Sensor-based calibrations to improve reliability of systems subject to multiple dependent competing failure processes

Author’s Accepted Manuscript Sensor-Based Calibrations to Improve Reliability of Systems Subject to Multiple Dependent Competing Failure Processes Dejing Kong, Chengwei Qin, Yong He, Lirong Cui www.elsevier.com/locate/ress

PII: DOI: Reference:

S0951-8320(16)30983-8 http://dx.doi.org/10.1016/j.ress.2016.12.007 RESS5714

To appear in: Reliability Engineering and System Safety Received date: 21 May 2016 Revised date: 14 December 2016 Accepted date: 15 December 2016 Cite this article as: Dejing Kong, Chengwei Qin, Yong He and Lirong Cui, Sensor-Based Calibrations to Improve Reliability of Systems Subject to Multiple Dependent Competing Failure Processes, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2016.12.007 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.

Sensor-Based Calibrations to Improve Reliability of Systems Subject to Multiple Dependent Competing Failure Processes Dejing Konga*, Chengwei Qinb, Yong Hec, and Lirong Cuia a

School of Management and Economics, Beijing Institute of Technology, Beijing, 100081, China

b

Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

c

Department of Statistics, Fudan University, Shanghai, 200433, China

[email protected] [email protected] [email protected] [email protected] *

Corresponding author.

Abstract

Reliability models with sensor-based calibrations are presented for systems or devices subject to dependent competing failure processes of soft failure due to degradation process and hard failure according to shock process. Shock magnitudes to indicate the hard failure are considered in terms of two patterns, inter-arrival times and arrival times. The soft failure is indicated by overall degradation measurement comprising of internal continuous degradation measurement and additional degradation damage caused by the shock process. By sensors, instant calibrations are carried out to rejuvenate the additional degradation damage. Then explicit reliability expressions are derived for the dependent competing failure system whose internal degradation measurement is described by the general degradation path model. And a simulation procedure is proposed to model the internal continuous degradation measurement indicated the Wiener process. Meanwhile, the usefulness of the proposed models is demonstrated by the numerical example. The results are illustrated that the proposed model with the instant calibrations are available to improve system reliability and make the system to operate in a higher safety in applications. Keywords: Dependent competing failure; Degradation; Poisson process; Calibrations; Inter-arrival time; Arrival time; Simulations 1. Introduction Imposed external damage created by outside stochastics shocks is a critical part leading to the failure of a system. In the past, the shocks could not always be tracked 1

and recorded due to limited inspections. However, sensor technology has brought an increased interest in prognostic health management and data collection. For some important parts, sensors are used to inspect vibration, temperature, and abrupt changing condition of the operating facilities and systems. They are commonly represented by the measureable performance of reliability. For example, a thermocouple converts temperature to an output voltage; and a mercury-in-glass thermometer is also a sensor which converts the measured temperature into expansion and contraction of a liquid shown on a calibrated glass tube. Hence, the sensors could detect the shocks exactly if they are installed in the system. From theoretical model to practice use, a plenty of literature on sensor technology, degradation, and competing failure mode have been reported. So far, the sensor technologies are generally utilized to obtain performance observations for reliability data analyses, maintenance, and so on. To specify, Inagaki [1] studied interdependence between safety-control policy and multiple-sensor schemes. In vehicle control, Agogino et al. [2] proposed to use intelligent sensor validation and sensor fusion to achieve reliability and safety enhancement. Recently, predictive maintenance management policy with sensor-based degradation models was reported in [3] and [4]. Pascale et al. [5] built wireless sensor networks for traffic management and road safety. The existing studies of sensor technology in reliability focused on detecting performance measurements without considering instant maintenance. Much more recently, Cui et al. [6] and Kong and Cui [7] studied that calibration can improve reliability of measurement systems. Actually, they pointed out that the calibration is one of important maintenance actions in many practical operating systems such as measure systems, position system, navigation systems, and guide systems. Meanwhile, some differences still exist between the calibration and the imperfect maintenance in real-world applications. Calibration focuses on correcting some biases for the systems due to degradation and shocks, but cannot reduce the underlying or essential faults of the system completely. Besides that, this action is easily carried out in operations compared to the maintenance which may cost too much time to reduce the underlying faults. It is known that hard failure and soft failure are generally modeled in a competing failure model for a complex system. Li and Pham [6] and [9] studied an inspection maintenance model for systems subject to multiple competing processes and a multi-state degraded system experienced multi-competing failures, respectively. Then a joint modelling of degradation and failure time data was derived by Lehmann [10]. 2

It is worth mentioning that Peng et al. [11], Peng et al. [12], Wang et al. [13], and Song et al. [14] studied the systems subject to multiple dependent competing failures. Based on the Gamma processes, Pan and Balakrishnan [15] assessed the reliability of degradation of products with multiple performance characteristics. Ye et al. [16] and [17] proposed a distribution-based system model under extreme shocks and natural degradation. In the hands of maintenance, Wang and Pham [18] proposed imperfect preventive maintenance policies for two-process cumulative damage model including degradation and random shocks. Systems subject to competing failure modes due to degradation and traumatic events were explored under periodic inspection and replacement policy in [19], [20], and [21]. Meanwhile, to reduce the risk of failure, Ye et al. [22] derived a degradation-based burn-in with preventive maintenance strategy. Xiang et al. [23] studied n subpopulations experiencing stochastic degradation: reliability modelling, burn-in, and preventive replacement optimization. Besides, a condition-based maintenance of a dependent degradation threshold shock mode in a system with multiple degradation processes was also derived by Caballé et al. [24]. Then Do et al. [25] listed a proactive condition-based maintenance strategy with both perfect and imperfect maintenance. Then, recently, calibration is a natural tool to rejuvenate degradation measurement, e.g., a Bayesian method was proposed to assess the reliability of a navigation system with calibrations under the general path degradation model studied by Kong and Cui [7]. Furthermore, the Wiener process degradation models with calibrations were studied by Cui et al. [6]. However, from the studies in the literature above, the studies of calibration focused on periodic or predetermined calibrations based on a pure degradation process without considering competing failures and instantaneous calibrations. And so, in this paper, it motivates us to tackle the related issues of calibrations in a dependent competing failure system wherein shocks produce additional damage on the overall degradation measurement of the system. But by the sensors, the shocks can be detected exactly as well as the damage can be rejuvenated completely or partly in terms of a calibration coefficient. There are several shock studies, for instance, extreme shock mode, m-shock mode,

