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STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY
R. K. Yadav1, N. Nandal2, S.C. Malik*
*12Department of Statistics, M.D. University, Rohtak 'yadav.ramesh546@gmail.com, 2nsinghnandal@gmail.com, *sc malik@rediffmail.com
Abstract
We describe the development of a stochastic model for a computer system with cold standby redundancy, priority and failure of service facility. A computer system (called a single unit) means the simultaneous working of its hardware and software components. The system has one more unit (called computer system) that can be used as and when required at the failure of any of the hardware/software components of the initially operative computer system. A single repair facility is made available to rectify the faults which occur due to the failure of hardware and software components. The failed hardware component undergoes for repair immediately while failed software is up-graded. The service facility is subjected to failure during hardware repair. The provision of perfect treatment has been made for the failed service facility. The components work as new after repair and up-gradation with the same life time distribution. The priority is given to the software up-gradation over the hardware repair. In steady state, the expressions for some important reliability measures have been derived using the well known semi-Markov process and regenerative point technique. The behavior of some useful reliability characteristics has been observed for particular values of the parameters related to failure times, repair and up-gradation times and treatment time which follow negative exponential distribution.
Keywords: Computer System, Unit Wise Redundancy, Priority, Failure of Service Facility and Stochastic Modelling
I. Introduction
Over the years an overwhelming transformation of the modern society into the digitalization World has been observed with the advent of advanced technology and frequent use of computer systems. As a result of which we are now in a position to complete the assigned jobs within time limits and perfectness. In the modern World of today the use of computer systems cannot be ignored completely or partially in order to survive in the competitive markets. On the other hand, the burden for the heavy use of computer systems grabs the attention of reliability engineers and scientists to identify all possible ways and means to improve the reliability and performance of these systems. The researchers in the field of reliability have succeeded somehow
R.K. Yadav, N. Nandal, S.C. Malik
STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM RT&A, No 4 (71)
WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY Volume 17, December 2022 in identifying the reliability improvement techniques. The provisions of standby redundancy in both parallel and cold standby have been frequently being used by the system developers. The other means such as priority in repair disciplines and proper repair facility have also been suggested by the researchers while analyzing profit of repairable and non repairable systems. The reliability can also be improved by giving priority to repair activities. Many researchers including Goel et al. [3], Leung et al. [11] and Malik [13] explained the model with the help of priority concept. Kumar and Saini [6] three models are developed under different priority policies. Kumar and Yadav [5] described a computer system with priority given to software up-gradation over hardware repair. Kumar et al. [7] the reliability of single unit system is calculated subject to arrival time of server. Kumar et al. [8] assumed the single server to handle the repair activities of computer system. Subramanian and Anantharaman [19] described the reliability analysis of a complex redundant system where standby unit is in cold state for a certain amount of time before it is allowed to become warm.
In most of the research work authors have analyzed the system models of repairable systems under a common assumption that the service facility cannot fail while performing jobs. This assumption seems to be unrealistic in case system has some complex failures and the service facility is very careless. In that situation the treatment to the failed service facility may be given in order to resume the jobs with full efficiency and perfectness. Kuo and Ke [9] compared system availability among three configurations with unreliable server and switching failure. Meng et al. [14] described a two unit cold standby system with switch failure and equipment maintenance. Nandal and Malik [15] evaluated reliability of a single unit system subject to arrival time of the server. Singh [18] evaluated the expected profit by taking repair man appearance and disappearance for a two unit cold standby system. Sridharan and Mohanavadivu [17] analyzed the two unit cold standby redundant system, two types of repairmen (regular and expert). It is also proved that component wise redundancy is better than that of unit wise redundancy so far as reliability of the system is concerned. Friedman and Tran [2] used the combined hardware/software systems. Gupta et al. [4] gave an idea of single server to determine the profit of two unit standby system model in which priority unit is in operation and ordinary unit is in cold standby. Lai et al. [10] determined the system availability for distributed hardware/software system. Mahmoud and Moshref [12] had taken the human error failure with hardware failure for cold standby system. Bhardwaj and Singh [1] considered the failure of server in steady state behavior of cold standby system. Poonam and Malik [16] analyzed a stochastic parallel system with the assumption of failure of service facility. Yadav and Malik [20] analyzed the computer system with unit wise cold standby redundancy.
