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A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE________________________________________________________________________
Analysis of Reliability Measures of Two Identical Unit System with One Switching Device and Imperfect
Coverage
Akshita Sharma and Pawan Kumar Department of Statistics
University of Jammu, Jammu-180006, J&K, India E-mail: [email protected]
Abstract
The present paper deals with the reliability analysis of a two identical unit system model with safe and unsafe failures, switching device and rebooting. Initially one of the units is in operative state and other is kept in standby mode. A single repairman is always available with the system for repairing and rebooting the failed units. In case of unsafe failure, repair cannot be started immediately but first rebooting is done which transforms the unsafe failure to safe failure and thereafter repair is carried out as usual. Switching device is used to put the repaired and standby units to operation. The failure time distributions of both the units and switch are also assumed to be exponential while the repair time distributions are taken general in nature. Reboot delay time is assumed to be exponentially distributed. Using regenerative point techniques, various measures of system effectiveness such as transition probabilities, availability, busy period, expected numbers of repairs etc. have been obtained, to make the study more informative some of them have been studied graphically.
Key Words: reliability, availability, regenerative point technique, rebooting, coverage probability
1. Introduction
In the context of global competition and paced development, it has become the foremost concern to make the apt decisions in order to increase the reliability and profit margin of every institution. With the advent of complexity of machines and more advancements in industrial sectors, the focus on increasing reliability and profit margins of any firm is increasing day by day as it is the sole aim on which most of the firms/industries are flourishing. It has become an important point which has to be kept in mind that the designs and layout of complex equipments should be in such a way that it enhances the reliability of the system and try to minimize the loopholes which are responsible for its degradation. Hence, designing the reliable systems and determining their availability have become the relevant steps in almost every sector.
In many situations from daily life, we find that the breakdown of the units' results into
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE__________________________________________________________________________
machine failure which results into huge losses and one of the ways to increase reliability is to introduce standby units which increase its reliability. There also arises certain situations when reason for the failure of unit is not detected immediately which leads to the situation of imperfect coverage, which is further tackled by reboot. Depending upon the complexity the timings of reboot delay vary from system to system.. In the last few decades, elaborated and comprehensive research work regarding the reliability, availability, standby systems, imperfect coverage, reboot etc has been carried out. The concept of reboot is discussed by Trivedi [8] in his book 'Probability and Statistics with Reliability, Queueing and Computer Science Applications'. Several empirical studies are proposed by P.A. Keiller and D.R. Miller [3] to increase the reliability of system. The imperfect coverage models with various status and trends were given by Amari, et al. [1]. Hsu, et al. [4] have studied the machine repair problem with standby system, repair and reboot delay. The reliability measure of repairable system with standby switching failures and reboot delay is studied by Jyh-Bin, et al. [5]. The other important contributions are made by Amari, et al. [2], Wang and Chen [9], Ke and Liu [7]. Ke, et al. [6] has also done the analysis by considering detection, imperfect coverage and reboot as major factor.
The present paper here deals with the reliability analysis of a system model with two identical units and one switching device. The switching device is used to turn the unit from standby or repaired state to operative state and is assumed to be in good condition when the system initially starts. The failure in any of the identical unit or switch may result into safe/ unsafe failures. Unsafe failure is the situation when reason for any of the breakdown is not known which is cleared by reboot first. Reboot delay time, failure time of both the units and switch are assumed to be exponentially distributed while the repair time distributions are general in nature. Other measures of system effectiveness such as mean time to system failure, reliability, availability, expected number of repair have been evaluated using regenerative point techniques.
2. System Description and Assumptions
1. The system comprises of two identical units N0 and Ns, one switch S is attached to it.
2. Initially one of the units is in operative state and other is kept in standby mode. Switching device is used to put the repaired and standby units to operation. In the initial phase, switching device is assumed to be in good condition.
3. The failures of units and switch in system might be safe and unsafe. Whenever any of the unit or switch of system results in safe failure, it may be immediately detected and located with coverage probability c and it will be repaired immediately if the repairman is available.
4. In case of unsafe failure, repair cannot be started immediately but first rebooting is done which transforms the unsafe failure to safe failure and thereafter repair is carried out as usual. Reboot delay times for units and switch are considered to be exponentially distributed random variables with different parameters.
5. A single repairman is always available with the system for repair and rebooting the failed units. Switch is always given preference over the failed units in the system for its repair.
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE_______________________________________________________________________
6. The failure time distributions of both the units and switch are assumed to be exponential while the repair time distributions are taken general in nature.
