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A TWO NON-IDENTICAL UNIT PARALLEL SYSTEM SUBJECT TO RT&A, No 2(62) TWO TYPES OF FAILURE AND CORRELATED LIFE TIMES_Volume 16, June 2021
A Two Non-Identical Unit Parallel System Subject to Two Types of Failure and Correlated Life Times
Pradeep Chaudhary, Lavi Tyagi
Department of Statistics Ch. Charan Singh University, Meerut-250004(India)
E-mail: pc25jan@gmail.com; lavityagi444@gmail.com
Abstract
The paper deals with the reliability and cost-benefit analysis of a two non-identical unit system with two types of failure. The units are named as unit-1 and unit-2 and they are arranged in a parallel configuration. Unit-1 can fail due to hardware or due to human error failure whereas unit-2 fails due to normal cause. A single repairman is considered with the system for all types of failure in the units and unit-1 gets priority in repair over the unit-2. The repair time distributions of unit-1 are taken as general with different c.d.fs and the repair time distribution of unit-2 is taken as exponential. Failure time distribution of unit-1 due to human error is taken exponential. Whereas the random variable denoting the failure time of unit-1 due to hardware failure and random variable denoting the failure time of unit-2 are assumed to be correlated random variables having their joint distribution as bivariate exponential (B.V.E.).
Keywords: Transition probabilities, mean sojourn time, bi-variate exponential distribution, regenerative point, reliability, MTSF, availability, expected busy period of repairman, net expected profit.
I. Introduction
Reliability is an important concept in the planning design and operation stages of various complex systems. Gupta et al. (2014) analysed a two non-identical unit parallel system with two independent repairmen-skilled and ordinary. A failed unit is first attended by skilled repairman to perform first phase repair and then it goes for second phase repair by ordinary repairman. Both types of repair discipline are FCFS. Chaudhary et al. (2015) analysed a two non-identical unit parallel system model assuming that an administrative delay occurs in getting the repairman available with the system whenever needed. Upon failure of a unit, the other operating unit shares the load of failed unit. Chopra and Ram (2017) analysed a two non-identical unit parallel system with two types of failures: common cause failure and partial
failure. The repairman is not always available with the system to repair a failed unit. Whenever a unit fails, the repairman is called to come at the system and he takes some significant time to reach at the system. This time is known as waiting time of repairman during which the failed unit waits for repair. Chandra et al. (2020) performed the reliability and cost benefit analysis of the two identical and non-identical unit parallel system models by using Semi - Markov Process in regenerative point. A study of comparison is made between the reliability characteristics for both the system models under study. In these papers, the authors did not consider the concept of human error failure. In all the above system models the authors have considered single cause of failure in a unit i.e. normal cause.
Mahmoud and Moshref (2010) analysed a two-unit cold standby system by considering two cause of failure in a unit namely-Due to hardware and Due to human error. It has also been assumed that an operating unit goes for preventive maintenance (PM) to increase the system effectiveness. All the distributions of random variables involved in the system model are taken to follow arbitrary distributions. Kumar and Malik (2011) carried out the profit analysis of a computer system model with software and hardware failure subject to maximum operation time (MOT) and maximum repair time (MRT). An operating unit goes for preventive maintenance (PM) after completing MOT, if it is not failed during this time. Further if a failed unit is not repaired during MRT, it is replaced by new one. The priority to software replacement is given over hardware repair. Singh et al. (2016) analysed a two-unit warm standby system with two types of repairman. The first type of repairman, usually called regular repairman who is always remains available with the system to attend a failed unit. If he might not be able to do some complex repairs within some tolerable (patience) time, an expert repairman is called from the outside to complete the repair of the failed unit and he takes some significant time to reach at the system. Further an operating unit may fail either due to hardware or due to human error. In all the above system models the common assumptions considered is that the failure and repair times of the units are taken to uncorrelated random variables.
Gupta and co-workers [2008,2018] analysed two unit parallel and standby system models under different sets of assumptions by taking the failure and repair times as correlated random variables having their joint distribution as bivariate exponential. They have considered only single type of failure in an operating unit. Some authors including [1999, 2013] analysed two-unit parallel system models by taking the joint distribution of life times of the units working in parallel as bivariate exponential. They have also considered the single type of failure in an operating unit. The objective of the present paper is to study a two non-identical unit parallel system subject to two causes of failure in an operating unit-Due to hardware and Due to human error. Human failure is defined as a failure to perform a prescribed task which could result in damage to the equipment and property. There exist a number of causes for human error; e.g., lack of good job environments, poor training or skill of the operating personnel and so on. On the other hand, hardware failure occurs due to flaws in design, poor quality control and poor maintenance.