 -shock mode, and consecutive shock mode. But recently, ones realize that inter-arrival time (arrival time) is a key factor to impact the magnitude of shocks. For example, a transportation system such as a public bus system experiences the random passenger flows at each stop. Once the inter-arrival time of the system getting the stop is longer, the number of passages is larger. In this case, it is obvious to see that the 3

shock magnitude of the system and the units in the system due to the sudden passages shock is related to the inter-arrival times [26] and [27]. In other words, when the inter-arrival time is longer, the corresponding latent shock magnitude will be larger. In addition, the phenomenon is common in other fields as well, e.g., the issues of inter-arrival time for congestion based on network packet studied by [28] and [29]. From the view of reliability, however, the corresponding shock modelling for an operating system has not been studied so far. Subsequently, in this article, we will propose two shock models and systematic methodologies. The shock magnitudes are associated with inter-arrival times and arrival times when the shock arrivals follow a Poisson process, respectively. And so, here, we focus on the following hands:1) analyzing a system subject to dependent competing failure including hard failure and soft failure with calibrations, 2) modelling the hard failure whose shock magnitudes are related to two patterns under homogeneous Poisson process (HPP) and nonhomogeneous Poisson process (NHPP), respectively, 3) modelling the soft failure indicated by the overall degradation performance composed of the internal continuous degradation measurement and the amount additional degradation increases caused by the shocks and calibrations, and 4) giving explicit reliability expressions under the general degradation path model as well as presenting a procedure to simulate the reliability results under the Wiener process. Due to the additional degradation damage produced, the overall degradation performance under the Wiener process will become a general jump diffusion process. Moreover, by simulations, we attempt to cover the gap focusing on the first passage time of the jump diffusion process in a dependent competing failure model. The remainder of the article is organized as follows. A dependent competing failure system with instant calibrations, the definition of the failure mode, and calibrations are proposed in Section 2. Based on the general degradation path model and the degradation model of the Wiener process, reliability modelling and simulation procedure are studied under the scenarios of HPP and NHPP in Section 3, respectively. In Section 4, numerical examples and simulations are given to demonstrate the proposed models. Section 5 concludes the paper with comments on pros and cons of the models. 2. System Descriptions For the complex systems subject to multiple dependent competing failure processes (MDCFP), the dependency among the failure processes presents challenging issues in 4

reliability modelling. The general degradation path model and the degradation model governed by the Wiener process have been applied to describe systems’ or devices’ degradation character in the practice of reliability engineering reviewed by Nikulin et al. [30] and Ye and Xie [31]. 2.1. Notation and Assumptions The notations are listed as, c.d.f., cumulative distribution function, p.d.f., probability density function, {N (t ), t  0} , a Poisson process, Ti , the i th arrival time conditioned on N (t )  n for i  1, 2,, n , and T0 =0 ,  i , i =Ti  Ti 1 , the i th inter-arrival time conditioned on N (t )  n , and 0=0

 , arrival rate of HPP,  (t ) , arrival rate function of NHPP, H i , the i th shock magnitude (load) associated with the i th arrival, and H 0  0 ,

Hi  N (I ( i ), I2 ( i )) , parameters associated with i   i , termed Pattern I,

Hi  N (II (ti ), II2 (ti )) , parameters associated with Ti  ti , termed Pattern II, DX , threshold for soft failure indicated by degradation measurement,

DS , threshold for hard failure indicated by shock mode,

Y (t ) , internal continuous degradation performance over time t , Yg (t ) , internal degradation performance under the general degradation path model, Yg (t ) ~ N ( g (t ), h(t )) , g (t ) and h(t ) are the mean and variance functions,

Yw (t ) , internal degradation performance governed by the Wiener process,

W (t ) , standard Brownian motion, Yw (t )  a(t )  b(t )W (t ) , a(t ) and b(t ) are the drift and diffusion functions,

 i , the i th calibration coefficient, Si , the i th additional degradation damage at the i th shock, and S0  0 , Ci : i Si , the i th rejuvenation measurement, and C0  0 ,

Vi : Si  Ci , the i th left sudden damage load after the i th shock and the i th

calibration, X S (t ) : i 0 Vi , cumulative additional degradation increase by time t , N (t )

X (t ) : Y (t )  X S (t ) , overall degradation performance measurement by time t .

Assumptions of the paper are given as, (1) Failure of a system depends on two dependent competing failures, soft failure and hard failure, whichever occurs first resulting in the failure of the system. 5

(2) Sensors which are completely reliable can detect the shock arrival instants precisely. (3) Calibrations are completed instantaneously. (4) Hard failure is described by an extreme shock mode when the shock magnitude H i over the threshold DS results in hard failure.

(5) Soft failure is described by the internal continuous degradation Y (t ) and the cumulative additional degradation increase X s (t ) . It occurs when the overall degradation volume X (t ) exceeds the threshold DX . 2.2. Dependent Competing Failure Systems with Calibrations Competing failure mode often exists in the highly reliable systems or devices such as inertial navigation system (INS), micro-engine, electrical engineering machines, and insulated gate bipolar translator (IGBT) mode. To be specific, for an INS used in the aerospace industry, its failure is usually described by a degradation process over a threshold for a certain indicator. Here, under monitoring indicators or by measuring the age of asset, we consider it operating in a complex and dynamic environment wherein the system experiences the random shocks and internal degradation. Thus the system can be described much well by a competing model. Besides, once the self-calibration device is installed, by using the instant calibrations, the reliability of the system could be improved either compared to the conventional periodic calibrations. This article focuses on a MDCFP system with instant calibrations to rejuvenate the additional degradation damage shown in Figure 1. sensors detecting shocks 2

3

system/ device Shocks Shocks

self-calibration

1

4

automatic machine

5

Output Output

The overall degradation performance

Figure 1. A system experiences MDCFP mode and calibrations.

A dependent competing failure system (device) installed with an automatic machine experiences both external random shocks and internal continuous degradation process 6

during field operation shown in Figure 1. The shocks (procedure 1) modeled by Poisson process not only lead to hard failure, but also cause additional degradation damage Si that contributes to the overall degradation process X (t ) . Sensors installed in the system can detect the shock arrivals accurately and provide a corresponding electrical or optical signal (procedures 2 and 3). And the shock signal is transformed to an automatic machine which can calibrate the additional degradation damage Si (procedure 4) in terms of the calibration coefficient  i . Then after calibration, the effective overall degradation measurement (procedure 5) which is the output degradation performance is displayed or measured finally. Consequently, the overall degradation measurement of soft failure includes the internal continuous degradation measurement Y (t ) and the cumulative additional degradation increase

X S (t ) due to the shocks. And so, taking the hard failure into account, it is a multiple dependent competing failure system with sensor-based calibrations (MDCFSSC) that is composed of internal factor leading to continuous degradation and the shocks resulting in hard failure in addition to the cumulative additional degradation increase. And so, in this manner, we can make the system operate with highly reliabilities in the uncertain environment, especially at the instants of shock arrivals. 2.3. Sensor-Based Calibrations In Figure 1, with the transformed signals from sensors, the automatic machine can rejuvenate the additional degradation damage due to shocks by self-calibrations. And so, we do not attempt to change the whole degradation trend of the internal degradation but focus on calibrating the additional degradation damage Si which can be regarded as the measurement errors due to shocks. So in this manner, the physical property of the internal degradation mechanism of the system will remain.