7. Once a component is repaired it is as good as new.
3. Notations And Symbols
a : Failure rate of identical units в : Failure rate of switching device
H1(.) : Repair rate of failed unit
H2(.) : Repair rate of switching device
c : Coverage probability
Y : Rebooting delay rate for unsafe failure for units S : Rebooting delay rate for unsafe failure for switching device
SYMBOLS FOR THE STATES OF THE SYSTEM
N0/Ns/Ng : Unit is in operative / standby / good condition.
Nusf/Susf : Unit/ switching device has undergone unsafe failure
Nr/Nwr : Unit is under repair / waiting for repair
Sr : Switch is under repair
With the help of the symbols defined above, the possible states of the system are: So = [No,Ns,Sg] Si = [Nr,No,Sg] S2 = [Nusf,Ng,Sg] S3 = \No,Ns,Sr]
S4 = [Ng, NS, SUsf] S5 = [Ng, Ng, Sr] S& = \NUSf, Ng, Sy ] Sj = \NWr, Ng, Sr]
S8 = \^r, NUsf, $9 = \^r, Nwr, ] S10 = \^r, Ng, 5^^
The transition diagram along with all transitions is shown in fig.1
TRANSITION DIAGRAM
О : Operative State
□
: Down-State
Fig.l
X: Non-Regenerative
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE_______________________________________________________________________
4. Transition Probabilities And Sojourn Times
Let Xn denotes the state visited at epoch Tn+ just after the transition at Tn, where T1, T2 .... represents the regenerative epochs. Then, Markov-Renewal process is constituted by [Xn, Tn} with state space E representing set of regenerative states and Qijit) = P[Xn+1 =j,Tn+i -Tn< t\Xn = i] is the semi Markov kernel over E.
Then the transition probability matrix of the embedded Markov chain is P = Pij = Qiji™) = Q(™)
First we obtain the following direct steady-state transition probabilities: p01 = ac f e-(a+^u du =
(a+p)
Similarly,
_ a(1-c)
P02 = (a+p)
P(1-c)
P04=l^ ^
Рп = т::Ь:,[1-!11(а + П\
P1,10
(a+P)
_ P(1-c)
[1-H1(a + P)]
— Pc
P03 = (a+p)
Pw = H1(a + p)
_ a(1-c)
P18 (a+p)
P30 = H2(a)
[l-H^a + P)]
(a+p) ^
P36 = (1- c)[l-H2(a)]
P21 = P45 = Pso = P67 = Pl1 = P89 = P91 = P10/7 = 1
The indirect transition probability may be obtained as follows:
Qu (t) = ac f e-(а+Юи H1( u)du f
= ac f* dH1(v) f” e-(a+P^u du
= ^fo(1-e-(a+P)V)dH1(v)
By taking t ^ ю, we obtain the following indirect steady-state transition probability:
p11 =^f('- e-(a+P)v)M1(v) =^[1- ^Л* + Ю]
Similarly,
Рз71 = C[1 - ^2(a)] (1)
From these steady state probabilities obtained above, it can be easily verified that the following results holds good:
(9)
P01 + P02 + P03 + P04 = 1 / P10 + P1t + P17 + P18 + P1,10 = 1
P30 + P36 + p371 = 1 , P21= P4S = Pso = P67 = P71 = P89 = P91 = Pw,7 = 1 (2)
Mean sojourn times
Mean sojourn time is defined as the expected time taken by the system in a state before making transition to any other state. Let Vi be the mean sojourn time for state Si, then to obtain mean sojourn time Vt in state St ,we observe that there is no transition from St to any other state as long as the system is in state St .If Tt denotes the sojourn time in state St then mean sojourn time Vt in state St is:
% = E[T] = fP(Ti>t)dt
Hence, using it following expressions for mean sojourn time is obtained:
1 1 1 ¥o=J^) ¥1 = (^f)(1-H1) Ч'2 = Ч'е = Ч'8=-
1
1Р3=~(1-Н2)
1
V4 = V^ = \
10
V9 = fH1(t) dt
s
Vs = ^7 = fH2(t)dt
(3)
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE_______________________________________________________________________
5. Analysis Of Reliability And MTSF
Let Ti be the random variable denoting time to system failure when system starts up from state St £ Et , then the reliability of the system is given by Ri(t) = P[Ti>t]
To obtain Ri(t), we consider failed states as absorbing states.
