Reliability analysis of body sensor networks with correlated isolation groups

Abstract

Body sensor networks (BSNs) are playing a crucial role in tackling arising challenges during the COVID-19 pandemic. This work contributes by modeling and analyzing the BSN reliability considering the effects of correlated functional dependence (FDEP) and random isolation time behavior. Particularly, the FDEP exists in BSNs where a relay is utilized to assist the communication between some biosensors and the sink device. When the relay malfunctions, the dependent biosensors may communicate directly with the sink for a limited, uncertain time. These biosensors then become isolated from the rest of the BSN when their remaining power depletes to the level insufficient to support the direct communication. Moreover, multiple biosensors sharing the same relay and a biosensor communicating with the sink via several alternative relays create correlations among different FDEP groups. In addition, the competition in the time domain exists between the local failure of the relay and the propagated failures of dependent biosensors. Both the correlation and competition complicate the reliability modeling and analysis of BSNs. This work proposes a combinatorial and analytical methodology to address both effects in the BSN reliability analysis. The proposed method is demonstrated using a detailed case study and verified using a continuous-time Markov chain method.

Acronyms

BDD

binary decision diagram

BSN

body sensor network

CTMC

continuous-time Markov chain

FDEP

functional dependence

FT

fault tree

LF

local failure

ITE

isolation time event

NDC

non-dependent component

PDF

probability density function

PF

propagated failure

PFGE

propagated failure with global effect

TE

trigger event

TTI

time-to-isolation

Notations

$A\to B$

event of the event A occurring before the event B

iLF

event of LF of component i

iI

event that component i is isolated from the system within the mission time

iPF

event of PFGE of component i

I_NDC

set of NDCs

SR

event that the system is reliable

t

mission time

CR(t)

conditional probability that the system is reliable given that at least one NDC has PFGE

f_iLF(t)

PDF of time-to-LF of component i

f_iPF(t)

PDF of time-to-PF of component i

f_iI(t)

PDF of time-to-isolation of the dependent component i given the failure of its relay

m

the number of triggers

P_u(t)

probability that at least one NDC has PFGE

q_iL(t)

conditional probability that component i has LF given that no PFGE occurs to component i

q_iLF(t)

LF occurrence probability of component i

q_iPF(t)

PFGE occurrence probability of component i

R(t)

system reliability

TE_j

occurrence of the trigger event j

D_j

set of affected dependent components under TE_j

n_j

the number of components in D_j

SPF_j,k

set of components with PFGEs in D_j under $T{E}_{j,k}$

p_j,k

the number of component in SPF_j,k

SNPF_j,k

set of components without PFGEs in D_j under $T{E}_{j,k}$

q_j,k

the number of components in SNPF_j,k

1. Introduction

Body sensor networks (BSNs), which monitor the physiological and metabolic parameters of the human body, play an essential role in tackling arising challenges during the COVID-19 pandemic [1,2]. For example, by detecting the health-related information (e.g., temperature, oxygen saturation (SpO₂), respiratory rate, and lung sounds), BSNs can monitor the populations at risk (such as caregivers and people working at airports and restaurants) and determine the priority of infected people [3]. A lower priority is determined for people who are in the early stages of coronavirus infection and need to be quarantined, and a higher one is determined for people who are with mild symptoms but may suddenly worsen and need emergency care [3,4]. Due to the life-critical nature of the BSN applications, reliability modeling and analysis are necessary when designing and operating a BSN system [3,5].

A BSN system contains several wearable biosensors for sensing and transmitting signals and a sink device for processing data acquired. The relayed communication, where biosensors usually communicate with the sink device through a relay, is typically used for the BSN system to improve the reliability, energy efficiency, and long-term sustainability of information delivery [6], [7], [8]. The relationship between the relay and its dependent biosensors is an example of the functional dependence (FDPE) behavior, which can lead to the competition in the time domain between the local failure (denoted by LF) of the relay and the propagated failures (denoted by PF) of corresponding dependent biosensors [[9], [10]]. This work assumes the propagated failure with global effect (PFGE), where a PF crashes the whole BSN system once occurring. Specifically, biosensors can be subject to PFs (which damage the biosensor itself and other BSN components, e.g., malicious cyber-attacks from intelligent eavesdropper [11]) and LFs (which damage the biosensor itself, e.g., malfunction due to aging [12]). The BSN system status is different according to the occurrence sequence of the relay's LF and the dependent biosensors’ PFs. If the LF of the relay occurs first, the failure isolation effect dominates, preventing PFs originating from the dependent biosensors from damaging the rest of the BSN system and at the same time preventing the isolated biosensors from contributing to the function of the BSN system. On the other hand, if the PFs of dependent biosensors occur first, the failure propagation effect dominates, causing extensive damages to the BSN system. In summary, the failure propagation effect and the failure isolation effect may take place in the actual BSN operation. The effect that dominates the status of the BSN system is dependent on the competitions between the relay's LF and the dependent biosensors’ PFs in the time domain. In addition to BSNs [13], such competitions have been explored in the operation of other systems, such as the flight control system of the aircraft [14], the electricity system [15], and the hydroelectric generation plant [16].

The system reliability analysis has been investigated for different types of competitions. For example, competitions between the degradation and the catastrophic failures were modeled based on different statistical failure modes (e.g., s-independent [17,18] or s-dependent [[19], [20], [21]]). Competing processes were investigated in the context of accelerated life tests in [22,23], system maintenance policies in [24,25], and the system size optimization in [26,27]. Competitions between the uncovered and covered fault modes of system components were studied as the imperfect fault coverage behavior in some studies [[28], [29], [30]]. Distinguishable from those works, the competition addressed in this paper is between the failure propagation and isolation effects.

Addressing the competing failure propagation and isolation effects has recently received lots of attentions. Combinatorial reliability analysis methods were proposed for binary state systems [[31], [32], [33]], multi-state systems [34], phased mission systems [[35], [36], [37]], and cascading systems [38,39]. A generalized stochastic Petri net method was proposed for modeling the reliability of a system where the stochastic dependencies exist among the system components [14]. However, most of the existing models assumed instantaneous isolation when the failure isolation effect takes place. In practice, the isolation may take random time to happen. Specifically, in the BSN, when the LF of the relay occurs first, the dependent biosensor can increase its transmission power to enable the direct communication with the sink. Before the biosensor's remaining power depletes to the level insufficient to support the direct contact, the biosensor can communicate with the sink device directly (i.e., is not isolated). The time that it takes to isolate the dependent biosensor is random and uncertain, depending on the battery's remaining power and the distance between the dependent biosensor and the sink device. With the consideration of the random isolation time, the failure propagation effect dominates if any PF of the dependent biosensor takes place before the relay's failure, or it happens after the relay's failure but before the remaining battery power in the biosensor becomes insufficient to support the direct communication to the sink node.

In [40], the random isolation time was first studied for BSN systems with independent FDEP groups. However, the correlated FDEP groups exist in many real-world systems [[41], [42], [43]], where multiple biosensors may share the same relay or a biosensor can communicate with the sink device via several alternative relays (for fault-tolerant routing), incurring correlations among multiple FDEP groups and thus further complicating the BSN reliability modeling and analysis. This paper advances the state of the art by suggesting a combinatorial and analytical methodology for modeling and analyzing the reliability of BSNs subject to competing failure propagation and isolation effects from multiple correlated FDEP groups and random isolation time.

The remainder of the paper is organized as follows: Section 2 introduces the preliminary method for addressing the single FDEP group with the random isolation time. Section 3 proposes a combinatorial approach for evaluating the reliability of BSN systems subject to correlated FDEP groups and random isolation time. Section 4 presents a step-by-step illustration of the proposed method using an example BSN system. Section 5 presents numerical results and analysis. Verification of correctness of the proposed method using the Markov-based method is also performed. Finally, Section 6 concludes the work and outlines future research directions.

2. Preliminary method for modeling FDEP with random isolation time

To model the FDEP relationship between the relay (referred to as the trigger) and the dependent biosensor with the random isolation time behavior, an FDEP gate with a tilde symbol is defined as shown in Fig. 1 , where T and D respectively model failure events of the trigger T and the dependent biosensor D, and $T\to D$ models that the biosensor D takes a random time to be isolated if the LF of the trigger T occurs.

Fig. 1.

Modeling FDEP with random isolation time.

As discussed in the Introduction, the competing behavior exists between the failure isolation and propagation effects. The failure isolation effect wins if T’s LF happens before D’s PFGE and the time needed for isolating D is less than the time difference between T’s LF and D’s PFGE. When the isolation effect wins, D is isolated and the BSN system may still be functional, depending on the components remaining in the system. The failure propagation effect wins if: i) D’s PFGE happens before T’s LF, or ii) D’s PFGE happens after T’s LF and the time needed for isolating D is greater than the time difference between them. When the failure propagation effect wins, D’s PFGE crashes the entire system. Meanwhile, when T’s LF occurs, it takes a random time for D to be isolated. Table 1 summarizes the occurrence of the isolation time behavior of D and the competing behavior based on the occurrence of T’s LF (denoted by TLF) and D’s PFGE (denoted by DPF). $\overline{TLF}$ and $\overline{DPF}$ are complementary events of TLF and DPF, respectively. Note that because we consider the isolation time behavior of the dependent biosensor D as the behavior that the dependent biosensor D takes a random time to be isolated given that TLF occurs and the competing behavior occurs when both TLF and DPF take place, in Table 1, neither the isolation time behavior nor the competing behavior happens under the combinations of $\overline{TLF}\cap \overline{DPF}$ and $\overline{TLF}\cap DPF$.

Table 1.

The system behavior based on the occurrence of TLF and DPF.

Combinations	Behavior involved
$\overline{TLF}\cap \overline{DPF}$	Neither the isolation time behavior nor the competing behavior
$\overline{TLF}\cap DPF$	Neither the isolation time behavior nor the competing behavior
$TLF\cap \overline{DPF}$	The isolation time behavior only
$TLF\cap DPF$	Both the isolation time behavior and the competing behavior

When the isolation time behavior of D happens, the system including the FDEP group may still function under three situations listed below.

• ■Situation (a): the system does not involve the competing behavior and the dependent component D is isolated after the mission time;
• ■Situation (b): the system does not involve the competing behavior and the dependent component D is isolated during the mission time; and
• ■Situation (c): the system involves the competing behavior and the dependent component D is isolated before DPF.

Fig. 2 illustrates the occurrence sequence of TLF, DPF, and DI under each situation, where DI denotes the event of D being isolated within the mission time given that TLF occurs (i.e., the failure isolation effect takes place).

Fig. 2.

Occurrence sequence of TLF, DPF, and DI under each situation.

The occurrence probability of each situation is calculated as follows. Under situations (a) and (b), no PFGE happens to D. Because the occurrence of the event DI depends only on the occurrence of the event TLF, the battery power remained in the dependent component D and the distance between D and the sink device, the relationship between the event DPF and the event DI is assumed to be independent. Note that while the failure propagation effect and the failure isolation effect are mutually exclusive, the relationship between the event DPF and the event TLF can be assumed to be independent since the two events occur to two different system components (i.e., the dependent component D experiencing the PFGE does not affect the trigger component T experiencing an LF and vice versa). Hence, the occurrence probability of situations (a) and (b), respectively denoted by $\text{Pr}\left(TLF\cap \overline{DPF}\cap \overline{DI}\right)$ and $\text{Pr}\left(TLF\cap \overline{DPF}\cap DI\right)$, can be evaluated as (1) and (2), where $\mathrm{Pr}\left(\overline{DPF}\right)$ is evaluated as $[1-q_{DPF}\left(t\right)]$ with $q_{DPF}\left(t\right)$ denoting the PFGE occurrence probability of the dependent component D.

$$\mathrm{Pr}\left(TLF\cap \overline{DPF}\cap \overline{DI}\right)=\mathrm{Pr}\left(\overline{DPF}\right)\times \mathrm{Pr}\left(TLF\cap \overline{DI}\right)$$ (1)

$$\mathrm{Pr}\left(TLF\cap \overline{DPF}\cap DI\right)=\mathrm{Pr}\left(\overline{DPF}\right)\times \mathrm{Pr}\left(TLF\cap DI\right)$$ (2)

Suppose that TLF happens at ${\tau }_{TLF}$ and $DI$ happens at ${\tau }_{DI}$, $0\le {\tau }_{TLF}\le t$ and ${\tau }_{TLF}\le {\tau }_{DI}$. The length of the time from the occurrence of TLF to the occurrence of DI, i.e., time-to-isolation (TTI), is denoted as ${\tau }_{TTI}$, ${\tau }_{TTI}={\tau }_{DI}-{\tau }_{TLF}$, and its interval is $[t-{\tau }_{TLF},\infty ]$ under situation (a) and is $[0,t-{\tau }_{TLF}]$ under situation (b). Based on the sequential event probability evaluation [[16], [44]], $\mathrm{Pr}\left(TLF\cap \overline{DI}\right)$ in (1) and $\mathrm{Pr}\left(TLF\cap DI\right)$ in (2) are obtained as (3), where $f_{TLF}\left({\tau }_{TLF}\right)$ is the probability density function (PDF) of time-to-LF of T at ${\tau }_{TLF}$, and $f_{DI}\left({\tau }_{TTI}\right)$ is the PDF of TTI of D at ${\tau }_{DI}$ (i.e., the time when the event DI occurs) given that TLF occurs at ${\tau }_{TLF}$.