7

X(t)

DX

i  0

Vi

 i  (0,1)

i  1

V2 V1 t1

t2

...

ti

t

Figure 2. Illustrations to degradation path with different calibration coefficients.

From Figure 2, we can gain an insight into calibrations. The displayed degradation path is different from the previous degradation models. In the system, by sensors, at the i th shock instant, the i th calibration is carried out whose rejuvenation measurement is less than the i th additional degradation damage up to time t for i  1, 2,

, N (t ) . By the notation, that is, Vi  Si  Ci  (1  i )Si is the effective

left sudden damage load on the overall degradation signal in the path as it is shown in Figure 2. In much more detail, in the figure, as  i =1, the i th calibration can make the system renew to a state before the i th shock occurs if the rejuvenation is finished completely like the thin solid curve (lower); the bold solid curve (middle) is the degradation path with calibration coefficient  i in (0, 1); in addition, the dashed curve (upper) is the upper limited degradation path without effective calibrations when  i  0 . Consequently, the calibration coefficient expresses the performance of rejuvenating additional degradation damage not calibrating the degradation function. And so, the effects of calibrations can be represented as the recovery of the degradation damage level completely or partly due to shocks. In other words, the coefficient,  i , it is used to represent the i th performance of calibration. Naturally, the rejuvenation measurement is Ci   i Si , where  i [0, 1] . And so, the larger  i is used, the more improvement of the system’s reliability can be obtained. When  i  1 , the calibration is a perfect rejuvenation which makes the system become the preceding one before the i th shock arrival. That is to say, the shock does not produce any effect on the overall degradation measurement. When

 i  0 , it means that the rejuvenation is completely failed. Other cases occur when 8

 i  (0,1) . And so, it reveals the effect of calibration in rejuvenation of the additional degradation damage Si . 2.4. Shock Magnitudes Associated with Two Patterns With the notation in Subsection 2.1, the shock magnitudes are associated with inter-arrival times and arrival times, respectively. Thus the following two shock models will be built according to the two patterns. In Pattern I, the shock magnitudes are related to inter-arrival times. That is to say, an undergoing system experiences shocks where the i th shock magnitude H i is associated with the ith inter-arrival time i   i . For example, the longer  i is, the larger H i is. Alternatively, based on the other real applications, the longer  i is, perhaps, the smaller H i is. Then Pattern II is respect with arrival time Ti  ti which determines the shock magnitude. To interpret the shock models, up to time t , the conditional p.d.f. of the shock magnitudes according to the inter-arrival times and arrival times should be derived. Pattern I. Shock Magnitudes Associated with Inter-Arrival Times In Pattern I, the p.d.f. of inter-arrival time i  (Ti  Ti 1 ) conditioned on N (t )  n shocks up to time t , is g denoted as fi |N (t ) ( i | n) for i  1, 2,, n . As the notation is in Subsection 2.1,

Hi

follows a mixture normal distribution, that is,

Hi  N (I ( i ), I2 ( i )) , whose c.d.f. is noted as FH (,  i ) , in which I () and  I2 () are related to  i . Without loss of generality, we could define I ( i )=1 i and

 I2 ( i )  (1 i )2 . 1 and 1 are assumed to be positive numbers. That is to say, when 1 is the positive number, the magnitude of H i grows with the increase of

 i , and vice versa. 2 Because I () and  I () are the variables related to  i , according to the

stress-strength model, the probability of the system surviving the applied stress Ds at the i th shock conditioned on N (t )  n , defined as PI ( Hi  DS | N (t )  n) , is a mixture model written as

PI  H i  DS | N (t )  n  



 i [0,t ] t

FH ( Ds , i   i ) f i |N (t ) ( i | n)d i



    Ds  I ( i )  0

where ( x)  1

2



x





(1)

 ( i ) f  | N (t ) ( i | n) d i , 2 I

i

e z 2 dz , i.e., ( x) is a distribution function of a standard 2

normal distribution. And then PI  H 0  DS | N (t )  0   1 . Pattern II. Shock Magnitudes Associated with Arrival Times 9

In Pattern II, the p.d.f. of the ith arrival time Ti conditioned on N (t )  n is noted as fTi |N (t ) (ti | n) . Also by the notation in Subsection 2.1, H i is mixture 2 normally distributed as well, noted as, Hi  N (II (ti ), II (ti )) , whose c.d.f. is noted as

2 2 FH (, ti ) where we can define II (ti )   2ti and  II (ti )  (2ti ) which are related

to the arrival time Ti  ti for i  1, 2,, n .  2 and  2 have the same performance on shock magnitudes as well as 1 and 1 in Pattern I. 2 Similarly, since II () and  II () are the variables related to ti , by the

stress-strength model, the probability of the system survives the applied stress Ds at the i th shock conditioned on N (t )  n , defined as P ( Hi  DS | N (t )  n) , is a mixture model given as

P  H i  DS | N (t )  n  



ti [0,t ] t

FH ( Ds , Ti  ti ) fTi |N (t ) (ti | n)dti



    Ds  II (ti )  0



(2)

 II2 (ti ) fT | N (t ) (ti | n)dti . i

where P  H 0  DS | N (t )  0   1 . Then in an extreme shock mode, the shock magnitudes are indicated by the two patterns in Eqns. (1) and (2). Subsequently, the system works over time t if all N (t ) shocks dose not exceed DS up to time t , i.e.,  N (t )  Rh ,k (t )  Pk   H i  DS    i 0    n   P( N (t )  0)   Pk   H i  DS | N (t )  n   P( N (t )  n) n 1  i 1    n   P( N (t )  0)     Pk  H i  DS | N (t )  n   P( N (t )  n), k  I, II. n 1  i 1 

(3)

wherein the third term in Eq. (3) holds naturally for Pattern I. And for Pattern II, it holds when the shock magnitudes related to the arrival times are assumed mutually independent. And so, the hard failure models due to shocks could be derived under the two patterns once the conditional p.d.f. and survival probability of the multiple shocks over time t are obtained in HPP and NHPP, respectively. 3. Reliability Modelling and Derivations The reliability modelling is then executed based upon a systematic methodology of calibrations to achieve improvement of reliability and safety of the system when the shock arrivals are obtained instantly. 10

