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A TWO NON-IDENTICAL UNIT PARALLEL SYSTEM WITH PRIORITY IN REPAIR AND CORRELATED LIFE
TIMES
Pradeep Chaudhary, Anika Sharma
Department of Statistics Ch. Charan Singh University, Meerut - 250004 (India) E-mail: [email protected]; [email protected]
Abstract
The paper analyses a two non-identical unit parallel system in respect of various measures of system effectiveness by using regenerative point techniques. It has been considered that the life times of both the units are correlated random variables and a single repairman is always available with the system to repair a failed unit.
Keywords: Transition probabilities, mean sojourn time, bi-variate exponential distribution, reliability, MTSF, availability, expected busy period of repairman, net expected profit.
I. Introduction
Various authors including Sridharan & Kalyani (2002), Mokaddis & Sherbeny (2008) in the field of reliability theory have been analyzed two unit parallel system models under different sets of assumptions using regenerative point technique. Some of the authors using the concept of giving the priority to one of the unit in repair and compare to other, Malik et al. (2010), Kumar et al. (2018, 2021) developed a reliability model for a system of non-identical units parallel system with priority to repair . In all these systems models it is assumed that the lifetimes are uncorrelated random variables, but in practical situations this seems to be unrealistic because in many cases there may be some sort of correlation between the lifetimes of operating units. Singh & Poonia (2019) introduced the concept of correlation between failure and their times in the analysis of a single server two unit cold standby system. Later various papers including those by Gupta et al. (2010) have been analyzed the correlated failure and repair time distribution of a unit.
Gupta and co-workers [2008,2018] analyzed 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] analyzed 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.
In the present paper we analyze a two non-identical unit parallel system model with priority in repair and correlated life times of the units working in parallel having their joint distribution as
bivariate exponential distribution with different parameters as the form of the joint p.d.f. given below.
f(xj,x2) =axa2(1 -r)e-a'Xl10(2^ja1a2rx1x2); xj,x2,ax,a2 >0; 0<r <1
Where, I0 = Y (z^2)2k 0 ¿0 (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. Transition probabilities and mean sojourn times in various states.
ii. Reliability and mean time to system failure (MTSF).
iii. Point-wise and steady-state availabilities of the system as well as expected up time of the system during time interval (0, t).
iv. Expected busy period of repairman in the repair of unit-1 and unit-2 during time interval (0, t).
v. Net expected profit earned by the system in time interval (0, t).
II. System Description and Assumptions
1. The system consists of two non-identical units (unit-1 and unit-2). Initially, both the units work in parallel configuration.
2. Each unit of the system has two possible modes-Normal (N) and total failure (F).
3. The first unit gets priority in repair.
4. System failure occurs when both the units stop functioning.
5. A single repairman is always available with the system to repair a totally failed unit and repair discipline is first come, first served (FCFS).
6. If during the repair of a failed unit the other unit also fails, then the later failed unit waits for repair until the repair of the earlier failed unit is completed.
7. The repair times of both the units are uncorrelated random variables, each having a general distribution with different parameters.
8. Each repaired unit works as good as new.
9. The joint distribution of lifetimes (failure times) of both the units is taken to be bivariate exponential having a joint density function of the form ,
f(xj,x2) =axa2(1 -r)e-a'Xl-aA10(2^a1a2rx1x2); xj,x2,ax,a2 >0; 0<r <1
Where, I0 = Y (z/2)2k 0 k o (k!)2
10. The arrival time distribution of repairman is general.
III. Notations and States of the System
We define the following symbols for generating the various states of the system-
N1O,N2O : Unit-1and unit-2 is in N-mode and operative in parallel.
F1r ,F2r : Unit-1 and unit-2 is in F-mode and under repair.
F2w : Unit-2 is in F-mode and under waiting for repair.
F\r : First unit is in failure mode and its repair is continued from state Sx.
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 S 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(xi,x2)
Set of regenerative states.
Random variables denoting the failure time of unit-1 N-mode and unit-
2 respectively for (i=1, 2)
Joint probability density function of xi; i = 1,2
= a1a2 (1 - r)e-a'Xl-aA10 (2^a1a2rx1x2) ; x1, x2, a1, a2 > 0; 0 < r < 1
__eu
Where, I0 (^a1a2rx1x2 ) = ^
(a1a2rx1x2)
k=o (k!)2 g (x) : Marginal p.d.f. of X = x
= ai (1 - r)e-ai (1-r)x ; x >0, a > 0 K; (.| X) : Conditional p.d.f. of X I Xj = x; i * j, j = 1,2
k (xj | X2 = x2 ) : Conditional p.d.f. of Xj | X2 = x
-(a.x. +a rx)T I ,
= a,je 1 1 2 10 (2Ja1a2rx1x) k2 (x2 | X1 = x1 ) : Conditional p.d.f. of X2 | Xj = x
= a2e-(a2x2+a1^)i aia2rxjx2)
gi ,Gj (•);i = 1,2 : The repair time probability distribution function and cumulative distribution function of x;
P.d.f. of transition time from state S to S and S to S via S. .
j') (•) Pj(') 'Pf (•)
Steady-state transition probabilities from state S; to Sj and S; to Sj via
Sk.
