- Email: [email protected]
New interval availability indexes for Markov repairable systems
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
New interval availability indexes for Markov repairable systems ⁎
Lirong Cui , Jianhui Chen, Bei Wu School of Management & Economics, Beijing Institute of Technology, Beijing 100081, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Markov repairable system Interval availability Index Laplace transform
In maintenance field, there are many indexes such as instantaneous availability, steady state availability, interval availability and others, which have been widely used to describe the various properties and performance of repairable systems on maintenance. However, all the present availability indexes still cannot cover some situations which people are interested in. In the paper, two new interval availability indexes are introduced for Markov repairable systems, which is named as an availability with a given window length and containing a specified point or interval. The new interval availability is the probability that the system is working during a given window length and this window must contains a specified point or interval. Their calculation formulas are presented in matrix forms by using the Laplace transform. Some properties on the two indexes are discussed briefly, and numerical examples are shown to illustrate the features of the two new indexes, finally conclusions are given. These two new indices can become the conventional interval availability when the given window length is zero or equal to the specified interval length. The results in the paper may be used to measure the performance of repairable system more deeply and detail.
1. Introduction In reliability field, the related measure indexes such as reliability, availability and safety etc. play an important role to describe system performances. Any theoretical research and engineering work in reliability must be related to the reliability measure indexes because these measure indexes can tell people if they have reached the prespecified target or if the state of system operating can meet the mission requirements. On the other hand, each system including that in reliability field has many performances, how to measure these performances of the system must rely on some indexes. As the technology developments, people want to know more deep and detail information on systems, new indexes are needed naturally. The similar situation in maintenance is faced at present, some new measure indexes are needed to describe the situation or information people are interested in. In fact, many indexes have been used in describing the various properties of repairable systems. For example, several reliability indexes for repairable systems were proposed in [1–3]. The reliability and availability indexes are an important topic, much literature can be found in this direction, for example, references [4,5]. Specially, on aggregated repairable systems, many availability indexes including the conventional pointwise and interval availabilities have been studied, for example, references [6–10]. The availability indexes are closely related to maintenance models because the maintenance models can describe the evolution processes of the repairable systems, while the availability ⁎
indexes can describe the performance levels of the repairable systems under given models. For example, Cui et al. [11] introduced a new changeable state repairable system, and then the conventional availability indexes were given in their paper. Recently, Liu et al. [12] and Cui et al. [13] studied some availability indexes under aggregated stochastic models. In fact, it is not an easy task to give the formulas for these indexes, and the related work is mainly for single-unit systems, Markov repairable systems and semi-Markov repairable systems, for example, Refs. [14–16]. Availability is an important measure of performance for repairable systems, which has been developed into more detail indexes such as pointwise availability, interval availability and joint availability and so forth. In general, the availability is a probability that the repairable system will be able to operate within the tolerances at a given instant time or interval, which are for point-wise and interval availabilities. The definition of interval availability has some different ways. The interval availability commonly refers to the expected proportion of time for which the repairable system is available over some interval of time. Sericola [17] defined it as the fraction of operation time over a finite observation period, which is a random variable. Finkelstein [18] defined a multiple availability as the probability of a system being in an operating state at each moment of demand. Csenki [19] also studied multiple availability, but in his work multiple availability was called joint availability. Csenki [20,21] also studied the related interval availability. Cui et al. [22] proposed the multi-point, multi-interval,
Corresponding author. E-mail address: [email protected] (L. Cui).
http://dx.doi.org/10.1016/j.ress.2017.03.016
0951-8320/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Cui, L., Reliability Engineering and System Safety (2017), http://dx.doi.org/10.1016/j.ress.2017.03.016
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
some intuitive understanding on usage of the two new availability indexes. Finally, the conclusions are summarized in Section 6.
and mixed multi-point-interval availabilities, which are extensions of the single point and interval availabilities. A similar concept is an interval reliability, Barlow and Proschan [23] defined it as the probability that at a specified time the system is operating and will continue to operate for an interval of duration. (The authors would like to call this interval reliability as interval availability not interval reliability especially for repairable systems, although the classical book and mathematical reliability pioneers named it. On the other hand, the name change for interval reliability can meet the feature of the pointwise availability, multiple availability and repairable systems, because when the interval length becomes into zero, multiple into single, the interval reliability becomes into the pointwise availability, but the multi-interval availability become into interval reliability.). Wu and Hillston [24] studied two kinds reliability measures under a semiMarkov process situation, one is the system must remain operational continuously for a minimum time within the given mission time interval, while the other required the total operational time of the system within the mission time window must be greater than a given value. Both reliability measures are defined by suing the fraction of working time to window length, and the first required the continuous working time, but the second is not required. As mentioned previously, there are many indexes for repairable systems, but they still cannot cover all practical situations, For example, people are interested in the probability that the system, which may be a computer system or an air condition system or service system etc., is available (working) at 9:00 o’clock or so within 15 min, or the probability that the system is available (working) during a window interval whose interval length is a half hour but it must cover the interval from 9:00 o’clock to 9:15 o’clock. This kind of problem is what our present paper will study, we name the first probability as an availability with window τ containing point x , which is the probability that a repairable system works throughout an interval window which has at least a length τ and contains a given point x . The second probability is called an availability with window τ containing interval [a, b], which is the probability that a repairable system works throughout an interval window which has at least a length τ and contains a given interval [a, b]. Both concepts are extensions of pointwise availability and interval availability, which definitely enrich the indexes measures in maintenance. For application examples of two indices introduced above, we can look at a water supply system. The requirement of customers on this system is: it must be working from 9:00 o’clock to 9:15 o’clock, but the manager wants to guarantee this customer requirement to be satisfied more likely, he/she requires the water supply system must be working at least a half hour which must also cover the time interval [9:00, 9:15]. This management strategy can provide more tolerance for the customer requirement. The author believe that the two new availability indices can be used in more real situations, particularly in increasing mission success probability of systems. In the paper, after doing the introduction of two new availability indexes, their calculation formulas are given by using the Laplace transform and matrix techniques, meanwhile, some basic results referenced in Colquhoun and Hawkes [25] are used in the process of derivations. Furthermore, some properties of the two new availability indexes are discussed briefly, which may provide more information and understanding on usage of the two new availability indexes. The rest of the paper is organized as follows. Section 2 provides the assumptions on Markov repairable model and definitions of two new availability indexes, including their mathematical expressions, and some basic results referenced in Colquhoun and Hawkes [25] are presented. In Section 3, the main results of the paper, the calculation formulas of the two new availability indexes, are given. Some properties and comparisons of the two new availability indexes are considered in Section 4, which includes the relationship between the two new availability indexes and the conventional pointwise and interval availabilities. Section 5 presents some numerical examples to illustrate the properties of the two new availability indexes, which may provide
2. Assumptions, definitions and preliminaries Suppose there is a repairable system following a homogeneous continuous time Markov process {X (t ), t ≥ 0} with finite state space S which contains a working subset and a failure subset, i.e., S = W ∪ F , where S = {1, 2, …, n} and F = {n + 1, n + 2, …, n + m}. The infinitesimal generator of the process {X (t ), t ≥ 0} is Q , in terms of working and failure states, which can be divided into four blocks, i.e.