As it is illustrated in Section 2, the overall degradation performance measurement X (t ) over the critical threshold DX leads to soft failure which is accumulated by

internal continuous degradation Y (t ) and left sudden damage load Vi . Meanwhile, the shocks also cause hard failure when the shock magnitude H i is over the threshold DS . Therefore, with the above analyses and illustrations on the system, we consider reliability models of the multiple dependent competing failure system with calibrations in the following hands: firstly, modelling the hard failure due to shocks for the two patterns under HPP and NHPP, separately; secondly, modelling the soft failure with calibrations under the general degradation path model and the Wiener process; and then based on the models, the modelling of MDCFSSC in terms of the two patterns could be completed. 3.1. Modelling of Hard Failure Due to Shocks 3.1.1. Shock Modelling under Pattern I Under the Pattern I, the shock magnitude is related to inter-arrival time. Then by Eqns. (1) and (3), the reliability of the system subject to hard failure over time t , noted as

Rh,I (t ) , is expressed as  N (t )  Rh ,I (t )  PI   H i  DS    i 0   D  I ( i )   P( N (t )  0)+ P( N (t )  n)    S  f i | N (t ) ( i | n)d i . n 1 i 1 0   I ( i )  

n

t

(4)

To be specific, in HPP, with the property of HPP, also termed memoryless, the p.d.f. of the i th inter-arrival time  i conditioned on N (t )  n is given as n  t  i  f i | N (t ) ( i | n)    , for 1  i  n and 0   i  t . t t  n-1

(5)

Then by Eqns. (4) and (5), the reliability of the system subject to hard failure over time t under HPP is given in Eq. (18) of Appendix A. And under NHPP, we suppose that  :[0, )  [0, ) is measurable, and define t

the mean function, m :[0, )  [0, ) by m(t )    ( s)ds . 0

In Pattern I, we implied the p.d.f. of inter-arrival time  i conditioned on N (t )  n

given as

11

 n ( )  m(t )  m( ) n i i i i  1,    , m ( t ) m ( t )     f i | N (t ) ( i | n)    t  i  i 2 n i  n !   ( y ) ( y   i )m( y ) (m(t )  m( y   i )) dy    0  , 2  i  n, n  m(t ) (i  2)!(n  i)! 

(6)

where 0   i  t for 1  i  n . And the proof is given in Appendix B. Then by Eqns. (4) and (6), the reliability of the system subject to hard failure over time t under NHPP is given in Eq. (19) of Appendix A. 3.1.2. Shock Modelling under Pattern II Similarly, in Pattern II, under the assumption in Eq. (3), when shock load is related to arrival time Ti , the reliability of the system subject to hard failure over time t , written as Rh,II (t ) , is expressed as

 N (t )  Rh,II (t )  PII   H i  DS    i 0   D  II (ti )   P( N (t )  0)+ P( N (t )  n)    S  fTi | N (t ) (ti | n)dti . n 1 i 1 0   II (ti )  

n

t

(7)

Then under HPP, the p.d.f. of the i th arrival time Ti conditioned on N (t )  n is expressed as fTi | N (t ) (ti | n) 

tii 1 (t  ti )n i n ! , for 1  i  n and 0  ti  t . t n (n  i)!(i  1)!

(8)

Then by Eqns. (7) and (8), the reliability expression of the system subject to hard failure over time t under HPP is given in Eq. (20) of Appendix A. In Pattern II, given the total number of arrivals, N (t )  n , by the results are derived by Kuniewski et al. [32] and Cocozza-Thivent [33], the instants of these n arrivals in the interval [0, t ] with conditional p.d.f. are written as, i 1

 m(ti )   m(ti )  n! fTi |N (t ) (ti | n)    1   (i  1)!(n  i)!  m(t )   m(t ) 

n i

 (ti ) m(t )

,

(9)

where for 1  i  n and 0  ti  t . Then by Eqns. (7) and (9), the reliability expression of the system subject to hard failure over time t under NHPP is given in Eq. (21) of Appendix A. Actually, when  (t )   , the results of HPP is a special case of HNPP both in 12

Pattern I and Pattern II. From the expressions above, the shock models related to the inter-arrival times and arrival times are obtained, respectively. As the notation, the longer of the shock duration is, the huger shock damage will be produced on the system which is consistent with the forms of 1 i and  2ti when both 1 and  2 are greater than 0. That is to say, the potential risk of a certain shock is much greater if the previous shocks do not occur for longer time. However, other alternative forms of

I () and II () could be used. 3.2. Modelling of Soft Failure Due to Degradation and Shocks with Calibrations As the description of the system is given in Subsection 2.2, the overall degradation measurement of the system includes internal continuous degradation measurement and cumulative additional degradation increase due to shocks. Although the shocks lead to abrupt damages, calibrations rejuvenate the additional degradation damages. 2 For the i th shock, Si is normally distributed here, noted as, Si  N (i ,  i ) . i

2 and  i are the known sequences according to the order of the i th shock.

By the i th calibration, the left sudden damage load produced on the overall degradation is Vi as it is shown in Figure 2. Therefore, with the notation, the cumulative additional degradation increase X S (t ) due to N (t ) shocks and N (t ) calibrations up to time t is expressed by

N (t )  0, 0,  N (t ) N (t ) X S (t )   V =  i  ( Si  Ci ), N (t )  0.  i 1  i 1

(10)

Furthermore, with the notation in Subsection 2.1, let Yg (t ) denote the degradation measurement emphasizing that degradation process is described by a model of the general degradation path, and Yw (t ) is the degradation measurement of the Wiener process. To specify, the overall degradation performance measurement, X (t ) ( X g (t ) or X w (t ) ) is derived which consists of internal degradation Y (t ) ( Yg (t ) or Yw (t ) ) and sudden increased measurement X S (t ) . And so, under the model of the general degradation path, the overall degradation measurement X g (t ) is expressed by N (t )

N (t )

i 0

i 0

X g (t )  Yg (t )  X S (t )  Yg (t )   Vi   g (t )      (1   i ) Si .

(11)

where random error term  is normally distributed, noted as,  ~ N (0, h(t )) , which is corresponding to the notation X g (t ) ~ N ( g (t ), h(t )) . 13

Thus up to time t , the reliability of system experiencing soft failure under the general degradation path model is given as N (t )  Rs (t )  P  X g (t )  DX   P  g (t )     (1   i ) Si  DX i 0 

 . 

