STOCHASTIC BEHAVIOUR OF AN ELECTRONIC SYSTEM SUBJECT TO MACHINE AND OPERATOR FAILURE Текст научной статьи по специальности «Техника и технологии»

STOCHASTIC BEHAVIOUR OF AN ELECTRONIC SYSTEM SUBJECT TO MACHINE AND OPERATOR

FAILURE

S. Malik1, Komal2, R. K. Yadav*, Anju3

123Department of Mathematics, Baba Mastnath University, Rohtak, India 'School of Sciences, Christ University, Bengaluru, India

1 [email protected], [email protected], '[email protected],

[email protected] *Corresponding Author

Abstract

A stochastic model is developed by assuming the human (operator) redundancy in cold standby. For constructing this model, one unit is taken as electronic system which consists of hardware and software components and another unit is operator (human being). The system can be failed due to hardware failure, software failure and human failure. The failed hardware component goes under repair immediately and software goes for upgradation. The operator is subjected to failure during the manual operation. There are two separate service facilities in which one repairs/upgrades the hardware/software component of the electronic system and other gives the treatment to operator. The failure rates of components and operator are considered as constant. The repair rates of hardware/software components and human treatment rate follow arbitrary distributions with different pdfs. The state transition diagram and transition probabilities of the model are constructed by using the concepts of semi-Markov process (SMP) and regenerative point technique (RPT). These same concepts have been used for deriving the expressions (in steady state) for reliability measures or indices. The behavior of some important measures has been shown graphically by taking the particular values of the parameters.

Keywords: Electronic System, Operator Failure, Reliability Measures, Stochastic Analysis, semi-Markov Process, Regenerative Point Technique

I. Introduction

The electronic systems containing h/w and s/w components are playing an impactful role in transforming modern society into a digitalization world. All types of modern industries have become dependent on these systems to furnish various jobs with higher accuracy timely. With electronic systems, the digital world can't ignore the impactful role of manpower in doing work. As these systems require hardware and software, the failure of these components during operation is undeniable. Due to these failures, the job cannot be completed in time or the losses can occur in terms of finances, development lives etc. Therefore, the reliability of these systems becomes very important for the completion of jobs.

Many techniques related to reliability improvement have been described by many researchers. The cold standby redundancy techniques have also been included by the engineers. In the modern society, many research works on reliability modelling of redundant systems have been done to do job accurately. In [14], Sridharan and Mohanavadivu checked the stochastic behavior of two-unit standby system having regular and expert repairmen. The aging properties of the residual life length of k-out-of-n system with independent non-identical components has been discussed in [5]. The idea of switch failure and equipment maintenance has been used in reliability analysis of a two-unit cold standby system [12]. The same type of system has been described in [6] with consideration of human error failure. In [13], Salah and Sherbeny assumed different types of failures in analyzing two non-identical unit system stochastically. The concept of Markov process has been used in reliability analysis of system [11].

Many reliability models for non-identical units have been discussed by engineers and researchers. In [7], Malik and Upma described the non-identical units under preventive maintenance. The reliability measures of two dissimilar units using Gumbel-Hougaard Family Copula have been determined in [1]. The provision of rest and switching device for two non-identical unit standby system have been elaborated [3]. In [2], Kadyan et al. studied non-identical repairable system with working of simultaneous cod standby units. The failure of repairman for system of two-non identical units has been considered by Kumar and Nandal [4] in their research work. During h/w repair, assumption of server failure has been considered in stochastic analysis of computer system by Malik and Yadav [8]. In [9], [10] and [15], the authors described the stochastic analysis of a computer system subject to failure of service facility. However, the human being (operator) has not been considered in redundancy in the above-mentioned research by discussing the reliability modelling of electronic system.

Thus, in this paper, the authors tried the use of manpower (operator) in reliability modelling of electronic system. Here, one unit is taken as electronic system which consists of hardware and software components and another unit is operator (human being). The system can be failed due to hardware failure, software failure and human failure. The failed hardware component goes under repair immediately and software goes for upgradation. The operator is subjected to failure during the manual operation. There are two separate service facilities in which one repairs/upgrades the hardware/software component of the electronic system and other gives the treatment to operator. The failure rates of components and operator are considered as constant. The repair rates of hardware/software components and human treatment rate follow arbitrary distributions with different pdfs. The state transition diagram and transition probabilities of the model are constructed by using the concepts of semi-Markov process (SMP) and regenerative point technique (RPT). These same concepts have been used for deriving the expressions (in steady state) for reliability measures or indices. The behavior of some important measures has been shown graphically by taking the particular values of the parameters.