⎛Q QWF ⎞ Q = ⎜ WW ⎟. ⎝ Q FW Q FF ⎠ The definitions of two new availability indexes to be discussed throughout the paper are given as follows. Definition 1. The probability that a repairable system works throughout an interval window, which has at least a length τ and contains a given point x , is called as an availability with window τ containing point x . The availability with window τ containing point x can be expressed in a formula way as
A (τ , x ): =P { ∃ c ≥ 0, system works in interval [c, c + τ ] and c ≤ x ≤ c + τ}, where τ is a required interval length, x is a given instant. The availability with window τ containing point x can be used to describe an interval availability in which the interval must have at least a length τ and contain a given time instant x , it is an extension of point availability. When the interval length τ = 0 , the availability with window τ containing point x becomes into a conventional point availability at time instant x . To understand the concept of availability with window τ containing point x , we can image that there is a window with length τ to move towards right or left containing point x and at least at one moment the repairable system works throughout the window, the probability for this situation is the availability with window τ containing point x . Definition 2. The probability that a repairable system works throughout an interval window which has at least a length τ and contains a given interval [a, b], is called as an availability with window τ containing interval [a, b]. The availability with window τ containing interval [a, b] can be expressed in a formula way as
A (τ , [a, b]): =P { ∃ c ≥ 0, system works in interval [c, c + τ ] and c ≤ a ≤ b ≤ c + τ}, where τ is a required interval length, [a, b] is a given interval. Similarly, the availability with window τ containing interval [a, b] can be used to describe an interval availability in which the interval must have at least a length τ and contain a given interval [a, b], this new availability concept is an extension of interval availability. When the interval length τ = b − a , the availability with window τ containing interval x becomes into a conventional interval availability at interval [a, b]. To understand the concept of availability with window τ containing point [a, b], we can image that there is a window with length τ to move towards right or left covering interval [a, b], and at least at one moment the repairable system works throughout the window, the probability for this situation is the availability with window τ containing interval [a, b]. Because of using Laplace transform throughout the paper, here we give its definition below and specify the Laplace transform on matrix for elementwise transforms. The Laplace transform for function f (t ) is defined as follows, 2
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
f * (s ) =
∫0
∞
fTx (t ) = ( fTx,1 (t ), fTx,2 (t ), …, fTx, n (t ))
e−st f (t ) dt .
=(h1 (t ), h2 (t ), …, hn (t )) eQWW τ = h (t ) eQWW τ ,
Colquhoun and Hawkes [25] introduced two important quantities in studying the stochastic properties of ion channel, which are given below. W
where fTx, i (t ) (i = 1, 2, …, n ) is a probability density function of Tx and X (T x+) = i ∈ W , and the Laplace transform of (h1 (t ), h2 (t ), …, hn (t )) is as follows.
pij (t ): =P {system remains within W throughout time 0 to time t ,
h* (s ): =(h1* (s ), h 2* (s ), …, hn* (s ))
and is in state j at time t | in state i at time 0),
∞ =ΦW ∑n =1 [G*WF (s ) G*FW (s )]n + ΦF G*FW (s )
i, j ∈ W .
∞
∑n =0 [G*WF (s ) G*FW (s )]n
After some manipulations, we can get, in matrix form,
=ΦW {[I − G*WF (s ) G*FW (s )]−1 − I}
⎞ ⎛ PWW (t ): =⎜W pij (t ) ⎟ = exp(QWW t ), ⎠|W |×| W | ⎝
+ ΦF G*FW (s ){[I − G*WF (s ) G*FW (s )]−1 =[ΦW + ΦF G*FW (s )][I − G*WF (s ) G*FW (s )]−1 − ΦW
where |W | denotes the cardinality of set W , and its Laplace transform can be given
=[ΦW + ΦF (s I − Q FF )−1Q FW ]
P*WW (s ) = (s I − QWW
[I − (s I − QWW )−1QWF (s I − Q FF )−1Q FW ]−1 − ΦW ,
)−1,
where ΦF is an initial failure condition probability vector for the repairable system. Note that the distribution formula of Tx contains the required window length working duration, which can be omitted in the following derivations. This formula is obtained because the Tx may consist of any number (1, 2, …, ∞) of cycles W → F → W if the repairable system begins at the working subset or the Tx may consist of a transition F → W followed by any number (0, 1, …, ∞) of cycles W → F → W if the repairable system begins at the failure subset. On the other hand, all these events are exclusive. Thus we can get the distribution vector of Tx . In terms of Tx = v , we can discuss the new interval availability in two cases, which are shown in Figs. 1 and 2. Let v be a realization of Tx , then we have, if 0 < v ≤ x − τ ,
where I is a unit matrix with proper dimension, and s is the Laplace variable. Note that PWW (t ) is not simply a submatrix of P(t ) which is the transition probability matrix for the Markov process. Another quantity defined by Colquhoun and Hawkes [25] is
gij (t ): = lim [P {system stays in W from time 0 to time t , and leave W Δt→0
for state j between t and t + Δt| in state i at time 0}/ Δt ],
i ∈ W , j ∈ F.
Similarly, in matrix form, we have
GWF (t ): =(gij (t ))|W |×| F | = PWW (t ) QWF ,
A (τ , x ) =
and its Laplace transform can be given as
∫0
(x − τ )
fTx (v ) eQWW (x − τ − v ) 1W dv,
and if x > v > x − τ , which is shown in Fig. 2. Based on Fig. 2, we have
G*WF (s ) = (s I − QWW )−1QWF . Similarly, we can define PFF (t ), P*FF (s ), GFW (t ) and G*FW (s ), which will be used in the paper. Note that gij (t ) is a probability density function, in general, gij (t ) is not, itself, a proper probability density function, without the initial and ending probabilities considered.
x
A (τ , x ) =
∫(x−τ )
+
fTx (v ) 1W dv,
where A (τ , x ) indicates the availability with window τ containing point x , and (x − τ )+ = max(x − τ , 0). The above result follows that the availability with window τ containing point x requires the repairable Markov system stays in the working subset at least duration (x − v ) or τ for the cases of 0 < v ≤ x − τ and x > v > x − τ , respectively, after time instant Tx . Summarizing Cases 1 and 2, we have gotten the closed form of A (τ , x ) as follows,
3. Main results First we consider the availability with window τ containing point x , whose formula is given as follows. 3.1. Availability with window τ containing point x There are two cases to be considered for the availability with window τ containing point x .
(x − τ )+
A (τ , x ) = ΦW eQWW (x ∨ τ ) 1W + ∫ 0
fTx (v ) eQWW (x − τ − v ) 1W dv
x
+ ∫ + fTx (v ) 1W dv (x − τ ) x
Case 1. The system works throughout the interval [0, x ]
+
=ΦW eQWW (x ∨ τ ) 1W + ∫ fTx (v ) eQWW (x − τ − v ) 1W dv, 0
A (τ , x ) = ΦW eQWW (x ∨ τ ) 1W ,
where (x − τ − v )+ = max(x − τ − v , 0).
where the symbol ∨ is a maximal operator, 1W is a unit vector (|W | × 1), ΦW is an initial working condition probability vector for the repairable system. The above formula can be obtained because of results presented in Section 2, considering the relationship of interval length τ and time instant x .