$$\begin{array}{c}\mathrm{Pr}\left(TLF\cap \overline{DI}\right)=\underset{0}{\overset{t}{\int }}\underset{t-{\tau }_{TLF}}{\overset{\infty }{\int }}{f}_{TLF}\left({\tau }_{TLF}\right)\times f_{DI}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{TLF}\\ \mathrm{Pr}\left(TLF\cap DI\right)=\underset{0}{\overset{t}{\int }}\underset{0}{\overset{t-{\tau }_{TLF}}{\int }}{f}_{TLF}\left({\tau }_{TLF}\right)\times f_{DI}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{TLF}\end{array}$$ (3)

Under situation (c), the competing failure behavior and the isolation time behavior occur. The FDEP model can be functional only if TLF occurs before DPF and the time needed for isolating D is less than the occurrence time between them (denotes by $\left\{\left(TLF\to DPF\right)\cap DI\right\}$). Suppose that TLF happens at ${\tau }_{TLF}$, DPF happens at ${\tau }_{DPF}$, and DI happens at ${\tau }_{DI}$, ${\tau }_{TTI}={\tau }_{DI}-{\tau }_{TLF}$, $0\le {\tau }_{TLF}\le {\tau }_{DPF}\le t$ and $0\le {\tau }_{TTI}\le {\tau }_{DPF}-{\tau }_{TLF}$. $\text{Pr}\left(TLF\cap DPF\cap DI\right)$ is evaluated as (4), where $f_{TLF}\left({\tau }_{TLF}\right)$ is PDF of time-to-LF of T at ${\tau }_{TLF}$, $f_{DPF}\left({\tau }_{DPF}\right)$ is the PDF of time-to-PFGE of D at ${\tau }_{DPF}$, and $f_{DI}\left({\tau }_{TTI}\right)$ is the PDF of time-to-isolation of D at ${\tau }_{DI}$ given that TLF occurs at ${\tau }_{TLF}$.

$$\begin{array}{c}\mathrm{Pr}\left(TLF\cap DPF\cap DI\right)=\mathrm{Pr}\left(\left\{\left(TLF\to DPF\right)\cap DI\right\}\right)\hfill \\ {\int }_0^t{\int }_{{\tau }_{TLF}}^t{\int }_0^{{\tau }_{DPF}-{\tau }_{TLF}}{f}_{TLF}\left({\tau }_{TLF}\right)\times f_{DPF}\left({\tau }_{DPF}\right)\times f_{DI}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{DPF}d{\tau }_{TLF}\hfill \end{array}$$ (4)

The model presented in this section aims to address the random isolation time behavior. Therefore, the FDEP group with a single trigger and a single dependent component is illustrated. This preliminary method is incorporated in the proposed methodology for addressing the FDEP model with multiple triggers and dependent components. Refer to Section 4 for specific illustrations.

3. Proposed combinatorial methodology

The combinatorial methodology proposed for the reliability analysis of BSNs with correlated FDEP groups and the random isolation time behavior contains five steps. Note that reliability is generally defined as “the probability that a product, system, process, or service will perform a required function under stated conditions for a stated period of time” [45]. In this work, the reliability of a BSN system is defined as the probability that the BSN system will perform its required monitoring function correctly using multiple biosensors and relays throughout an interval of time [0, t], where biosensors are used to measure physiological and/or motion information and relays are used to assist communications between biosensors and the sink device.

3.1. Step 1: separate PFGEs of non-dependent components

PFGEs originating from components whose functions are not dependent on any other system components’ function cannot be isolated. Such components are referred to as non-dependent components (NDCs). Applying the total probability law, we can separate PFGEs of NDCs from the system reliability (denoted by R(t)) as (5), where SR represents that the system is reliable.

$$\begin{array}{c}R\left(t\right)=\mathrm{Pr}\left(SR|atleastoneNDChavingPFGE\right)\times \mathrm{Pr}\left(atleastoneNDChavingPFGE\right)\hfill \\ +\mathrm{Pr}\left(SR|noNDCshavingPFGEs\right)\times \mathrm{Pr}\left(noNDCshavingPFGEs\right)\hfill \end{array}$$ (5)

In (5), Pr(SR│at least one NDC having PFGE) is 0 since any PFGEs originating from NDCs can lead to the failure of the whole system. Denoting Pr(SR│no NDCs having PFGEs) by CR(t) and Pr(no NDCs having PFGEs) by P_u(t), (5) is simplified to (6).

$$R\left(t\right)=CR\left(t\right)\times P_u\left(t\right)$$ (6)

In (6), $P_u\left(t\right)$ is calculated by (7), where $I_{NDC}$ denotes the set of NDCs subject to PFGEs and $q_{iPF}\left(t\right)$ denotes the PFGE occurrence probability of component i at time t and is evaluated based on the input time-to-PFGE parameters of component i.

$$P_u\left(t\right)=\prod _{\forall i\in I_{NDC}}\left[1-q_{iPF}\left(t\right)\right]$$ (7)

The calculating procedure of $CR\left(t\right)$ is presented in steps 2 to 5 in Sections 3.2 – 3.5. Note that $CR\left(t\right)$ is a conditional probability. Therefore, the corresponding component i’s failure probability used in calculating $CR\left(t\right)$, denoted by $q_{iL}\left(t\right)$, is a conditional failure probability given that component i does not experience PFGE. Considering the s-relationship between LF and PFGE of component i, $q_{iL}\left(t\right)$ is calculated by (8), where iLF and iPF are events of component i having an LF and a PFGE, respectively.

$$\begin{array}{c}{q}_{iL}\left(t\right)=\mathrm{Pr}\left(iLF|\overline{iPF}\right)=\frac{\mathrm{Pr}\left(iLF\cap \overline{iPF}\right)}{\mathrm{Pr}\left(\overline{iPF}\right)}\hfill \\ =\{\begin{array}{c}\frac{\mathrm{Pr}\left(iLF\right)\times \mathrm{Pr}\left(\overline{iPF}\right)}{\mathrm{Pr}\left(\overline{iPF}\right)}=q_{iLF}\left(t\right)independentLFandPFGE,\\ \frac{\mathrm{Pr}\left(iLF\right)}{\mathrm{Pr}\left(\overline{iPF}\right)}=\frac{q_{iLF}\left(t\right)}{1-q_{iPF}\left(t\right)}mutuallyexclusiveLFandPFGE.\end{array}\hfill \end{array}$$ (8)

In (8), the independent relationship between LF and PFGE implies that the incidence of one event does not affect the occurrence probability of the other event, thus we have $\mathrm{Pr}\left(iLF\cap \overline{iPF}\right)=\mathrm{Pr}\left(iLF\right)\times \mathrm{Pr}\left(\overline{iPF}\right)$. The mutually exclusive relationship represents that LF and PFGE are disjoint and cannot occur at the same time. Thus, we have $\mathrm{Pr}\left(iLF\cap \overline{iPF}\right)=\mathrm{Pr}\left(iLF\right)$ [46]. Note that each biosensor considered in this work is subject to LFs that may compromise its sensing function (referred to as sensing LF) or transmission function (referred to as transmission LF) [13]. If the PFGE of the same biosensor is caused by the jamming attacks that are based on the transmission function by transmitting illegal signals without compromising the sensing function of the biosensor, the relationship is independent between PFGE and sensing LF and is mutually exclusive between PFGE and transmitting LF.

3.2. Step 2: generate trigger event space

In the BSN system having m triggers, which are denoted as $T_1$, …, $T_w$, …, $T_m$, a trigger event space of $2^m$ disjoint sets can cover all possible combinations of the occurrence or non-concurrence of LFs of triggers. Denoting the occurrence and non-concurrence of LF of $T_w$ by $T_{w}LF$ and $\overline{T_{w}LF}$ respectively, one can enumerate each combination of trigger events (TEs) as shown in (9).

$$\begin{array}{c}\begin{array}{c}T{E}_1=\overline{T_{1}LF}\cap \overline{T_{2}LF}\cap \dots \cap \overline{T_{m-1}LF}\cap \overline{T_{m}LF}\\ T{E}_2=T_{1}LF\cap \overline{T_{2}LF}\cap \dots \cap \overline{T_{m-1}LF}\cap \overline{T_{m}LF}\end{array}\\ ⋮\\ T{E}_{2^m}=T_{1}LF\cap T_{2}LF\cap \dots \cap T_{m-1}LF\cap T_{m}LF\end{array}$$ (9)

Applying the total probability law, $CR\left(t\right)$ is calculated as

$$CR\left(t\right)=\sum _{j=1}^{2^m}\mathrm{Pr}\left(SR\cap T{E}_j\right).$$ (10)

To calculate $\mathrm{Pr}\left(SR\cap T{E}_j\right)$ in (10), two cases (Case 1 and Case 2) are considered based on the logic input of the FDEP gate.

Case 1

The inputs of all the FDEP gates are logic false (‘0′), implying that either none of the triggers’ LFs happen or the triggers’ LFs happen but have no effect on the dependent components. Under Case 1, neither the competing failure behavior nor the isolation time behavior affects the BSN system. Applying the total probability law, $\text{Pr}\left(SR\cap T{E}_j\right)$ under this case is calculated as (11), where $\mathrm{Pr}\left(T{E}_j\right)$ is determined by its definition in (9). $\text{Pr}\left(SR\right|T{E}_j)$ is determined by the reduced fault tree (FT) model under TE_j, which is obtained by reducing the original system FT model through removing all the trigger failure events with the efficient binary decision diagram (BDD) based method [47]. $\mathrm{Pr}\left(SR\cap T{E}_j\right)$ under Case 1 is evaluated in this step.

$$\mathrm{Pr}\left(SR\cap T{E}_j\right)=\mathrm{Pr}\left(SR|T{E}_j\right)\times \mathrm{Pr}\left(T{E}_j\right)$$ (11)

Case 2

At least one of the inputs of the FDEP gates is logic true (‘1′), i.e., the LF of the trigger happens and affects the related dependent components. Under this case, the competing failure behavior and/or the isolation time behavior occur(s) to the BSN system. Applying the divide-and-conquer principle, we divide $\mathrm{Pr}\left(SR\cap T{E}_j\right)$ under Case 2 by addressing these two behaviors in Step 3.

3.3. Step 3: analyze competing failure behavior and isolation time behavior under Case 2

Under $T{E}_j$, suppose that the set of affected dependent components is $D_j$ and the number of them is $n_j$, i.e., $D_j=\left\{D_{j1},\dots D_{jx},\dots D_{j{n}_j}\right\}$. Based on the occurrence or non-concurrence of $D_{jx}$’s PFGE (respectively denoted by $D_{jx}PF$ and $\overline{D_{jx}PF}$), a dependent component event space is generated. Each combination (denoted by $T{E}_{j,k},k=1,2,\cdots ,2^{n_j}$) is enumerated as shown in (12).

$$\begin{array}{c}\begin{array}{c}T{E}_{j,1}=\overline{D_{j1}PF}\cap \overline{D_{j2}PF}\cap \dots \cap \overline{D_{j(n_j-1)}PF}\cap \overline{D_{j{n}_j}PF}\\ T{E}_{j,2}=D_{j1}PF\cap \overline{D_{j2}PF}\cap \dots \cap \overline{D_{j(n_j-1)}PF}\cap \overline{D_{j{n}_j}PF}\end{array}\\ ⋮\\ T{E}_{j,2^{n_j}}=D_{j1}PF\cap D_{j2}PF\cap \dots \cap D_{j(n_j-1)}PF\cap D_{j{n}_j}PF\end{array}$$ (12)

Applying the total probability law, $\mathrm{Pr}\left(SR\cap T{E}_j\right)$ under Case 2 can be calculated by

$$\mathrm{Pr}\left(SR\cap T{E}_j\right)=\sum _{k=1}^{2^{n_j}}\mathrm{Pr}\left(SR\cap T{E}_{j,k}\right).$$ (13)

Consider $T{E}_{j,k}$. Denote the number of affected dependent components (i.e., components in $D_j$) that have PFGEs as $p_{j,k}$ and they constitute the set $SP{F}_{j,k}$, and denote the number of components in $D_j$ that have no PFGEs as $q_{j,k}$ and they constitute the set $SNP{F}_{j,k}$. Based on the definition of $T{E}_{j,k}$ in (12), $p_{j,k}$, $q_{j,k}$, $SP{F}_{j,k}$, and $SNP{F}_{j,k}$ are summarized in Table 2 .

