Distribution and availability for aggregated second-order semi-Markov ternary system with working time omission

Author’s Accepted Manuscript Distribution and Availability for Aggregated Second-Order Semi-Markov Ternary System with Working Time Omission He Yi, Lirong Cui www.elsevier.com/locate/ress

PII: DOI: Reference:

S0951-8320(16)30888-2 http://dx.doi.org/10.1016/j.ress.2016.11.025 RESS5707

To appear in: Reliability Engineering and System Safety Received date: 30 April 2016 Revised date: 23 September 2016 Accepted date: 27 November 2016 Cite this article as: He Yi and Lirong Cui, Distribution and Availability for Aggregated Second-Order Semi-Markov Ternary System with Working Time O m i s s i o n , Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2016.11.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Distribution and Availability for Aggregated Second-Order Semi-Markov Ternary System with Working Time Omission He Yia & Lirong Cuib. (a.School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China b.

School of Management and Economics, Beijing Institute of Technology, Beijing, China)

Abstract: Discrete repairable degradation systems can be modeled by a homogeneous discrete-time second-order semi-Markov chain with finite state space for theoretical and practical sake. In the present paper, the state space can be divided into three subclasses, namely excellent states, good states (middle states) and failure states. The transitions among states may be driven by degradations and inside shocks of the system, repair actions, self-healings, energy supplement and other recovery ways. The Z-transform is used to give distributions for some interesting problems such as distributions of an I-period (oscillation & working time) and a gap between I-periods, and instantaneous and steady availabilities of the system. Finally, some numerical examples are given to illustrate the results obtained in the paper. The work may be used in reliability and maintenance analysis of discrete time repairable systems. Keywords Second-order semi-Markov, multi-state system, discrete repairable degradation system, time omission, distribution and availability

Notation：

Qik ,hj (n)

Elements in the (i  k  k ) th row and (h  k  j ) th column of semi-Markov kernel Q(n) of a system.

QXY ,YZ (n)

Sub-matrixes of Q(n) whose element Qik ,hj (n) satisfies i  X , k , h Y ,

j  Z and X , Y , Z {A , B , D} .

 A , B ,D

a

State space of a second-order semi-Markov ternary system. Subsets of  and   A

B

Corresponding author: Email: [email protected] 1

D.

k A , kB , kD

Number of states in subsets A , B , D .

N

Set of nature numbers and N  {0,1, 2,3 } .

N*

Set of positive integers and N *  N /{0}  {1, 2,3 } . 2

2

pikX,hj, X (n)

Conditional probabilities that the system is in state h after the l  1 th jump, then jump to state j at time n and the system doesn’t leave subset X in

[0, n] condition on the initial state pair (i, k ), i, k , h, j  X , X {A , B , D} .

PX 2 , X 2 (n)

Matrixes whose elements are pikX,hj, X (n), i, k , h, j  X , X {A , B , D} .

gikXY,hj,YZ (n)

Conditional probabilities that the system is in state h after the l  1 th jump

2

2

and the system doesn’t leave subset Y until the l th jump to state j at time

n condition on the initial state pair (i, k ), i  X , k , h Y , j  Z where X , Y , Z {A , B , D}, Y  Z .

GXY ,YZ (n)

XY ,YZ Matrixes whose elements are gik ,hj (n), i  X , k , h  Y , j  Z .

G*XY ,YZ ( z )

The Z-transform of matrix G XY ,YZ (n), G *XY ,YZ ( z ) 

S *XY ,YZ ( z ), L*XY ,YZ ( z)



G n 0

nY

S *XY ,YZ ( z )   GXY ,YZ (n) z  n , L*XY ,YZ ( z )  n 0

GXY ,YZ , S XY ,YZ , LXY ,YZ

*

*

XY ,YZ





n  nY 1

(n) z  n .

GXY ,YZ (n) z  n .

*

Abbreviations of GXY ,YZ (1), S XY ,YZ (1), LXY ,YZ (1) .

Initial-state probability of the state pair (i, j ), i, j Ω.

pij (0)

Vector whose elements pij (0) satisfies i  X , j  Y and X , Y {A , B , D} .

PXY (0)

Conditional probabilities that the system jumps to a state in A from D

PB A , PDA

or B condition on that it’s the beginning of an I-period.

PAB , PAD

Conditional probabilities that the system jumps to a state in D or B from

A condition on that it’s the beginning of a gap between two I-periods. uXY

Vector with dimension k X kY 1 whose elements are all ones, X , Y {A , B , D} .

2

1. Introduction Multi-state system is a hot topic in the field of reliability and great achievements have been made recently [1-8]. Aggregated Markov repairable system which was first proposed [9] and refined [10] by Colquhoun and Hawkes takes an important role in the Multi-state systems. Cui et al. [11] considered changeable states depending on the immediately preceding state and applied aggregated Markov repairable system to the field of reliability. Hawkes et al. [12] further studied Markov system operating under alternating regimes which is obviously reasonable in practice. Maintenance behaviors of a single-unit system were also investigated by Cui et al. [13] under assumption that maintenance subsystem can also fail and be repaired at constant rates. Liu et al. [14] further discussed repairman's multiple behaviors which are partly decided by operation conditions of cold standby systems. Without require of memoryless, semi-Markov systems can cover more practical situations than Markov system. For, example, Csenki [15] further developed the renewal-theoretical approach and joint distribution of sojourn times were given for semi-Markov processes. Ball et al. [16] extended aggregation Markov process in ion channel theory to corresponding semi-Markov process with definition of empirical burst which has more practical value than a theoretical one. Limnios and Oprisan [17] generalized semi-Markov processes in discrete state space to general state space with applications in reliability and more general processes were presented. Barbu et al. [18] defined a discrete-time semi-Markov process with proof of properties, algorithm for renewal equations and expressions for reliability measures. Wang and Cui [19] discussed performance measures of aggregated semi-Markov repairable system considering changeable states as in [11]. Limnios [20] investigated reliability measures of continuous and discrete time semi-Markov processes with general state space. Yi et al. [21] extended continuous Markov process in [10] to discrete time semi-Markov process and some distributions which can be applied to the field of reliability were given. In recent years, high-order Markov and semi-Markov systems have also been taken into consideration. Berchtold and Raftery [22] reviewed a mixture transition distribution model for high-order Markov chains and described in detail how the model can be applied not only to analysis of wind directions, DNA sequences and social behaviors, but also to modeling of bilinear time series, financial and economics time series and some biological data. They also 3