In view of the above facts and observations here we describe the stochastic modeling of a computer system with cold standby redundancy (unit wise), priority in repair discipline and failure of service facility. A computer system (called a single unit) means the simultaneous working of its hardware and software components. The system has one more unit (called computer system) that can be used as and when required at the failure of any of the hardware/software components of the initially operative computer system. A single repair facility is made available to rectify the faults which occur due to the failure of hardware and software components. The failed hardware component undergoes for repair immediately while failed software is up-graded. The service facility is subjected to failure during hardware repair. The provision of perfect treatment has been made for the failed service facility. The components work as new after repair and up-gradation with the same life time distribution. The priority is given to the software up-gradation over the hardware repair. In steady state, the expressions for some important reliability measures including MTCSF, availability and profit function have been
R.K. Yadav, N. Nandal, S.C. Malik
STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM RT&A, No 4 (71)
WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY Volume 17, December 2022 derived using the well known semi-Markov process and regenerative point technique. The behavior of some useful reliability characteristics has been observed for particular values of the parameters related to failure times, repair and up-gradation times and treatment time which follow negative exponential distribution.
In section 2, notations and abbreviations are explained. In section 3, assumptions and state descriptions are described. In section 4, the reliability measures are calculated. Section 5 determines the profit analysis. The particular values are given in section 6. Section 7 describes the graphical behavior of reliability measures. The numerical example is illustrated in section 8. Section 9 comprises of conclusion of the present study. In final, the relevant references are incorporated.
II. Assumptions and State Descriptions
1. There is a computer system comprises hardware & software components which function independently.
2. The hardware and software components fail independently.
3. The system is a cold standby in which one unit (called computer system) is initially operative and the other unit (computer system) is kept as spare.
4. There is a single service facility that repairs the hardware and upgrades the software.
5. The service facility (server) can fail during hardware repair.
6. The h/w repairs, s/w up-gradation and treatments are perfect.
7. The h/w and s/w failures (s/w failure occurs when it fails to furnish the jobs as per the instructions) are assumed to be constant.
8. The distributions for repair, up-gradation and treatment rates are considered as arbitrary.
9. So is an initial state in which one unit (computer system) is in operation and another unit (computer system) is in cold standby.
10. Si is the operative state in which one unit is in operation and second unit's failed h/w component is under repair.
11. S2 is the failed state in which one unit's h/w component is continued under repair from state S1 while second unit's h/w component is waiting for repair.
12. S3 is the operative state in which one unit is in operation and second unit's failed s/w component is under up-gradation.
13. S4 is the operative state in which the failed server is under treatment, one unit is in operation and second unit's h/w component is waiting for repair.
14. S5 is the failed state in which the failed server is under treatment, one unit's h/w component is continued waiting for repair from state S2 while second unit's h/w component is waiting for repair.
15. S6 is the failed state in which the failed server is under continued treatment from state S4 while one unit's h/w component is continued waiting for repair from state S4 and second unit's h/w component is waiting for repair.
16. S7 is the failed state in which one unit's h/w component is continued waiting for repair from state S5 and second unit's h/w component is under repair.
17. S8 is the failed state in which the failed server is under continued treatment from state S4 while one unit's h/w component is continued waiting for repair from state S4 and second unit's s/w component is waiting for up-gradation.
18. S9 is the failed state in which one unit's h/w component is waiting for repair from while second unit's s/w component is under up-gradation.
19. S10 is the failed state in which one unit's s/w component is under continued up-gradation from state S3 while second unit's s/w component is waiting for up-gradation.