II. Abbreviations and Notations

MTSF

SMP

RPT

MST

a/b

Mean Time to System Failure Semi-Markov Process Regenerative Point Technique Mean Sojourn Time

Probability of hardware/software failure Hardware/software/ operator failure rates

Hardware repair/software up-gradation/operator treatment rates pdf/cdf of hardware repair time

Xl/X2/

a/ß/Y f(t)/F(t)

g(t)/G(t)

s(t)/S(t)

pdf/cdf

%(py

Qij(t)

q.ij.kr(t)/

Qij.kr(t)

Pij/Pij.kr

ßi

™ij

№) Ai(t)

Mi(t)

Ri(t)

W)

Ti(t)

©/©

p

zjz2 z*

pdf/cdf of software repair time pdf/cdf of human treatment time

Probability density function/Cumulative density function

pdf/cdf of passage time from regenerative state St to a regenerative state Sj

or to a failed state Sj without visiting any other regenerative state in (0, t]

pdf/cdf of direct transition time from regenerative state St to a regenerative

state Sj or to a failed state Sj visiting states Sk and Sr once in (0, t]

Steady state probability of transition from state St to state Sj directly/via

states Sk and Sr once

MST in state S£ which is given by fc = E(Ti) = f™ P(Ti > t) dt where Ti denotes the sojourn time in state St.

Contribution to MST(fcj) in state Si when system transits directly to state Sj so that ßi = Y.j mij and my = f tdQy (t) = -qfj(0) cdf of first passage time from regenerative state Si to a failed state Probability that the system is in up-state at instant 'f given that the system entered in regenerative state St at t = 0

Probability that the system up initially in regenerative state Si is up at time t without visiting any other state

Expected number of hardware repairs in the interval (0, t] given that the system entered in regenerative state St at t = 0.

Expected number of software up-gradations in the interval (0, ] given that the system entered in regenerative state Si at t = 0.

Expected number of treatments given to the human in the interval (0, ] given that the system entered in regenerative state St at t = 0. Standard notation for Laplace-Stieltjes convolution/Laplace convolution Symbol for Laplace Transform (LT)/Laplace Stieltjes Transform (LST) Profit function of system System revenue per unit up-time

Repair/up-gradation cost per unit time due to hardware failure/software failure

Operator Treatment cost of per unit time

III. Assumptions and State Descriptions

To describe the system the following assumptions are made:

a) There are two units in which one is an electronic system made up of hardware and software and other unit is a human being (operator).

b) The components of electronic system and operator are repaired by separate servers.

c) The h/w repairs, s/w up-gradation and treatments are perfect.

d) The failure rates of components and operator are constant.

e) The arbitrary distributions are taken for h/w repair, s/w up-gradation and operator treatment rates.

The description of all states is given in the Table 1

Table 1: Description of all states in the system model

States Description

So Electronic system is in working and operator is in spare

Si h/w component is under repair and operator is working manually

S2 s/w is under up-gradation and operator is working manually

S3 h/w component is under repair continuously and operator is under treatment

S4 s/w is under up-gradation continuously and operator is under treatment

S5 Electronic system is in working and operator is under treatment continuously

Se h/w component is under repair continuously and operator is working manually

S7 s/w is under up-gradation continuously and operator is working manually

S8 h/w component is under repair and operator is under treatment continuously

S9 s/w is under up-gradation continuously and operator is under treatment continuously

All the states are presented in the state transition diagram as shown in Figure 1.