first introduced the basic principle and presented some extensions. Siu et al. [23] applied high-order Markov chain to describe the risk of portfolio and comparison with first-order model was given to explain how first-order model underestimated the VaR if the “true” model is the second order one. D’Amico et al. compared first and second order semi-Markov models with real wind speed data [24] and some performance and reliability measures was given for second [25] and higher [26] order semi-Markov chain. There are different kinds of high-order semi-Markov chains. For high-order semi-Markov chains in state and duration defined in [25], the knowledge of the several (  2) previous visited states and durations between those states is needed for probabilistic forecasting. For indexed high-order semi-Markov chains in [26], the indexed knowledge of those states and durations is needed. For high-order semi-Markov chains in state in [17], only the knowledge of those states is needed. Those high-order models also have many applications in real word, for example, they can be used to predict wind speeds, share prices, electricity consumption demands and they can also be used to describe time series data of device performances in the field of reliability. Time interval omission problems have received great attention for many practical situations. Hawkes et al. [27] considered open times and shut times too short to be detected by device in ion channel theory and derived exact distributions of some durations for the first time. Ball [28] developed two methods to investigate the degree of temporal clustering of bursts for single channel gating incorporating time interval omission. The and Timmer [29] discussed the dead-time of minimal detectable interval length whose wrong chosen was proved to lead to biased estimates. Time interval omission was introduce to the field of reliability by Zheng et al. [30] and reliability measures were given for Markov repairable system with both constant and random critical omission time. Bao and Cui [31] further studied delayed failure which means repair interval of the system can be only detected after exceeding the critical value. Cui et al. [32] investigated availability and interval availability for aggregated Markov systems without omitted failures, with omitted failures and with omitted or delayed failures. Liu et al. [33] first proposed interval reliability and interval unreliability for aggregated Markov system with time interval omission. Zhang et al. [34] derived point and interval availability indexes for the semi-Markov repairable system with neglected working and repair 4

times. In the present paper, a repairable degradation model will be considered in which the evolution of the process is assumed to be a discrete time semi-Markov chain second-order in state, this evolution may result from repair actions, self-recovery and deteriorations of the system. The degradation states are defined as some subclasses of states, that is to say, the state space can be divided into three subclasses, namely excellent states, good states (middle states) and failure states. This definition for deterioration processes can cover many realistic systems with degradations. For example, health conditions of human can be described by healthy, sub healthy and sick, water pollutions can be described by lower or no pollution, middle pollution and serious pollution, states of equipment can be described by perfect functioning, partial failure and complete failure. In order to know the behaviors of deterioration process, I-periods, which include oscillations and working times, and gaps between I-periods, are defined. An I-period is a duration that the system is in excellent or good states which starts from the moment of transition to an excellent state from a failure state directly or through some middle states and ends at the moment of transition from an excellent state to a failure state directly or through some middle states. A gap is the duration of time between two consecutive I-periods. In practice, many devices need some time to recovery from failures and don’t work steadily immediately after repaired, for example, our computer needs some minutes to start after closed. This kind of system state is different from working but obviously not failure and it can be described by oscillation in this paper which is similar to working time interval omission happening only after a failure. The current work is based on Colquhoun and Hawkes [7], but in the present paper, we consider discrete time rather than continuous one because data in the field of reliability are mostly discrete and discrete models are much easier to handle for engineers and we also employ semi-Markov second-order model in state rather than first-order Markov model for practical sake. For example, to predict a wind speed in the future, we need not only the present wind scale, but also the previous wind scale. Future share prices depend not only on present price, but also the previous price. Health condition predictions for human need not only health condition at present, but also the previous condition. Performance predictions of 5

equipments need not only present performance, but also the previous performance level. We also considered working time interval omission because enough long working time is a good way to testify the systems’ recovery from failures that means if the system fails very soon after it’s repaired, it’s reasonable for us to believe that the system is not really repaired. Results in this paper can be applied to many real word systems in the field of reliability. For example, a power station often have multi performance levels and future performance levels are often determined not only by the present performance level, but also by the past performance records. Those levels can be divided to three parts, levels higher than a demand level (which mean perfect functioning of the power station), levels less than a demand level (which mean partial failure of the power station), and the zero level (which means complete failure of the station). Once a power station fails completely, the users won’t believe it’s repaired until it functions perfectly for enough long time. In an I-period which will be defined later, the power station never performs at the zero level and is thought to work steadily and efficiently after an oscillation during which perfect functioning periods are always too short. Only during the working time after an oscillation in each I-period, the power system is thought to be available. In a gap between two I-periods, performance levels of the power station never meet the demand level and users often complain during this period. Similar analysis can also be given for network systems whose performances can be described by traffic data and heating systems whose performances can be described by heating capacity. The remainder of this paper is organized as follows: In Section 2, the model and its assumptions are given explicitly and some important definitions are presented which will be used throughout the paper. In Section 3, some stochastic properties of discrete repairable degradation systems under three deterioration subclass sets are discussed. In Section 4, instantaneous availability and steady availability of the discrete repairable degradation system are given. In Section 5, some numerical examples are showed to illustrate the results obtained in Sections 3 and 4. Section 6 provides conclusions finally.

2. Models and Important Terms It is assumed that the degradation process of the system follows a time-homogeneous, 6

irreducible, connected, discrete-time, semi-Markov chain {Z k , k  0,1, } second order in state on the finite state space Ω {1, 2,

, kΩ} . Then we consider the following chains

associated to {Z k , k  0,1, } : The embedded chain {J k , k  0,1, } with state space Ω {1, 2,

, kΩ} , where J k is

the state of the system at the k th jump time. The chain {Sk , k  0,1, },{X k , k  1, 2, } with state space N  {0,1, we

suppose

(0  S0 

that

Sk

 Sk  Sk 1 

is

the

state

of

the

system

at

the k th

) , and X 0  0 , X k  Sk  Sk 1 (k  1, 2,

, } , where jump

time

) is the sojourn

time in state J k 1 before the k th jump. Then the stochastic chain {( J k , Sk ), k  0,1, } has a homogeneous discrete time semi-Markov process, whose semi-Markov kernel can be defined as follows,

Q(n) : (Qi j (n))k where

k 2 Ω2 Ω

,

Ω2 : {(i, j ) : i, j  Ω} is the state space whose elements are state pairs

( J k 1 , J k )  (i, j ) of the original stochastic chain {( J k , Sk ), k  0,1, }. For i  (i, k ),

j   (h, j ) , Qi j (n) is in the (i  kΩ  k ) th row and (h  kΩ  j ) th column of matrix Q(n) which can be defined as follows,

Qi j (n) : Qik ,hj (n)  P( J l  h, J l 1  j, X l 1  n| J l 1  i, J l  k )  P( J 0  h, J1  j , X 1  n| J 1  i, J 0  k ) =P( J1  j, X 1  n| J 1  i, J 0  k )1{h  k }. The state space can be divided into three subsets, that is,

Ω A

B

D,

where the subsets A , B and D denote excellent state subset, good state subset, and failure state subset, and the numbers of their states are k A , kB and kD , respectively. If the state space is divided into three subsets, the corresponding discrete time second order semi-Markov kernel can be divided as follows, 7