20. S11 is the failed state in which one unit's s/w component is under up-gradation while h/w
RT&A, No 4 (71) Volume 17, December 2022
component of second unit is continued waiting for repair from state S8. 21. S12 is the failed state in which one unit's h/w component is waiting for repair while second unit's s/w component is under continued up-gradation from state S3. The state transition diagram shown in the figure 1 as:
: Point O : Operative State □ : Failed i
Figure 1: State Transition Diagram
a) Notations and Abbreviations
MTCSF Mean Time to Computer System Failure
SMP Semi-Markov Process
RPT Regenerative Point Technique
MST Mean Sojourn Time
O/Cs The unit is operative/ in cold standby
a/b Probability of hardware/software failure
x1/x2/ |д Hardware/software/ server failure rates
HFUr/HFWr The failed hardware is under/waiting for repair
HFUR/HFWRThe failed hardware is continuously under/waiting for repair from prior state SFUg/SFWg The failed software is under/waiting for up-gradation
SFUG/SFWG The failed software is continuously under/waiting for up-gradation from prior state SUt The failed server (service facility) is under treatment
SUT The failed server (service facility) is continuously under treatment from prior state
h(t)/H(t) pdf/cdf of hardware repair time
u(t)/U(t) pdf/cdf of software repair time
s(t)/S(t) pdf/cdf of server treatment time
m(t)/M(t) pdf/cdf of hardware preventive maintenance time
4ij/Qij pdf/cdf of first passage time
m#$ Contribution to MST (p) in state Si when system transits directly to state Sj
Mi (t) Probability that the system up initially in regenerative state Si is up at time t without
visiting any other regenerative state W% (t) Probability that the server is busy in the state Si due to hardware failure up to time
't' without making any transition to any other regenerative state or returning to the same state via one or more non-regenerative states
WjS (t) Probability that the server is busy in the state Si due to software up-gradation up to
time 't' without making any transition to any other regenerative state or returning to the same state via one or more non-regenerative states
©/© Standard notation for Laplace-Stieltjes convolution/Laplace convolution
*/** Symbol for Laplace Transform (LT)/Laplace Stieltjes Transform (LST)
P Profit function by considering busy period cost of the server per unit time due to
hardware repair/ software up-gradation and treatment cost of the server per unit time
Zi System revenue per unit up-time
Z2/Z3 Busy period cost of the server per unit time due to hardware repair/ software up
gradation
Z4 Treatment cost of the server per unit time
III. Reliability Measures of the System a) Transition Probabilities
The differential transition probabilities for state So are given by
dQo1(t) = axie-(%&i+(&2)tdt, dQm(t) = 6x2e-(%&i+(&2)t¿t Taking LST of above equations and using the following results
Py = lim 0**(s) = 0»(O) = JJ" dQ- (t) = J"" q- (t)dt, we get po1 = raXle-(%&i+(&2)2 dt = p = r¿X2e-(a&1+(&2)t dt =
Foi J0 1 a*! + b*2 ' Fo2 J0 2 a&i+bX"
Similarly, the other transition probabilities for remaining states are given by
Pio = h*(axi + bx+ + u),Pi2 = {1 - h*(axi + bx+ + ¿»l P21 = P71 = h*(u)
p14 =---{1 - h*(ax1 + bx2 + u)},p15 = ——-{1 - h*(ax1 + bx2 + u)},
P21 = P71 = h*(u), P25 = P75 = 1- h*(u),p3o = u*(axi + bx2),p4i = s*(axi + bx^) Ps,io = P33.10 = a&(+(&2 {1 - u*(axi + bx2)},P3,i2 = P31.12 = ^ {1 - u*(axi + bx2)}, P4i = s*(axi + bx2),P46 = a;X2 {1 - s*{axi + bx2)},pS7 = p8,ii = P67 = P48 = P41.8,11 = {1 - s*{axi + bx2)}, p9i = Pio,3 = Pii,i = Pi2,i = u*(0),
P11.2 = Pl2P21, P41.67 = P46P71, Pi1.2(5,7)n = Pl2P25, P41.67(5,7)n = P46P75 From the above transition probabilities, the following relations are obtained as follows: Poi + P03 = Pio + P12 + Pi4 + Pi9 = P21 + P25 = P30 + P3,10 + P3,12 = P41 + P46 + P48 = 1 P71 + P75 P57 = P67 = P8,11 = Pg,1 = Pi0,3 = Pii,1 = Pi2,1 = P30 + P33.10 + P31.12 = 1, Pio + Pi4 + Pii.2 + Pii.2(5,7)n + Pig = P41 + P41.67 + P41.67(5,7)n + P41.8,11 = 1