Pijlx (*), (•) : Steady-state transition probabilities from state S; to Sj and S; to Sj via Sk when it is known that the unit has worked for time x before its failure.
bj : Repair time of failed unit-2
* : Symbol for Laplace Transform i.e. g* (s) = J e-stq1J (t)dt
~ : Symbol for Laplace Stieltjes Transform i.e. Q°(s) = J e-stdQ1J (t)t
© : Symbol for ordinary convolution i.e.
t
A(t)©B(t)=J A(u)B(t-u)du
fThe limits of integration are 0 to oo whenever they are not mentioned.
TRANSITION DIAGRAM
So
S4
: Up State
Failed State
• : Regenerative Point ^^ : Non Regenerative Point Figure. 1
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 = {S0toS4}.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
a,
a, =-
a2 +0!
—-? u
a-i+ßi
as follows-
p01 = fa1(l-r)e~<Xl(1~r)te~°'2(1~r)tdt = ——— J ax+a2
p02 = {a.CL-De-^^V^-^dt =
J ax + a2
p42=JdG1(t) = l
CO CO
p10, x = JdG, (t)K2(t I x) = JdG.wJa.e-^^Io (2,/a^ixy )dy
jdGl(t) J
;-(a2y+aint)£(aia2rXy)]
J=0
(j!)2
du
(aja2rx)J
' : i'a \
è (jo
CO
je^V [G,(y)]dy
Similarly,
P20 | x = 1 -ale
-a2rx(1-a')
= 1 -m|x 1
a 2e
j=0
(aja 2rx)j
(j!)2
P24 | x = a1e
¥
j e-a2V[G!(y)] dy
-a2rx(1-a!)
è 0
The unconditional transition probabilities with correlation coefficient from some of the above conditional transition probabilities can be obtained as follows:
P10 = J Pl0|x gl(x) dx
1 (a1a2rx)j f" - ~ ^
W
Similarly,
,,(1 -r) j ea2e^ £ j e ^V^y) ] dy
j=0
dx
P20 =1 -
ai(i- r) (1 -a1r)'
P24 =
a1(1- r) (1 -a^r)
=a,(l -r)J e-a
-aj (l-r)x
1-
± j1 ÇJ e-MGi(y) ] dy"
It can be easily verified that,
P01 + P02 =1,
(3) 1 P10 + PI2 = 1
j=o (i!)'
P42 = 1 P20 + P24 =1
dx
(1-4)
V. Mean Sojourn Time
The mean sojourn time y in state S; is defined as the expected time taken by the system in state S; before transiting into any other state. If random variable U; denotes the sojourn time in state S; then,
y = j P [ U > t ]dt
Therefore, its values for various regenerative states are as follows-
y0 = f e-ai(1-r)te-a2(1-r)tdt = ---
J ( a1 + a2
) (1 - r)
(a1a2rx)J
I x = j Gl (t)K2 (11 x) dt = a2e-airx £ -
j=o (J!)
So that,
Vi = Jvi|x gi(x) dx
= Jyi|x ai(1 -r)e-ai(1-r)x dx
j e-a2yyj j Gi(t)dt
ö ö dy
0 0
= aia2(1 -r) ^
(aia2r)J
j=o (J!) = J e-a2(1-r)tGi (t)dt
2
A ö
è 0
Je-a2yyJ JGi(t)dt dy (Je-a'ixe-a'(1-r)xxJdx)
/ 0
y 2|~ = ¡51
So that, 1
y 2 =
bi
1 -
1 -aie-a2rx(1-a1)
a1(1 -(1
V4 =
j Gi (t) dt
(5)
(6)
(7)
(8)
VI. Analysis of Characteristics
I. 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 S2 and S4 as absorbing. By simple probabilistic arguments, the value of R0(t) in terms of its Laplace Transform (L.T.) is given by
R0 (s) = Z0 + q0lZ* + q02Z2 (9)
1 - q01q10-q02q20
We have omitted the argument's from q*(s) and Z*(s) for brevity. Z*(s); i = 0, 1, 2 are the L. T. of Z0 (t ) = e-(a+a2)(i-r)t, Z1 (t )= Gi(t), Z2 (t ) = e-blt
Taking the Inverse Laplace Transform of (9), one can get the reliability of the system when system initially starts from state S0 . The MTSF is given by,
E (To ) = JRo (t) = lim R* (s) = y°+ P0l¥l+ P02y2 (10)
J s®0 1 - P01P10 - P02P20
II. 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. A0 (s) given as follows, ^ N1 (s)
A0 (s)=DM (11)
Where,
Ni (s) = Z0 [1- q*24q42 ]+Zi*qoi f1 - q24q42 ]+Z2 and
* * * * / * * \ * / * (3)* * \
Di(s) =1 - q24q42- qioqoi (i- q24q42)- q20 (qoiq12 + qo2) (12) Where, Z; (t), i=0,1,2 are same as given in section VI(I). The steady-state availability of the system is given by
A0 = lim A0 (t) = lim s AO (s) (13)
t®¥ S®0 V '
We observe that Di (0) = 0
Therefore, by using L. Hospital's rule the steady state availability is given by N1 (s) N,
A0 = li^-^f = -?- (14)
-CD, (s) D,
Where,
Ni = y [l- P24P42 ] + ¥i P01 [l- P24P42 ] + ¥2 and