The life time of the units working in parallel form are taken to be correlated random variables having their joint distribution as bivariate exponential with different parameters as the form of joint p.d.f. given below.
f(xx,x2) = axa2 (1 -r)e-a[Xl-ttiX! I0 (2<Ja1a2rx1x2); xx,x2,al5a2 > 0; 0 < r < 1
W / / '")\2k
where, Io (z) = V (z 2 V (k!)2
is the modified Bessel function of type-I and order zero.
By using regenerative point technique, the following measures of system effectiveness are
obtained-
i. Transient-state and steady-state transition probabilities.
ii. Mean sojourn time in various regenerative states.
iii. Reliability and mean time to system failure (MTSF).
iv. Point-wise and steady-state availabilities of the system as well as expected up time of the system during time interval (0, t).
v. The expected busy period of repairman in time interval (0, t).
vi. Net expected profit earned by the system in time interval (0, t) and in steady-state.
II. System Description and Assumptions
1. The system comprises of two non-identical units-unit-1 and unit-2. Initially, both the units are operative in parallel configuration.
2. Each unit has two modes-Normal (N) and Total failure (F).
3. Unit-1 can fail either due to hardware or human error. Whereas unit-2 can fail only due to its normal cause.
4. The system failure occurs when both the units are totally failed.
5. A single repairman is always available to repair the failed unit-1 either due to hardware or human error and the failed unit-2. The unit-1 gets priority in repair over the unit-2.
6. Failure time of unit-1 due to human error is taken exponential distribution whereas the failure time of unit-1 due to hardware and failure time of unit-2 due to normal cause are assumed to be correlated random variables having their joint distribution as bivariate exponential (B.V.E.) with density function as follows-
f(x1,x2) =axa2 (1-r)e-"1*1 -aiXl I0 (2^a1a2rx1x2 ); xx,x2,al5a2 > 0; 0 < r < 1
where, I0 (z ) = Y ^^
, 0 ( ) ¿0 (k!)2
7. The repair time distribution of unit-1 failed either due to hardware or due to human error are taken as general with different c.d.fs whereas the repair time distribution of unit-2 failed due to normal cause is taken as exponential.
8. A repaired unit always works as good as new.
III. Notations and States of the System We define the following symbols for generating the various states of the system-
No1, N22 : Unit-1 and Unit-2 in normal (N) mode and operative.
Fr\ : Unit-1 is in failure (F) mode and repair which is failed due to hardware
failure.
Fr12 : Unit-1 is in failure (F) mode and repair which is failed due to human
error.
Fr2 ,Fw;r : Unit-2 is in failure (F) mode and under repair/waits for repair.
Considering the above symbols in view of assumptions stated in section-2, the possible states of the system are shown in the transition diagram represented by Figure. 1. It is to be noted that the epochs of transitions into the state S4 from S1, S5 from S2 are non-regenerative, whereas all the other
entrance epochs into the states of the systems are regenerative. The other notations used are defined as follows:
E
X (i = 1,2) f (x1,x2 )
gi ( x )
k1 ( Xl|X2 = X2 )
k2 (X2|X1 = X)
K (• | X)
X p
jj)
pij,p(k)
p.., ,p<k) ijlx ijlx
Set of regenerative states.
Random variables representing the failure time of uni1-1 in N-mode and unit-2 respectively for i=1,2. Joint p.d.f. of (x1;x2).
f(x1?x2) = a1a2 (1-r)e a'Xl I0 (^a1a2rx1x2 )
; x1;x2, a1; a2 > 0 ; 0 < r < 1
2k
where Io (z) = jj (Z/2)r k=0 (k !)
Marginal p.d.f. of X1 = x
= ai (1 -ii )e-ai(1-r)x
Conditional p.d.f. of X^X2 = x.
= -(a1x1 + a2rx )T
= a, e ' 1 1 2 110 (2^/a1a2rxx1 )
Conditional p.d.f. of X2|X1 = x.
-(ax +arx)T in. I \
= a2e ( 2 2 1 % (2Va1a2rxx2 ) Conditional c.d.f. of Xi | Xj = x, i ^ j ; i, j = 1,2 .
Constant failure rate of unit-1 due to Human error.