(12)

Similarly, under the degradation model governed by the Wiener process, the overall degradation measurement X w (t ) is given as, N (t )

N (t )

i 0

i 0

X w (t )  Yw (t )  X S (t )  Yw (t )   Vi  a(t )  b(t )W (t )   (1   i ) Si ,

(11’)

where Yw (t )  X S (t )  Yw (t )  i 0 Vi  a(t )  b(t )W (t )  i 0 (1  i )Si is a general N (t )

N (t )

jump diffusion process with calibrations on the jump loads. Particularly, it reduces to a generalized Wiener process when  i  1 . Thus up to time t , the reliability of the soft failure under the degradation model mainly governed by the Wiener process could be modeled as,

Rs (t )  P Td  inf  s  0 : X w (s)  DX | X w (0)=xw   t  ,

(12’)

where X w (0)=xw is the initial degradation measurement value. Moreover, with the soft failure model derived above and the hard failure model in Subsection 3.1, the dependent competing failure modelling will be yielded. 3.3. Modelling of Dependent Competing Failure System For the dependent competing failure system, over time t , it fails once one shock magnitude H i is over DS , or X (t ) exceeds the critical threshold DX . Model 1. Dependent Competing Models under General Degradation Path Model Then under the general degradation path model, by Eqns. (3) and (11), the reliability of the system is N (t )   Rg ,k (t ) P  X g (t )  DX ,  H i  DS   i 0   n        P  X (t )  DX ,  H i  DS  | N (t )  n   P( N (t )  n) n 0  i 0    n     n    P( N (t )  n) P  Yg (t )   (1   i ) Si  DX | N (t )  n   Pk ( H i  DS | N (t )  n)  n 1  i 1   i 1 

 P Yg (t )  DX | N (t )  0  P( N (t )  0), k  I, II.

(13) To specify, in Pattern I, by Eqns. (3), (4), and (11)-(13), Rg , (t ) is given as 14

n      DX  ( g (t )   (1   i ) i )  n  D  g (t )   exp(t )(t )   i 1  Rg , (t )    X  exp(t )     n   n ! h(t )  n 1 2 2   h ( t )  (1   )    (14)  i i   i 1    

 t  D  I ( i )        S  f i | N (t ) ( i | n)d i   0   I ( i )  

n

 .  

where the analytical expression is obtained once the mean and variance functions are 2 2 known, for example, I ( i )  1 i ,  I ( i )  (1 i ) , g (t )   t , and h(t )  (  t )2 .

The dependent competing model in HPP is easily obtained by Eq. (5). If the shock arrivals follow a NHPP with arrival rate  (t ) , the reliability model can be addressed as well. Suppose that  (t )  ct , by (6), we can derive fi |N (t ) ( i | n)d i explicitly written as follows,

 2n  t 2   2 n i  2i 2 i  , i  1,  t  t   f i | N (t ) ( i | n)    t  i  2( i  2) 2 (t  ( y   i ) 2 ) n i dy   4n !  y ( y   i ) y    0  , 2  i  n. n  m ( t ) ( i  2)!( n  i )! 

(15)

Naturally, by substitution Eq. (15) into Eq. (14), the explicit reliability formula is obtained easily for Pattern I under NHPP. And in Pattern II, under HPP, by Eqns. (3), (7), and (11)-(13), we can also calculate the reliability Rg , (t ) explicitly as n      DX  ( g (t )   (1   i ) i )  n  D  g (t )   exp(t )(t )   i 1  Rg , (t )   X  exp(t )     n   n! h(t )  n 1 2 2   h ( t )  (1   )    (16)  i i   i 1    

  D  II (ti )     S  fTi | N (t ) (ti | n)dti  , i 1 0   II (ti )   n

t

where the explicit expression can be obtained once the mean and variance functions 2 2 are known, e.g., II (ti )   2ti and  II (ti )  (2ti ) .

The dependent competing model in HPP is easily obtained by Eq. (8). If the shock arrivals follow a NHPP with arrival rate  (t ) , the reliability model can be addressed as well. If  (t )  ct , by Eq. (9), we can derive fTi |N (t ) (ti | n)dti explicitly. By substitution Eq. (16) into Eq. (15), the explicit reliability formula is obtained easily 15

for NHPP under Pattern I. Remark: When  (t )   which is a constant, the reliability expression of HPP is special case of NHPP, respectively. Moreover, the models of the general degradation path have been yielded for the two patterns under HPP and NHPP which are valuable to real applications in reliability modelling and assessments. In fact, it is known that Cui et al. [6] and Kong and Cui [7] applied periodic calibrations to degradation systems and defined calibration degree wherein the degradation function was calibrated in the form of a series of piecewise functions. Besides, we can also extend the models to the piecewise degradation function models under stochastic calibrations as long as the mean and variance functions are defined by gi (t ) and hi (t ) for i  1, 2,

,n.

Model 2. Dependent Competing Models under Wiener Process In terms of the dependent competing failure system, then the reliability for the internal continuous degradation following the Wiener process can be modeled as N (t )      Rw (t )  P Tc  min Td  inf  a(t )  b(t )W (t )   (1   i ) Si  DX |X w (0)  xw  , i 1   (17)   

T

sc

= min  ti , H i  DS , i  1, 2,

 

, N (t )   t .

As we all know, although Eq. (17) could be written in a simple expression, the explicit distribution of the first passage time for Eqns. (11’) and (12’) dose not exist. Naturally, an effective procedure is proposed to simulate the results of the models. Based on the notation and assumptions, we consider getting the simulation results under the two patterns sequentially. Firstly, we define the operating time [0, Ts ] with the discretization size l under Euler discretization for simulating the Wiener process, where Ts is a termination time of operating stage. Secondly, consider the ith simulated interval, [ti 1 , ti ) , with the property of jump diffusion process, the stochastic process x(t ) in the ith period [ti 1 , ti ) is a Wiener process with drift function a(t ) , and diffusion function b(t ) started at an initial value x(ti 1 ) , where x(t0 )  X w (0)  xw and i 



. Then by using the method to simulate the degradation

model of the Wiener process, the degradation path in interval [ti 1 , ti ) is obtained. Finally, we combine the simulated shock magnitudes according to the shock arrivals and simulated degradation measurement to get the dependent competing failure times. And so, a specified simulation procedure is described as follows. 16

Simulation_procedure Step 1: For  =1 to  , perform the following Monte Carlo runs (Steps 2-5); Step 2: Obtain ti by generating the inter-arrival time  i  ti  ti 1 according to the given density in the following two cases, (a) Generate inter-arrival times in the case of HPP, then from t0  0 , produce inter-arrival time which is exponential distributed; if  i  Ts , we could criticize and get  shock instants until





t  Ts ,

i 1 i

(b) Generate inter-arrival times in the case of NHPP, then from t0  0 , inter-arrival time has the c.d.f. noted as F ( i )  1  exp(m(ti 1   i )  m(ti 1 )) , if  i  Ts , then we could criticize and get  shock instants until





t  Ts ;

i 1 i

Step 3: For i  1 to  , simulate the interval between the (i  1) th shock instant and i th shock instant in [0, Ts ] ,

(a) Generate x(ti )  x(ti )  a(t )dt  b(t ) dt   (1  i )i , where x(ti ) is the last simulation achievement value of interval [ti 1 , ti ) ,  ~ N (0,1) , i ~ N ( i ,  i2 ) , and x(t0 )  x(0) .