Figure 1: State transition Diagram of system model

IV. Reliability Measures

I. Transition Probabilities

The arbitrary distributions are considered as: f(t) = ae at, g(t) = fie and s(t) = ye yt. Using probabilistic arguments, the differential transition probabilities for state So are given by dQ01(t) = ax1e-(axi+bx2)tdt , dQ02(t) = bx2e-(axi+bx2)t dt

The following results: py = lim0**(s) = 0ÏJ(O) = f™ dQtj (t) = J™ q^ (t)dt, have been used to determine the transition probabilities. These are obtained as

_ ax1 _ b%2 _ _ a _ _ ^ _ _ P

V01 = ax1 + b%2 , V02 = axi + bxj Vl0 = P&0 = ~yl Pl3 = P&3 = P20 = P?0 = J+Y

— — Y — — a — — Y — — P — — Y V24 = P74 = p+Y, P35 = Pss = a+v P36 = ps6 = a+v P45 = P95 = p+Yf p47 = p97 = P+Y

__Y _ axi _ b%2 _______

P50 = axi + b%2+Yf PS8 = axi + b%2+Yf ?59 = , Pl53 = P65 3 = Pl3P35 Pl6 3 = P663 = Pl3P36

P25.4 = P75.4 = P24P45, P27.4 = P77.4 = P24P47, P55.8 = P58P85, P56.8 = P58P86, P55.9 = P59P95,

P57.9 = P59P97

From these, we have

P01 + P02 = Pl0 + Pl3 = P20 + P24 = P35 + P36 = P45 + P47 = P50 + P58 +P59 = 1

Peo + Pb3 = Pl0 + P?4 = P85 + P86 = P95 + P97 = P10 + Pl53 + P16.3 = 1

Pso + P55.8 + P56.8 + P55.9 + P57.9 = Peo + P653 + P663 = P70 + P75.4 + P77.4 = 1

II. MST

The MSTs for all states are determined as follows:

11111 1

^1= — = He, U2=p+y = V7, H3= — = ^,H4=—y = V9, Us = aXl+bX2+V

, _ a+y+Y _ , , _ p + y+Y _ , , _ (p+Y) (a+Y+axi) + bX2 (a+Y) U1 (a+y)(a+Y) U6, U2 (P + y)(P+Y) (a+Y)(axi + b%2+Y)(P+Y)

III. Reliability and MTSF

Let 0 1 (t) be the c.d.f. of first passage time from regenerative state Si to a failed state. Regarding the failed state as absorbing state, we have following recursive relations for 0 ¿( t): 01(t) = Qij(t)®0j(t) + Qik(t)

where Sj is an un-failed regenerative state to which the given regenerative state St can transit and Sk is a failed state to which the state St can transit directly. Thus, the following equations are obtained as: 0o(t) = Qo1(t)®01(t) + Qo2(t)®02(t)

01(t)= Q1o(t)®0o(t) + Q13(t)

02 (t)= Q2o(t)®0o(t) + Q24OO Taking Laplace Stieltjes Transform of above equations, we get 0O*(5) = QOi(s)01*(s) + Qo2(s)07(s) 0l'(s) = Qro(s)0'o'(s) + Q00(s) 02*(s) = QO'o(s)0'oo(s) + Q04(s) Solving for 0*o° (s) by Cramer Rule, we have

0*oo(s)=^

Where A =

A =

- QM - Qo2(S> Q%(s) 1 0

Q20(s> 0 1 0 - QÔi(s) - Q02(s>

— 1

— 1

and

QïKs) 1 0

QZ4(s> 0 1

_ 1 — 0p* (s)

Now, we have R*(s>

The reliability of the system model can be obtained by

R(t)=L-1[R*(s)]

The MTSF is given by

MTSF = limR*(s) = R*(0) = —, where ^ = + p01ß1 + p02ß2 and Di = P01P13 + P02P24

IV. Availability

Let Ai(t) be the probability that the system is in up-state at epoch't' given that the system entered regenerative state S^ at t = 0. The recursive relations for Ai(t> are given as Ai(t) = Mi(t) + Zjq(f>(t)©Aj(t)

where Sj is any successive regenerative state to which the regenerative state St can transit through n transitions. Thus, the following equations are obtained as: Ao(t> = Mo(t> + qoi(t>©Ai(t> + qo2(t>©A2(t>

Ai(t> = Mi(t> + qio(t>©Ao(t> + qi5.3 (t>©As(t> + qi63(t>©A6(t> A2(t> = M2(t> + q2o(t>©Ao(t> + q25A(t>©A5(t> + q27A(t>©A7(t>