By referring to the state transition diagram, the recursive relations among Ri(t) can be formulated on the basis of probabilistic arguments. Taking their Laplace Transform and solving the resultant set of equations for Rl(s), we get
R*0(s) = N1(s)/D1(s) (4)
where,
Ni(s) = Z0 + qUl + q*03Z*3 and
Di(s) = 1 4oi4lo ЧогЧго
Taking inverse Laplace Transform of (4), we get reliability of the system. To get MTSF, we use the well known formula E(T0) = f R0(t)dt = limRl(s) = N1(0)/D1(0)
where,
Ni(0) = Wi + pnWi + pi3W3
and
Di(0) = 1 — PiiPii — P13P31
Since, we have 4lj(0) = Pij and limZl(s) = f Zi(t)dt =
6. Availability Analysis
Define At (t) as the probability that the system is available at time У given that initially started from state St e Et. Point wise availability is another measure of system effectiveness and is defined as the probability that the system will be able to work satisfactorily within tolerances at any instant of time. By using simple stochastic arguments, the recurrence relations among different point wise availabilities are obtained and taking the Laplace transforms and solving the resultant set of equations for d0(s), we have
A1(s) = N2(s)/D2(s) (5)
where,
N2(s) = (1
(9)
4W
4i848949i 4i,il4i1,747i
4l34364h47i + 4l24li)
4l74li)(Zl + 4I3ZI) + Zl(qli + q*i3q(37i> +
(6)
*
and,
^2(s) = (1 4li 4l848949i 4l,il4l1,747i 4r74’7i)(1 40343l 4*144*154*51) 4li4l1
(7)
(7)l
4il4l343i — 4il4l343646747i — 4il4l242i The steady state availability is given by A0 = limAl(t) = lims dl(s) = N2(0)/D2(0)
s —^1
as we know that, qij(t) is the pdf of the time of transition from state St to Sj and qij(t)dt is the probability of transition from state St to Sj during the interval (t, t + dt), thus 4ij(s) s=l= 4i*j(0) = Pij Also we know that
limZl(s) = f Zi(t)dt = Wi
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT COVERAGE
Therefore,
N2(0) = (1
9
-Pi i
P02P21 + P03P36P67P71)
PisPsiPii - Р17Р71 - Pi,10Pl0,7P71)(^0 + P03^3) + Vi(P01 + Р03Р31 +
(8)
^2(0) = (1 Pli P18P89P91 Pi7P71 Pi,10Pi0,7P71)(1 P03P30 P04P45P50) PioPoi
P10P03P31 - РюРо3Р3бРбзР71 - P10P02P21 (9)
The steady state probability that the system will be up in the long run is given by d0 = dmd0(t) = (jmsA0(s)
s^0
Um sW2( ^ = (jmN2(s)dm—7-
S^0 ^2(s) S^0 S^0°2(s)
Since as s ^ 0, B2(s) becomes zero.
Hence, on using L'Hospital's rule, d0 becomes
do = ^2(0)/D2(0) (10)
where,
02(0) = PioW> + ^2P02 + ^3P03) + PioP04^4 + ^5) + РюР03?3б(^6 + ^з) + (1-P03P30 - P04)[^1 + Pi10(^7 + ^10) + ^7Pi7] + (1 - P03P30 - P04)Pi8^8 + ^l)
(11)
Using (8) and (11) in (10), we get the expression for d0.
The expected up time of the system during (0, t] is given by PupOO = Jo^o(u) du So that,
Pup(s) = d0(s)/s.
7. Busy Period Analysis
Bj (t) is defined as the probability that the system having started from regenerative state Sj e E at that t=0, is under repair at time t due to failure of the unit. Now to determine these probabilities, we use simple probabilistic arguments and further taking the Laplace transform and solving the resultant set of equations for #0 (s), we have
ВД = ВД/ад (12)
where,
ад = (1- q(i)o - q^si^i - 4*,io4iV40i - qi*7?3i)[qM + 4*3^ + 4*4^4 + 445^*) +
403^6] + [q0i + 402^21 + 403(4(i)o + д3б^6з^71)] ft + чМ + 4*9^*) + 4*,Л] +
* * * * * * * * * * 4n4li - 418489491 - 4i,104i0,7471) + (4i,104i0,7 +
(13)
(9)*
{4034*64б7(1 4ii
+
ч1з) [qSi + 4*242i + 4*3(4(i)o + 4*64674*1)]}
and, B2(s) is same as given by (7).