0 0 0 0 0 0  QA 2 , A 2 (n) QAA , AB (n) QAA , AD (n)    0 0 0 Q ( n ) Q ( n ) Q ( n ) 0 0 0   AB ,B A AB ,B B AB ,B D  0 0 0 0 0 0 QAD ,DA ( n) QAD ,DB ( n) QAD ,DD ( n)    0 0 0 0 0 0  QB A , AA (n) QB A , AB (n) QB A , AD (n)    0 0 0 Q ( n ) Q ( n ) Q ( n ) 0 0 0 Q ( n)   B B ,B A B B ,B D B 2 ,B 2 ,  0 0 0 0 0 0 QB D ,DA (n) QB D ,DB ( n) QB D ,DD (n)    0 0 0 0 0 0  QDA , AA (n) QDA , AB (n) QDA , AD (n)    0 0 0 QDB ,B A (n) QDB ,B B ( n) QDB ,B D ( n) 0 0 0    0 0 0 0 0 0 QDD ,DA ( n) QDD ,DB ( n) QD 2 ,D 2 ( n)  

where, take QAB ,B D (n) for example, Qik ,hj (n) is an element of QAB ,B D (n) when,

i  A , k  B , h  B , j D . Then the probability that the system remains within a specified subset of states, A say, throughout the time from 0 to n (n  N  , N  =N /{0}) as follows,

pikA,hj, A (n) : P( J l 1  h, J l  j, Sl  n, l  n, J1 , J 2 , 2

2

, J l  A | J 1  i, J 0  k ),

i, k , h, j  A . (2.1)

And easily we have, A 2 ,A 2 ik , hj

p

n

(n)    P( J l 1  h, J l  j , Sl  n, l  n, J1  s, X 1  m, J1 , J 2 , sA m 1 n

   P( J l 1  h, J l  j , Sl  n, l  n, J 2 , sA m 1

, J l  A | J 1  i, J 0  k )

, J l  A | J 1  i, J 0  k , J1  s, X 1  m)

 P( J1  s, X 1  m| J 1  i, J 0  k ) n

   P( J l 1  h, J l  j , Sl  n, l  n,, J 2 , sA m 1

, J l  A | J 0  k , J1  s , X 1  m)

 P( J1  s, X 1  m| J 1  i, J 0  k ) n

   pksA ,hj, A (n  m)  QikA,ks, A (m), 2

2

2

2

i, k , h, j  A .

sA m 1

(2.2) In matrix notation, we have, for equation (2.2),

PA 2 , A 2 (n)  QA 2 , A 2 (n)  PA 2 , A 2 (n), n  N * ,

(2.3)

where pikA,hj, A (n) is an element of PA 2 , A 2 (n) when i, k , h, j  A , and for matrices 2

2

F1 (n) and F2 (n) whose elements are functions, " " is a symbol for a convolution operation defined as follows,

8

n

F1 (n)  F2 (n)   F1 (n  m) F2 (m). m 1

Similarly, for n  N * we have,

PB 2 ,B 2 (n)  QB 2 ,B 2 (n)  PB 2 ,B 2 (n),

PD 2 ,D 2 (n)  QD 2 ,D 2 (n)  PD 2 ,D 2 (n).

Another important quantity is defined as the probability that describes the probability of the system staying within the subset of states A



for a duration n (n  N ) after a

transition from a state outside A (from subset D , say) and then leaving A in the next step for a state outside A (for subset B , say), i.e.

gikDA,hj, AB (n) : P( J l 1  h, J l  j , Sl  n, l  n, J1 , J 2 ,

, J l 1 A | J 1  i, J 0  k ), i D , k , h  A , j  B . (2.4)



It is clear that for n  N , we have

gikDA,hj, AB (n) 

n

  P( J

sA { j } m 1

l 1

 h , J l  j , S l  n , J 1  s , X 1  m, J 2 ,

n

   P( J l 1  h, J l  j , Sl  n, J 2 , sA m 1

, J l 1 A | J 1  i, J 0  k )

, J l 1 A | J 1  i, J 0  k , J1  s, X 1  m)

 P( J1  s, X 1  m| J 1  i, J 0  k )  P ( J 0  h, J1  j , X 1  n| J 1  i, J 0  k ) n

   P( J l 1  h, J l  j , Sl  n, J 2 , sA m 1

, J l 1 A | J 0  k , J1  s, X 1  m)

 P( J1  s, X 1  m| J 1  i, J 0  k )  P ( J1  j , X 1  n| J 1  i, J 0  k )1{k  h} n

   g ksAA,hj, AB (n  m)  QikDA,ks, AA (m)  QikDA,kj, AB (n)1{k h} ,

i D, k , h  A , j  B .

sA m 1

(2.5) In matrix form, we have,

GDA , AB (n)  QDA , AA (n)  GAA , AB (n)  QDA , AB (n),

(2.6)

AA , AB where GAA , AB (n) is a matrix with element gik ,hj (n) which means the system staying



within the subset of states A for a duration n (n  N ) after a transition from a state in

A and then leaving A in the next step for a state outside A (in subset B , say), i.e. gikAA,hj, AB (n) : P( J l 1  h, J l  j, Sl  n, l  n, J1, J 2 , 9

, J l 1 A | J 1  i, J 0  k ) i, k , h  A , j  B .

(2.7) 

It is clear that for n  N , we have, n

gikAA,hj, AB (n)    P( J l  j , Sl  n, l  n, J l 1  A | J 1  i, J 0  k , J l 2  s, J l 1  h, Sl 1  m, sA m 1

l  1  m, J1 , J 2 ,

, J l  2 A )  P ( J l  2  s, J l 1  h, Sl 1  m, l  1  m, J1 ,

, J l  2 A | J 1  i, J 0  k )

J2 , n

   P( J l  j , Sl  n, l  n, J l 1  A | J l  2  s, J l 1  h, Sl 1  m, J l 1 , J l  2  A )  sA m 1

P( J l  2  s, J l 1  h, Sl 1  m, l  1  m, J1 , J 2 ,

, J l  2 A | J 1  i, J 0  k )

n

   P( J l  j , X l  n  m,| J l  2  s, J l 1  h)  P ( J l  2  s, J l 1  h,, Sl 1  m, l  1  m, sA m 1

J1 , J 2 , , J l  2 A | J 1  i, J 0  k ) n

   pikA, sh, A (m)  QshAA,hj, AB (n  m), 2

i, k , h  A , j  B .

2

sA m 1

(2.8) In matrix form, we have,

GAA , AB (n)  PA 2 , A 2 (n)  QAA , AB (n). In the sequel, the Z-transform will be used, which is defined as, for a series {an , n  1, 2, } , 

an ( z )   an z  n , (| z | R0 ) . n 0

The Z-transform for PA 2 , A 2 (n) is

PA 2 , A 2 ( z)  [ I  QA 2 , A 2 ( z)]1. 