Constant repair rate of unit-2 due to normal cause.
c.d.f. of repair time of unit-1 failed due to hardware failure and unit-1
failed due to human error.
p.d.f. of transition time from state Si to Sj and Si to Sj via Sk . Steady-state transition probabilities from state Si to Sj and Si to Sj
via Sk .
Steady-state transition probabilities from state Si to Sj and Si to Sj via Sk when it is known that the unit has worked for time x before its failure.
+Symbol for Laplace Transform i.e. q*j (s) = Je~(t) dt
*
©
Symbol for Laplace Stieltjes Transform i.e. Q- (s) = je stdQ1| (t)
Symbol for ordinary convolution i.e.
t
A (t) ©B (t ) = J A (u )B (t - u ) du
+The limits of integration are 0 to ro whenever they are not mentioned.
TRANSITION DIAGRAM
X,
S«
o
Up State
X
g2(.)
X
No1 ' N oO
X
MO
I I : Failed State
X,
G.W
G.(-)
# : Regenerative Point Figure. 1
S,
X2Xi
^^ : Non-Regenerative point
IV. Transition Probabilities and Sojourn Times
Let X (t) be the state of the system at epoch t, then |X (t) ;t > 0} constitutes a continuous parametric Markov-Chain with state space E = {S0 to S5} .The various measures of system effectiveness are obtained in terms of steady-state transition probabilities and mean sojourn times in various states. First we obtain the direct conditional and unconditional transition probabilities in terms of
a1 , a 2
aj =-1-, a2 =-2—
a1 + ^ + p a2 +01
0
S
S
2
0
ß
S
S
3
4
as follows-
P01 = Ja (1 - r ) e-!^ <1-r >- <1-r«'dt = °1('u r) ■
j x+a(1- r )+a2(1 - r )
Similarly,
a 2 (1 - r )
P02 =■
X + a1 (1 - r ) + a2 (1 - r )
P20 =G2[a2(l-r)],
P43 =J dG1 ( t ) = 1, P10IX =JdGi (u)K (u|x)
P03 =-
x+a (1 - r )+a2 (1 - r )
P23 = 1 ~ G2 [a2 (l ~ r)]
P53 =J dG2 (t) = 1
Similarly,
P(4)x =J G ( u ) dK ( u|x )
P30IX = JPe-(X+p)u
(a1a2rxy)J j (J!)2
f -(a1y+a,rxv--1--2 — -J j j
Ja1e(1y 2 ) j—- ~ dy
du = ■
x+p
1 -aie-a2rx(1-a1)
r -arx(1-al)
P34X =aie ,
P35|X
X
x+p
1 -ale-a2rx(1-a1
The unconditional transition probabilities with correlation coefficient from some of the above conditional transition probabilities can be obtained as follows:
P10 =J P10XS1 (x ) dx =J P10x{a1(1 - r)e-a1(1-r)x}dx
Similarly,
P^lP^a-^«^, P30
al (1 - r)
al (1 - r)
P34 = n , P35 =■
C - rai)
It can be easily verified that,
P01 + P02 + P03 =1, P10 + P(3J=1,
P30 + P34 + P35 =1, P43 = P53 =1
1 -
X + p[ (1 - ral)
( 5 ) ,
P20+p2S =1
(1-5)
V. Mean Sojourn Times
The mean sojourn time in state Si is defined as the expected time taken by the system in state Si before transiting into any other state. If random variable Ui denotes the sojourn time in state Si then, =J p[U > t]dt
Therefore, its values for various regenerative states are as follows-
Vo = J e-i^+a1(1-r )+a2 (1-r)it dt =-—1-— (6)
J X + a1 (1 - r) + a2 (1 - r)
Vi|x =JGi (t)K (t|x)dt =JG! (t) Ja2e
(a2u+atrx(a1a2rxu)j ^ ^
J=0
2
( J!)