As(t> = Ms(t> + qSo(t>©Ao(t> + [q55.g(t> + q55.9(t>]©A5(t> + qS6,g(t>©A6(t> + qS7.9(t>©A7(t>

RT&A, No 4(80)

S. Malik, Komal, R K. Yadav, Anju Volume 19, December, 2024 STOCHASTIC BEHAVIOUR OF AN ELECTRONIC SYSTEM_;_

A6(t) = M6(t) + q6o(t)©Ao(t) + q6S3(t)©As(t) + q663(t)©A6(t)

A7(t) = M7(t) + q7o(t)©Ao(t) + q75A(t)©A5(t) + q77A(t)©A7(t)

where M0(t) = e-(axi+bx2)t, M1(t) = M6(t) = e-^tF(t),M2(t) = M7(t) = e-^tG(t), and

M5(t) = e-(axi+bx2)tS(t)

Taking LT of above equations and solving for ^0(s), the steady state availability is calculated by ¿o(") = limits) = ^

O2

where N2 = Poi[P57.9(ßlP20 + ^2Pl5.3) + (1 - P27.4)iVi(Ps0 + P56.8) + VsPlS.s)] + P02[P56.8(ß2Pi0 + ßlP25A) + (1- Pl6.3){ß2(P50 + Ps7.9) + PsP25a}] + ß0[P57.9P20(1 - Pl6.3) + (1 - P27.4){PS0(1 - Pl6.3) + PS6.8Pl0}l

D2 = (Poiß'l +ß'o+ P02ß2)[P57.9P20(1 - Pl6.3) + (1 - P27a){P50(1 - Pl6.3) + P56.8Pio}] + ß'siPl5.3P01(1 - P27A) - P25AP02(1 - Pl6.3)} + ß'l[Pl6.3P0liP57.9P20 + (1 - P27a)(P50 + PS8)} + P25aP02P56.8] + ß'2[P27AP02{P56.8Pl0 + (1 - Pl6.3)(Ps0 + Ps9)] + Pl5.3PoiPs7.9]

and Pi = M*(0),i = 0,1,2,5

V. Expected Number of Hardware Repairs

Let Ri(t) be the expected number of the hardware repairs by the server in the interval (0, t]given that the system entered regenerative state St at t = 0. The expected number of the hardware repairs is given by R0(= limsRg*(s)

The recursive relations for Rt(t) are given as: Ri(t) = ZjQlf(t)©[8j+Rj(t)]

Where Sj is any regenerative state to which the regenerative state St can transit through n transitions and Sj = 1, if Sj is the regenerative state where server does job afresh, otherwise Sj = 0. Thus, the following equations are obtained as: Ro(t) = Qoi(t)®Ri(t) + Qo2(t)®R2(t)

Ri(t) = Qio(0®[1 + Ro(t)] + QiS.3(t)®[1 + Rs(t)] + Qi6.3(t)®R6(t) R2(t) = Q2o(t)®Ro(t) + Q2sA(t)®RS(t) + Q27A(t)®R7(t)

Rs(t) = QSo(t)®Ro(t) + Qss.8(t)®[1 + Rs(t)] + Q55.9(t)®R5(t) + QS6Ät)®R6(t) + QS7Ät)®R7(t) R6(t) = Q6o(t)®[1 + Ro(t)] + Q65.3(t)®[1 + Rs(t)] + Q66.3(t)®R6(t) R7(t) = Q7o(t)®Ro(t) + Q7sA(t)®R5(t) + Q77.4(t)®R7(t)

Taking Laplace Stieltjes Transform of above relation and solving for R0*(s). The expected number of the hardware repairs is given by Ro(™) = limsR*o*(s) = ^

s^0 O2

where N3 = Poi[P57.9P20(1 - Pi6.3) +(1- P27.4){PS0(1 - Pi6.3) + Ps6.8Pio)] + Pz8{Pi5.3Poi(1 - P27.4) + p25,4p02(1 - pi6.3)} and D2 is same as calculated in availability.