(2.7)

A ,A To show that I  QA 2 , A 2 ( z ) is invertible, first we have 0  pik ,hj (n)  1, i, k , h, j  2

2



p

 A , then by taking Z-transforms, we get PA 2 , A 2 ( z ) , whose elements are

n 0

A 2 ,A 2 ik , hj

( n) z  n .

For z  1 , we have, 



 pikA,hj, A (n) z n   pikA,hj, A (n) z n 0

2

2

2

n 0



2

n



 z n 0

n



z z 1

.

Therefore, PA 2 , A 2 ( z ) exists for z  1 . Then by convolution theorem of the Z-transform, we have, 10

PA 2 , A 2 ( z)[ I  QA 2 , A 2 ( z)]  I , 

which means I  QA 2 , A 2 ( z ) is invertible for z  1 . Similarly, for the Z-transforms of GAA , AB (n), GDA , AB (n) , we have   1  GAA , AB ( z )  [ I  QA 2 , A 2 ( z )] QAA , AB ( z ),

(2.8)

    GDA , AB ( z )  QDA , AA ( z )G AA , AB ( z )  QDA , AB ( z )

(2.9)

  1    QDA , AA ( z )[ I  QA 2 , A 2 ( z )] QAA , AB ( z )  QDA , AB ( z ),

 with elements gik ,hj ( z ) , say, 



n 1

n 1

 gikAA,hj, AB (n)   P( Jl 1  h, Jl  j, Sl  n, J1, J 2 ,

, J l 1 A | J 1  i, J 0  k )  gikAA,hj, AB  (1),

which are the probabilities allowing for any number of transitions within A states before  the system eventually leaves A for B . The matrix GAA , AB (1) with elements that are AA , AB  (1) will be denoted simply as GAA , AB throughout the the transition probabilities gik ,hj

paper, for brevity. The matrix, Q (1) with elements that are the transition probabilities 

Qi j (1)   Qi j (n) will be denoted simply as Q , throughout the paper, for brevity. If the n 1

state space is divided into three subsets, Q can be divided as follows,  QA 2 , A 2   0  0   QB A , AA    Q  Q (1)   Q (n)   0 n 1  0   QDA , AA  0   0 

QAA , AB

QAA , AD

0

0

0

0

0

0

QAB ,B A

QAB ,B B

QAB ,B D

0

0

0

0

QB A , AB

QB A , AD

0 0

0 0

0 0

QAD ,DA 0

QAD ,DB 0

0

0

QB B ,B A

QB 2 ,B 2

QB B ,B D

0

0

0

0

0

0

0

QB D ,DA

QB D ,DB

QDA , AB

QDA , AD

0

0

0

0

0

0

0

QDB ,B A

QDB ,B B

QDB ,B D

0

0

0

0

0

0

0

QDD ,DA

QDD ,DB

  0  QAD ,DD   0  0  . QB D ,DD   0  0   QD 2 ,D 2 

0

0

Thus we have, from (2.8) and (2.9),  1 * 1 GAA , AB  [ I  QAA , AA (1)] QAA , AB (1)  ( I  QAA , AA ) QAA , AB ,

(2.10)

GDA , AB  QDA , AB GAA , AB  QDA , AB  QDA , AB ( I  QAA , AA )1QAA , AB  QDA , AB , (2.11) AA , AB  gikAA,hj, AB * (1), where elements of GAA , AB , GAA , AB are denoted respectively as gik ,hj

11

gikDA,hj, AB  gikDA,hj, AB * (1), which are transition probabilities from A to B and transition probabilities from D to A and then to B .

3. Stochastic Properties of Discrete Repairable Degradation Ternary Systems It is well-known that the initial probability and one-step transition probability matrix can determine all properties and movements of a Markov chain. Let PΩ2 (k )  ( PAA (k ), PAB (k ),

PAD (k ), PB A (k ), PB B (k ), PB D (k ), PDA (k ), PDB (k ), PDD (k )), k  N

be

the

probability

vector whose element pij (k ) is the probability that state pair (i, j ) occurs at the k th jump which means that the system is in state i after the k  1 th jump and then jumps to state

j at the k th jump. Provided that the initial probability vector PΩ2 (0) and

semi-Markov

kernel

Q ( n)

are

PΩ2 (k )

given,

can

be

calculated

easily

by

PΩ2 (k )  PΩ2 (0)Q k . Specially, PΩ2 () means steady-state vector which must exists for the embedded finite-state Markov chain, and it can be solved in terms of the following set of equations,

 PΩ2 ()Q  PΩ2 (),   PΩ2 ()u  1,

(3.1)

where u is a unit vector with dimension (kΩ2 1) . Some systems such as ion channel systems in [10] tend to reach steady states very soon after they start, so we often investigate properties of these systems in their steady states. For the other systems which tend to reach steady states slowly, we choose to discuss their properties from the very beginning or from a certain step k . In the paper we focus mainly on two issues: (1) the duration of a specified sequence of events, (2) some reliability measures. In the sequel, these distributions of durations and reliability measures will be discussed respectively. First we define an important term called type I period, referred to briefly as “I-period”, which will be used throughout this paper. Definition 1. An I-period is defined as a duration in which the system starts at subset A after leaving subset D , may be via subset B , and has some durations within subset 12

EA

B , and ends at subset A before entering into subset D , may be via subset B ,

i.e., an I-period is a sojourn time within subset E since leaving subset D for subset A to leaving subset A , may be via subset B , entering subset D , which can be depicted in Figures 1 and 2 Definition 2. An oscillation within A is defined as a possible duration in the beginning of an I-period during which the system transits between A and B frequently, which means the system stays in A in less than nA units of time as shown in Figure 2. Some other terms are also given in Figure 2. Remark. The definition of oscillation is similar to time omission problem in ion channel theory. The subsets A , B and D denote excellent state subset, good state subset, and failure state subset. States in B can be regarded as intermediate states of A and D which have the same performance levels as the states in A or D before them. However, if the system stays in A in no more than nA units of time in the beginning of an I-period, it’s hard for us to believe that the system is really repaired. Every time the system is in failure states in D , we believe that it doesn’t work until staying in A for more than nA units of time.

A

E

A

D

D B

B I-period

Figure 1. Possible transition paths for an I-period.

(a)

A state subset B

nA

D

(b)

Working time

Up

Oscillation Oscillation

Down

I-period

gap between I-periods

I-period

Figure 2. The possible transitions of system between “up” and “down” states, some I-periods and gaps within and 13

between I-periods.

In Figure 2, the upper part shows the possible transitions of system between the three subsets of states defined in Section II. A are “excellent” and “I-period” states, B are “good” and “gap within I-period” states, D are “failure” and “gap between I-period” states,

E= A

B are “excellent or good” states; the lower part shows possible transitions

between “up” and “down” of the system and other concepts too. As shown in Figure 2, at any time, the system is either in an I-period or a gap between two I-periods and an I-period consists of an oscillation and a period of working time. The system is thought to be in failure states in gaps between two I-periods, and it works effectively in periods of working time while functions unsteadily in oscillations. In the following, some stochastic properties on the semi-Markov degradation system second order in state will be discussed in terms of I-periods and other concepts given above.