3.2. Availability with window τ containing interval [a, b] Now we consider the availability with window τ containing interval [a, b], whose formula is given as follows. Like considering the availability with window τ containing point x , two cases are considered similarly.
Case 2. The system has at least one failure in the interval [0, x ] (including at the failure states initially). Let Tx : =sup{t : t < x , X (t ) ∈ F}, which is the last failure time instant before time x . Now we find the distribution of Tx . Its distribution density function is, because the end of Tx must be followed by at least a working duration excessing the window length τ ,
Fig. 1. The case of 0 < v ≤ x − τ .
3
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
interval [a, b]. 4. Properties of two new indexes In this section, we shall simply discuss some properties and do some comparisons between two new availability indexes, which may provide some useful information on understanding them completely in both theoretical and practical sides. We can now have the following equalities and inequalities.
Fig. 2. The case of x > v > x − τ .
Case 1. The system works throughout interval [0, a],
A (τ , [a, b]): =ΦW eQWW (b ∨ τ ) 1W , where A (τ , [a, b]) indicates the availability with window τ containing interval [a, b].
(1) If x ∈ [a, b], then A (τ , x ) ≥ A (τ , [a, b]). (2) [a1, b1] ⊆ [a2 , b 2], then A (τ , [a1, b1]) ≥ A (τ , [a2, b2]). Both inequalities above can easily be followed by the inclusiveness of events, and which indicate the availability with window τ containing point x is greater than A (τ , [a, b]) for x ∈ [a, b]. Meanwhile, the second inequality tells us that the specified interval is less, the interval availability is larger. b − a = τ , then A (τ , x ) ≥ A ([a, b]). (3) If a < x < b , (4) A (τ , x ) and A (τ , [a, b]) are decreasing functions in τ , respectively. Both above can be obtained by considering the formulas presented in Section 3. (5) When x = a = b , then A (τ , x ) = A (τ , [a, b]). This equality is gotten by letting x = a = b in the formula of A (τ , [a, b]), which provides the relationship between two new availability indexes, i.e., the availability with window τ containing interval [a, b] will reduce to the availability with window τ containing point x when x = a = b . In fact, the availability with window τ containing interval [a, b] is an extension of the availability with window τ containing point x . (6) When τ = b − a , then A (τ , [a, b]) = A ([a, b]). The above formula holds because the given window length τ does not play any role, that is to say, the availability with window τ containing interval [a, b] will reduce to the conventional interval availability when τ = b − a . fory ∈ [a, a + τ ]. (7) If τ ≥ b − a , then A (τ , [a, y]) = const. The above formula holds because
Case 2. The system has at least one failure in interval [0, a] (including at the failure states initially), in this situation, we can take 3 cases into consideration: (1) When v < a < b < v + τ , i.e. we have (b − τ )+ < v < a , a
A (τ , [a, b]) =
∫(b−τ )
+
fTa (v ) 1W dv.
(3) When v < a < v + τ < b , i.e. we have (a − τ )+ < v < {a ∧ (b − τ )+}, a ∧ (b − τ )+
A (τ , [a, b]) =
∫(a−τ )
+
fTa (v ) eQWW (b − τ − v ) 1W dv,
where the symbol ∧ is a minimal operator. (4) When v < v + τ < a < b, i.e. we have 0 < v < (a − τ ),
A (τ , [a, b]) =
∫0
(a − τ )
fTa (v ) eQWW (b − τ − v ) 1W dv.
All above results follow that the availability with window τ containing interval [a, b] requires the repairable Markov system sojourns in the working subset at least duration τ or (b − v ) for the cases of (b − τ )+ < v < a and 0 < v < a ∧ (b − τ )+, respectively, after time instant Ta . Summarizing Cases 1 and 2, we have gotten the closed form of A (τ , [a, b]) as follows. When b − a ≤ τ , we have
a
+
A (τ , [a, y]) = ΦW eQWW ( y ∨ τ ) 1W + ∫ fTa (v ) eQWW ( y − τ − v ) 1W dv 0
a
A (τ , [a, b]) = ΦW eQWW (b ∨ τ ) 1W + ∫ fTa (v ) 1W dv (b − τ )+
=ΦW eQWW τ 1W a
(b − τ )+
+ ∫ fTa (v ) 1W dv = const, 0
+ ∫ fTa (v ) eQWW (b − τ − v ) 1W dv (a − τ )+ +∫
(a − τ )+
0
when y ∈ [a, a + τ ]. (8) When τ = 0 , A (τ , x ) = A (x ).
fTa (v ) eQWW (b − τ − v ) 1W dv
=ΦW eQWW (b ∨ τ ) 1W a
+
This holds obviously, which indicates the availability with window τ containing point x will reduce to the conventional point availability when the window length τ = 0 . We can use another way to prove this
+ ∫ fTa (v ) eQWW (b − τ − v ) 1W dv, 0 which is a formula for the availability with window τ containing
Fig. 3. Some curves for A (τ , x ) .
4
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
Fig. 4. Some curves for A (τ , [a, b]) .
ΦW (s I − QWW )−11W + f *Tx (s )(s I − QWW )−11W = [ΦW + f *Tx (s )](s I − QWW )−11W = [ΦW + h* (s )](s I − QWW )−11W = [ΦW + ΦF (s I − Q FF )−1Q FW ][I − (s I − QWW )−1QWF (s I − Q FF )−1Q FW ]−1 (s I − QWW )−11W = [ΦW + ΦF (s I − Q FF )−1Q FW ][s I − QWW − QWF (s I − Q FF )−1Q FW ]−11W = A* (s ), which tells us that the Laplace transform of A (0, t ) is the same that of A (t ), this is to say, A (τ , x ) = A (x ) when τ = 0 . 5. Numerical examples A repairable system may be a computer center consisting of servers, may be an air condition system or a water pump system etc., which are repairable systems. For the applications, we can think the real situation is as follows. A customer requires the computer system being available at 10:00 o’clock exactly, a manager wants to guarantee to satisfy the customer's requirement more likely, she requires the computer center being available at least 30 min which must cover 10:00 o’clock. The probability of the manager's requirement is the availability with (x = 6, if we set 9:00 o’clock window τ (τ = 3) containing point x as an initial time, i.e., t = 0 ). The similar questions may be raised more in real life of the world. In the following, we just focus our attention in numerical examples, which may bring some intuitive understanding for two new indices in some contents. Suppose the repairable system can be modeled by a homogeneous continuous time Markov process with finite space, its infinitesimal generator of the Markov repairable system is
Fig. 5. Some curves for A (τ , [a, b]) for different b − a .