VI. Expected Number of Software Up-gradations

Let Ui (t) be the expected number of the software up-gradations by the server in the interval (0, t] given that the system entered regenerative state St at t = 0. The expected number of the software up-gradations is given by U0(x>) = limsU0*(s) The recursive relations for Ut(t) are given as: Ui(t) = ZjQlf(t)®[Sj + Uj(t)]

Where Sj is any regenerative state to which the regenerative state St can transit through n transitions

and Sj = 1, if Sj is the regenerative state where server does job afresh, otherwise Sj = 0. Thus, the

following equations are obtained as:

Uo(t) = Qoi(t)®Ui(t) + Qo2(t)®U2(t)

Ui(t) = Qio(t)®Uo(t) + Qi53(t)®U5(t) + Qi6,3(t)®U6(t)

U2(t) = 02o(0®[1 + Uo(t)] + Q25.4(0®[1 + U5(t)] + Q27.4(t)®U7(t)

US(t) = Qso(t)®Uo(t) + Q55,8(t)®U5(t) + Q55,9(t)®[1 + Us(t)] + Q56,8(t)®U6(t) + Q57.9(t)®U7(t) U6(t) = Q6o(t)®To(t) + Q6s3(t)®Us(t) + Q66.3(t)®U6(t)

Uy(t) = Qyo(0©[1 + Uo(t)] + Q7SA(t)®[1 + Us(t)] + Q77A(t)®U7(t)

Taking Laplace Stieltjes Transform of above relation and solving for Ro*(s). The expected number of the software up-gradationsis given by Uo(^) = \imsU0o(s) = ¡4

where N4 = Po2[Ps7.9P2o(1 - P16.3) + (1 - P27a){Pso(1 - P16.3) + P56.8Pw}] + Ps9{P15.3Po1(1 - P27.4) + p25,4Po2(1 - P16.3)} and D2 is same as calculated in availability.

VII. Expected Number of Treatments given to Operator

Let Ti (t) be the expected number of the treatments given to human by the server in the interval (0, t] given that the system entered regenerative state Si at t = 0. The expected number of the treatments given to operator is given by To(x>) = limsTo*(s) The recursive relations for Ut(t) are given as: Ti(t) = ZjQlf(tm8j + Tj(t)]

Where Sj is any regenerative state to which the regenerative state St can transit through n transitions and Sj = 1, if Sj is the regenerative state where server does job afresh, otherwise Sj = 0". Thus, the following equations are obtained as: To(t) = Qo1(t)®T1(t) + Qo2(t)®T2(t)

T1(t) = Qw(t)®To(t) + Q15.3 (t)®Ts(t) + Q16.3 (0®[1 + T6(t)] T2(t) = Q2o(t)®To(t) + Q2S.A(t)®Ts(t) + Q27.A(t)®[1 + T7O:)]

Ts(t) = Qso(t)®[1 + To(t)] + [Qssx(t) + QsS.9(t)]®Ts(t) + QS6.B(t)®[1 + T6(t)] + Q57.9(t)®[1 + T7(t)]

T6(t) = Q6o(t)®To(t) + Q65.3(t)®T5(t) + Q66.3(t)®[1 + T6(t)] T7 (0 = Q7o(t)®To(t) + Q7S.A(t)®Ts(t) + Q77.A(t)®[1 + T7Q;)]

Taking Laplace Stieltjes Transform of above relation and solving for Ro*(s). The expected number of the treatments given to human is given by To(^) = limsT0*(s) = ¡5

where Ns = Po1iPs7.9(P15

.3 + P16.3 P2o) + P16 .3(1 P27.A )(Pso + Psa^)} + Po2{P56 .8(P2A - P27.AP13) +

P27A(1 - P16.3)(Pso + Ps9)} and D2 is same as calculated in availability.