3.1 The length of I-period From definition 1, it’s easy to find that in I-periods, the system is in either excellent or good states and an I-period can begin from D to A directly or via B and end from A to

D directly or via B . Then there will be four kinds of I-period to be considered and we define them as I1，I 2，I3，I 4 , respectively, for any I-period, it must be just one kind of the four. Then the Z-transforms of the four kinds of I-period length distribution, which are denoted as

f I1 ( z)，f I2 ( z)，f I3 ( z)，f I4 ( z)， can be summed over together to get the

Z-transform of I-period length distribution. The conditional probabilities that the system jumps to a state in A from D or B condition on that it’s the beginning of an I-period which begins at the k th jump and steady states are given respectively as follows,

PDA

PDA (k )  , k 1    PDA (k )uDA   PDB ( j )GDB ,B A (k  j )uB A  j 0  PDA () ,  P (  ) u + P (  ) G u  DA DA DB DB , B A B A 

14

at the kth jump (k  0),

at steadystate,

PB A

k 1  PDB ( j )GDB ,B A (k  j )   j 0  k 1  P ( k ) u  PDB ( j )GDB ,B A (k  j )uB A    DA DA j 0   PDB ()GDB ,B A ,  P (  ) u + P (  ) G u DA DB DB ,B A B A  DA

at the kth jump (k  0),

at steadystate.

Note that when k  0 , an I-period begins at the k th jump means an I-period begins at the k 1

initial states and we assume that

P j 0

DB

( j )GDB ,B A (k  j )uB A  0 for k  0 .

An I-period may consist of any number (1, 2,

, ) of transitions from A to B and

back to A except the last time when the system leaves B for D rather than A . We should sum over these probability and not set z to one for periods in subsets A and B that constitute the I-period. The four cases are considered as follows, Case 1: the I-period begins with transition D  B  A and ends in transition

A  B D 

 i   f I1 ( z )   PB A [GB A , AB ( z )GAB ,B A ( z )] GB A , AB ( z )G AB ,B D ( z ) uB D i 0

 1    PB A [ I  GB A , AB ( z )G AB ,B A ( z )] GB A , AB ( z )G AB ,B D ( z ) uB D

 PB A f I1 ( z )uB D , 





1





where f I1 ( z)  [ I  GB A , AB ( z)GAB ,B A ( z)] GB A , AB ( z)GAB ,B D ( z) is a matrix to represent 

main part of f I1 ( z ) and similar matrixes will be defined in the following. Case 2: the I-period begins with transition D  B  A and ends in transition A  D  1   f I2 ( z)  PB A [ I  GB A , AB ( z)GAB ,B A ( z )] GB A , AD ( z )uAD  PB A f I 2 ( z )uAD ;

Case 3: the I-period begins with transition D  A and ends in transition A  B  D    1  f I3 ( z)  PDA GDA , AB ( z )[ I  GAB ,B A ( z )GB A , AB ( z )] GAB ,B D uB D  PDA f I3 ( z )uB D ;

Case 4: the I-period begins with transition D  A and ends in transition A  D    1   f I4 ( z )  PDA GDA , AB ( z )[ I  G AB ,B A ( z )GB A , AB ( z )] GAB ,B A ( z )GB A , AD ( z ) uAD    PDA GDA , AD ( z ) uAD  PDA f I 4 ( z ) uAD .

Then the Z-transform of I-period length distribution is given by

f I ( z)  f I1 ( z)  f I2 ( z)  f I3 ( z)  f I4 ( z), 15

(3.2)

and the distribution of I-period length f I (n) can be given by the inverse Z-transform of

d  f I ( z) . z 1 dz

f I ( z ) , and the mean I-period length is mI   lim

3.2 The length of oscillation An oscillation is made up with commutative short sojourns in excellent states and sojourns in good states. In oscillations, the system is in either excellent or good states. According to the four kinds of I-period, there will be four kinds of oscillation to be considered. The kind of oscillation is decided by which kind of I-period it’s in and we define them as O1 , O2 , O3 , O4 , respectively. For any oscillation, it must be just one kind of the four. An I-period may consist of any number (1, 2,

, ) of transitions from A to B and

back to A except the last time when the system leaves B

for D rather than A .

Sojourn in A in an oscillation must be no more than nA units of time and then the end of an oscillation may not happen (if the I-period don’t have sojourn in A more than nA units of time), happen at the beginning of the last opening in the I-period (if the I-period have only one sojourn in A more than nA units of time and it’s the last sojourn in A ), or happen within the I-period (if the I-period have at least one sojourn in A and the first one more than nA units of time is not the last sojourn in A ). We should sum over these probability and not set z to one for periods in subsets A and B that constitute the oscillation. GAB ,B D is the probability that condition on previous state in A and present state in B , the system leaves B

for D after some transition in B

and uB D is a

kB kD 1 vector whose elements are all 1 . The oscillation may consist of any number (1, 2,

, ) of sojourn times in A in less

than nA units of time which must be summed over together and the Z-transforms of the distributions of sojourn time in A no more and more than nA units of time can be written as follows, respectively, 16

S

 B A , AB

nA

( z )=  GB A , AB (n) z , S n

n 0

LB A , AB ( z )=





n  nA 1

 DA , AB

nA

( z )=  GDA , AB (n) z  n , n 0

GB A , AB (n) z  n , LDA , AB ( z )=





n  nA 1

GDA , AB (n) z  n .

Similarly, the probability of sojourn in A no more and more than nA units of time can be written as follows, respectively,  DA , AB

SB A , AB =S

nA

(1)   GB A , AB (n), SDA , AB  S

LB A , AB =LDA , AB (1) 

 DA , AB

n 0





n  nA 1

nA

(1)=  GDA , AB ( n), n 0

GB A , AB (n), LDA , AB  LDA , AB (1)=





n  nA 1

GDA , AB ( n).