property as follows. It is well- known that
P (t ): =(P {X (t ) = j|X (0) = i}) = exp(Qt ), its Laplace transform is
⎛ s I − QWW − QWF ⎞−1 ⎛ UWW (s ) UWF (s )⎞ P*(s ): =(s I − Q)−1 = ⎜ ⎟. ⎟ : =⎜ s I − Q FF ⎠ ⎝ UFW (s ) UFF (s ) ⎠ ⎝ − Q FW In terms of the well-known results on the inverse of a partitioned matrix, we have
UWW (s ) = [s I − QWW − QWF (s I − Q FF )−1Q FW ]−1 ,
⎛− 4 1 2 1 ⎞ ⎜ 2 −3 1 0 ⎟, Q=⎜ ⎟ 1 1 − 4 2 ⎟ ⎜ ⎝ 0 1 1 − 2⎠
UFW (s ) = (sI − Q FF )−1Q FW UWW (s ). The Laplace transform of point availability at time t is
A* (s ) = ΦW U*WW (s ) 1W + ΦF U*FW (s ) 1W =[ΦW + ΦF (sI − Q FF )−1Q FW ] U*WW (s ) 1W =[ΦW + ΦF (sI − Q FF )−1Q FW ]
and the state space S = W ∪ F , where W = {1, 2} and F = {3, 4}, the initial probability vector of the repairable system is Φ = [ΦW , ΦF ] = [0.4, 0.5, 0.0, 0.1], ΦW = {0.5, 0.4} i.e., and ΦF = {0.0, 0.1}. By using the Maple, some curves for A (τ , x ) are presented as in Fig. 3. It is clear that A (τ , x ) is decreasing in the window length τ for fixed x, (x = 0.5 or 2.5) points (see Fig. 3(a)), and A (τ , x ) (τ = 0.5 and τ = 1) varies as x increases from 0 to 8, and they reach fixed values as x increases (see Fig. 3(b)). Some curves for A (τ , [a, b]) are also presented in Fig. 4. Fig. 4(a) shows the property of A (τ , [a, b]) when τ ≥ b − a the
.
[s I − QWW − QWF (s I − Q FF )−1Q FW ]−11W . On the other hand, based on the formula on the availability with window τ containing point x , we have, for τ = 0 ,
A (0, x ) = ΦW eQWW x 1W +
∫0
x
fTx (v ) eQWW (x − v ) 1W dv,
its Laplace transform is 5
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
gineering. Springer-Verlag London Limited; 2008. p. 789–805, [Chapter 48]. [3] Shen JY, Cui LR. Reliability performance for dynamic systems with cycles of K regimes. IIE Trans 2016;48:398–402. [4] Cui LR, Xie M. Availability analysis of periodically inspected systems with random walk model. J Appl Probab 2001;38:860–71. [5] Cui LR, Xie M. Availability of a periodically inspected system with random repair or replacement times. J Stat Plan Inference 2005;131:89–100. [6] Liu BL, Cui LR, Wen YQ, Shen JY. A performance measure for Markov system with stochastic supply patterns and stochastic demand patterns. Reliab Eng Syst Saf 2013;119:300–7. [7] Bao XZ, Cui LR. An analysis of availability for series Markov repairable system with neglected or delayed failures. IEEE Trans Reliab 2010;59:734–43. [8] Wang LY, Cui LR. Aggregated semi-Markov repairable systems with historydependent up and down states. Math Comput Model 2011;53:883–95. [9] Hawkes AG, Cui LR, Zheng ZH. Modeling the evolution of system reliability performance under alternative environments. IIE Trans 2011;43:1–12. [10] Cui LR, Du SJ, Hawkes AG. A study on a single-unit repairable system with state aggregations. IIE Trans 2012;44:1022–32. [11] Cui LR, Li HJ, Li JL. Markov repairable systems with history-dependent up and down states. Stoch Models 2007;23:665–81. [12] Cui LR, Du SJ, Zhang AF. Reliability measures for two-part partition of states for aggregated Markov repairable systems. Ann Oper Res 2014;212:93–114. [13] Liu BL, Cui LR, Wen YQ. Interval reliability for aggregated Markov repairable system with repair time omission. Ann Oper Res 2014;212:169–83. [14] Zheng ZH, Cui LR, Hawkes AG. A study on a single-unit Markov repairable system with repair time omission. IEEE Trans Reliab 2006;55:182–8. [15] Du SJ, Cui LR, Li HJ, Zhao XB. A study on joint availability for k out of n and consecutive k out of n points and intervals. Qual Technol Quant Manag 2013;10:179–91. [16] Csenki A. Joint availability of systems modeled by finite semi-Markov processes. Appl Stoch Models Data Anal 1994;10:279–93. [17] Sericola B. Interval-availability distribution of 2-state systems with exponential failures and phase-type repairs. IEEE Trans Reliab 1994;43:335–43. [18] Finkelstein MS. Multiple availability on stochastic demand. IEEE Trans Reliab 1999;48:19–24. [19] Csenki A. Joint availability of systems modeled by finite semi-Markov processes. Appl Stoch Models Data Anal 1994;10:279–93. [20] Csenki A. An integral equation approach to the interval reliability of systems modeled by finite semi-Markov processes. Reliab Eng Syst Saf 1995;47:37–45. [21] Csenki A. Joint interval reliability for Markov systems with an application in transmission line reliability. Reliab Eng Syst Saf 2007;92:685–96. [22] Cui LR, Du SJ, Liu BL. Multi-point and multi-interval availabilities. IEEE Trans Reliab 2013;62:811–20. [23] Barlow RE, Proschan F. Mathematical theory of reliability. New York: Wiely; 1965. [24] Wu XY, Hillston J. Mission reliability of semi-Markov systems under generalized operational time requirements. Reliab Eng Syst Saf 2015;140:122–9. [25] Colquhoun, D D, Hawkes AG. On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Philos Trans R Soc Lond – Ser B: Biol Sci 1982;300:1–59.
A (τ , [0.3, 0.5]) decrease when τ ∈ [0.2, 4] and then A (1, [x, x + 0.5]) varies as x increases in interval [0, 4]. Some curves for A (τ , [a, b]) are also presented in Fig. 5 for different b − a. Fig. 5 shows that A (τ , [a, b]) does not increase as b − a increases, note that b − a ≤ τ (0.1 < 1 and 0.3 < 1) must be satisfied, because of the requirement in Definition 2. 6. Conclusion As the increasing of various performance features of the systems, the indexes to describe these performance features are needed naturally, especially in reliability field. However, how to define and calculate them make these indexes become some very interesting and difficult questions. In the paper, two new indexes for repairable systems on measuring availability are proposed, which extend the conventional point and interval availability indexes, respectively. The authors believe that the two new invented indexes can be widely used in repairable systems, which enrich definitely our understanding on availability of repairable systems. The calculation formulas for the two indexes are given for Markovian repairable systems, and some properties and the relationship between the two indexes and conventional point and interval availability indexes are discussed, the numerical examples and discussions on some figures appeared in the examples are shown to illustrate some features of two new indexes. The similar problems on semi-Markov repairable systems will be our future work, and we hope them to be finished under our hard work. Acknowledgements This work is supported by the NSF of China under Grants 71631001 and 71371031. References [1] Cui LR, Xie M, Loh H. Inspection schemes for general systems. IIE Trans 2004;36:817–25. [2] Cui LR. Maintenance models and optimizationHandbook of performability en-
6
Contents lists available at ScienceDirect
Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
New interval availability indexes for Markov repairable systems ⁎
Lirong Cui , Jianhui Chen, Bei Wu School of Management & Economics, Beijing Institute of Technology, Beijing 100081, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Markov repairable system Interval availability Index Laplace transform
In maintenance field, there are many indexes such as instantaneous availability, steady state availability, interval availability and others, which have been widely used to describe the various properties and performance of repairable systems on maintenance. However, all the present availability indexes still cannot cover some situations which people are interested in. In the paper, two new interval availability indexes are introduced for Markov repairable systems, which is named as an availability with a given window length and containing a specified point or interval. The new interval availability is the probability that the system is working during a given window length and this window must contains a specified point or interval. Their calculation formulas are presented in matrix forms by using the Laplace transform. Some properties on the two indexes are discussed briefly, and numerical examples are shown to illustrate the features of the two new indexes, finally conclusions are given. These two new indices can become the conventional interval availability when the given window length is zero or equal to the specified interval length. The results in the paper may be used to measure the performance of repairable system more deeply and detail.