Table 2.

$p_{j,k},q_{j,k},SP{F}_{j,k},$ and $SNP{F}_{j,k}$ for $T{E}_{j,k},k=1,2,\cdots ,2^{n_j}.$

	$p_{j,k}$	$q_{j,k}$	$SP{F}_{j,k}$	$SNP{F}_{j,k}$
$k=1$	0	$n_j$	$\varnothing$	$\{D_{j1},D_{j2},\cdots ,D_{j{n}_j}\}$
$k=2$	1	$n_j-1$	$\left\{D_{j1}\right\}$	$\{D_{j2},\cdots ,D_{j{n}_j}\}$
$\cdots$	$\cdots$	$\cdots$	$\cdots$	$\cdots$
$k=2^{n_j}$	$n_j$	0	$\{D_{j1},D_{j2},\cdots ,D_{j{n}_j}\}$	$\varnothing$

According to the values of $p_{j,k}$ and $q_{j,k}$, Case 2 can be further divided into three scenarios: (a)$p_{j,1}=0$, $q_{j,1}=n_j$ (i.e.,$T{E}_{j,1}$); (b)$p_{j,k}=n_j,q_{j,k}=0$ (i.e., $T{E}_{j,2^{n_j}}$); and (c)$p_{j,k}\ne 0,q_{j,k}\ne 0$ (i.e., $T{E}_{j,k},k=2,\cdots ,2^{n_j}-1$).

The isolation time behavior is involved when calculating $\mathrm{Pr}\left(SR\cap T{E}_{j,k}\right)$. Specifically, it is incorporated with the competing failure behavior for components in $SP{F}_{j,k}$ and is considered without the competing failure behavior for components in $SNP{F}_{j,k}$. The analysis procedures for $T{E}_{j,1}$, $T{E}_{j,2^{n_j}}$, and $T{E}_{j,k},k=2,\cdots ,2^{n_j}-1$ are performed as follows.

$T{E}_{j,1}$: $p_{j,1}=0$, $q_{j,1}=n_j$, $SP{F}_{j,1}=\varnothing$, and $\mathit{SNP}{F}_{j,1}=\left\{D_{j,1,x}\right|D_{j1},D_{j2},\cdots ,D_{j{n}_j},x=1,2,\cdots ,n_j$}. Since $SP{F}_{j,1}$ is empty, the competing failure behavior does not exist. The isolation time behavior is addressed for $D_{j,1,x}$ in $SNP{F}_{j,1}$. According to the isolation status of $D_{j,1,x}$, an isolation time event (ITE) space is generated. It contains $2^{n_j}$ mutually exclusive ITEs as shown in (14), where $D_{j,1,x}I$ and $\overline{D_{j,1,x}I}$ represents $D_{j,1,x}$’s isolation and $D_{j,1,x}$’s non-isolation, respectively. Applying the total probability law, $\mathrm{Pr}\left(SR\cap T{E}_{j,1}\right)$ in (13) can be calculated by (15).

$$\begin{array}{c}\begin{array}{c}IT{E}_{j,1,1}=\overline{D_{j,1,1}I}\cap \overline{D_{j,1,2}I}\cap \dots \cap \overline{D_{j,1,(n_j-1)}I}\cap \overline{D_{j,1,n_j}I}\\ IT{E}_{j,1,2}=D_{j,1,1}I\cap \overline{D_{j,1,2}I}\cap \dots \cap \overline{D_{j,1,(n_j-1)}I}\cap \overline{D_{j,1,n_j}I}\end{array}\\ ⋮\\ IT{E}_{j,1,2^{n_j}}=D_{j,1,1}I\cap D_{j,1,2}I\cap \dots \cap D_{j,1,(n_j-1)}I\cap D_{j,1,n_j}I\end{array}$$ (14)

$$\begin{array}{c}\mathrm{Pr}\left(SR\cap T{E}_{j,1}\right)=\sum _{l=1}^{2^{n_j}}\mathrm{Pr}\left(SR\cap T{E}_{j,1}\cap IT{E}_{j,1,l}\right)\hfill \\ ={\sum }_{l=1}^{2^{n_j}}\mathrm{Pr}\left(SR|T{E}_{j,1}\cap IT{E}_{j,1,l}\right)\times \mathrm{Pr}\left(T{E}_{j,1}\cap IT{E}_{j,1,l}\right)\hfill \end{array}$$ (15)

$T{E}_{j,2^{n_j}}$: According to Table 2, $p_{j,2^{n_j}}=n_j$, $q_{j,2^{n_j}}=0$, $\mathit{SP}{F}_{j,2^{n_j}}=\left\{D_{j,2^{n_j},y}\right|D_{j1},D_{j2},\cdots ,D_{j{n}_j},y=1,2,\cdots ,n_j\}$, and $SNP{F}_{j,2^{n_j}}=\varnothing$. Since $SP{F}_{j,2^{n_j}}$ is not empty, the competing failure behavior occurs, which is caused by the different occurrence sequences of $T{E}_j$ (defined in (9)) and the PFGE of $D_{j,2^{n_j},y}$ in the time domain. The competing results and related requirements are introduced in Section 2. Note that under $T{E}_{j,2^{n_j}}$, D and T in Section 2 are revised to $D_{j,2^{n_j},y}$ and $T{E}_j$, respectively. Denote the event of the failure propagation dominating the BSN system as $T{E}_{j,2^{n_j},p}$ with subscript p being the propagation effect and the event of the failure isolation dominating as $T{E}_{j,2^{n_j},i}$ with subscript i being the isolation effect. $\mathrm{Pr}\left(SR\cap T{E}_{j,2^{n_j}}\right)$ in (13) is calculated and simplified as

$$\begin{array}{c}\mathrm{Pr}\left(SR\cap T{E}_{j,2^{n_j}}\right)=\text{Pr}\left(SR\cap T{E}_{j,2^{n_j},p}\right)+\text{Pr}\left(SR\cap T{E}_{j,2^{n_j},i}\right)\hfill \\ =\mathrm{Pr}\left(SR\right|T{E}_{j,2^{n_j},p})\times \mathrm{Pr}\left(T{E}_{j,2^{n_j},p}\right)+\mathrm{Pr}\left(SR\right|T{E}_{j,2^{n_j},i})\times \text{Pr}\left(T{E}_{j,2^{n_j},i}\right)\hfill \\ =\text{Pr}\left(SR\right|T{E}_{j,2^{n_j},i})\times \text{Pr}\left(T{E}_{j,2^{n_j},i}\right).\hfill \end{array}$$ (16)

$T{E}_{j,k},k=2,3,\cdots ,2^{n_j}-1$: According to Table 2, $p_{j,k}\ne 0,q_{j,k}\ne 0$, $SP{F}_{j,k}\ne \varnothing$, and $SNP{F}_{j,k}\ne \varnothing$. Consider $SP{F}_{j,k}=\left\{D_{j,k,y}\right|D_{j1},\dots ,D_{jy},y=1,\cdots ,p_{j,k}\}$ and $SNP{F}_{j,k}=\left\{D_{j,k,x}\right|D_{j1},\dots ,D_{jx},x=1,\cdots ,q_{j,k}\}$. The competing failure behavior and the isolation time behavior affect $D_{j,k,y}$ and the isolation time behavior affects $D_{j,k,x}$. The competing result is either the failure propagation (denoted by $T{E}_{j,k,p}$) or the failure isolation (denoted by $T{E}_{j,k,i}$). An ITE space is generated to cover $2^{q_{j,k}}$ mutually exclusive combinations. Combining the analyzing procedure of $T{E}_{j,1}$ and $T{E}_{j,2^{n_j}}$, $\mathrm{Pr}\left(SR\cap T{E}_{j,k}\right)$ in (13) is obtained by (17).

$$\begin{array}{c}\mathrm{Pr}\left(SR\cap T{E}_{j,k}\right)=\mathrm{Pr}\left(SR\cap T{E}_{j,k,p}\right)+\text{Pr}\left(SR\cap T{E}_{j,k,i}\right)\hfill \\ ={\sum }_{m=1}^{2^{q_{j,k}}}\mathrm{Pr}\left(SR\cap T{E}_{j,k,p}\cap IT{E}_{j,k,m}\right)+{\sum }_{m=1}^{2^{q_{j,k}}}\mathrm{Pr}\left(SR\cap T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right)\hfill \\ ={\sum }_{m=1}^{2^{q_{j,k}}}\left[\mathrm{Pr}\left(SR|T{E}_{j,k,p}\cap IT{E}_{j,k,m}\right)\times \mathrm{Pr}\left(T{E}_{j,k,p}\cap IT{E}_{j,k,m}\right)\right]\hfill \\ +{\sum }_{m=1}^{2^{q_{j,k}}}\left[\mathrm{Pr}\left(SR|T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right)\times \mathrm{Pr}\left(T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right)\right]\hfill \\ ={\sum }_{m=1}^{2^{q_{j,k}}}\left[\mathrm{Pr}\left(SR|T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right)\times \mathrm{Pr}\left(T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right)\right]\hfill \end{array}$$ (17)

$\mathrm{Pr}\left(T{E}_{j,1}\cap IT{E}_{j,1,l}\right),l=1,2,\cdots ,2^{n_j}$ in (15), $\text{Pr}\left(T{E}_{j,2^{n_j},i}\right)$ in (16), and $\mathrm{Pr}\left(T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right),m=1,2,\cdots ,2^{q_{j,k}}$ in (17) can be evaluated by applying the preliminary method in Section 2 with components’ input time-to-LF/PFGE parameters and components’ input isolation time parameters. Refer to Section 4.2.3 for illustrations. $\mathrm{Pr}\left(SR|T{E}_{j,1}\cap IT{E}_{j,1,l}\right),l=1,2,\cdots ,2^{n_j}$ in (15), $\text{Pr}\left(SR|T{E}_{j,2^{n_j},i}\right)$ in (16), and $\mathrm{Pr}\left(SR|T{E}_{j,k,i}\cap IT{E}_{j,k,m}\right),m=1,2,\cdots ,2^{q_{j,k}}$ in (17) are evaluated in the next step.

3.4. Step 4: evaluate conditional system reliability

$\mathrm{Pr}\left(SR|T{E}_{j,1}\cap IT{E}_{j,1,l}\right)$ denotes the conditional system reliability given that the dependent components indicated by $IT{E}_{j,1,l}$ are isolated by the occurrence of $T{E}_{j,1}$. For example, when l = 2, according to (14), $D_{j,1,1}$ is isolated. Similarly, $\text{Pr}\left(SR|T{E}_{j,2^{n_j},i}\right)$ in (16) and $\text{Pr}\left(SF\right|T{E}_{j,k,i}\cap IT{E}_{j,k,m})$ in (17) denote the conditional system reliability given the isolation of the dependent components under $T{E}_{j,2^{n_j},i}$ and $T{E}_{j,k,i}\cap IT{E}_{j,k,m}$, respectively. These conditional system reliabilities can be evaluated by applying the efficient BDD-based method on the reduced FT model [47], which is obtained by removing the trigger failure events and replacing all isolated dependent components with constant ‘1′ (True). Refer to Section 4.2.4 for illustrations.

3.5. Step 5: integrate for the entire system reliability

$\text{Pr}\left(SR\cap T{E}_j\right)$ under Case 1 is obtained by (11) in Step 2. According to (14), $\text{Pr}\left(SR\cap T{E}_j\right)$ under Case 2 is obtained in Step 3 by (13) via integrating $\text{Pr}\left(SR\cap T{E}_{j,1}\right)$ evaluated by (15), $\text{Pr}\left(SR\cap T{E}_{j,2^{n_j}}\right)$ evaluated by (16), and $\mathrm{Pr}\left(SR\cap T{E}_{j,k}\right),k=2,\cdots ,2^{n_j}-1$ evaluated by (17). Having $\text{Pr}\left(SR\cap T{E}_j\right)$ under both Cases 1 and 2, $CR\left(t\right)$ is consecutively obtained by (10). Finally, with $P_u\left(t\right)$ obtained by (7), the entire BSN system reliability is integrated using (6).

Based on discussions in Sections 3.1–3.5, Fig. 3 shows the flowchart of the proposed BSN reliability evaluation algorithm.

Fig. 3.

Flowchart for the proposed BSN reliability evaluation method.

4. Illustrative example

Fig. 4 shows an example patient monitoring BSN system, which consists of three biosensors ($S_1$, $S_2$, and $S_3$) and two relays ($R_1$ and $R_2$). Specifically, sensors S ₁ and S ₂ are used to measure patient's physiological information like heart rates and oxygen saturation ($\text{Sp}{\mathrm{O}}_2$); sensor S ₃ is the motion sensor and is used to track patient's body positions. Sensor S ₁ can communicate with the sink device directly. Sensor S ₂ can deliver sensed data to the sink device via $R_1$ or $R_2$. Sensor S ₃ can deliver sensed data to the sink device via $R_2$. The sink device gathers and processes the data acquired by all the sensors, facilitating caregivers’ involvement in action taking. The sink device is assumed to be perfect during the considered mission time.