With z not set to one for periods in subsets A and B that constitute the oscillation, we have the Z-transform of the length of oscillation in four cases: Then the Z-transforms of the four kinds of oscillation length distribution which are denoted     as fO1 ( z )，fO2 ( z )，fO3 ( z )，fO4 ( z )，can be summed over together to get the Z-transform of

oscillation length distribution. The four cases are considered as follows, Case 1: the I-period begins with transition D  B  A and ends in transition

A  B D 



 i   fO1 ( z )   PB A [ SB A , AB ( z )G AB ,B A ( z )] SB A , AB ( z )G AB ,B D uB D   PB A [ SB A , AB ( z ) i 0

i 0





 i   i  G AB ,B A ( z )] LB A , AB G AB ,B D uB D   PB A [ SB A , AB ( z )G AB ,B A ( z )]  i 0 k 0

k

LB A , AB G AB ,B A (GB A , AB GAB ,B A ) GB A , AB GAB ,B D uB D  1   PB A [ I  SB A , AB ( z )G AB ,B A ( z )] [ SB A , AB ( z )  LB A , AB  LB A , AB G AB ,B A 

( I  GB A , AB G AB ,B A ) 1 GB A , AB ]GAB ,B D uB D  PB A f O1 ( z )uB D ; Case 2: the I-period begins with transition D  B  A and ends in transition A  D  1  fO2 ( z )  PB A [ I  SB A , AB ( z )GAB ,B A ( z )] [ SB A , AD ( z )  LB A , AD  LB A , AB GAB ,B A

( I  GB A , AB GAB ,B A )1GB A , AD ]uAD  PB A fO2 ( z )uAD ; Case 3: the I-period begins with transition D  A and ends in transition A  B  D    1 fO3 ( z )  PDA SDA , AB ( z )[ I  G AB ,B A ( z ) SB A , AB ( z )] [G AB ,B D  G AB ,B A LB A , AB ( I  G AB ,B A

 GB A , AB ) 1 G AB ,B D ]uB D  PDA LDA , AB ( I  GAB ,B A GB A , AB ) 1GAB ,B D uB D  PDA fO3 ( z )uB D ; Case 4: the I-period begins with transition D  A and ends in transition A  D 17

   1   fO4 ( z )  PDA SDA , AB ( z )[ I  G AB ,B A ( z ) SB A , AB ( z )] {G AB ,B A ( z ) SB A , AD ( z )  G AB ,B A LB A , AD  + GAB ,B A LB A , AB ( I  GAB ,B A GB A , AB )1GAB ,B A GB A , AD }uAD  PDA [ SDA , AD ( z ) 

LDA , AD +LDA , AB ( I  GAB ,B A GB A , AB ) 1GAB ,B A GB A , AD ]uAD  PDA f O4 ( z )uAD . Then the Z-transform of oscillation length distribution is as follows,

fO ( z)  fO1 ( z)  fO2 ( z )  fO3 ( z )  fO4 ( z ).

(3.3)

The distribution of oscillation length fO (n) can be given by the inverse Z-transform of

d  fO ( z ) . z 1 dz

fO ( z ) , and the mean oscillation length is mO   lim

3.3 The length of working time (non-oscillation time in I-period) A period of working time is made up with a long sojourn in excellent states and commutative sojourns in excellent states and good states. In working time, the system is in either excellent or good state. According to the four kinds of I-period, there will be four kinds of working time to be considered. The kind of working time is decided by which kind of I-period it’s in and we define them as W1,W2 ,W3 ,W4 , respectively. For any period of working time, it must be just one kind of the four. Then the Z-transforms of the four kinds of     working time length distribution, which are denoted as fW1 ( z )，fW2 ( z )，fW3 ( z )，fW4 ( z )，

can be summed over together to get the Z-transform of working time length distribution. Similarly to oscillation, with z not set to one for periods in subsets A and B that constitute the working time, we have the Z-transform of the length of working time in four cases: Case 1: the I-period begins with transition D  B  A and ends in transition

A  B D 



fW1 ( z )   PB A ( SB A , AB G AB ,B A )i SB A , AB GAB ,B D uB D   PB A ( SB A , AB GAB ,B A )i i 0

i 0





  LB A , AB ( z )G AB ,B D uB D   PB A ( SB A , AB GAB ,B A )i LB A , AB ( z )GAB ,B A ( z ) i 0 k 0

 B A , AB

[G

( z )G

 AB ,B A

( z )] GB A , AB ( z )GAB ,B D uB D k

18

  PB A ( I  SB A , AB GAB ,B A )1[ SB A , AB  LB A , AB ( z )  LB A , AB ( z )GAB ,B A ( z )   1   [ I  GB A , AB ( z )GAB ,B A ( z )] GB A , AB ( z )]GAB ,B D uB D  PB A fW1 ( z ) uB D ;

Case 2: the I-period begins with transition D  B  A and ends in transition A  D  fW2 ( z )  PB A ( I  SB A , AB GAB ,B A )1{SB A , AD  LB A , AD ( z )  LB A , AB ( z )GAB ,B A ( z )   1   [ I  GB A , AB ( z )GAB ,B A ( z )] GB A , AD ( z )}uAD  PB A fW2 ( z ) uAD ;

Case 3: the I-period begins with transition D  A and ends in transition A  B  D  fW3 ( z )  PDA SDA , AB ( I  GAB ,B A SB A , AB ) 1{G AB ,B D  G AB ,B A LB A , AB ( z )[ I  GAB ,B A ( z )   1 GB A , AB ( z )]1 GAB ,B D }uB D  PDA LDA , AB ( z )[ I  GAB ,B A ( z )GB A , AB ( z )] G AB ,B D uB D

 PDA fW3 ( z )uB D ; Case 4: the I-period begins with transition D  A and ends in transition A  D

fW4 ( z ) PDA SDA , AB ( I  GAB ,B A SB A , AB ) 1{GAB ,B A SB A , AD  GAB ,B A LB A , AD ( z )  GAB ,B A   1    LB A , AB ( z )[ I  GAB ,B A ( z )GB A , AB ( z )] G AB ,B A ( z )GB A , AD ( z )}uAD  PDA {SDA , AD   1    LDA , AD ( z )  LDA , AB ( z )[ I  GAB ,B A ( z )GB A , AB ( z )] G AB ,B A ( z )GB A , AD ( z )}uAD

 PDA fW4 ( z )uAD . Then Z-transform of working time length distribution is as follows,

fW ( z )  fW1 ( z )  fW2 ( z )  fW3 ( z )  fW4 ( z ).

(3.4)

The distribution of working time length fW (n) can be given by the inverse Z-transform of

fW ( z ) , and the mean working time length is mW   lim z 1

d  fW ( z ) . dz

3.4 The length of gap between I-periods A gap between I-periods is made up with commutative sojourns in good states and failure states. In gap between I-periods, the system is in either good or failure states. A gap between I-periods can begin from A to D directly or via B and end from D to A directly or via B . Then there will be four kinds of gap to be considered and we define them as

G1,G2 ,G3 ,G4 , respectively, for any gap between I-periods, it must be just one kind of the four. The conditional probabilities that the system jumps to a state in D or B from A 19

condition on that it’s the beginning of a gap between two I-periods which begins at the k th jump and steady state are given respectively as follows,

PAD

PAB

PAD (k )   P (k )u  P ( k )G AD AB AB ,B D uB D  AD  PAD ()    PAD ()uAD +PAB ()G AB ,B D uB D PAB (k )   P (k )u  P ( k )G AD AB AB ,B D uB D  AD  PAB ()    PAD ()uAD +PAB ()G AB ,B D uB D

,

at the kth jump (k  0),

,

at steadystate,

,

at the kth jump (k  0),

,

at steadystate.