1. Introduction In reliability field, the related measure indexes such as reliability, availability and safety etc. play an important role to describe system performances. Any theoretical research and engineering work in reliability must be related to the reliability measure indexes because these measure indexes can tell people if they have reached the prespecified target or if the state of system operating can meet the mission requirements. On the other hand, each system including that in reliability field has many performances, how to measure these performances of the system must rely on some indexes. As the technology developments, people want to know more deep and detail information on systems, new indexes are needed naturally. The similar situation in maintenance is faced at present, some new measure indexes are needed to describe the situation or information people are interested in. In fact, many indexes have been used in describing the various properties of repairable systems. For example, several reliability indexes for repairable systems were proposed in [1–3]. The reliability and availability indexes are an important topic, much literature can be found in this direction, for example, references [4,5]. Specially, on aggregated repairable systems, many availability indexes including the conventional pointwise and interval availabilities have been studied, for example, references [6–10]. The availability indexes are closely related to maintenance models because the maintenance models can describe the evolution processes of the repairable systems, while the availability ⁎
indexes can describe the performance levels of the repairable systems under given models. For example, Cui et al. [11] introduced a new changeable state repairable system, and then the conventional availability indexes were given in their paper. Recently, Liu et al. [12] and Cui et al. [13] studied some availability indexes under aggregated stochastic models. In fact, it is not an easy task to give the formulas for these indexes, and the related work is mainly for single-unit systems, Markov repairable systems and semi-Markov repairable systems, for example, Refs. [14–16]. Availability is an important measure of performance for repairable systems, which has been developed into more detail indexes such as pointwise availability, interval availability and joint availability and so forth. In general, the availability is a probability that the repairable system will be able to operate within the tolerances at a given instant time or interval, which are for point-wise and interval availabilities. The definition of interval availability has some different ways. The interval availability commonly refers to the expected proportion of time for which the repairable system is available over some interval of time. Sericola [17] defined it as the fraction of operation time over a finite observation period, which is a random variable. Finkelstein [18] defined a multiple availability as the probability of a system being in an operating state at each moment of demand. Csenki [19] also studied multiple availability, but in his work multiple availability was called joint availability. Csenki [20,21] also studied the related interval availability. Cui et al. [22] proposed the multi-point, multi-interval,
Corresponding author. E-mail address: [email protected] (L. Cui).
http://dx.doi.org/10.1016/j.ress.2017.03.016
0951-8320/ © 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: Cui, L., Reliability Engineering and System Safety (2017), http://dx.doi.org/10.1016/j.ress.2017.03.016
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
some intuitive understanding on usage of the two new availability indexes. Finally, the conclusions are summarized in Section 6.
and mixed multi-point-interval availabilities, which are extensions of the single point and interval availabilities. A similar concept is an interval reliability, Barlow and Proschan [23] defined it as the probability that at a specified time the system is operating and will continue to operate for an interval of duration. (The authors would like to call this interval reliability as interval availability not interval reliability especially for repairable systems, although the classical book and mathematical reliability pioneers named it. On the other hand, the name change for interval reliability can meet the feature of the pointwise availability, multiple availability and repairable systems, because when the interval length becomes into zero, multiple into single, the interval reliability becomes into the pointwise availability, but the multi-interval availability become into interval reliability.). Wu and Hillston [24] studied two kinds reliability measures under a semiMarkov process situation, one is the system must remain operational continuously for a minimum time within the given mission time interval, while the other required the total operational time of the system within the mission time window must be greater than a given value. Both reliability measures are defined by suing the fraction of working time to window length, and the first required the continuous working time, but the second is not required. As mentioned previously, there are many indexes for repairable systems, but they still cannot cover all practical situations, For example, people are interested in the probability that the system, which may be a computer system or an air condition system or service system etc., is available (working) at 9:00 o’clock or so within 15 min, or the probability that the system is available (working) during a window interval whose interval length is a half hour but it must cover the interval from 9:00 o’clock to 9:15 o’clock. This kind of problem is what our present paper will study, we name the first probability as an availability with window τ containing point x , which is the probability that a repairable system works throughout an interval window which has at least a length τ and contains a given point x . The second probability is called an availability with window τ containing interval [a, b], which is the probability that a repairable system works throughout an interval window which has at least a length τ and contains a given interval [a, b]. Both concepts are extensions of pointwise availability and interval availability, which definitely enrich the indexes measures in maintenance. For application examples of two indices introduced above, we can look at a water supply system. The requirement of customers on this system is: it must be working from 9:00 o’clock to 9:15 o’clock, but the manager wants to guarantee this customer requirement to be satisfied more likely, he/she requires the water supply system must be working at least a half hour which must also cover the time interval [9:00, 9:15]. This management strategy can provide more tolerance for the customer requirement. The author believe that the two new availability indices can be used in more real situations, particularly in increasing mission success probability of systems. In the paper, after doing the introduction of two new availability indexes, their calculation formulas are given by using the Laplace transform and matrix techniques, meanwhile, some basic results referenced in Colquhoun and Hawkes [25] are used in the process of derivations. Furthermore, some properties of the two new availability indexes are discussed briefly, which may provide more information and understanding on usage of the two new availability indexes. The rest of the paper is organized as follows. Section 2 provides the assumptions on Markov repairable model and definitions of two new availability indexes, including their mathematical expressions, and some basic results referenced in Colquhoun and Hawkes [25] are presented. In Section 3, the main results of the paper, the calculation formulas of the two new availability indexes, are given. Some properties and comparisons of the two new availability indexes are considered in Section 4, which includes the relationship between the two new availability indexes and the conventional pointwise and interval availabilities. Section 5 presents some numerical examples to illustrate the properties of the two new availability indexes, which may provide
2. Assumptions, definitions and preliminaries Suppose there is a repairable system following a homogeneous continuous time Markov process {X (t ), t ≥ 0} with finite state space S which contains a working subset and a failure subset, i.e., S = W ∪ F , where S = {1, 2, …, n} and F = {n + 1, n + 2, …, n + m}. The infinitesimal generator of the process {X (t ), t ≥ 0} is Q , in terms of working and failure states, which can be divided into four blocks, i.e.