Fig. 4.

Example patient monitoring BSN system.

4.1. FT modeling

Fig. 5 illustrates the dynamic FT model of the example BSN system. To obtain a good diagnostic certainty, the caregiver requires at least two types of information; in other words, the example BSN system fails when any two out of three biosensors malfunction, which is modeled using the top 2-out-of-3 gate in the system dynamic FT.

Fig. 5.

Dynamic FT model of the example BSN system.

The FDEP behavior with random isolation time exists among relays ($R_1$ and $R_2$) and biosensors ($S_2$ and $S_3$) in the illustrative example. In the case where $R_1$ and $R_2$ undergo LFs, $S_2$ and $S_3$ can communicate with the sink directly within a certain period (modeled by random isolation time); as time goes on, remaining power in respective biosensor depletes to the level insufficient to support the direct link, resulting in biosensor(s) becoming inaccessible/isolated from the system. To model the FDEP behavior, two FDEP gates are used for the two correlated FDEP groups. The left FDEP group involves $R_1$, $R_2$, $S_2$, and a top AND gate connecting $R_1$ and $R_2$, which models $S_2$ depending on either $R_1$ or $R_2$. The right FDEP group involves $R_2$ and $S_3$. To model the random isolation time behavior, $R_1\cap R_2\to S_2$ denotes that $S_2$ takes a random time to be isolated when $R_1$ and $R_2$ fail, and $R_2\to S_3$ denotes that $S_3$ takes a random time to be isolated when $R_2$ fails.

Note that various approaches (e.g., Markov models [48], Monte Carlo simulations [49], stochastic approaches [50], and fuzzy approaches [51]) may be applied to analyze dynamic FT models (without considering the competing failure isolation and propagation effects). In this work, we perform the dynamic FT analysis using the proposed combinatorial methodology to evaluate the reliability of BSN systems subject to the competing behavior.

4.2. Step by step analysis

To evaluate the example BSN reliability, i.e., $R\left(t\right)$ in (6), the procedure proposed in Section 3 is applied. The detailed analysis is performed step by step in Sections 4.2.1 – 4.2.5.

4.2.1. Step 1: separate PFGEs of non-dependent biosensors

Applying (6), the PFGE effect of NDCs is separated. For the illustrated BSN system, $I_{NDC}=\left\{R_1,R_2,S_1\right\}$. Based on (7), $P_u\left(t\right)$ is evaluated as (18). The evaluating procedure of $CR\left(t\right)$ in (6) is elaborated in the following subsections (Sections 4.2.2 – 4.2.5).

$$P_u\left(t\right)=\left(1-q_{R_{1}PF}\left(t\right)\right)\times \left(1-q_{R_{2}PF}\left(t\right)\right)\times \left(1-q_{S_{1}PF}\left(t\right)\right)$$ (18)

4.2.2. Step 2: generate trigger event space

In the example BSN system, there are two triggers. Therefore, $T_1=R_1$, $T_2=R_2$, and $m=2$. The example system contains $2^2=4$ TEs. Applying (9), each TE is defined as shown Table 3 . Applying (10), $CR\left(t\right)$ is obtained as (19).

$$CR\left(t\right)=\sum _{j=1}^4\mathrm{Pr}\left(SR\cap T{E}_j\right)$$ (19)

Table 3.

Definition and classification of TE_j, j = 1,2,3,4.

	Definition	Inputs of FDEP groups	Classification
$j=1$	$\overline{R_{1}LF}\cap \overline{R_{2}LF}$	‘0′ for both FDEP groups	Case 1
$j=2$	$R_{1}LF\cap \overline{R_{2}LF}$	‘0′ for both FDEP groups	Case 1
$j=3$	$\overline{R_{1}LF}\cap R_{2}LF$	‘0′ for the left FDEP group and ‘1’ for the right FDEP group	Case 2
$j=4$	$R_{1}LF\cap R_{2}LF$	‘1′ for both FDEP groups	Case 2

According to logic inputs of each FDEP group, $T{E}_1$ and $T{E}_2$ are classified into Case 1, and $T{E}_3$ and $T{E}_4$ are classified into Case 2. $\mathrm{Pr}\left(SR\cap T{E}_1\right)$ and $\mathrm{Pr}\left(SR\cap T{E}_2\right)$ are evaluated in this step; $\mathrm{Pr}\left(SR\cap T{E}_3\right)$ and $\mathrm{Pr}\left(SR\cap T{E}_4\right)$ are evaluated in Step 3.

$T{E}_1$: Applying (11), $\mathrm{Pr}\left(SR\cap T{E}_1\right)$ is evaluated as (20), where $\mathrm{Pr}\left(T{E}_1\right)$ is determined by its definition in Table 3 as (21).

$$\mathrm{Pr}\left(SR\cap T{E}_1\right)=\mathrm{Pr}\left(SR|T{E}_1\right)\times \mathrm{Pr}\left(T{E}_1\right)$$ (20)

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_1\right)=\mathrm{Pr}\left(\overline{R_{1}LF}\cap \overline{R_{2}LF}\right)=\mathrm{Pr}\left(\overline{R_{1}LF}\right)\times \mathrm{Pr}\left(\overline{R_{2}LF}\right)\hfill \\ =\left[1-q_{R_{1}L}\left(t\right)\right]\times \left[1-q_{R_{2}L}\left(t\right)\right]\hfill \end{array}$$ (21)

Under $T{E}_1$, both $R_1$ and $R_2$ function correctly, and neither the competing failure behavior nor the isolation time behavior occurs to $S_2$ and $S_3$. The reduced system FT model for evaluating $\mathrm{Pr}\left(SR|T{E}_1\right)$ is given in Fig. 6 (a), which is generated by removing $R_1$, $R_2$, and all the FDEP gates from the FT model in Fig. 5. The reduced FT model under TE ₁ contains the biosensors undergoing PFGEs, therefore, the separable method formulated in (6) is applied to calculate $\mathrm{Pr}\left(SR|T{E}_1\right)$ as (22), where $P_{u-T{E}_1}\left(t\right)$ denotes the occurrence probability of no PFGEs originating from biosensors under TE ₁ and is evaluated as (23). Note that $S_1$’s PFGE effect is excluded when calculating $P_{u-T{E}_1}\left(t\right)$ since it has been considered in (18).

Fig. 6.

Reduced system models under $T{E}_1.$

$C{R}_{T{E}_1}\left(t\right)$ in (22) denotes the conditional system reliability given that no PFGEs originate from biosensors in the reduced FT model. To convert the FT model to the BDD model, the 2-out-of-3 gate is expanded to a set of AND and OR gates as shown in Fig. 6(b). Using the operations rules in [47], the corresponding BDD model is gendered as shown in Fig. 6(c), where the 0-edge of each non-sink node is marked with the solid green line, representing that the corresponding component is reliable; the 1-edge is marked with the dashed red line, representing that the corresponding component is failed. Therefore, $C{R}_{T{E}_1}\left(t\right)$ is evaluated as (24) by adding all paths’ probabilities from the root node $S_1$ to the sink node ‘0′ in the BDD model.

$$\mathrm{Pr}\left(SR|T{E}_1\right)=R_{T{E}_1}\left(t\right)=C{R}_{T{E}_1}\left(t\right)\times P_{u-T{E}_1}\left(t\right)$$ (22)

$$P_{u-T{E}_1}\left(t\right)=\left[1-q_{S_{2}PF}\left(t\right)\right]\times \left[1-q_{S_{3}PF}\left(t\right)\right]$$ (23)

$$\begin{array}{c}C{R}_{T{E}_1}\left(t\right)=\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{2}L}\left(t\right)\right]+\left[1-q_{S_{1}L}\left(t\right)\right]\times q_{S_{2}L}\left(t\right)\times \left[1-q_{S_{3}L}\left(t\right)\right]\hfill \\ +q_{S_{1}L}\left(t\right)\times \left[1-q_{S_{2}L}\left(t\right)\right]\times \left[1-q_{S_{3}L}\left(t\right)\right]\hfill \end{array}$$ (24)

$T{E}_2$: Applying (11), $\mathrm{Pr}\left(SR\cap T{E}_2\right)$ is evaluated as (25), where $\mathrm{Pr}\left(T{E}_2\right)$ is determined by its definition in Table 3 as $q_{R_{1}L}\left(t\right)\times \left[1-q_{R_{2}L}\left(t\right)\right]$.

$$\mathrm{Pr}\left(SR\cap T{E}_2\right)=\mathrm{Pr}\left(SR|T{E}_2\right)\times \mathrm{Pr}\left(T{E}_2\right)$$ (25)

Under $T{E}_2$, $R_1$ fails locally and $R_2$ functions correctly. Since $R_1$’s LF has no effect on $S_2$ and $S_3$, neither the competing failure behavior nor the isolation time behavior occurs to the BSN system. The reduced system FT model for evaluating $\mathrm{Pr}\left(SR|T{E}_2\right)$ is obtained by removing $R_1$, $R_2$, and all the FDEP gates, which is the same as that in Fig. 6(a). Therefore, $\mathrm{Pr}\left(SR|T{E}_2\right)$ is the same as $\mathrm{Pr}\left(SR|T{E}_1\right)$, i.e., $\mathrm{Pr}\left(SR|T{E}_2\right)=\mathrm{Pr}\left(SR|T{E}_1\right)$.

4.2.3. Step 3: analyze competing failure behavior and isolation time behavior under Case 2

According to Table 3, $T{E}_3$ and $T{E}_4$ are under Case 2. The competing failure behavior and the isolation time behavior under $T{E}_3$ and $T{E}_4$ are analyzed in Sections 4.2.3.1 and 4.2.3.2 as follows.

4.2.3.1. $\mathrm{T}{\mathrm{E}}_3$

$R_1$ functions correctly and $R_2$ fails locally, resulting in $S_3$ being affected. Therefore, the set of the affected dependent biosensor under $T{E}_3$ is $D_3=\left\{S_3\right\}$ and the number of the affected dependent biosensor is $n_3=1$. Based on (12), the dependent biosensor event space has $2^1=2$ combinations, which are $T{E}_{3,1}=\overline{S_{3}PF}$ and $T{E}_{3,2}=S_{3}PF$. Applying (13), $\mathrm{Pr}\left(SR\cap T{E}_3\right)$ in (19) is obtained by

$$\mathrm{Pr}\left(SR\cap T{E}_3\right)=\sum _{k=1}^2\mathrm{Pr}\left(SR\cap T{E}_{3,k}\right).$$ (26)

The isolation time behavior of $S_3$ under $T{E}_{3,1}$ and $T{E}_{3,2}$ is addressed. Specifically, under $T{E}_{3,1}$, no PFGE originates from $S_3$; therefore, the number of affected dependent biosensors with PFGEs is 0, and the number of affected dependent biosensor without PFGEs is 1 and it forms the set $SNP{F}_{3,1}$, i.e., $p_{3,1}=0$, $q_{3,1}=1$, $SP{F}_{3,1}=\varnothing$, and $SNP{F}_{3,1}=\left\{S_3\right\}$. Under $T{E}_{3,2}$, PFGE originates from $S_3$, therefore, $p_{3,2}=1$, $q_{3,2}=0$, $SP{F}_{3,2}=\left\{S_3\right\}$, and $SNP{F}_{3,2}=\varnothing$. Table 4 summarizes the definition, $p_{3,k}$, $q_{3,k}$, $SP{F}_{3,k}$, and $SNP{F}_{3,k}$ under $T{E}_{3,k}$. The competing failure behavior and the isolation time behavior under each $T{E}_{3,k}$ are analyzed as follows.

Table 4.

Definition, $p_{3,k},q_{3,k},SP{F}_{3,k}$, and $SNP{F}_{3,k}$ under $T{E}_{3,k},k=1,2.$

	Definition	$p_{3,k}$	$q_{3,k}$	$SP{F}_{3,k}$	$SNP{F}_{3,k}$
$k=1$	$\overline{S_{3}PF}$	0	1	$\varnothing$	$\left\{S_3\right\}$
$k=2$	$S_{3}PF$	1	0	$\left\{S_3\right\}$	$\varnothing$

$T{E}_{3,1}$: No PFGE originates from $S_3$ and R ₂ fails locally; therefore, the competing failure behavior does not exist and the isolation time behavior of $S_3$ is addressed. Applying (14), the ITE space under $T{E}_{3,1}$ is generated and shown in Table 5 . Applying (15), $\mathrm{Pr}\left(SR\cap T{E}_{3,1}\right)$ in (26) is evaluated as (27), where $\mathrm{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,l}\right)$ is evaluated in this step and $\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right)$ is evaluated in Step 4 (refer to Section 4.2.4.1).

$$\mathrm{Pr}\left(SR\cap T{E}_{3,1}\right)=\sum _{l=1}^2\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right)\times \mathrm{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,l}\right)$$ (27)

Table 5.