Then the Z-transforms of the four kinds of gap between I-periods length distribution, which     are denoted as fG1 ( z )，fG2 ( z )，fG3 ( z )，fG4 ( z )，can be summed over together to get the

Z-transform of gap length distribution. The four cases are considered as follows, Case 1: the last I-period ends in transition A  B  D and the next one begins with transition D  B  A 

   i   fG1 ( z )   PAB G AB ,B D ( z )[GB D ,DB ( z )GDB ,B D ( z )] GB D ,DB ( z )GDB ,B A ( z ) uB A i 0

   1    PAB GAB ,B D ( z )[ I  GB D ,DB ( z )GDB ,B D ( z )] GB D ,DB ( z )GDB ,B A ( z ) uB A

 PAB fG1 ( z )uB A ; Case 2: the last I-period ends in transition A  B  D and the next one begins with transition D  A    1   fG2 ( z)  PAB GAB ,B D ( z )[ I  GB D ,DB ( z )GDB ,B D ( z )] GB D ,DA ( z )uDA  PAB f G2 ( z )uDA ;

Case 3: the last I-period ends in transition A  D and the next one begins with transition

D B  A    1  fG3 ( z)  PADGAD ,DB ( z )[ I  GDB ,B D ( z )GB D ,DB ( z )] GDB ,B A ( z )uB A  PAD f G3 ( z ) uB A ;

Case 4: the last I-period ends in transition A  D. and the next one begins with transition

DA    1   fG4 ( z )  PAD GAD ,DB ( z )[ I  GDB ,B D ( z )GB D ,DB ( z )] GDB ,B D ( z )GB D ,DA ( z ) uDA    PAD GAD ,DA ( z )uDA  PAD f G4 ( z ) uDA .

Then the Z-transform of distribution the length of gap between I-periods is as follows, 20

fG ( z)  fG1 ( z )  fG2 ( z )  fG3 ( z )  fG4 ( z ).

(3.5)

The distribution of gap length between I-periods fG (n) can be given by the inverse

d  fG ( z ) . z 1 dz

 Z-transform of fG ( z ) , and the mean gap length between I-periods is mG   lim

4. Availability of Discrete Repairable Degradation Ternary Systems To analyze the reliability measures such as instantaneous availability and steady availability of our semi-Markov system second order in state from the beginning of an I-period, a new stochastic process can be defined as follows,

1 Yn   0

the system is in working period at n, the system is in oscillation or gap at n .

Denote the event that an I-period in steady state begins with state i*  (i, j ), i  A , j  A , at n  0 as H i* . I-periods, oscillations, working periods and gaps between I-periods from the I-period we choose as beginning can be denoted as {I n },{On },{Wn },{Gn }, n  N  , respectively, and obviously I n  On  Wn which can be shown in Figure 3. O1

W1

G1

O2

I1

W2

O2

G2

I2

Wn

Gn

In

Figure 3. I-periods, oscillation periods, working periods and gaps between I-periods.

4.1 Instantaneous availability Instantaneous availability of the system at time n is

A(n)  P{Yn  1} 



i*  ( i , j ),iA , jA

P{H i* }P{Yn  1 H i* }   PB A

 A ( n)  PDA   B A ,  ADA (n) 

where PB A , PDA are row vectors given in 3.1 whose elements P{H i* }, i  (i, j ) satisfy *

i  B , j  A and i D , j  A , respectively. AB A (n), ADA (n) are column vectors whose * elements Ai* (n), i  (i, j ) satisfy i  B , j  A and i D , j  A , respectively.

21

By using total probability formula and simple analysis, for i*  (i, j ), i D , j D , we have,

Ai* (n)  P{Yn  1 H i* }  P{Yn  1, n  O1 H i* }  P{Yn  1, O1  n  O1  W1 H i* }  P{Yn  1, n  O1  W1 H i* }  P{O1  n  O1  W1 H i* }  P{Yn  1, n  O1  W1  G1 H i* }. The two terms can be given as follows,

P{O1  n  O1  W1 H i* }  P{O1  n H i* }  P{O1  W1  n H i* }, n

P{Yn  1, n  O1  W1  G1 H i* }   P{Yn  1 O1  W1  G1 =k , H i* }P{O1  W1  G1  k H i* }. k 0

Then we have, 2

2

i 1

i 1

AB A (n)   FOi (n)   FIi (n)  [ f I1 (n)  fG1 (n)  f I 2 ( n)  f G3 ( n)]  AB A ( n),  [ f I1 (n)  f G2 (n)  f I 2 (n)  f G4 (n)]  ADA ( n), 4

4

i 3

i 3

ADA (n)   FOi (n)   FIi (n)  [ f I3 (n)  f G1 ( n)  f I 4 ( n)  f G3 ( n)]  AB A ( n),  [ f I3 (n)  fG2 (n)  f I 4 (n)  fG4 (n)]  ADA (n), where

f I*1 ( z), f I*2 ( z), f I*3 ( z), f I*4 ( z), fG*1 ( z), fG*2 ( z), fG*3 ( z), fG*4 ( z) are matrixes defined in

section 3. And FOi (n), FIi (n),(i  1, 2,3, 4) are column vectors whose Z-transforms are given

z z , where is used z 1 z 1 to change the Z-transforms of probability density function f (n) to the Z-transforms of * * by replacing PB A or PDA in fOi ( z ), f Ii ( z ),(i  1, 2,3, 4) by

* distribution function F (n) because F ( z ) 



n 0



= f (k ) k 0



n





 F (n) z n   f (k )z n   f (k )z n n 0 k 0

k 0 n  k

z k z z fO ( z )uB D . By the Z-transform,  f * ( z ). For example, FO*1 ( z )  1 1 z z 1 z 1 1

we have, 2  2 *  F ( z )  FI*i ( z )    Oi  *  f I1 ( z )  AB A ( z )  i 1 i 1 *  A ( z)   *  4 4    f I3 ( z ) A ( z ) * *  DA    FOi ( z )   FIi ( z )  i 3  i 3 

Then we have, 22

f I2 ( z )  fG1 ( z )   f I4 ( z )   fG3 ( z )

fG2 ( z )  *  A ( z ). fG4 ( z ) 

  f I ( z ) A ( z )   I   1   f I3 ( z )

fG2 ( z )    fG4 ( z )  

f I2 ( z )  fG1 ( z )   f I4 ( z )   fG3 ( z )

*

1

2  2 *  *   FOi ( z )   FIi ( z )  i 1  i 1 . 4 4   * *   FOi ( z )   FIi ( z )  i 3  i 3 

Then the Z-transform of instantaneous availability at time n is as follows,

A* ( z )   PB A

  f ( z) PB A   I      f I3 ( z )  I1

f ( z )  f ( z )   f I4 ( z )   fG3 ( z )  I2

 G1

f ( z)   fG4 ( z )    G2

1

2  2 *  F ( z )  FI*i ( z )     Oi i 1  i 1 . 4 4   * *   FOi ( z )   FIi ( z )  i 3  i 3 

(4.1) Instantaneous availability A(n) (n  1, 2,

) can be given by taking the inverse Z-transform

*

of A ( z ) .