V. Profit Analysis

The profit function in the time t is given by

P(t) = Expected revenue in (0, t] - expected total cost in (0, t]

In steady state, the profit of the system model can be obtained by the following formula:

P = Z0A0(œ) - Z1R0(œ) - Z2U0(œ) - Z3T0(œ)

where Z0 = ^5000, Z1 = ^2000, Z2 = ^ 1500,Z3 = ^1000

VI. Application

The application of the present research work is described in the car washing machines. In modern era, most of the people have at least one vehicle in their home for easy service. Therefore, the Vehicle Washing Shops/Vehicle Service Shops are opened within 2KM circle in most of cities. The vehicle is washed with the help of automatic washing machine as shown in the Figure 2, just standing the vehicle under that machine. Due to short circuit, hardware failure or sudden error in software, the machine can be stopped. A server is facilitated for hardware repair and/or software up-gradation of automatic washing machine. After the failure of the machine, the vehicle is washed by a human. During the washing of vehicle, there is possibility that human can be hurt by any part of vehicle; therefore, another service facility has been given for treatment of human being.

Figure 2: Automatic Car Washing Machine VII. Numerical Illustration

Suppose that in a vehicle washing shop there is an automatic car washing machine and a labor. It is obvious that the vehicle washing machine can fail due to h/w component or s/w component with probabilities 'a' or 'b' respectively. The respective failure rates of h/w, s/w and human being are taken as x1, x2 and The repair rates of h/w and s/w are assumed as a and p respectively. The labor undergoes treatment with rate y.

The reliability measures are determined for arbitrary values of the following parameters:

x1 = 0.15,x2 = 0.003, ^ = 0.002, a = 2, p = 3,y = 6, a = 0.6 and b = 0.4

Z„ = ^ 5000,Zx = ^ 2000,Z2 = ^ 1500, Z3 = ^ 1000.

The particular values of the transition probabilities are as follows:

Poi = 0.98 = Pso,Po2 = 0.02, p^ = 0.99 = P20,Pi3 = 0.01 = P24 = P58 = P59, P35 = 0.25

P36 = 0.75, P45 = 0.33, P47 = 0.67

The particular values of the MST's are as follows:

,u0 = 10.96, = 0.5 = = 0.33 = = 0.12, ;U4 = 0.11,^5 = 0.16,^5 = 0.17 Thus, MTSF = 11456.66, Availability = 0.999848 and Profit = ^ 4886.86

VIII. Graphical Study of Reliability Measures

Some important reliability measures such as MTSF, availability and profit have been studied w.r.t h/w failure rate. The graph of MTSF vs h/w failure rate has been shown in the Figure 3. The behavior of availability vs h/w failure rate has been presented in Figure 4. In the similar way, profit analysis has been shown in Figure 5.

Figure 3: MTSF Vs Hardware Failure Rate (x1)

Figure 4: Availability Vs Hardware Failure Rate (x1)

4920 1 —♦— x2=.003, n=.001. a=l, ß=3, T=5,

4900 - a=.6, b=.4

4880 - —■— x2=.004, n=.001, a=.6, b=.4 a=l, ß=3, T=5,

S o La 4860 4840 - —A— x2=.003, n=.002, a=.6, b=.4 - — x2=.003, ^=.001, a=l, ß=3, T=5, ct=2, ß=3, T=5,

OH a=.6, b=.4

4820 - —*—x2=.003, n=.001. a=l, ß=4, T=5,

4800 - a=.6, b=.4 —•-x2=.003, n=.001. a=l, ß=3, T=6,

4780 - a=.6, b=.4 —1—x2=.003, ^=.001, a=l, ß=3, T=5,

4760 a=.4, b=.6

0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Hardware Failure Rate (xx)

Figure 5: Profit Vs Hardware Failure Rate (x1)

IX. Conclusion

A reliability model of an electronic system with operator failure has been analyzed in present study. The use of SMP and RPT has been incorporated in determining the transition probabilities and various reliability measures. These reliability measures have been explored graphically for the different values of parameters. Figure 3 concludes that MTSF decline according to inclined behavior of components failures and operator failure and MTSF increases with increments in component repair rates and treatment rate. Also, MTSF is very high when hardware repair rate increases from |j=1 to |j=2. Availability and Profit function shows the approximate same behavior as MTSF shows. This nature can be seen if the Figure 4 and Figure 5. There are various future scopes of the present study so that we can make model impactful. The inspection policy for machine can be considered before going under hardware repair as well as software upgradation. The replacement of hardware/software can be done if hardware is not repairable and software is not working. The concept of power failure and timing of power restoration can be considered during the stochastic model. The idea of arbitrary distributions of failure rates of components can be considered.

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