Definition of $IT{E}_{3,1,l},l=1,2.$

	Definition
$l=1$	$\overline{S_{3}I}$
$l=2$	$S_{3}I$

$T{E}_{3,1}\cap IT{E}_{3,1,1}$: $S_3$ is not isolated from the system. According to defined $T{E}_3$ (Table 3), $T{E}_{3,1}$ (Table 4), and $IT{E}_{3,1,1}$ (Table 5), $\text{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,1}\right)$ denotes the occurrence probability of the event $\left\{\overline{R_{1}LF}\cap R_{2}LF\cap \overline{S_{3}PF}\cap \overline{S_{3}I}\right\}$. Since events $\overline{R_{1}LF}$, $\overline{S_{3}PF}$, and ($R_{2}LF\cap \overline{S_{3}I}$) are assumed to be independent, $\text{Pr}\left\{\overline{R_{1}LF}\cap R_{2}LF\cap \overline{S_{3}PF}\cap \overline{S_{3}I}\right\}$ can be calculated as $\left[\mathrm{Pr}\left(\overline{R_{1}LF}\right)\times \mathrm{Pr}\left(\overline{S_{3}PF}\right)\times \mathrm{Pr}(R_{2}LF\cap \overline{S_{3}I}\}\right]$. Applying the first equation of (3) in Section 2 with the time-to-LF parameters of R ₂ and the isolation time parameters of S ₃, $\text{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,1}\right)$ is obtained as

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,1}\right)=\left[1-q_{R_{1}L}\left(t\right)\right]\times \left[1-q_{S_{3}PF}\left(t\right)\right]\hfill \\ \times {\int }_0^t{\int }_{{\tau }_{R_{2}L}}^{\infty }{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{3}I}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{R_{2}L}\hfill \end{array}$$ (28)

$T{E}_{3,1}\cap IT{E}_{3,1,2}$: $S_3$ is isolated from the system. $\text{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,2}\right)$ denotes the occurrence probability of the event $\left\{\overline{R_{1}LF}\cap R_{2}LF\cap \overline{S_{3}PF}\cap S_{3}I\right\}$. Applying the second equation of (3), it is obtained as

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{3,1}\cap IT{E}_{3,1,2}\right)=\left[1-q_{R_{1}L}\left(t\right)\right]\times \left[1-q_{S_{3}PF}\left(t\right)\right]\hfill \\ \times {\int }_0^t{\int }_0^{t-{\tau }_{R_{2}L}}{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{3}I}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{R_{2}L}\hfill \end{array}$$ (29)

$T{E}_{3,2}$: Since PFGE originates from $S_3$, the competing failure behavior and the isolation time behavior take effect on $S_3$. Based on Table 4, $p_{3,2}=1$, $q_{3,2}=0$, $SP{F}_{3,2}=\left\{S_3\right\}$, and$SNP{F}_{3,2}=\varnothing$. According to the competing result, two disjoint events are distinguished:

• ■$T{E}_{3,2,i}$: R₂LF happens before S ₃ PF, and the time for isolating S ₃ is less than the difference between them. The example BSN system may still be alive depending on S ₂ and S ₃.
• ■$T{E}_{3,2,p}$: S₃PF happens before R ₂ LF, or S ₃ PF happens after R ₂ LF and the time needed for isolating S ₃ is greater than the time difference between the occurrence time of R ₂ LF and S ₃ PF. The entire BSN system fails.

Applying (16), we have (30), where $\text{Pr}\left(T{E}_{3,2,i}\right)$ is evaluated in this step and $\text{Pr}\left(SR\right|T{E}_{3,2,i})$ is evaluated in Step 4 (refer to Section 4.2.4.1.).

$$\mathrm{Pr}\left(SR\cap T{E}_{3,2}\right)=\text{Pr}\left(SR\right|T{E}_{3,2,i})\times \text{Pr}\left(T{E}_{3,2,i}\right)$$ (30)

According to the definition of $T{E}_3$ (Table 3) and $T{E}_{3,2,i}$, $\text{Pr}\left(T{E}_{3,2,i}\right)$ denotes the occurrence probability of the event $\{\overline{R_{1}LF}\cap [(R_{2}LF\to S_{3}PF)\cap S_{3}I]\}$. Since the events $\overline{R_{1}LF}$ and $[(R_{2}LF\to S_{3}PF)\cap S_{3}I]$ are assumed to be independent, $\text{Pr}\{\overline{R_{1}LF}\cap [(R_{2}LF\to S_{3}PF)\cap S_{3}I]\}$ is evaluated as $\text{Pr}\left\{\overline{R_{1}LF}\right\}\times \text{Pr}\left\{\left(R_{2}LF\to S_{3}PF\right)\cap S_{3}I\right\}$. Applying (4) with the input time-to-LF parameters of R ₂, the input time-to-PFGE and isolation time parameters of S ₃, $\text{Pr}\left(T{E}_{3,2,i}\right)$ is evaluated as

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{3,2,i}\right)=\left[1-q_{R_{1}L}\left(t\right)\right]\hfill \\ \times \left[{\int }_0^t{\int }_{{\tau }_{R_{2}L}}^t{\int }_0^{{\tau }_{S_{3}PF}-{\tau }_{R_{2}L}}{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{3}PF}\left({\tau }_{S_{3}PF}\right)f_{S_{3}I}\left({\tau }_{TTI}\right)d{\tau }_{TTI}d{\tau }_{S_{3}PF}d{\tau }_{R_{2}L}\right]\hfill \end{array}.$$ (31)

4.2.3.2. $\mathrm{T}{\mathrm{E}}_4$

Both $R_1$ and $R_2$ fail locally, resulting in $S_2$ and $S_3$ being affected. Therefore, $D_4=\left\{S_2,S_3\right\}$ and $n_4=2$. Based on (12), the dependent biosensor event space has four combinations. The definition, $p_{4,k}$, $q_{4,k}$, $SP{F}_{4,k}$, and $SNP{F}_{4,k}$ of each combination are given in Table 6 . Applying (13), $\mathrm{Pr}\left(SR\cap T{E}_4\right)$ in (19) is calculated as (32). The competing failure behavior and the isolation time behavior under each $T{E}_{4,k}$ are analyzed as follows.

$$\mathrm{Pr}\left(SR\cap T{E}_4\right)=\sum _{k=1}^4\mathrm{Pr}\left(SR\cap T{E}_{4,k}\right)$$ (32)

Table 6.

Definition, $p_{4,k},q_{4,k},SP{F}_{4,k}$, and $SNP{F}_{4,k}$ under $T{E}_{4,k},k=1,2,3,4.$

	Definition	$p_{4,k}$	$q_{4,k}$	$SP{F}_{4,k}$	$SNP{F}_{4,k}$
$k=1$	$\overline{S_{2}PF}\cap \overline{S_{3}PF}$	0	2	$\varnothing$	$\left\{S_2,S_3\right\}$
$k=2$	$S_{2}PF\cap \overline{S_{3}PF}$	1	1	$\left\{S_2\right\}$	$\left\{S_3\right\}$
$k=3$	$\overline{S_{2}PF}\cap S_{3}PF$	1	1	$\left\{S_3\right\}$	$\left\{S_2\right\}$
$k=4$	$S_{2}PF\cap S_{3}PF$	2	0	$\left\{S_2,S_3\right\}$	$\varnothing$

$T{E}_{4,1}$: R₁ and R ₂ fail locally and no PFGE originates from S ₂ or S ₃; therefore, the isolation time behavior of $S_2$ and $S_3$ is addressed but the competing failure behavior does not exist. Applying (14) with p _4,1, q _4,1, SPF _4,1, and SNPF _4,1 given in Table 6, the ITE space under $T{E}_{4,1}$ is generated and shown in Table 7 . Applying (15), $\mathrm{Pr}\left(SR\cap T{E}_{4,1}\right)$ is obtained as (33), where $\mathrm{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right)$ is evaluated as follows and $\mathrm{Pr}\left(SR|T{E}_{4,1}\cap IT{E}_{4,1,l}\right)$ is evaluated in Step 4 (refer to Section 4.2.4.2.).

$$\mathrm{Pr}\left(SR\cap T{E}_{4,1}\right)=\sum _{l=1}^4\mathrm{Pr}\left(SR|T{E}_{4,1}\cap IT{E}_{4,1,l}\right)\times \mathrm{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right)$$ (33)

Table 7.

Definition of $IT{E}_{4,1,l},l=1,2,3,4.$

	Definition
$l=1$	$\overline{S_{2}I}\cap \overline{S_{3}I}$
$l=2$	$S_{2}I\cap \overline{S_{3}I}$
$l=3$	$\overline{S_{2}I}\cap S_{3}I$
$l=4$	$S_{2}I\cap S_{3}I$

$T{E}_{4,1}\cap IT{E}_{4,1,l}$: The definition and the formula for evaluating $\text{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right)$ are summarized in Table 8 . Take $\text{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,1}\right)$ (i.e., $\text{Pr}\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap \overline{S_{3}PF}\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$) as an example. Since no PFGEs originate from $S_2$ and $S_3$ under TE _4,1, it is evaluated as $\text{Pr}\left\{\overline{S_{2}PF}\right\}\times \text{Pr}\left\{\overline{S_{3}PF}\right\}\times \text{Pr}\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$ with $\mathrm{Pr}\left\{\overline{S_{2}PF}\right\}=\left[1-q_{S_{2}PF}\left(\mathrm{t}\right)\right]$ and $\mathrm{Pr}\left\{\overline{S_{3}PF}\right\}=$[$1-q_{S_{2}PF}\left(\mathrm{t}\right)]$. $\text{Pr}\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$ is evaluated by the preliminary method. Specifically, there are two possible occurrence sequences between R ₁ LF and R ₂ LF, which are either R ₁ LF before R ₂ LF (denoted by $R_{1}LF\to R_{2}LF$) or R ₂ LF before R ₁ LF (denoted by $R_{2}LF\to R_{1}LF$). Therefore, $\text{Pr}\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$ equals to the summation of $[\mathrm{Pr}\left\{(R_{1}LF\to R_{2}LF)\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$ and $\text{Pr}\left\{\left(R_{2}LF\to R_{1}LF\right)\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}]$. Suppose that R ₁ LF happens at ${\tau }_{R_{1}L}$, R ₂ LF happens at ${\tau }_{R_{2}L}$, $S_{2}I$ happens at ${\tau }_{S_{2}I}$, and $S_{3}I$ happens at ${\tau }_{S_{3}I}$. In terms of the event $(R_{1}LF\to R_{2}LF)\cap \overline{S_{2}I}\cap \overline{S_{3}I}$, we have $0\le {\tau }_{R_{1}L}\le {\tau }_{R_{2}L}\le t$, ${\tau }_{R_{2}L}\le {\tau }_{S_{2}I}$, and ${\tau }_{R_{2}L}\le {\tau }_{S_{3}I}$. According to the BSN system configuration, the event S ₂ I happens when $R_{1}LF\cap R_{2}LF$ occurs and the event S ₃ I happens when $R_{2}LF$ occurs. Based on (3), the interval is $[t-{\tau }_{R_{2}L},\infty ]$ of ${\tau }_{TTI-S_2}$ (which is the difference of ${\tau }_{S_{2}I}$ and ${\tau }_{R_{2}L}$) and is $[t-{\tau }_{R_{2}L},\infty ]$ of ${\tau }_{TTI-S_3}$ (which is the difference of ${\tau }_{S_{3}I}$ and ${\tau }_{R_{2}L}$). In terms of the event $(R_{2}LF\to R_{1}LF)\cap \overline{S_{2}I}\cap \overline{S_{3}I}$, $0\le {\tau }_{R_{2}L}\le {\tau }_{R_{1}L}\le t$, ${\tau }_{R_{1}L}\le {\tau }_{S_{2}I}$, and ${\tau }_{R_{2}L}\le {\tau }_{S_{3}I}$, the interval is $[t-{\tau }_{R_{1}LF},\infty ]$ of ${\tau }_{TTI-S_2}$ (which is the difference of ${\tau }_{S_{2}I}$ and ${\tau }_{R_{1}L}$) and is $[t-{\tau }_{R_{2}LF},\infty ]$ of ${\tau }_{TTI-S_3}$ (which is the difference of ${\tau }_{S_{3}I}$ and ${\tau }_{R_{2}L}$). In summary, $\text{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,1}\right)$ is evaluated as (34). $\mathrm{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right),l=2,3,4$ can be obtained via the same analysis procedure.