⎛Q QWF ⎞ Q = ⎜ WW ⎟. ⎝ Q FW Q FF ⎠ The definitions of two new availability indexes to be discussed throughout the paper are given as follows. Definition 1. The probability that a repairable system works throughout an interval window, which has at least a length τ and contains a given point x , is called as an availability with window τ containing point x . The availability with window τ containing point x can be expressed in a formula way as
A (τ , x ): =P { ∃ c ≥ 0, system works in interval [c, c + τ ] and c ≤ x ≤ c + τ}, where τ is a required interval length, x is a given instant. The availability with window τ containing point x can be used to describe an interval availability in which the interval must have at least a length τ and contain a given time instant x , it is an extension of point availability. When the interval length τ = 0 , the availability with window τ containing point x becomes into a conventional point availability at time instant x . To understand the concept of availability with window τ containing point x , we can image that there is a window with length τ to move towards right or left containing point x and at least at one moment the repairable system works throughout the window, the probability for this situation is the availability with window τ containing point x . Definition 2. The probability that a repairable system works throughout an interval window which has at least a length τ and contains a given interval [a, b], is called as an availability with window τ containing interval [a, b]. The availability with window τ containing interval [a, b] can be expressed in a formula way as
A (τ , [a, b]): =P { ∃ c ≥ 0, system works in interval [c, c + τ ] and c ≤ a ≤ b ≤ c + τ}, where τ is a required interval length, [a, b] is a given interval. Similarly, the availability with window τ containing interval [a, b] can be used to describe an interval availability in which the interval must have at least a length τ and contain a given interval [a, b], this new availability concept is an extension of interval availability. When the interval length τ = b − a , the availability with window τ containing interval x becomes into a conventional interval availability at interval [a, b]. To understand the concept of availability with window τ containing point [a, b], we can image that there is a window with length τ to move towards right or left covering interval [a, b], and at least at one moment the repairable system works throughout the window, the probability for this situation is the availability with window τ containing interval [a, b]. Because of using Laplace transform throughout the paper, here we give its definition below and specify the Laplace transform on matrix for elementwise transforms. The Laplace transform for function f (t ) is defined as follows, 2
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
f * (s ) =
∫0
∞
fTx (t ) = ( fTx,1 (t ), fTx,2 (t ), …, fTx, n (t ))
e−st f (t ) dt .
=(h1 (t ), h2 (t ), …, hn (t )) eQWW τ = h (t ) eQWW τ ,
Colquhoun and Hawkes [25] introduced two important quantities in studying the stochastic properties of ion channel, which are given below. W
where fTx, i (t ) (i = 1, 2, …, n ) is a probability density function of Tx and X (T x+) = i ∈ W , and the Laplace transform of (h1 (t ), h2 (t ), …, hn (t )) is as follows.
pij (t ): =P {system remains within W throughout time 0 to time t ,
h* (s ): =(h1* (s ), h 2* (s ), …, hn* (s ))
and is in state j at time t | in state i at time 0),
∞ =ΦW ∑n =1 [G*WF (s ) G*FW (s )]n + ΦF G*FW (s )
i, j ∈ W .
∞
∑n =0 [G*WF (s ) G*FW (s )]n
After some manipulations, we can get, in matrix form,
=ΦW {[I − G*WF (s ) G*FW (s )]−1 − I}
⎞ ⎛ PWW (t ): =⎜W pij (t ) ⎟ = exp(QWW t ), ⎠|W |×| W | ⎝
+ ΦF G*FW (s ){[I − G*WF (s ) G*FW (s )]−1 =[ΦW + ΦF G*FW (s )][I − G*WF (s ) G*FW (s )]−1 − ΦW
where |W | denotes the cardinality of set W , and its Laplace transform can be given
=[ΦW + ΦF (s I − Q FF )−1Q FW ]
P*WW (s ) = (s I − QWW
[I − (s I − QWW )−1QWF (s I − Q FF )−1Q FW ]−1 − ΦW ,
)−1,
where ΦF is an initial failure condition probability vector for the repairable system. Note that the distribution formula of Tx contains the required window length working duration, which can be omitted in the following derivations. This formula is obtained because the Tx may consist of any number (1, 2, …, ∞) of cycles W → F → W if the repairable system begins at the working subset or the Tx may consist of a transition F → W followed by any number (0, 1, …, ∞) of cycles W → F → W if the repairable system begins at the failure subset. On the other hand, all these events are exclusive. Thus we can get the distribution vector of Tx . In terms of Tx = v , we can discuss the new interval availability in two cases, which are shown in Figs. 1 and 2. Let v be a realization of Tx , then we have, if 0 < v ≤ x − τ ,
where I is a unit matrix with proper dimension, and s is the Laplace variable. Note that PWW (t ) is not simply a submatrix of P(t ) which is the transition probability matrix for the Markov process. Another quantity defined by Colquhoun and Hawkes [25] is
gij (t ): = lim [P {system stays in W from time 0 to time t , and leave W Δt→0
for state j between t and t + Δt| in state i at time 0}/ Δt ],
i ∈ W , j ∈ F.
Similarly, in matrix form, we have
GWF (t ): =(gij (t ))|W |×| F | = PWW (t ) QWF ,
A (τ , x ) =
and its Laplace transform can be given as
∫0
(x − τ )
fTx (v ) eQWW (x − τ − v ) 1W dv,
and if x > v > x − τ , which is shown in Fig. 2. Based on Fig. 2, we have
G*WF (s ) = (s I − QWW )−1QWF . Similarly, we can define PFF (t ), P*FF (s ), GFW (t ) and G*FW (s ), which will be used in the paper. Note that gij (t ) is a probability density function, in general, gij (t ) is not, itself, a proper probability density function, without the initial and ending probabilities considered.
x
A (τ , x ) =
∫(x−τ )
+
fTx (v ) 1W dv,
where A (τ , x ) indicates the availability with window τ containing point x , and (x − τ )+ = max(x − τ , 0). The above result follows that the availability with window τ containing point x requires the repairable Markov system stays in the working subset at least duration (x − v ) or τ for the cases of 0 < v ≤ x − τ and x > v > x − τ , respectively, after time instant Tx . Summarizing Cases 1 and 2, we have gotten the closed form of A (τ , x ) as follows,
3. Main results First we consider the availability with window τ containing point x , whose formula is given as follows. 3.1. Availability with window τ containing point x There are two cases to be considered for the availability with window τ containing point x .
(x − τ )+
A (τ , x ) = ΦW eQWW (x ∨ τ ) 1W + ∫ 0
fTx (v ) eQWW (x − τ − v ) 1W dv
x
+ ∫ + fTx (v ) 1W dv (x − τ ) x
Case 1. The system works throughout the interval [0, x ]
+
=ΦW eQWW (x ∨ τ ) 1W + ∫ fTx (v ) eQWW (x − τ − v ) 1W dv, 0
A (τ , x ) = ΦW eQWW (x ∨ τ ) 1W ,
where (x − τ − v )+ = max(x − τ − v , 0).
where the symbol ∨ is a maximal operator, 1W is a unit vector (|W | × 1), ΦW is an initial working condition probability vector for the repairable system. The above formula can be obtained because of results presented in Section 2, considering the relationship of interval length τ and time instant x .