dt
So that,
Vi = J^ixSi (x)dx = JV|xai (1-r)e-ai(1-r)xdx
a2 (!-r )t
V2 =J G2 (t) e—a(1-r )ldt
(7)
(8)
V3X {!-a;e-a2rX(1-a:)}
So that,
V3 =■
1
p+x V4 =J Gi (t) dt V5 =J G2 (t) dt
! aK1 - r)
(i - ra;)_
(9)
(10) (11)
VI. Analysis of Characteristics
a) Reliability and MTSF
Let R; (t) be the probability that the system operates during (0, t) given that at t=0 system starts from S; e E. To obtain it we assume the failed states S4 and S5 as absorbing. By simple probabilistic arguments, the value of R0 (t) in terms of its Laplace Transform (L.T.) is given by
(s) _ Z0 + q0iZ1 + q02Z2 + q03Z3
0\/i 11 11 11
1 — q01q10 — q02 q20 — q03q30
We have omitted the argument's from q*:(s) and Z*(s) for brevity. Z*(s);i = 0,1,2,3 are the L. T. of
(12)
Z0 (t) = Z2 (t )= e
_ —{X+at (1—r )+a2 (1—r )}t
Zi (t) = G (t)J K2 (11 x) g! (x) dx
lG2 (t) ,
Z3 (t) = e—(x+p)t JKi (11 x)g2 (x)d
x) dx
Taking the Inverse Laplace Transform of (12), one can get the reliability of the system when system initially starts from state S0. The MTSF is given by,
V0 + P01V1 + P02V2 + P03V3
E(T0 ) = JRq (t)dt = limR; (s) = V
-p01p10 p02p20 p03p30
(13)
b) Availability Analysis
Let A (t) be the probability that the system is up at epoch t, when initially it starts operation from state S; e E . Using the regenerative point technique and the tools of Laplace transform, one can obtain the value of A0 (t) in terms of its Laplace transforms i.e. A* (s) given as follows-N1 (s)
AO (s) =
Di (s)
(14)
where,
-VT / \I~1 11 1 1 1 i r-r1 1 rr1 1 r-r1 \ 1 (4 )* 1 (5 )* 1 r-r1
Ni (s) = [(—q34q43— q35q53 J( Zo + qoiZi + qo2Z2 )+[qoiqi3 + qo2q2s + qo3 JZ3
and
11 11~l/i 11 1 1 \ 1 I 1 (4 )* 1 (5 )* 1 I /-1 r-\
Di (s) = L(—q34q43— q35q53 JJ(— qoiqio— qo2q2o)—q3o [qoiqi3 + qo2q2s + qo3 J (15) where, Z (t) , i=0,1,2,3 are same as given in section VI(a). The steady-state availability of the system is given by
A0 = lim A0 (t) = limsAQ (s) (16)
We observe that
Di (0) = 0
Therefore, by using L. Hospital's rule the steady state availability is given by
Aq = lim—) = (17)
0 s-m>D;( s) D; ( )
where,
Ni = p30 ( Vo + poi Vi + po2 V2 ) + (! — poipio — p02p20 ) V3
and
D; = p30 L Vo + poi (Vi + pi4 V4 ) + p02 (V2 + p25 V5 )J + (! — po^io — p02p20 ) [V3 + p34 V4 + p35 V5 ] (18) The expected up time of the system in interval (0, t) is given by
Mup (t) = J Ao (u) du
0
So that, M1up (s ) = ^ (19)
c) Busy Period Analysis
Let Bi (t) , B2 (t) and B3 (t) be the respective probabilities that the repairman is busy in the repair of
unit-1 failed due to hardware, unit-1 failed due to human error and unit-2 failed due to normal cause at epoch t, when initially the system starts operation from state S; e E. Using the regenerative point technique and the tools of L. T., one can obtain the values of above three probabilities in terms of their L. T. i.e. Bi1 (s), B21 (s) and B31 (s) as follows-
b;1(s ) =N2M, Bi21( s ) = and Bf1(s ) =—iM (20-22)
' W Di (s) iW Di (s) iW Di (s) ( )
where,
N2(s) = qOi [(—q3iq43— q35q53 J JZ1 + q^)+q3i |_q0iq((+q02q253)*+q>3J Z'
4
1 11 11 _|/r_1 1 r-r1\ 1(1 (4)1 1 (5)* 1 l r-r1
N3(s) = qo2 L1—q34q43— q15q53 J (Z2 + q25Z5)+q35 (qoiqV + qo2q23) + qo3) Z5
1 (4)1 1 (5)1 1
qoiq(( 3 + qo2q23) + qo3
Z1 Z3
N4(s) =
and D( (s) is same as defined by the expression (15) of section VI(b).
Also Z* and Z* are the L. T. of
Z4 (t ) = G (t), z5 (t ) = G2 (t) The steady state results for the above three probabilities are given by-
BO = limsB0 (s)= N2/D;, b2 = N3/d; and B0 = N4/d; (23-25)
N2 = P30P01 (V + P14V4 ) + L (1- P0lPl0 - P02P20 )_|V4 N3 = P30P02 ( Vb + Pb5 V5 ) + LP35 (l - P0lPl0 - P02P20 )J V5 N4 =[1 - P01P10 - P02P20 ]V3 and DJ is same as given in the expression (18) of section VI(b).