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,1}\right)=\left[1-q_{S_{2}PF}\left(\mathrm{t}\right)\right]\times \left[1-q_{S_{3}PF}\left(\mathrm{t}\right)\right]\times \hfill \\ [{\int }_0^t{\int }_{{\tau }_{R_{1}L}}^t{\int }_{t-{\tau }_{R_{2}L}}^{\infty }{\int }_{t-{\tau }_{R_{2}L}}^{\infty }{f}_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)\times \hfill \\ f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}d{\tau }_{TTI-S_2}d{\tau }_{R_{2}L}d{\tau }_{R_{1}L}\hfill \\ {\int }_0^t{\int }_{{\tau }_{R_{2}L}}^t{\int }_{t-{\tau }_{R_{1}L}}^{\infty }{\int }_{t-{\tau }_{R_{2}L}}^{\infty }{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)\times \hfill \\ f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}d{\tau }_{TTI-S_2}d{\tau }_{R_{1}L}d{\tau }_{R_{2}L}]\hfill \end{array}$$ (34)

Table 8.

Definition and formula for evaluating $\text{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right),l=1,2,3,4.$

Event	Definition	$\mathrm{Pr}\left(T{E}_{4,1}\cap IT{E}_{4,1,l}\right)$
$T{E}_{4,1}\cap IT{E}_{4,1,1}$	$\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap \overline{S_{3}PF}\cap \overline{S_{2}I}\cap \overline{S_{3}I}\right\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \mathrm{Pr}\left(\overline{S_{3}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}\{(R_{1}LF\to R_{2}LF)\cap \overline{S_{2}I}\cap \overline{S_{3}I}\}+\hfill \\ \mathrm{Pr}\{R_{2}LF\to R_{1}LF)\cap \overline{S_{2}I}\cap \overline{S_{3}I}\}\hfill \end{array}\right]$
$T{E}_{4,1}\cap IT{E}_{4,1,2}$	$\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap \overline{S_{3}PF}\cap S_{2}I\cap \overline{S_{3}I}\right\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \mathrm{Pr}\left(\overline{S_{3}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}\{(R_{1}LF\to R_{2}LF)\cap S_{2}I\cap \overline{S_{3}I}\}+\hfill \\ \mathrm{Pr}\{(R_{2}LF\to R_{1}LF)\cap S_{2}I\cap \overline{S_{3}I}\}\hfill \end{array}\right]$
$T{E}_{4,1}\cap IT{E}_{4,1,3}$	$\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap \overline{S_{3}PF}\cap \overline{S_{2}I}\cap S_{3}I\right\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \mathrm{Pr}\left(\overline{S_{3}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}\{(R_{1}LF\to R_{2}LF)\cap \overline{S_{2}I}\cap S_{3}I\}+\hfill \\ \mathrm{Pr}\{(R_{2}LF\to R_{1}LF)\cap \overline{S_{2}I}\cap S_{3}I\}]\hfill \end{array}\right]$
$T{E}_{4,1}\cap IT{E}_{4,1,4}$	$\left\{R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap \overline{S_{3}PF}\cap S_{2}I\cap S_{3}I\right\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \mathrm{Pr}\left(\overline{S_{3}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}\{(R_{1}LF\to R_{2}LF)\cap S_{2}I\cap S_{3}I\}+\hfill \\ \mathrm{Pr}\{(R_{2}LF\to R_{1}LF)\cap S_{2}I\cap S_{3}I\}\hfill \end{array}\right]$

$T{E}_{4,2}$: R₁ and R ₂ fail locally, PFGE originates from $S_2$, and no PFGE originates from S ₃; therefore, the competing failure behavior and the isolation time behavior affect $S_2$ and the isolation time behavior of $S_3$ is addressed. Based on the competing result, two disjoint events ($T{E}_{4,2,p}$ and $T{E}_{4,2,i}$) are distinguished. According to the isolation status of S ₃, two ITEs are generated and shown in Table 9 . Applying (17), $\mathrm{Pr}\left(SR\cap T{E}_{4,2}\right)$ is obtained as (35), where $\mathrm{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right),m=1,2$ is evaluated in this step and $\mathrm{Pr}\left(SR|T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right)$ is evaluated in Step 4.

$$\mathrm{Pr}\left(SR\cap T{E}_{4,2}\right)=\sum _{m=1}^2\left[\mathrm{Pr}\left(SR|T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right)\times \mathrm{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right)\right]$$ (35)

Table 9.

Definition of $IT{E}_{4,2,m},m=1,2.$

	Definition
$m=1$	$\overline{S_{3}I}$
$m=2$	$S_{3}I$

$T{E}_{4,2,i}\cap IT{E}_{4,2,m}$: When m = 1, $\mathrm{Pr}(T{E}_{4,2,i}\cap IT{E}_{4,2,1})$ denotes the occurrence probability of the event $\left\{R_{1}LF\cap R_{2}LF\cap S_{2}PF\cap \overline{S_{3}PF}\cap S_{2}I\cap \overline{S_{3}I}\right\}$, and it is calculated as $\text{Pr}\left\{\overline{S_{3}PF}\right\}\times \text{Pr}\left\{R_{1}LF\cap R_{2}LF\cap S_{2}PF\cap S_{2}I\cap \overline{S_{3}I}\right\}$ with $\mathrm{Pr}\left\{\overline{S_{3}PF}\right\}=\left[1-q_{S_{3}PF}\left(\mathrm{t}\right)\right]$ and $\mathrm{Pr}\left\{R_{1}LF\cap R_{2}LF\cap S_{2}PF\cap S_{2}I\cap \overline{S_{3}I}\right\}=[\mathrm{Pr}\left\{(R_{1}LF\to R_{2}LF\to S_{2}PF)\cap S_{2}I\cap \overline{S_{3}I}\right\}+\text{Pr}\left\{\right(R_{2}LF\to R_{1}LF$ $\to S_{2}PF)\cap S_{2}I\cap \overline{S_{3}I}\left\}\right]$. Applying the preliminary method, $\text{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,1}\right)$ is evaluated as (36). Similarly, $\text{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,2}\right)$ is evaluated as (37).

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,1}\right)=\left[1-q_{S_{3}PF}\left(\mathrm{t}\right)\right]\hfill \\ \times [\underset{0}{\overset{t}{\int }}\underset{{\tau }_{R_{1}L}}{\overset{t}{\int }}\underset{{\tau }_{R_{2}L}}{\overset{t}{\int }}\underset{0}{\overset{{\tau }_{S_{2}PF}-{\tau }_{R_{2}L}}{\int }}\underset{t-{\tau }_{R_{2}L}}{\overset{\infty }{\int }}{f}_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{2}PF}\left({\tau }_{S_{2}PF}\right)\hfill \\ \times f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}\cdots d{\tau }_{R_{1}L}\hfill \\ +\underset{0}{\overset{t}{\int }}\underset{{\tau }_{R_{2}L}}{\overset{t}{\int }}\underset{{\tau }_{R_{1}L}}{\overset{t}{\int }}\underset{0}{\overset{{\tau }_{S_{2}PF}-{\tau }_{R_{1}L}}{\int }}\underset{t-{\tau }_{R_{2}L}}{\overset{\infty }{\int }}{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{S_{2}PF}\left({\tau }_{S_{2}PF}\right)\hfill \\ \times f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}\cdots d{\tau }_{R_{2}L}]\hfill \end{array}$$ (36)

$$\begin{array}{c}\mathrm{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,2}\right)=\left[1-q_{S_{3}PF}\left(\mathrm{t}\right)\right]\hfill \\ \times [\underset{0}{\overset{t}{\int }}\underset{{\tau }_{R_{1}L}}{\overset{t}{\int }}\underset{{\tau }_{R_{2}L}}{\overset{t}{\int }}\underset{0}{\overset{{\tau }_{S_{2}PF}-{\tau }_{R_{2}L}}{\int }}\underset{0}{\overset{t-{\tau }_{R_{2}L}}{\int }}{f}_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{S_{2}PF}\left({\tau }_{S_{2}PF}\right)\hfill \\ \times f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}\cdots d{\tau }_{R_{1}L}\hfill \\ +\underset{0}{\overset{t}{\int }}\underset{{\tau }_{R_{2}L}}{\overset{t}{\int }}\underset{{\tau }_{R_{1}L}}{\overset{t}{\int }}\underset{0}{\overset{{\tau }_{S_{2}PF}-{\tau }_{R_{1}L}}{\int }}\underset{0}{\overset{t-{\tau }_{R_{2}L}}{\int }}{f}_{R_{2}L}\left({\tau }_{R_{2}L}\right)f_{R_{1}L}\left({\tau }_{R_{1}L}\right)f_{S_{2}PF}\left({\tau }_{S_{2}PF}\right)\hfill \\ \times f_{S_{2}I}\left({\tau }_{TTI-S_2}\right)f_{S_{3}I}\left({\tau }_{TTI-S_3}\right)d{\tau }_{TTI-S_3}\cdots d{\tau }_{R_{2}L}\hfill \end{array}$$ (37)

$T{E}_{4,3}$: R₁ and R ₂ fail locally, no PFGE originates from S ₂, and PFGE originates from $S_3$; therefore, the isolation time behavior of $S_2$ is addressed and the competing failure behavior and the isolation time behavior affect $S_3$. Two disjoint events ($T{E}_{4,3,i}$ and $T{E}_{4,3,p}$) are distinguished and two ITEs are generated ($IT{E}_{4,3,1}=\overline{S_{2}I}$ , and $IT{E}_{4,3,2}=S_{2}I$). Applying (17), $\mathrm{Pr}\left(SR\cap T{E}_{4,3}\right)$ is obtained as (38). The analysis procedure of $\mathrm{Pr}\left(T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)$ is similar to that of $\mathrm{Pr}\left(T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right)$. Table 10 summarizes the formula for calculating $\mathrm{Pr}\left(T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)$. $\mathrm{Pr}\left(SR|T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)$ is evaluated in Step 4.

$$\mathrm{Pr}\left(SR\cap T{E}_{4,3}\right)=\sum _{m=1}^2\left[\mathrm{Pr}\left(SR|T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)\times \mathrm{Pr}\left(T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)\right]$$ (38)

Table 10.

Formula for evaluating $\text{Pr}\left(T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right),m=1,2.$

Event	Definition	$\text{Pr}\left(T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)$
$T{E}_{4,3,i}\cap IT{E}_{4,3,1}$	$R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap S_{3}PF\cap \overline{S_{2}I}\cap S_{3}I\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}(R_{1}LF\to R_{2}LF\to S_{3}PF)\cap S_{3}I\cap \overline{S_{2}I})+\hfill \\ \mathrm{Pr}(R_{2}LF\to R_{1}LF\to S_{3}PF)\cap S_{3}I\cap \overline{S_{2}I})+\hfill \\ \mathrm{Pr}(R_{2}LF\to S_{3}PF\to R_{1}LF)\cap S_{3}I\cap \overline{S_{2}I})\hfill \end{array}\right]$
$T{E}_{4,3,i}\cap IT{E}_{4,3,2}$	$R_{1}LF\cap R_{2}LF\cap \overline{S_{2}PF}\cap S_{3}PF\cap S_{2}I\cap S_{3}I\}$	$\mathrm{Pr}\left(\overline{S_{2}PF}\right)\times \left[\begin{array}{c}\mathrm{Pr}(R_{1}LF\to R_{2}LF\to S_{3}PF)\cap S_{3}I\cap S_{2}I)+\hfill \\ \mathrm{Pr}(R_{2}LF\to R_{1}LF\to S_{3}PF)\cap S_{3}I\cap S_{2}I)+\hfill \\ \mathrm{Pr}(R_{2}LF\to S_{3}PF\to R_{1}LF)\cap S_{3}I\cap S_{2}I)\hfill \end{array}\right]$

$T{E}_{4,4}$: R₁ and R ₂ fail locally and PFGEs originate from S ₂ and S ₃; therefore, the competing failure behavior and the isolation time behavior are addressed for S ₂ and S ₃. According to the competing result, two disjoint events ($T{E}_{4,4,i}$, and $T{E}_{4,4,p}$) are distinguished. Applying (16), we have (39), where $\mathrm{Pr}\left(SR|T{E}_{4,4,i}\right)$ equals to 0 for the illustrative BSN system since both S ₂ and S ₃ are isolated under $T{E}_{4,4}$.

$$\mathrm{Pr}\left(SR\cap T{E}_{4,4}\right)=\mathrm{Pr}\left(SR|T{E}_{4,4,i}\right)\times \mathrm{Pr}\left(T{E}_{4,4,i}\right)=0$$ (39)

4.2.4. Step 4: evaluate the conditional system reliability

This step illustrates the procedure for analyzing the conditional system reliabilities under TE ₃ and TE ₄, which are $\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right),l=1,2$ in (27), $\text{Pr}\left(SR\right|T{E}_{3,2,i})$ in (30), $\mathrm{Pr}\left(SR|T{E}_{4,1}\cap IT{E}_{4,1,l}\right),l=1,2,3,4$ in (33), $\mathrm{Pr}\left(SR|T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right),m=1,2$ in (35), $\mathrm{Pr}\left(SR|T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right),m=1,2$ in (38), and $\text{Pr}\left(SR\right|T{E}_{4,4,i})$ in (39).