D1 = V0P20 +yiP0lP20 +V2 (1-P10P01 ) + V4P24 (1-PloPoi ) (15)
The expected up time of the system in interval (0, t) is given by
* (3)* *
q>iql2 + qo2
(3)
P01PI2 +P02
t
^up (t ) = { A0 (u ) du
0
So that, ^ (s) = ^ (16)
III. Busy Period Analysis
Let Bj (t) and B2 (t)be the respective probabilities that the repairman is busy in the repair of unit-1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair 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 two probabilities in terms of their L. T. i.e. Bf (s) and Bf (s) as follows-
B*(s) = ^ B2*(s) =
Di(s) D1(s) (17-18)
Where,
(* * * \ * / * * \ * I * (3)* * * * \
Zi + qi3Z3)qoi (1-^42^24)-Z4 (qoiql2 q24 + qo2q24)
and
N3O) = (q0iqi2)* + q02) and Dx (s) is same as defined by the expression (12) of section VI(II). The steady state results for the above two probabilities are given by-
B0 = lim s B0* (s) = N2\D' and B^ = lim s Bf (s) = N3\Dj (19-20)
s®0 s®0
Where,
N2 (0)=(yi + P13V3 ) P01 (l - P42P24 ) -V4 (P0lPl3)P24 + P02P24 ) (21)
N3 (0) = ^2 (P0lPi32)+PG2 ) (22)
and Dx is same as given in the expression (15) of section VI(II).
The expected busy period in repair of unit-1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair during time interval (0, t) are respectively given byt t ^ (t) = { B0 (u) du, vl (t) = { B0 (u) du
0 and 0
So that,
(s ) = ^IM and tf (s ) = (23-24)
IV. 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)
= KoH-Up (t)-KlWb (t) -K2Hb2 (t) (25)
Where, K0 is the revenue per- unit up time by the system during its operation. K and K2 are the amounts paid to the repairman per-unit of time when the system is busy in repair of unit-1 failed due to first repair with priority of unit-1 and unit-2 failed due to second repair respectively.
The expected total profit incurred in unit interval of time is P = K0A0 - K1B^) - K2B(
VII. Particular Case
Let, Gi(y) = e-0iy
In view of above, the changed values of transition probabilities and mean sojourn times.
¥ (a1a2rx)j (¥ - - ^ j=o j
p =a e-aiixY_
Pl0|x ~tJ-2C /L
¥
J e-a2yyj[Gi(y)] dy
-airx(1-a2)
1
Pl0 =
y =
01 + «2(1 - r)'
y 4 =
a2(1 - r) (1 -a2r);
1
p(3) =!
(1 _a2r)
01
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. at for three different values of correlation coefficient a2 =0.1, 0.5, 0.9 and two different values of repair parameter r =0.25, 0.6 while the other parameters are pt =0.085, 0j = 0.6. It is clearly observed from Fig. 2 that MTSF increases uniformly as the value of a2 and r increase and it decrease with the increase in at. Further, to achieve MTSF at least 17 units we conclude for smooth curves that the values of at must be less than 0.13, 0.23 and 0.45 respectively for a2 =0.1, 0.5, 0.9 when r =0.6. Whereas from dotted curves we conclude that the values of at must be less than 0.12, 0.14 and 0.31 for a2 =0.1, 0.5, 0.9 when r =0.25.
Similarly, Fig.3 reveals the variations in profit (P) with respect to at for three different values of a2 = 0.3, 0.6, 0.9 and two different values of r =0.3, 0.6, when the values of other parameters pt =0.95, 0j = 0.09, Ko=200, Ki=95 and K2=175. Here also the same trends in respect of at, a2 and r are observed in case of MTSF. Moreover, we conclude from the smooth curves that the system is profitable only if at is less than 0.22, 0.42 and 0.66 respectively for a2 = 0.3, 0.6, 0.9 when r =0.6. From dotted curves, we conclude that the system is profitable only if at is less than 0.19, 0.3 and 0.5 respectively for a2 = 0.3, 0.6, 0.9 when r =0.3.
Behaviour of MTSF w.r.t. at for different values of a2 and r
Figure.2
Behaviour of PROFIT (P) w.r.t. </., for different values of a2 and r
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