The expected busy period in repair of unit-1 failed due to hardware, unit-1 failed due to human error and unit-2 failed due to normal cause during time interval (0, t) are respectively given byt t t Mb (t) = J B0 (u) du, ^ (t) = J B2 (u) du and (t) = J B0 (u) du
00 0
So that,
wb*(s ) = B0* (s )/s Mb*(s ) = B0 •( s )/s and Mb*(s ) = B|*(s )/s (26-28)
d) Profit Function Analysis
The net expected total cost incurred in time interval (0, t) is given by P (t) = Expected total revenue in (0, t) - Expected cost of repair in (0, t)
= KiM-UP (t) - K^b (t) - K^b (t) - K^ (t) (29)
Where, K0 is the revenue per- unit up time by the system during its operation. K1 , K2 and K3 are the amounts paid to the repairman per-unit of time when he is busy in repair of unit-1 failed due to hardware, unit-1 failed due to human error and unit-2 failed due to normal cause respectively.
The expected total profit incurred in unit interval of time is P = K0A0 - K^ - K2Bb - K2b3
VII. Particular Case
When the repair time of unit-1 failed due to hardware and human error also follow exponential with p.d.fs as follows-
gi(t) = e1e-01t, g2(t) = e2e-0bt
The Laplace Transform of above density function are as given below-
g* (s) = Gj (s) = - A-, g*2 (s) = G2 (s) = - 02
S+6] S+62
Here, Gj (s) and G2(s)are the Laplace-Stieltjes Transforms of the c.d.fs G, (t) and G2(t) corresponding to the p.d.fs g1(t) and gb(t).
In view of above, the changed values of transition probabilities and mean sojourn times.
ab(1 -r) (4^ _ a2(1 -r) _ eb
p10 = 1 „ p13 = p20 = "
(1 - rab)' 13 (1 - tab/ 20 ab(1 - r) +eb
(5) ab(1 - r) 1 1
PW =-, V1 =-, V 2 =-
23 a2(1 - r)+e2 a(1 - r)+e a(1 - r)+e2
VIII. Graphical Study of Behaviour and Conclusions
For a more clear view of the behaviour of system characteristics with respect to the various parameters involved, we plot curves for MTSF and profit function in Fig. 2 and Fig. 3 w.r.t. a for three different values of correlation coefficient r =0.25, 0.35 and 0.45 and two different values of repair parameter 8t =0.7 and 0.9 while the other parameters are kept fixed as 1 = 0.09, a2 = 0.045, p = 0.8, 02 = 0.7 .From the curves of Fig. 2, we observe that MTSF increases uniformly as the values of r and 8j increase and it decreases with the increase in a1. Further, to achieve MTSF at least 94 units we conclude from smooth curves that the value of a1 must be less than 0.118, 0.190 and 0.332 respectively for r = 0.25 , 0.35 , 0.45 when 01 = 0.9. Whereas from dotted curves we conclude that the value of a1 must be less than 0.100, 0.171, 0.294 for r = 0.25 , 0.35 and 0.45 when 0 = 0.7.
Similarly, Fig. 3 reveals the variations in profit (P) w.r.t. aj for varying values of r and 0t, when the values of other parameters are kept fixed as 1 = 0.09, a2 = 0.045, p = 0.8, 02=0.7, K0 =160, Kj = 400, K2 = 250 and K3 = 350 . Here also the same trends in respect of a1, r and 01 are observed as in case of MTSF. Moreover, we conclude from the smooth curves that the system is profitable only if a1 is less than 0.581, 0.700 and 0.850 respectively for r = 0.25 , 0.35, 0.45 when 0 = 0.9. From dotted curves, we conclude that the system is profitable only if a1 is less than 0.520, 0.612 and 0.759 respectively for r = 0.25 , 0.35 and 0.45 when 0 = 0.7.
Behaviour of MTSF w.r.t. a for different values of r and 0t
I 80
60 40 20 0
-■ r=0.25,6i=0.7 -■ r=0.35,6i=0.7 -■ r=0.45,6i=0.7
■r=0.25,6i=0.9 ■r=0.35,6i=0.9 ■r=0.45,6i=0.9
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
ai
Figure.2
Behaviour of PROFIT (P) w.r.t. at for different values of r and 0t
IX. Acknowledgment
Authors are highly thankful to Prof. Rakesh Gupta, Department of Statistics, Senior most Professor of CCS University, Meerut for helpful discussion for preparing this paper.
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