3) (1) = 0.098, M (3) (2) = (28) Finally, considering (10), (17), and (19), system unconditional reliability is given by R(t, ) = [1, R(t, 1), R(t, 2)] R(t, 1) = 0.684R (1) 2,129 (t, 1) R(2) 1,129 (t, 1) R (3) 1,129 (t, 1) R(t, 2) = 0.684R (1) 2,129 (t, 2) R(2) 1,129 (t, 2) R (3) 1,129 (t, 2) R (1) 2,129 (t, 1), R(2) 1,129 (t, 1), R(3) 1,129 (t, 1) and (t, 2), R(2) (t, 2), R(3) (t, 2) are respectively R (1) 2,129 1,129 1,129 given by (20), (23), (26), and by (21), (24), and (27). Hence, applying (11), (18), (22), (25) and (28), we obtain the mean values of system unconditional lifetimes in the reliability state subsets given by µ(1) = = years µ(2) = = years and, according to (13), the mean values of system unconditional lifetimes in particular states: µ(1) = µ(1) µ(2) = years µ(2) = µ(2) = years. If a critical reliability state of the system is r = 2, then from (8), its risk function takes the form: r(t) = 1 R(t, 2). Hence, from (9), the moment at which risk will exceed the critical level δ = 0.05 is τ = r 1 = years.

8 206 International Journal of Automation and Computing 2 (2006) Conclusions This paper proposes an approach to the solution of the practical and important problem of linking a systems reliability and its operational processes. To involve the interactions between a system operational process and its time varying reliability structures, a semi-markov model of system operation processes and system limit reliability functions was applied. The interactions semi-markovian linking and applied limit reliability function are an innovative and unique aspect of this paper. References [1] F. Grabski. Semi-Markov Models of Systems Reliability and Operations. Systems Research Institute, Polish Academy of Sciences, Warsaw, [2] K. Kolowrocki. Asymptotic Approach to System Reliability Analysis. Monograph, System Research Institute, Polish Academy of Sciences, Warsaw, [3] K. Kolowrocki. Reliability of Large Systems. Elsevier, [4] J. Soszyńska. Reliability of Multi-state Systems in Variable Operation Conditions. Report Maritime University, Gdynia, [5] K. Kolowrocki, A. Blokus, A. Cichocki, B. Kwiatuszewska- Sarnecka, B. Milczek, J. Soszyńska. Asymptotic Approach to Reliability Analysis and Optimisation of Complex Transport Systems. Gdynia Maritime University. Project founded by the Polish Committee for Scientific Research, Gdynia, (in Polish) Joanna Soszyńska received her M.Sc degree in physics and mathematics at Gdansk University, Poland, in She is currently an assistant at the Mathematics Department of the Faculty of Navigation at Gdynia Maritime University. She has published a number of reports and papers in journals and conference proceedings. Her research interests include mathematical modelling of safety and reliability in complex systems (related to their operation). She is also interested in physics.