4.2.4.1. TE₃

$T{E}_{3,1}\cap IT{E}_{3,1,l},l=1,2$: The reduced FT and BDD models for evaluating $\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right)$ are obtained as follows. Under $T{E}_{3,1}\cap IT{E}_{3,1,1}$, the isolation time behavior of S ₃ is addressed. But because the power in S ₃ is sufficient to support the direct link between the sink and S ₃ within the mission time, S ₃ is not isolated (refer to the Situation (a) in Section 2), the failure of R ₂ cannot cause the isolation of S ₃ and S ₃ remains in the BSN system. The reduced FT and BDD models under $T{E}_{3,1}\cap IT{E}_{3,1,1}$ are the same as those in Fig. 6. Under $T{E}_{3,1}\cap IT{E}_{3,1,2}$ (refer to Situation (b) in Section 2), the failure of R ₂ causes the isolation of S ₃. Therefore, the reduced FT model is obtained by replacing S ₃ with ‘1′, which is shown as Fig. 7 (a). The corresponding BDD model is given in Fig. 7(b).

Fig. 7.

Reduced system models under $T{E}_{3,1}\cap IT{E}_{3,1,2}.$

Using the similar separable method formulated in (6), $\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right)$ is evaluated as (40), where $P_{u-T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$ denotes the occurrence probability of no PFGEs originating from biosensors appearing in the reduced FT model (excluding S ₁ considered in (18) and S ₃ considered in (28) or (29)). $C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$ denotes the conditional system reliability given that no PFGEs originate from biosensors in the reduced FT model under $T{E}_{3,1}\cap IT{E}_{3,1,l}$ and can be obtained by the BDD-based method. The evaluation expressions of $C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$ and $P_{u-T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$ are summarized in Table 11 . Note that $C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,1}}\left(t\right)$ equals to $C{R}_{T{E}_1}\left(t\right)$ as they have the identical reduced FT and BDD models.

$$\mathrm{Pr}\left(SR|T{E}_{3,1}\cap IT{E}_{3,1,l}\right)=C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)\times P_{u-T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$$ (40)

Table 11.

$P_{u-T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$ and $C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right),l=1,2.$

l	$P_{u-T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$	$C{R}_{T{E}_{3,1}\cap IT{E}_{3,1,l}}\left(t\right)$
1	$\left[1-q_{S_{2}PF}\left(t\right)\right]$	$C{R}_{T{E}_1}\left(t\right)$
2	$\left[1-q_{S_{2}PF}\left(t\right)\right]$	$\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{2}L}\left(t\right)\right]$

$T{E}_{3,2,i}$: The competing failure behavior occurs between R₂LF and S ₃ PF, and the competing result is S ₃ being isolated from the BSN system. Therefore, the reduced FT model is generated by removing all the relays and the FDEP gates from the dynamic FT model and replacing S ₃ with ‘1′. The reduced FT and BDD models under $T{E}_{3,2,i}$ are the same as those in Fig. 7. Using the similar separable method formulated in (6), $\mathrm{Pr}\left(SR|T{E}_{3,2,i}\right)$ is evaluated as

$$\begin{array}{c}\mathrm{Pr}\left(SR|T{E}_{3,2,i}\right)=C{R}_{T{E}_{3,2,i}}\left(t\right)\times P_{u-T{E}_{3,2,i}}\left(t\right)\hfill \\ =\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{2}L}\left(t\right)\right]\times \left[1-q_{S_{2}PF}\left(t\right)\right].\hfill \end{array}$$ (41)

4.2.4.2. TE₄

$T{E}_{4,1}\cap IT{E}_{4,1,l},l=1,2,3,4$: The reduced FT models for evaluating $\text{Pr}\left(SR\right|T{E}_{4,1}\cap IT{E}_{4,1,l})$ are shown in Fig. 8 . Using the same analysis procedure, the expressions for evaluating $C{R}_{T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$and $P_{u-T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$ are summarized in Table 12 . Correspondingly, we have $\mathrm{Pr}\left(SR\right|T{E}_{4,1}\cap IT{E}_{4,1,l})=C{R}_{T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)\times P_{u-T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$.

Fig. 8.

Reduced system FT models under $T{E}_{4,1}\cap IT{E}_{4,1,l},l=1,2,3,4.$

Table 12.

$P_{u-T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$ and $C{R}_{T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right),l=1,2,3,4.$

l	$P_{u-T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$	$C{R}_{T{E}_{4,1}\cap IT{E}_{4,1,l}}\left(t\right)$
1	1	$C{R}_{T{E}_1}\left(t\right)$
2	1	$\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{3}L}\left(t\right)\right]$
3	1	$\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{2}L}\left(t\right)\right]$
4	–	0

$T{E}_{4,2,i}\cap IT{E}_{4,2,m},m=1,2$: The reduced system FT models under events $T{E}_{4,2,i}\cap IT{E}_{4,2,1}$ and $T{E}_{4,2,i}\cap IT{E}_{4,2,2}$ are identical to the models in Fig. 8(b) and (d), respectively. Based on the reduced FT models, $C{R}_{T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$ and $P_{u-T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$ are summarized in Table 13 . We have $\mathrm{Pr}\left(SR|T{E}_{4,2,i}\cap IT{E}_{4,2,m}\right)=C{R}_{T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)\times P_{u-T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$.

Table 13.

$P_{u-T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$ and $C{R}_{T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right),m=1,2.$

m	$P_{u-T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$	$C{R}_{T{E}_{4,2,i}\cap IT{E}_{4,2,m}}\left(t\right)$
1	1	$\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{3}L}\left(t\right)\right]$
2	–	0

$T{E}_{4,3,i}\cap IT{E}_{4,3,m},m=1,2$: The reduced FT models under events $T{E}_{4,3,i}\cap IT{E}_{4,3,1}$ and $T{E}_{4,3,i}\cap IT{E}_{4,3,2}$ are identical to the models in Fig. 8(c) and 8(d), respectively. Also, we have $\mathrm{Pr}\left(SR|T{E}_{4,3,i}\cap IT{E}_{4,3,m}\right)=C{R}_{T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)\times P_{u-T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)$ with $C{R}_{T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)$ and $P_{u-T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)$ summarized in Table 14 .

Table 14.

$P_{u-T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)$ and $C{R}_{T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right),m=1,2.$

m	$P_{u-T{E}_{4,3,i}\cap IT{E}_{4,3,m}}\left(t\right)$	$C{R}_{T{E}_{4,3,i}\cap IT{E}_{4,3,m}}$
1	1	$\left[1-q_{S_{1}L}\left(t\right)\right]\times \left[1-q_{S_{2}L}\left(t\right)\right]$
2	–	0

$T{E}_{4,4,i}$: The reduced system FT model under $T{E}_{4,4,i}$ is generated by removing all the relay failure events and the FDEP gates from the dynamic FT of the example BSN system and replacing S ₂ and S ₃ with ‘1′, which is identical to that under $T{E}_{4,1}\cap IT{E}_{4,1,4}$ (Fig. 8(d)). Therefore, $C{R}_{T{E}_{4,4,i}}\left(t\right)=0$, resulting in $\mathrm{Pr}\left(SR\right|T{E}_{4,4,i})=C{R}_{T{E}_{4,4,i}}\left(t\right)\times P_{u-T{E}_{4,4,i}}\left(t\right)=0$.

4.2.5. Step 5: integrate for the entire BSN system reliability

In Step 3, according to (20), we integrate $\mathrm{Pr}\left(SR\cap T{E}_1\right)$ with $\mathrm{Pr}\left(T{E}_1\right)$ and $\mathrm{Pr}\left(SR|T{E}_1\right)$ evaluated by (21) and (22). According to (25), we integrate $\mathrm{Pr}\left(SR\cap T{E}_2\right)$ with $\mathrm{Pr}\left(SR|T{E}_2\right)$ and $\mathrm{Pr}\left(T{E}_2\right)$. According to (26), we integrate $\mathrm{Pr}\left(SR\cap T{E}_3\right)$ with $\mathrm{Pr}\left(SR\cap T{E}_{3,1}\right)$ and $\mathrm{Pr}\left(SR\cap T{E}_{3,2}\right)$ evaluated by (27) and (30). And according to (32), we integrate $\mathrm{Pr}\left(SR\cap T{E}_4\right)$ with $\mathrm{Pr}\left(SR\cap T{E}_{4,k}\right),k=1,2,3,4$ evaluated by (33), (35), (38), and (39). In Step 2, we integrate $CR\left(t\right)$ with $\mathrm{Pr}\left(SR\cap T{E}_j\right),j=1,2,3,4$. According to (6), we integrate the example BSN reliability, R _BSN(t), with $P_u\left(t\right)$ evaluated by (18) and $CR\left(t\right)$ evaluated by (19).

5. Numerical results and analysis

The Weibull distribution with the scale parameter ($\alpha$) and the shape parameter ($\beta$) is adopted for modeling biosensors’ time-to-LF/PFGE and isolation time (IT) [13]. The PDF of random variable W with the Weibull distribution is given as (42). Particularly, the Weibull distribution reduces to the exponential distribution when $\beta =1$. Table 15 illustrates the values of input time-to-LF/PFGE parameters.

$$f_W\left(t\right)=\beta {\alpha }^{\beta }{t}^{\left(\beta -1\right)}{e}^{-{\left(\alpha t\right)}^{\beta }}$$ (42)

Table 15.

Time-to-LF/PFGE parameters of the example BSN biosensors.

Biosensor	Set 1 (Exponential)		Set 2 (Weibull)
	αiLF/hrs	αiPF/hrs	(αiLF/hrs, βiLF)	(αiPF/hrs, βiPF)
S1	2.78e-3	5.00e-5	(2.78e-3, 1.0)	(5.00e-5, 2.0)
S2	4.17e-3	5.00e-5	(4.17e-3, 2.0)	(5.00e-5, 2.5)
S3	4.17e-3	5.00e-5	(4.17e-3, 1.5)	(5.00e-5, 1.7)
R1	5.21e-3	1.00e-4	(5.21e-3, 2.3)	(1.00e-4, 1.2)
R2	8.33e-3	1.00e-4	(8.33e-3, 2.5)	(1.00e-4, 3.0)

Note that in this paper, we have focused on system-level reliability modeling and evaluation with the assumption that these component-level failure parameters are known input parameters. In practice, estimation approaches based on collected failure data (e.g., statistical inference [52], Bayesian estimation [53]) are often applied to estimate component failure time distribution functions and related parameters.

In this work, we discuss the IT performance of the dependent biosensors (i.e., S ₂ and S ₃) on the reliability of the example BSN system based on the different relationships between LF and PFGE of the same biosensor (s-independent in Section 5.1 and disjoint in Section 5.2). The analysis of the results is performed in Section 5.3. In Section 5.4, the correctness of the proposed method is verified using the continuous-time Markov chains (CTMC) method.

5.1. Impact of isolation time under the s-independent relationship

Using parameters of Set 1, Table 16 presents the BSN reliabilities at different mission time (t = 24, 48, and 72 h) when S ₂ IT and S ₃ IT follow the exponential distribution with changing scale parameters (${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$).

Table 16.

Impacts of ${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$ on R_BSN under Set 1 (Exponential).

Using parameters of Set 2, Tables 17 and 18 present the BSN reliabilities when S ₂ IT and S ₃ IT follow the Weibull distribution. Specifically, Table 17 addresses the impact of S ₂ IT (${\alpha }_{S_{2}IT}$, ${\beta }_{S_{2}IT}$) where S ₃ IT is fixed as (${\alpha }_{S_{3}IT},{\beta }_{S_{3}IT}$)=(8.33e-3/hour, 1). Table 18 addresses the impact of S ₃ IT (${\alpha }_{S_{3}IT}$, ${\beta }_{S_{3}IT}$) where S ₂ IT is fixed as (${\alpha }_{S_{2}IT},{\beta }_{S_{2}IT}$)=(1.04e-2/hour, 1).

Table 17.