In the steady state, the probability that the repairman will be busy is given by B0 = BmB0(t) = dms B0(s) = N3(0)/D2(0)
t^ro
s^0
(14)
where,
^3(0) = (1 - pi? - P17P71 - P18P89P91 - Pi,10Pi0,7P7i)[ P02P04 + P03^3 + P04(P04 + P45^5) + P03P36^6] + [Poi + P02P21 + P03(P(i) + P36P67P71)] [^1 + Pi8(^8 + P89^9) + Pi,10^10] + ^7 {P03P36P67(1 - Pi1) - P17P71 - P18P89P91 - Pi,10Pi0,7P71) + (Pi,10Pi0,7 +
Pi7) [Poi + P02P21 + P03(P(i) + P36P67P71)]} (15)
and D2(0) is same as obtained in (11).
The expected busy period of the repairman during (0, t] is given by
Rb(0 = j0^Bo(u)du So that, Mi(s) = #0 (s)/s
A. Sharma, P. Kumar
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT COVERAGE
8. Expected number of Repairs
Vi (t) is defined as the expected number of repairs of the failed units during the time interval (0, t] when the system initially starts from regenerative state S;. Further, using the definition of F;(t) and by probabilistic reasoning the recurrence relations are easily obtained and taking their Laplace- Stieltjes transforms and solving the resultant set of equations for Vo(s), we get
Vo(s) = N4(s)/D3(s) (16)
where,
N4(S) = (l Qll Q17Q71 $18$89$91 Ql,loQl0jQ7l)(Qo3Q30 + ^?04(?45(35o) +
(?10((?01 + Q02Q21 + (?03(?3l + (?0з(?3б(?67(?71) (17)
and D3(s) can be written on replacing qF- and q((jc)* by Qij and <3l(jc) respectively in the equation (7) .
In the steady state, the expected number of repairs per unit time is given by
V0 = lim[V0(t)/t] = Ums F0(s) = N4(0)/D2(0) (18)
where,
s^0
N4(0) = (1 - pH - P17P71 - P18P89P91 - Pl,l0Pl0,7P7l)(P03P30 + P04P45P50)+Plo(P0l +
(7)
P02P21 + P03Pil + P03P36P67P71)
(19)
9. Profit Function Analysis
With the help of reliability characteristics obtained, the profit function P(t) for the system can be obtained easily. Profit is excess of revenue over the cost, therefore, the expected total profits generated during (0, f] is:
P(t) = Expected total revenue in (0,t]- Expected total expenditure in (0,f]
= Ko№uP(t) - Kly.b(t) - K2Vo (20)
where,
K0 : Revenue per unit up time of the system.
K1 : Cost per unit time for which repair man is busy in repairing the failed unit.
K2 : Cost of repair per unit.
In steady state, the expected total profits per unit time, is P = lim[P(t)/t] = lims2P*(s)
s^0
Therefore, we have
P = K0A0 - K1B0 - K2V0 (21)
10. Graphical Study Of The System Model
Graphical study of the system model gives a more perceived picture of system behaviour. So, for more concrete study, we plot MTSF and Profit function with respect to a, failure rate of identical unit for different values of hl, repair rate of identical unit.
Fig. 2 represents the variations in MTSF with respect to a for different values of hlas 0.05,
A. Sharma, P. Kumar RT&A, No 1 (52)
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT Volume 14, March 2019
SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT
COVERAGE_______________________________________________________________________
0.40, 0.80 by keeping all the other parameters fixed at = 0.03, h = 0.20, у = 0.35, S =
0.45 . The coverage probability c for the system is set at 0.70. It can be clearly seen from the graph of MTSF that it decreases continuously with increase in failure rate a and by increasing repair rate h1, the value for MTSF also increases, thereby concluding that repair facility increases the lifetime of the system.
Fig. 3 represents the change in Profit function P with respect to a for different values of h as 0.05, 0.40, 0.80 by keeping all the other parameters fixed at = 0.03, h = 0.20, у = 0.35, S = 0.45 . The coverage probability c for the system is set at 0.70. Clearly, it is observed that profit function decreases with increase in failure rate a but increases with increase in repair rate hx. Hence, repairing the system from time to time will result in better performance of the system.
A. Sharma, P. Kumar
ANALYSIS OF RELIABILITY MEASURES OF TWO IDENTICAL UNIT SYSTEM WITH ONE SWITCHING DEVICE AND IMPERFECT COVERAGE________________________________________________
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