(b) Generate shock size H i according to the given normal distribution, Hi ~ N (  ( i ), 2 ( i )) and H 0  0 ,

(c) Generate a sample path of the Wiener process started with initial value x(ti 1 ) in [ti 1 , ti ) under the discretization size l , If H i 1  DS , or if there is a generated t w in the interval satisfied x(t )  DX for the first time, then we obtain the failure time ti or t w , denoted as t f , . Then perform another cycle from Steps 2-3, Else if H i 1  DS and x(t )  DX for the discretization points th in [ti 1 , ti ) , continue (a) to (c) in Step 3, (d) Else if there is no failure in [0, Ts ] for this generation cycle, we let t f , : Ts ; Step 4: According to Step 3, get the failure time t f , = min tw , th , Ts  , define the successful operation time interval as [0, t f , ] . Then perform another cycle from Steps 2-4; Step 5: If    , we have completed all iterations of the simulation, observing the estimate for the reliability of the model, Rw, (t )  1 I {t  t f , ]  calculated 

by the indicator function. End_simulation_procedure

17

To simulate Rw, (t ) in Pattern II wherein the shock magnitudes are according to arrival times, we use the generated ti from Step 2 to get Hi ~ N ( (ti ),  2 (ti )) in Step 3. Then as the procedure is shown above, the reliability under the Wiener process for the two patterns under HPP and NHPP can be derived as well. To simulate the HPP (NHPP), here, we applied the method of generating inter-arrival times by the integration method to get the Poisson processes in Step 2. In addition to it, some other generating methods are available to the processes studied by Cocozza-Thivent [33], Lee et al. [34], Ogata [35], and Lewis and Shedler [36], [37]. And so, based on the extreme shock model under the two patterns, the reliability results of the dependent competing failure model with calibrations are addressed. 3.4. Related Derivations With the reliability models in Eqns. (13)-(17), some other analyses can be explored, 



t

t

e.g., the mean residual lifetime MRL(t )   R( x | t )dx   R( x)dx R(t ) and the

nth moment of lifetime of the system under the given models. Sometimes, the explicit expressions can be derived but the most of the statistics are too complex to be implied. Then we can approximate the results by simulations once the parameters are obtained in the applications. Apart from the method to extreme shock mode, other shock modes could also be studied such as consecutive shock mode, cumulated shock mode, and  shock mode. Furthermore, the reliability formulas and numerical examples could also be yielded which are omitted here. 4. Numerical Examples and Simulations With the example reported by Tanner and Dugger [38], Peng et al. [11], [12], and Rafiee et al. [39], we shall extend to use the parameters and some assumptions to illustrate the proposed models. Meanwhile, the sensitivity analyses and simulations to the corresponding models are obtained as well. 4.1. Illustrated Examples For simplicity, we consider a classic instance used commonly on micro-engines in the previous works in addition to some assumptions. A micro-engine consists of orthogonal linear comb drive actuators that mechanically connected to rotating gear (see Tanner and Dugger [38]). It experiences two dependent competing failure processes: soft failures due to aging degradation and debris shock magnitudes. 18

Consider the micro-engines are used in a load-sharing redundant system wherein the shocks are self-announcing and degradation random error due to shocks. And the external shock damage could be rejuvenated by instant calibrations. Assume that the sensor-based calibration is executed to enhance the system’s performance. Table 1. Parameter values for the system. Parameter

Value

Source

DX

1.25e3  m3

Tanner and Dugger [38]

DS

1.5GPa

Tanner and Dugger [33]



8.4823e9  m3

Tanner and Dugger [38]



6.0016e10  m3

Tanner and Dugger [33]



2.5e5

Tanner and Dugger [38]

i

1e4  m3

Peng et al. [12]

i

2e5  m3

Peng et al.[12]

1

1e5 GPa

Assumption

1

1e6 GPa

Assumption

2

1e5 GPa

Assumption

2

1e6 GPa

Assumption

i

 i  0, 1 i , (i  1) i , and 1

Assumption

Under the linear degradation path model, we suppose that g (t )   t and h(t )  (  t )2 . The parameters of random shocks are assumed for the purpose of

illustration, including the arrival rate  , shock damage size H i , shock loads size Si and some other parameters listed in Table 1. By the models proposed above and the parameters in Table 1, we could make use of the models to assess the system reliability. As the parameters are listed in Table 1, Yg (t ) follows a linear degradation path process, Yg (t ) ~ N (8.4823e9t ,(6.0016e10t )2 ) . The shock magnitude Si caused by the

i th shock is normally distributed, Si  N (1e4 ,(2e5 )2 ) , where the mean and variance are the constants in this instance. In Pattern I of Model 1, by Eq. (4), we know that 2 2 f i |N (t ) ( i | n)  n t  (t   i ) t  . Besides, I ( i )  1 i and  I ( i )  (1 i ) . n -1

19

Figure 3. Reliability of the system in Patterns I (a) and II (b) of Model 1 under HPP.

And so, by the parameters in Table 1 and Eqns. (5), (14), and (16), the reliability formula of the system in Pattern I of Model 1 is obtained easily. The degree of the i th calibration is represented by  i  0 , (i  1) i , and 1, respectively. They indicate the performance of calibrations on the system lifetimes displayed in the figures below. With the corresponding parameters given in Table1 and the analyses above, the reliability results are shown in Figure 3(a). Meanwhile, as an extended study, by using Eq. (17) in Pattern II of Model 1, assuming that  2 =1 and  2 =1 , other parameters are the same as those listed in Table 1. The reliability are calculated under  i  0 , (i  1) i , and 1 shown in Figure 3(b) as well. Because the shock magnitudes of Pattern II are larger than those in Pattern I under the same parameters used, consequently, the reliability in the former one is greater than the later one through the results in Figures 3(a) and 3(b). For the purpose of comparison, we can also derive the reliability under coefficient 1 =2.0e5 of the shock magnitudes in Figures 3(a) and 3(b). To be mentioned, by the results obtained in Figure 3, it is clear to conclude that the calibrations are effective. It is valuable to ensure the system to operate with much higher reliability in the middle to late period of the operating stage.