Impacts of $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$ on R_BSN under Set 2 (Weibull) with $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$=(8.33e-3, 1).

t (hrs)	${\alpha }_{S_{2}IT}$	${\beta }_{S_{2}IT}$
		0.5	1.0	1.5	2.0
24	3600	0.99625759	0.99625759	0.99625759	0.99625759
	1	0.99626165	0.99626057	0.99626038	0.99626035
	1.04e-2	0.99627417	0.99627683	0.99627751	0.99627769
	1.00e-9	0.99627777	0.99627777	0.99627777	0.99627777
48	3600	0.97778415	0.97778412	0.97778412	0.97778412
	1	0.97790569	0.97786070	0.97785454	0.97785361
	1.04e-2	0.97852760	0.97867541	0.97872813	0.97874813
	1.00e-9	0.97876260	0.97876268	0.97876268	0.97876268
72	3600	0.93013347	0.93013335	0.93013334	0.93013334
	1	0.93084126	0.93054733	0.93051130	0.93050555
	1.04e-2	0.93586248	0.93711128	0.93764795	0.93789459
	1.00e-9	0.93816123	0.93816210	0.93816210	0.93816210

Table 18.

Impacts of $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$ on R_BSN under Set 2 (Weibull) with $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$=(1.04e-2, 1).

t (hrs)	${\alpha }_{S_{3}IT}$	${\beta }_{S_{3}IT}$
		0.5	1.0	1.5	2.0
24	3600	0.99508553	0.99508549	0.99508549	0.99508567
	1	0.99527294	0.99520745	0.99519790	0.99519648
	8.33e-3	0.99610152	0.99627683	0.99632557	0.99633970
	1.00e-9	0.99634600	0.99634610	0.99634610	0.99634609
48	3600	0.96639659	0.96639640	0.96639638	0.96639638
	1	0.96754405	0.96705044	0.96699188	0.96698237
	8.33e-3	0.97652341	0.97867541	0.97952194	0.97987073
	1.00e-9	0.98015184	0.98015334	0.98015334	0.98015333
72	3600	0.89507901	0.89507860	0.89507856	0.89507855
	1	0.89779649	0.89652825	0.89639152	0.89636863
	8.33e-3	0.92924305	0.93711128	0.94087306	0.94276758
	1.00e-9	0.94513426	0.94514111	0.94514112	0.94514111

5.2. Impact of isolation time under the disjoint relationship

Using the same parameters as those in Section 5.1, the BSN reliabilities for disjoint LF and PFGE are collected in Table 19, Table 20, Table 21 .

Table 19.

Impacts of ${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$ on R_BSN under Set 1 (Exponential).

Table 20.

Impacts of $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$ on R_BSN under Set 2 (Weibull) with $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$=(8.33e-3, 1).

t (hrs)	${\alpha }_{S_{2}IT}$	${\beta }_{S_{2}IT}$
		0.5	1.0	1.5	2.0
24	3600	0.99625755	0.99625755	0.99625755	0.99625755
	1	0.99626161	0.99626053	0.99626034	0.99626031
	1.04e-2	0.99627414	0.99627680	0.99627748	0.99627766
	1.00e-9	0.99627774	0.99627774	0.99627774	0.9962774
48	3600	0.97778197	0.97778195	0.97778194	0.97778194
	1	0.97790375	0.97785869	0.97785251	0.97785158
	1.04e-2	0.97852669	0.97867474	0.97872753	0.97874756
	1.00e-9	0.97876205	0.97876213	0.97876213	0.97876213
72	3600	0.93010875	0.93010863	0.93010862	0.93010862
	1	0.93081889	0.93052405	0.93048791	0.93048214
	1.04e-2	0.93585381	0.93710593	0.93764391	0.93789110
	1.00e-9	0.93815824	0.93815911	0.93815911	0.93815911

Table 21.

Impacts of $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$ on R_BSN under Set 2 (Weibull) with $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$=(1.04e-2, 1).

t (hrs)	${\alpha }_{S_{3}IT}$	${\beta }_{S_{3}IT}$
		0.5	1.0	1.5	2.0
24	3600	0.99508550	0.99508546	0.99508546	0.99508546
	1	0.99527291	0.99520742	0.99519787	0.99519645
	8.33e-3	0.99610149	0.99627680	0.99632554	0.99633967
	1.00e-9	0.99634598	0.99634607	0.99634607	0.99634606
48	3600	0.96639565	0.96639547	0.96639545	0.96639545
	1	0.96754312	0.96704950	0.96699095	0.96698143
	8.33e-3	0.97652269	0.97867474	0.97952129	0.97987009
	1.00e-9	0.98015121	0.98015271	0.98015271	0.98015270
72	3600	0.89507121	0.89507080	0.89507076	0.89507076
	1	0.89778874	0.89652045	0.89638372	0.89636084
	8.33e-3	0.92923720	0.93710593	0.94086799	0.94276266
	1.00e-9	0.94512953	0.94513638	0.94513638	0.94513638

5.3. Analysis of numerical results

5.3.1. Scale parameter analysis

It can be seen from Table 16, Table 17, Table 18, Table 19, Table 20, Table 21 that, when the shape parameters are fixed, the example BSN reliability (R _BSN) shows a growing trend as the scale parameters of S ₂ IT and S ₃ IT (${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$) decrease. This phenomenon demonstrates that, compared with the improvement effect of the failure isolation, the deterioration effect caused by the same failure isolation dominates the system performance. Specifically, the failure isolation of S ₂ and S ₃ leads to a two-sided effect on system performance: preventing PFGEs of S ₂ and S ₃ from affecting other parts of the BSN system (the improvement effect); meanwhile, S ₂ and S ₃ becoming inaccessible (the deterioration effect). In the example BSN system with considered parameter settings, when ${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$ increase, the isolation probabilities of S ₂ and S ₃ are increasing, i.e., there are more likely for S ₂ and S ₃ to be isolated from the BSN system when relays fail. As the deterioration effect is dominating, the BSN system performance becomes worse.

5.3.2. Shape parameter analysis

The impact of the shape parameter on the example BSN reliability is less obvious than that of the scale parameter. In the extreme cases, where ${\alpha }_{s_{2}IT}$ and ${\alpha }_{s_{3}IT}$ are 1.0e-9/hours or 3600/hours (corresponding to the case that biosensors take almost infinite or instant time to be isolated), the impact of ${\beta }_{S_{2}IT}$ and ${\beta }_{S_{3}IT}$ on the BSN reliability is trivial for the considered parameter settings.

In addition, the example BSN reliability's trend as the shape parameter (${\beta }_{S_{2}IT}$/${\beta }_{S_{3}IT}$) increases depends on the value of the scale parameter. Take results at t = 48 h in Tables 17 and 18 as examples. Fig. 9 depicts how the BSN reliability changes as ${\beta }_{S_{2}IT}$ and ${\beta }_{S_{3}IT}$ increase from 0.5 to 2.0. When the scale parameter (α) is 1/hour, the example BSN reliability decreases as the shape parameter (β) increases; whereas, when α is 1.04e-2/hour or 8.33e-3/hour, the example BSN reliability increases as β increases. This phenomenon is caused by the dominating deterioration effect. When α is 1/hour, the isolation probability increases with increasing β, i.e., it is more likely that dependent biosensors are isolated from the BSN system. As the deterioration effect dominates, the BSN system performs worse. When α is 1.04e-2/hour or 8.33e-3/hour, the isolation probability decreases with increasing β, leading to a better reliability performance of the example BSN system.

Fig. 9.

R_BSN under s-independent case (a) Impact of $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$ when $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$=(8.33e-3, 1); (b) Impact of $({\alpha }_{S_{3}IT},{\beta }_{S_{3}IT})$ when $({\alpha }_{S_{2}IT},{\beta }_{S_{2}IT})$=(1.04e-2, 1).

5.3.3. LF and PFGE relationship impact

Results show that the BSN reliability under the case of LF and PFGE being s-independent is higher than that under the case of LF and PFGE being disjoint. For example, using the parameters given in Set 1 and (${\alpha }_{S_{2}IT}$, ${\alpha }_{S_{3}IT}$) = (3600/hour, 3600/hour), the system reliability is 0.929843 under the s-independent relationship (in the cell marked with shadow of Table 16) and is 0.929633 under the disjoint relationship (in the cell marked with shadow of Table 19). This is because the biosensor's reliability (i.e., the occurrence probability that biosensor experiences neither LFs nor PFGEs) under the s-independent case (calculated as $\left[1-q_{iLF}\left(t\right)\right]\times \left[1-q_{iPF}\left(t\right)\right]$ for biosensor i) is higher than that under the disjoint case (calculated as $\left[1-q_{iLF}\left(t\right)-q_{iPF}\left(t\right)\right]$).

5.3.4. System reliability's sensitivity to S₂IT and S₃IT

Take the impact of ${\alpha }_{S_{2}IT}$ and ${\alpha }_{S_{3}IT}$ as illustration. In Table 16, at t = 72 h and ${\alpha }_{S_{3}IT}$=1.00e-9/hour, changing ${\alpha }_{S_{2}IT}$ from 3600/hour to 1.00e-9/hour, the BSN reliability increases by 4.37% (computed as (0.841565–0.806354)/0.806354 × 100%); however, at ${\alpha }_{S_{2}IT}$=1.00e-9/hour, taking the same decrement of ${\alpha }_{S_{3}IT}$, the BSN reliability increases by 15.41% (computed as (0.841565–0.729170)/0.729170 × 100%).

The reason of this phenomenon is that isolating S ₂ requires both R ₁ and R ₂ to fail whereas isolating S ₃ only requires R ₂ to fail, i.e., S ₃ is more easily isolated from the example BSN system. Take an extreme case where R ₁ is assumed being perfect (i.e., R ₁ is replaced by ‘0′ in Fig. 5) as an example. The battery of S ₂ is impossible to support the direct communication between S ₂ and the sink device; however, the battery of S ₃ may be sufficient to support the direct communication once R ₂ fails locally. In this extreme case, the example BSN performance is not affected by the IT parameters of S ₂ but is by the IT parameters of S ₃. In summary, for the example BSN system, the BSN reliability is more sensitive to the IT parameters of S ₃ than S ₂.

5.4. Verification

To verify the correctness of the proposed method, we use a special case of the example BSN system that can be analyzed using the CTMC method.

The special case requires that parameters of Set 1 (Exponential distribution) are used, LF and PFGE of the same biosensor are s-independent, and the isolation of the dependent biosensor upon the LFs of relays is instantaneous (by setting ${\alpha }_{S_{2}IT}=1.00e5/hour$ and ${\alpha }_{S_{3}IT}=1.00e5/hour$). To reduce the size of CTMC, we set PFGE parameters of NDCs to be 0, i.e., ${\alpha }_{S_{1}PF}=0$, ${\alpha }_{R_{1}PF}=0$, and ${\alpha }_{R_{2}PF}=0$. Fig. 10 shows the Markov model for the example BSN system.

Fig. 10.

Markov model for calculating the example BSN reliability.

Using the Laplace transform-based method, we obtain the BSN reliability as 0.935438 at t = 24 h, which exactly matches the result obtained using the proposed method.

Note that the CTMC-based method cannot analyze cases where the LF and PFGE of the same biosensor are dependent or disjoint. Moreover, the CTMC-based method is limited to the exponential distribution and has the size explosion issue (its model size grows exponentially with the number of system components). In contrast, the proposed approach has no limitation on the types of distributions characterizing biosensors’ time-to-LF/PFGE and isolation time. In addition, the proposed approach practices the “divide-and-conquer” strategy to decompose the original complex reliability problem into independent reduced problems that can be analyzed in parallel, given available computing resources.

6. Conclusion and future work

This work proposes a combinatorial and analytical method for the reliability analysis of BSNs with multiple correlated FDEP groups and random isolation time. The proposed method is a five-step procedure, which practices the divide-and-conquer principle to decompose a complex reliability problem into several reduced problems that can be efficiently solved. The proposed method has no limitation on the distribution types of time-to-LF/PFGE and IT of the BSN biosensors. An example of a BSN system is modeled and analyzed to demonstrate the proposed method and impacts of random isolation time parameters of dependent biosensors. Numerical results reveal that the isolation time parameters, the relationship between LF and PFGE, and the configuration of dependent biosensors can significantly affect the example BSN reliability. The correctness of the method is verified using the CTMC method.

Note that the addressed problem and the proposed solution procedure center around the BSN application. However, the problem of competing with correlated isolation groups and the proposed reliability analysis method are applicable to other applications that adopt relay-based wireless sensor networks to achieve efficient planning and management solutions [41], such as smart homes [54], smart grids [55], intelligent transportation systems [56], etc.

This work assumes that the BSN system and its biosensors have binary states (operation or failure) and are not repairable during the considered mission time. As one direction of the future work, it should be interesting to extend the proposed methodology to consider the repair behavior and the multi-state behavior where both the system and components may have multiple performance levels. Another direction is to model, analyze, and design resilience of BSN systems, which is concerned with the system's ability to survive from failures or hazards [57]. Resilience metrics, models, analysis methods, and design techniques will be explored for BSN systems while considering the competing failure behavior and random isolation time.

CRediT authorship contribution statement

Guilin Zhao: Writing – original draft, Methodology, Funding acquisition. Liudong Xing: Writing – review & editing, Supervision, Methodology, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

• No data was used for the research described in the article.