4.2 Steady availability The

Z-transform

of



A* ( z )   A(n) z  n， then n 1

instantaneous

availability

at

time

n (n  1, 2, )

is



A* ( z 1 )   A(n) z n , from which we have

A(n) 

n 0

1 d n [ A* ( z 1 )] (n  1, 2, ). Then steady availability of the system is given by z 0 n ! dz n

lim

A  lim A(n) which should be the same as limitation of A(n) which is given by taking n 

*

inverse Z-transform of A ( z ) in (4.1).

5. Numerical Example 5.1 The semi-Markov kernel and the transition diagram. The general results will now be illustrated for an engine system. Future performances of the engine system are determined not only by the present performance but also by the past performance records. Assume that performance records of the engine system can be described by a discrete time homogeneous semi-Markov chain second order in state and semi-Markov kernel of its state pairs can be fitted approximately by (5.1). As it is shown in Figure 5, excellent state 1 constitutes the subset A and it means that 23

the system works at its highest level; good states, 2 and 3 , which happen because of degradation constitute the subset B and state 2 means the system works at medium level and state 3 means it works at its lowest level; failure state 4 , which happens because of inside shocks caused by high performance level of the system, constitutes the subset D and it means the system fails completely. Assume that the performance level in state 3 is lower than the level to cause an inside shock and the system in state 3 can only be repaired to medium level after one repair. Possible transitions of the system states can be shown in Figure 4.             Q ( n)               

0.4  0.5 n

0.8  0.2n

0.1 0.8n 0.4  0.2n

0.1 0.8

n

0.6  0.5

0.9  0.5n

n

0.5 n 0.8  0.2n

0.4  0.5 n

0.1 0.8n 0.9  0.5n

0.6  0.5n

0.1 0.8n 0.8  0.2n

0.8  0.2n

0.6  0.5n

0.4  0.2n

                         

(5.1)

1 4

2

3

Figure 4. The transition diagram for the engine system.

Assume that the engine system tends to reach steady states very soon after it starts, so we investigate its properties in its steady states. By solving functions in (3.1), we get the steady-state probability vector of the system PΩ2 ()  (0,0.1377,0,0.1247,0.0987,0,0,

0.0894,0.0889,0,0.1681,0,0.1638,0.1291,0,0). To investigate performances of the engine system, we calculate length distributions of I-periods, oscillations, working time periods and gaps between I-periods when the system is in the steady states as follows.

5.2 The I-period length The probability distribution f I (n) (n  1, 2 24

) of I-period length given by the inverse

Z-transform of (3.2) can be shown in Figure 5. f I (n) (n  1, 2

) is a decreasing function

of n and the mean length of per I-period is mI  5.1259. In an I-period, the engine system always works and the performance levels vary from its highest level to lowest level.

5.3 The length of oscillation The probability distribution fO (n) (n  0,1,

) of the length of oscillation period for

nA =1 given by the inverse Z-transform of (3.3) can be shown in Figure 5. fO (0)  0.6200 and fO (1)  0.3296 take almost all the probability and the mean length of oscillation is

mO  0.5717. In an oscillation, performances of the engine system are unsteadily and perfect-functioning periods are always too short.

5.4 The length of working time The probability distribution fW (n) (n  0,1,

) of the length of working time for nA =1

given by the inverse Z-transform of (3.4) can be shown in Figure 5. fW (n) (n  0,1,

) is a

decreasing function of n except that fW (1)  0 and the mean length of per working time is

mW  4.3502. In working time, the engine system recovers from the last inside shock and begins to work steadily and efficiently.

5.5 The length of gap between I-periods The probability distribution fG (n) (n  1, 2,

) of the length of gap between I-periods

given by the inverse Z-transform of (3.5) can be shown in Figure 5. fG (n) (n  1, 2,

) is

an increasing function of n before n  4 and a decreasing function of n after n  4 ,

fG (4)  0.1159 and the mean length of per gap between I-periods is mG  7.4492. In a gap, the engine system is in complete failure because of a inside shock.

25

Figure 5. The probability distributions of I-period length, oscillation length, working time length and gap length.

5.6 The instantaneous availability of the system The instantaneous availability A(n) (n  1, 2,

) of the system given by the inverse

z-transform of (4.1) can be shown in Figure 6. In Figure 6, A(n) (n  1, 2,

) is a decreasing function of n before n  9 and then

approximately tends to be the steady-state availability A()  0.36216 which is exactly the same as A  0.36216 given by (4.2). That means system availability decreases after each recovery from an inside shock and tends to be steady at about 0.36261 in each time.

Figure 6. The instantaneous availability of the system.

26

6. Conclusion For more complex systems in practice, first-order Markov and semi-Markov system are not always suitable to describe their performances, and then more general models are needed. In this paper, we assume that the underlying repairable degradation system follows a homogeneous discrete-time semi-Markov chain second-order in state. With the state space divided into three subsets for excellent states, good states and failure states, stochastic properties for the repairable degradation systems have been discussed. Working time omission is considered in this paper because enough long working time is important for us to believe degradation systems’ recoveries from failures. I-periods, oscillations, periods of working time, and gaps between I-period are defined to describe some sojourn times people may be interested in when studying discrete repairable degradation system. In I-periods, the system is in excellent states or good states and an I-period consists of an oscillation and a period of working time. In oscillations, the system doesn’t stay in excellent states for enough long time, so it’s thought to perform inefficiently and unsteadily which means it’s unavailable during that time. A period of working time begins at the beginning of an enough long sojourn time in excellent states after an oscillation and the system is available in periods of working time. In gaps between I-periods, the system is in good states or failure states and it’s thought to be unavailable during that time. The Z-transform has been used to give these sojourn time distributions and instantaneous and steady availabilities of the system. Numerical examples are provided to check theorems given in the paper and some illustrations are given. The results obtained, we believe, can be used in the analysis of discrete repairable degradation systems in the field of reliability and many other fields. For example, the models can describe not only data of wind directions, DNA sequences, social behaviors, share prices, electricity consumption demands, health conditions of human, but also performance levels of power stations, network systems, heating systems and many other systems whose reliability measures are very important in practice. Acknowledgment This work was supported by the National Natural Science Foundation of China (No.71371031, 71631001). 27

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Highlights    

We investigate a repairable aggregated semi-Markov ternary system second-order in state. The working time omission problem in which oscillation and working period are given is considered. Some distributions for I-periods, oscillations, working periods and gaps are given by using the Z-transform. Instantaneous and steady availabilities of the system are presented.

30