3.2. Availability with window τ containing interval [a, b] Now we consider the availability with window τ containing interval [a, b], whose formula is given as follows. Like considering the availability with window τ containing point x , two cases are considered similarly.
Case 2. The system has at least one failure in the interval [0, x ] (including at the failure states initially). Let Tx : =sup{t : t < x , X (t ) ∈ F}, which is the last failure time instant before time x . Now we find the distribution of Tx . Its distribution density function is, because the end of Tx must be followed by at least a working duration excessing the window length τ ,
Fig. 1. The case of 0 < v ≤ x − τ .
3
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
interval [a, b]. 4. Properties of two new indexes In this section, we shall simply discuss some properties and do some comparisons between two new availability indexes, which may provide some useful information on understanding them completely in both theoretical and practical sides. We can now have the following equalities and inequalities.
Fig. 2. The case of x > v > x − τ .
Case 1. The system works throughout interval [0, a],
A (τ , [a, b]): =ΦW eQWW (b ∨ τ ) 1W , where A (τ , [a, b]) indicates the availability with window τ containing interval [a, b].
(1) If x ∈ [a, b], then A (τ , x ) ≥ A (τ , [a, b]). (2) [a1, b1] ⊆ [a2 , b 2], then A (τ , [a1, b1]) ≥ A (τ , [a2, b2]). Both inequalities above can easily be followed by the inclusiveness of events, and which indicate the availability with window τ containing point x is greater than A (τ , [a, b]) for x ∈ [a, b]. Meanwhile, the second inequality tells us that the specified interval is less, the interval availability is larger. b − a = τ , then A (τ , x ) ≥ A ([a, b]). (3) If a < x < b , (4) A (τ , x ) and A (τ , [a, b]) are decreasing functions in τ , respectively. Both above can be obtained by considering the formulas presented in Section 3. (5) When x = a = b , then A (τ , x ) = A (τ , [a, b]). This equality is gotten by letting x = a = b in the formula of A (τ , [a, b]), which provides the relationship between two new availability indexes, i.e., the availability with window τ containing interval [a, b] will reduce to the availability with window τ containing point x when x = a = b . In fact, the availability with window τ containing interval [a, b] is an extension of the availability with window τ containing point x . (6) When τ = b − a , then A (τ , [a, b]) = A ([a, b]). The above formula holds because the given window length τ does not play any role, that is to say, the availability with window τ containing interval [a, b] will reduce to the conventional interval availability when τ = b − a . fory ∈ [a, a + τ ]. (7) If τ ≥ b − a , then A (τ , [a, y]) = const. The above formula holds because
Case 2. The system has at least one failure in interval [0, a] (including at the failure states initially), in this situation, we can take 3 cases into consideration: (1) When v < a < b < v + τ , i.e. we have (b − τ )+ < v < a , a
A (τ , [a, b]) =
∫(b−τ )
+
fTa (v ) 1W dv.
(3) When v < a < v + τ < b , i.e. we have (a − τ )+ < v < {a ∧ (b − τ )+}, a ∧ (b − τ )+
A (τ , [a, b]) =
∫(a−τ )
+
fTa (v ) eQWW (b − τ − v ) 1W dv,
where the symbol ∧ is a minimal operator. (4) When v < v + τ < a < b, i.e. we have 0 < v < (a − τ ),
A (τ , [a, b]) =
∫0
(a − τ )
fTa (v ) eQWW (b − τ − v ) 1W dv.
All above results follow that the availability with window τ containing interval [a, b] requires the repairable Markov system sojourns in the working subset at least duration τ or (b − v ) for the cases of (b − τ )+ < v < a and 0 < v < a ∧ (b − τ )+, respectively, after time instant Ta . Summarizing Cases 1 and 2, we have gotten the closed form of A (τ , [a, b]) as follows. When b − a ≤ τ , we have
a
+
A (τ , [a, y]) = ΦW eQWW ( y ∨ τ ) 1W + ∫ fTa (v ) eQWW ( y − τ − v ) 1W dv 0
a
A (τ , [a, b]) = ΦW eQWW (b ∨ τ ) 1W + ∫ fTa (v ) 1W dv (b − τ )+
=ΦW eQWW τ 1W a
(b − τ )+
+ ∫ fTa (v ) 1W dv = const, 0
+ ∫ fTa (v ) eQWW (b − τ − v ) 1W dv (a − τ )+ +∫
(a − τ )+
0
when y ∈ [a, a + τ ]. (8) When τ = 0 , A (τ , x ) = A (x ).
fTa (v ) eQWW (b − τ − v ) 1W dv
=ΦW eQWW (b ∨ τ ) 1W a
+
This holds obviously, which indicates the availability with window τ containing point x will reduce to the conventional point availability when the window length τ = 0 . We can use another way to prove this
+ ∫ fTa (v ) eQWW (b − τ − v ) 1W dv, 0 which is a formula for the availability with window τ containing
Fig. 3. Some curves for A (τ , x ) .
4
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
Fig. 4. Some curves for A (τ , [a, b]) .
ΦW (s I − QWW )−11W + f *Tx (s )(s I − QWW )−11W = [ΦW + f *Tx (s )](s I − QWW )−11W = [ΦW + h* (s )](s I − QWW )−11W = [ΦW + ΦF (s I − Q FF )−1Q FW ][I − (s I − QWW )−1QWF (s I − Q FF )−1Q FW ]−1 (s I − QWW )−11W = [ΦW + ΦF (s I − Q FF )−1Q FW ][s I − QWW − QWF (s I − Q FF )−1Q FW ]−11W = A* (s ), which tells us that the Laplace transform of A (0, t ) is the same that of A (t ), this is to say, A (τ , x ) = A (x ) when τ = 0 . 5. Numerical examples A repairable system may be a computer center consisting of servers, may be an air condition system or a water pump system etc., which are repairable systems. For the applications, we can think the real situation is as follows. A customer requires the computer system being available at 10:00 o’clock exactly, a manager wants to guarantee to satisfy the customer's requirement more likely, she requires the computer center being available at least 30 min which must cover 10:00 o’clock. The probability of the manager's requirement is the availability with (x = 6, if we set 9:00 o’clock window τ (τ = 3) containing point x as an initial time, i.e., t = 0 ). The similar questions may be raised more in real life of the world. In the following, we just focus our attention in numerical examples, which may bring some intuitive understanding for two new indices in some contents. Suppose the repairable system can be modeled by a homogeneous continuous time Markov process with finite space, its infinitesimal generator of the Markov repairable system is
Fig. 5. Some curves for A (τ , [a, b]) for different b − a .