20

Figure 4. Sensitivity analyses of R(t ) on  and threshold DS under HPP with i  1 i in Patterns I (a) and II (b) of Model 1.

Figure 5. Reliability of the system in Patterns I (a) and II (b) of Model 1 under NHPP.

By Figure 4, the sensitivity analyses are consistent with the dependent competing failure model wherein both shock rate and threshold could determine the failure rate. For the generalized studies, under Patterns I and II of Model 2, suppose that  (t ) is an increasing rate function, for example,  (t )  2.5e10t . Other parameters are the same as those listed in Table 1. And so, fi |N (t ) ( | n)d can be obtained through Eqns. (8) and (18). Then by using the models in Eqns. (19) and (20), the reliability formulas are derived as well. And so, the detailed reliability assessments are displayed in Figure 5 under the two patterns by different coefficients of calibrations, for instance,

 i  0, 1 i , and 1 conditioned on N (t )  n . From Figures 3 and 4, they indicate that we would be better to calibrate with much more rejuvenation in the operating period 21

because the performance of i  (i  1) i is better than that of  i  1 i . 4.2. Simulations for the Models under Wiener Process The studies are also developed to the simulations of the reliability analyses under MDCFSSC for jump diffusion process and extreme shock mode in Model 2. Assume that the parameters are given as, DX  30 , DS  3.5 , i  2.0 ,  i  1.0 , a(t )  0.9t , b(t )  0.3t ,   5 , 1 =0.3 , 1 =0.03 , X w (0)  xw  0 , and l  0.001

for HPP. By using the simulation procedure proposed in Subsection 3.3 to iterate 10,000 generations, the empirical p.d.f. of the failure time under the model are derived shown in Figures 6(a) and 7(a), respectively. From Figures 6(b) and 7(b), by the reliability results, calibrations play an important role in improving reliability. Compared to the results of the two patterns, with the same parameters, the reliability of arrival time is much lower than the case of inter-arrival time because the former one has the higher probability to hard failure because  2Ti is stochastic greater than 1i when 1   2 . Similarly, besides the parameters listed in the case of HPP, we denote  2 = 1 ,

2 = 1 , and  (t )  8.0e3t for NHPP. By using the procedure proposed in Subsection 3.3 to iterate for 10,000 generations, the empirical p.d.f. of the failure time under the model is derived as well. The specified results are displayed in Figures 8 and 9. The results are similar with those obtained in the case of HPP. And so, from Figures 6-9, we not only verify that calibrations contribute to the improvement of system reliability compared with the models without calibrations when  i  0 , but also know that the threshold level and the shock rate play a key role in the model. In particular, they could determine the shape and the rate of the first passage time under the models. In fact, the empirical density functions could be appeared as a single peak curve in Figures 7(a) and 9(a) or a bimodal curve in Figures 6(a) and 8(a) according to  (t ) and the thresholds. Through lots of simulations, an empirical conclusion reveals that either the soft failure or the hard failure is the main reason leading to failure that the empirical density will be displayed in terms of a single peak curve. Otherwise, it will be shown in a bimodal curve. That is to say, under the case of the single peak, one of the two failure mechanisms occurs much more often than another one or their failure times are almost the same under the parameters.

22

Figure 6. In Pattern I of Model 2, the empirical curves of p.d.f. of the failure time and reliability under the Wiener process by simulations (HPP).

Figure 7. In Pattern II of Model 2, the empirical curves of p.d.f. of the failure time and reliability under the Wiener process by simulations (HPP).

23

Figure 8. In Pattern I of Model 2, the empirical curves of p.d.f. of the failure time and reliability under the Wiener process by simulations (NHPP).

Figure 9. In pattern II of Model 2, the empirical curves of p.d.f. of the failure time and reliability under the Wiener process by simulations (NHPP).

In addition to the similar summary obtained in Subsection 4.1, by Figures 7 and 9, the reliabilities are almost equal because the system failed commonly due to hard failure when the calibrations have not been carried out before hard failure. In other words, the hard failure due to large shock magnitude often occurs earlier than the soft failure in the most of the simulations. 24

Figures 3-9 are given as illustrations to Models 1 and 2. Moreover, based on other real problems, the influence of arrival times on the shock load could be changed to another opposite case where the probability of failure will be lower if the inter arrival time or arrival time is much longer. Apart from the coefficients used, based on the empirical knowledge, other calibration coefficients could be used as well. In the view of the improvement of system safety and reliability, it concludes that the calibrations play an effective role in applications by the results shown in the figures. In addition, with the figures illustrated, the calibration is much more effective at the beginning of the operating stage; and with time goes by, the degradation cannot be rejuvenated too much since the soft failure arises in a much higher probability. Hence, MDCFSSC is meaningful for engineer to make decisions, such as maintenance and calibrations. 5. Conclusions The sensor-based calibration proposed in this paper is an extended method to improve the system reliability according to the MDCFSSC due to the rejuvenation of additional degradation damage on the overall degradation measurement. On the one hand, the proposed shock models based on the two patterns are valuable to the real problems. On the other hand, from the dependent competing failure models based on the general degradation path model and the Wiener degradation process model for HPP and NHPP, it is concluded that the calibration associated with the shock process can enhance safety of the system operating in an uncertain dynamic environment. Besides, from the reliability results and simulations, the obtained result from the two patterns and conclusion are useful and meaningful in engineering. In summary, if we would like to eliminate the risk, this is a safe situation where risks of injury or property damage are low and manageable with the way given here. However, some other difficult works need to be done in the future, for instance, the lower bound of the confidence interval of the reliability, the reliability model in the case of shocks following a general renewal process, and the prediction of residual useful lifetime in real applications. Besides, the conditional p.d.f. of (inter) arrival time is perhaps difficult to be derived in other cases, e.g., a general renewal process and the internal continuous degradation indicted by a general stochastic degradation process.