property as follows. It is well- known that
P (t ): =(P {X (t ) = j|X (0) = i}) = exp(Qt ), its Laplace transform is
⎛ s I − QWW − QWF ⎞−1 ⎛ UWW (s ) UWF (s )⎞ P*(s ): =(s I − Q)−1 = ⎜ ⎟. ⎟ : =⎜ s I − Q FF ⎠ ⎝ UFW (s ) UFF (s ) ⎠ ⎝ − Q FW In terms of the well-known results on the inverse of a partitioned matrix, we have
UWW (s ) = [s I − QWW − QWF (s I − Q FF )−1Q FW ]−1 ,
⎛− 4 1 2 1 ⎞ ⎜ 2 −3 1 0 ⎟, Q=⎜ ⎟ 1 1 − 4 2 ⎟ ⎜ ⎝ 0 1 1 − 2⎠
UFW (s ) = (sI − Q FF )−1Q FW UWW (s ). The Laplace transform of point availability at time t is
A* (s ) = ΦW U*WW (s ) 1W + ΦF U*FW (s ) 1W =[ΦW + ΦF (sI − Q FF )−1Q FW ] U*WW (s ) 1W =[ΦW + ΦF (sI − Q FF )−1Q FW ]
and the state space S = W ∪ F , where W = {1, 2} and F = {3, 4}, the initial probability vector of the repairable system is Φ = [ΦW , ΦF ] = [0.4, 0.5, 0.0, 0.1], ΦW = {0.5, 0.4} i.e., and ΦF = {0.0, 0.1}. By using the Maple, some curves for A (τ , x ) are presented as in Fig. 3. It is clear that A (τ , x ) is decreasing in the window length τ for fixed x, (x = 0.5 or 2.5) points (see Fig. 3(a)), and A (τ , x ) (τ = 0.5 and τ = 1) varies as x increases from 0 to 8, and they reach fixed values as x increases (see Fig. 3(b)). Some curves for A (τ , [a, b]) are also presented in Fig. 4. Fig. 4(a) shows the property of A (τ , [a, b]) when τ ≥ b − a the
.
[s I − QWW − QWF (s I − Q FF )−1Q FW ]−11W . On the other hand, based on the formula on the availability with window τ containing point x , we have, for τ = 0 ,
A (0, x ) = ΦW eQWW x 1W +
∫0
x
fTx (v ) eQWW (x − v ) 1W dv,
its Laplace transform is 5
Reliability Engineering and System Safety xxx (xxxx) xxx–xxx
L. Cui et al.
gineering. Springer-Verlag London Limited; 2008. p. 789–805, [Chapter 48]. [3] Shen JY, Cui LR. Reliability performance for dynamic systems with cycles of K regimes. IIE Trans 2016;48:398–402. [4] Cui LR, Xie M. Availability analysis of periodically inspected systems with random walk model. J Appl Probab 2001;38:860–71. [5] Cui LR, Xie M. Availability of a periodically inspected system with random repair or replacement times. J Stat Plan Inference 2005;131:89–100. [6] Liu BL, Cui LR, Wen YQ, Shen JY. A performance measure for Markov system with stochastic supply patterns and stochastic demand patterns. Reliab Eng Syst Saf 2013;119:300–7. [7] Bao XZ, Cui LR. An analysis of availability for series Markov repairable system with neglected or delayed failures. IEEE Trans Reliab 2010;59:734–43. [8] Wang LY, Cui LR. Aggregated semi-Markov repairable systems with historydependent up and down states. Math Comput Model 2011;53:883–95. [9] Hawkes AG, Cui LR, Zheng ZH. Modeling the evolution of system reliability performance under alternative environments. IIE Trans 2011;43:1–12. [10] Cui LR, Du SJ, Hawkes AG. A study on a single-unit repairable system with state aggregations. IIE Trans 2012;44:1022–32. [11] Cui LR, Li HJ, Li JL. Markov repairable systems with history-dependent up and down states. Stoch Models 2007;23:665–81. [12] Cui LR, Du SJ, Zhang AF. Reliability measures for two-part partition of states for aggregated Markov repairable systems. Ann Oper Res 2014;212:93–114. [13] Liu BL, Cui LR, Wen YQ. Interval reliability for aggregated Markov repairable system with repair time omission. Ann Oper Res 2014;212:169–83. [14] Zheng ZH, Cui LR, Hawkes AG. A study on a single-unit Markov repairable system with repair time omission. IEEE Trans Reliab 2006;55:182–8. [15] Du SJ, Cui LR, Li HJ, Zhao XB. A study on joint availability for k out of n and consecutive k out of n points and intervals. Qual Technol Quant Manag 2013;10:179–91. [16] Csenki A. Joint availability of systems modeled by finite semi-Markov processes. Appl Stoch Models Data Anal 1994;10:279–93. [17] Sericola B. Interval-availability distribution of 2-state systems with exponential failures and phase-type repairs. IEEE Trans Reliab 1994;43:335–43. [18] Finkelstein MS. Multiple availability on stochastic demand. IEEE Trans Reliab 1999;48:19–24. [19] Csenki A. Joint availability of systems modeled by finite semi-Markov processes. Appl Stoch Models Data Anal 1994;10:279–93. [20] Csenki A. An integral equation approach to the interval reliability of systems modeled by finite semi-Markov processes. Reliab Eng Syst Saf 1995;47:37–45. [21] Csenki A. Joint interval reliability for Markov systems with an application in transmission line reliability. Reliab Eng Syst Saf 2007;92:685–96. [22] Cui LR, Du SJ, Liu BL. Multi-point and multi-interval availabilities. IEEE Trans Reliab 2013;62:811–20. [23] Barlow RE, Proschan F. Mathematical theory of reliability. New York: Wiely; 1965. [24] Wu XY, Hillston J. Mission reliability of semi-Markov systems under generalized operational time requirements. Reliab Eng Syst Saf 2015;140:122–9. [25] Colquhoun, D D, Hawkes AG. On the stochastic properties of bursts of single ion channel openings and of clusters of bursts. Philos Trans R Soc Lond – Ser B: Biol Sci 1982;300:1–59.
A (τ , [0.3, 0.5]) decrease when τ ∈ [0.2, 4] and then A (1, [x, x + 0.5]) varies as x increases in interval [0, 4]. Some curves for A (τ , [a, b]) are also presented in Fig. 5 for different b − a. Fig. 5 shows that A (τ , [a, b]) does not increase as b − a increases, note that b − a ≤ τ (0.1 < 1 and 0.3 < 1) must be satisfied, because of the requirement in Definition 2. 6. Conclusion As the increasing of various performance features of the systems, the indexes to describe these performance features are needed naturally, especially in reliability field. However, how to define and calculate them make these indexes become some very interesting and difficult questions. In the paper, two new indexes for repairable systems on measuring availability are proposed, which extend the conventional point and interval availability indexes, respectively. The authors believe that the two new invented indexes can be widely used in repairable systems, which enrich definitely our understanding on availability of repairable systems. The calculation formulas for the two indexes are given for Markovian repairable systems, and some properties and the relationship between the two indexes and conventional point and interval availability indexes are discussed, the numerical examples and discussions on some figures appeared in the examples are shown to illustrate some features of two new indexes. The similar problems on semi-Markov repairable systems will be our future work, and we hope them to be finished under our hard work. Acknowledgements This work is supported by the NSF of China under Grants 71631001 and 71371031. References [1] Cui LR, Xie M, Loh H. Inspection schemes for general systems. IIE Trans 2004;36:817–25. [2] Cui LR. Maintenance models and optimizationHandbook of performability en-
6