25

Appendix A By Eqns. (1), (3), (4), and (5), under extreme shock model, the reliability of the system subject to hard failure over time t for HPP, noted as Rh,I (t ) , is expressed as n

  t  D  I ( ))  n  t   n-1   N (t )  PI   H i  DS   P( N (t )  0)+ P( N (t )  n)     S    d  . (18)   (  ) t t   n 1  i 0  I  0  

By Eqns. (1), (3), (4), and (6), the reliability of the system subject to hard failure at time t for NHPP, Rh,I (t ) is then explicitly written as  t  DS  I ( 1 )   ( 1 )   N (t )  PI   H i  DS   P( N (t )  0)+P( N (t )  1)     d 1    i 0   0   I ( 1 )  m(t )    t  D   ( )  n ( )  m(t )  m( ) n 1  I 1 1 1   P( N (t )  n)      S d 1        0   I ( 1 )  m(t )  m(t ) n2    

n  t  D  I ( i )    n!       S   n i2  0   I ( i )   m(t ) (i  2)!(n  i )! 

  t  i      ( y ) ( y   i )m( y )i  2 (m(t )  m( y   i )) n i dy  d i  .     0  

(19) By Eqns. (2), (3), (7), and (8), the reliability of the system subject to hard failure over time t for HPP, written as Rh,II (t ) , is given as n t   D  II (ti )  tii 1 (t  ti )ni n !  N (t )  PII   H i  DS    P( N (t )  0)  P( N (t )  n)    S dti .  n  ( t ) t ( n  i )!( i  1)! n  1 i  1 i  0   II i   0

(20) And so, similarly, by Eqns. (2), (3), (7), and (9), the reliability of the system subject to hard failure at time t for NHPP, Rh,II (t ) , is presented as n  t   D  II (ti )   N (t )  PII   H i  DS    P( N (t )  0)   P( N (t )  n)     S  n 1 i 1  0  i 0    II (ti )  i 1

 m(ti )   m(ti )  n!    1   (i  1)!(n  i)!  m(t )   m(t ) 

n i

 dti  . m(t ) 

 (ti )

(21)

Appendix B Generally, we denote the arrival times of NHPP by T0  0,T1 , T2 , , Tn by time t . The corresponding inter-arrival times are 1  T1 , 2  T2  T1 , , n  Tn  Tn1 . Let t  (t ) be a known, deterministic function of time and let m(t )    (u )du be the 0 mean function which is the expected number of the NHPP in the time interval (0, t). Let N (t ) be the number of arrivals in the interval (0, t ) . Then the probability of 26

having i arrivals on the interval (t , t  x) is given by

P  N (t  x)  N (t )  i  

(m(t  x)  m(t ))i exp((m(t  x)  m(t ))) . i!

Firstly, up to time t , we write an equation of the probability of the first inter-arrival time 1   1 and the total number of events N (t )  n ,

(m( 1 )  m(0))0 exp((m( 1 )  m(0))) 0! (m(t )  m( 1 )) n  exp((m(t )  m( 1 ))) n! (m(t )  m( 1 )) n  exp(m(t )), n!

P  1   1 N (t )  n  

(22)

where the first factor is the probability to have no arrivals on interval (0, 1 ) , and the second one is the probability to have exactly n arrivals on interval ( 1 , t ) . These probabilities should be multiplied as N (1  0) and N (t  1 ) are independent variables on the disjoint intervals.

P  1   1 N (t )  n   m(t )  m(1 )  Then P  1   1| N (t )  n     . P( N (t )  n) m(t )   n

(23)

Next, we write an expression for the probability of the kth arrival time tk  ( y, y  dy) for 1  k  (n  1) and the (k +1)th inter-arrival time k 1   k 1 and the total number of events N (t )  n , P Tk  ( y, y  dy )

m( y )k 1 exp(m( y )) ( y )dy (k  1)!  exp(m( y   k +1 )  m( y ))

 k +1   k +1



N (t )  n  

(m(t )  m( y   k +1 )) (n  k )!

nk

(24)

exp(m(t )  m( y   k +1 )),

where the first factor is the probability to have the kth arrival time Tk  ( y, y  dy) , the second factor is the probability to have no arrivals on interval ( y, y   k 1 ) , and the last one is the probability to have (n  k ) arrivals on the interval ( y   k 1 , t ) . Again, these probabilities should be multiplied due to mutual independence. Then by Eq. (24), we implied, P   k +1   k 1

N (t )  n  

t  k 1



Pr Tk  ( y, y  dy)

 k +1   k 1

N (t )  n  ,

(25)

0

where the upper limit of integration is (t   k 1 ) because for any given  k 1  t , ( y   k 1 ) must be less than t . Furthermore, by Eq. (25), the (k  1)th inter-arrival time conditional probability is

27

P   k +1   k +1 | N (t )  n  

P   k +1   k +1 N (t )  n  Pr( N (t )  n)

t  k +1

 P T

k





 k +1   k +1

 ( y, y  dy )

N (t )  n 

0

(26)

m(t ) n exp(m(t )) n! n! n m(t ) (k  1)!(n  k )!

t  k +1



 ( y )m( y ) k 1 (m(t )  m( y   k +1 )) n  k dy.

0

Subsequently, by Eqns. (23) and (26), we obtain

 m(t )  m( ) n k +1 k  0,   , m(t )    P   k +1   k +1 | N (t )  n     t  k +1  k 1 nk  n !   ( y )m( y ) (m(t )  m( y   k +1 )) dy    0  ,1 k  (n  1). n  m(t ) (k  1)!(n  k )!  By Eq. (4), it is known that P(i   i | N (t )  n)   (t   i ) t 

n

for i  1, 2,

, n in

HPP. Then with Eqns. (23) and (26) when  (t )   , the conditional c.d.f. is obtained which is coherent with that under HPP. Besides, the distribution can also be obtained by the joint distribution of the adjacent arrivals which is much easier and shorter.

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Acknowledgments This work was supported by the NSF of China Grants 71631001 and 71371031. China Scholarship Council supported Dejing Kong to study under the supervision of Prof. N. Balakrishnan in McMaster University. Meanwhile, the authors appreciate the editors and the anonymous reviewers for their supportive and constructive comments. References [1] Inagaki T. (1991). Interdependence between safety-control policy and multiple-sensor schemes via Dempster-Shafer theory. Reliability, IEEE Transactions on, 40(2), 182-188. [2] Agogino A, Goebel K, Alag S. Intelligent sensor validation and sensor fusion for reliability and safety enhancement in vehicle control. California Partners for Advanced Transit and Highways (PATH). 1995. [3] Elwany A H, Gebraeel N Z. Sensor-driven prognostic models for equipment replacement and spare parts inventory. IIE Transactions, 2008; 40(7): 629-639. [4] Kaiser K A, Gebraeel N Z. Predictive maintenance management using sensor-based degradation models. Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, 2009; 39(4): 840-849. [5] Pascale A, Nicoli M, Deflorio F, Dalla Chiara B, Spagnolini U. Wireless sensor networks for traffic management and road safety. IET Intelligent Transport 28

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