b) Mean Sojourn Times
The expected time taken by the system in a particular state before transiting to any other state is known as mean sojourn time or mean survival time in the state. If T, be the sojourn time in the state i, then the mean sojourn time in the state i is The MST in state Si are calculated by the following relations
= \-= -Qîj'(0) and ft = Z-rn0- where q;;(s) = J""e"stdÇ0- (t). Thus, we have
Vo = ™01 + ™03, = ™10 + ™12 + ™14 + ™19,^3 = ™30 + ™33.10 + ™31.12, ^3 = ™30 + ™3,10 + ™3,12, ^4 = ™41 + ™46 + ™48, fa = ™91,
ß'i = mw + m112 + m112(57)n + m14 + m19,ß4 = m41 + m4167 + m4167(5 7)n + m41.811
c) Reliability and MTCSF
Let 0, (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 for0, (t):
0i(t) = X- Qi-(t)®0j(t) + Xk Qik(t) (1)
where Sj is an un-failed regenerative state to which the given regenerative state Si can transit and Sk is a failed state to which the state Si can transit directly. Thus, the following equations are obtained by using (1) as:
0o(t) = Qol(t)®0"(t) + QO3(t)®03(t)
0i(t) = Qw(t)®0o(t) + Qi+(t) + Qu(t)®04(t) + Q±g(t)
03(t) = Q3o(t)®0o(t) + Q3.io(t) + Q3.i+(t) 04(t) = Q4i(t)®0"(t) + Q46(t) + Q48(t) Taking LST of above equations, we get
0Ö(s) = Q5l(s)0V(s) + QO3(S)03*(S) 04* (s) = QOO(s)0Ö(s) + QOO (s) + QO 4 (s)04*(s) + QOOO (*) 0*3 * (s) = Q3O(s)0O4 (s) + Q3 4io(s) + Q34i+(s) 044(s) = Q4l(s)0l*(s) + Q£(s) + Q4*8(s) By using Cramer Rule, 0*o*(s) is calculated as
A"
Where
and
A, =
A =
1
-Q10 (s) -Q30(s) 0
0
0V(s)=-a1 -QOKs) -QOKs) 0 1 0 -Q14(s)
0 10 -Q41(s) 0 1
Q1+(s) + Q15(s)
QZio(s) + QZi+(s)
Q4l(s) + Q4;(s)
-Qöi(s) -Q!7(s) 0
1 0 -G14(s)
0
-Q41(s)
Now, we have
R°(s) =
1-0*0* (s)
The reliability of the computer system model can be obtained by
R(t) = L~"[R*(s)]
The MTCSF is given by
MTCSF = limR*(s) = R*(0) = Bp where N± = (1 - p14p41)(p03^3 + p0) + p01(p14^4 + fri) and D±
S—0 L>!
(1 - Pl4P4l)(1 - P03P30) - P0lPl0
d) Steady State Availability
Let A, (t) be the probability that the system is in up-state at epoch 't' given that the computer system entered regenerative state S, at t = 0. The recursive relations for A, (t) are given as
A,(t) = M,(t) + X- qf (t)©A-(t) (2)
where Sj is any successive regenerative state to which the regenerative state Si can transit through n transitions. Thus, the following equations are obtained by using (2) as:
A0(t) = M0(t) + q0i(t)©Ai(t) + q03(t)©A3(t) Ai(t) = Mi(t) + qw(t)©Ao(t) + [qn,2(t) + q^+^n (t)]©M0 + qi4(t)©A4(t)+qi9(t)©A9(t) Mt) = M3(t) + q3o(t)©Ao(t) + q3i.i+(t)©Ai(t) + q33.io(*)©A3(t)
A4(t) = M4(t) + [q4i(t) + q41.67(t) + R41.67(5,7)n(t) + q4i.8,ii(t)]©Ai(t)
A9(t) = q9i(t)©Ai(t)
Where,
R.K. Yadav, N. Nandal, S.C. Malik
STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM RT&A, No 4 (71)
WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY Volume 17, December 2022 Mo(t) = e-(%&i+(&2)£, Ml(t) = e-(%&i+(&2+^)tH(t), Ms(t) = e-(%&i+(&2)tu(t), M4(t) = e-(axi+bx2)tS(t)
Taking LT of above equations and solving for ^!(s), the steady state availability is calculated by
N+
¿0(<x>) = lims^(s) = —
s^o D+
Where N+ = (Pi4^4 + fli){l - p3,io - PosPso) + Pio[Vo(l - Ps,lo) + ^3P03]
D+ = (Pl4^4 +l*'l + Pl9^9)(l - Ps,io - PosPso) + Pio[Vo(l - P3io) + ^3Pos]
and
Hi = M*(0), i = 1,2,3,4
e) Busy Period of the Repairman Due to Repairs
Let BR(t) be the probability that server is busy in repairing the unit at epoch 't' given that the system entered state Si at t = 0. The recursive relations for BR (t) are given as:
bRR (t) = WR (t) + qf (t)©B? (t) (3)
where Sj is any successive regenerative state to which the regenerative state Si can transit through n transitions. Thus, the following equations are obtained by using (3) as:
i)Repair of Hardware
BF(t) = qoi^BF^+qos^BF^) B?(t) = WiH(t) + qw(t)©B» (t) + [qi12(t) + qu,(s,7)n (t)]©B? (t) + qi4(t)©B?(t) + qi9(t)©B9F (t) BF (t) = qso(t)©BH (tnqsi^tyOBH^) + fe.io (0©^)
BH(t) = [R4l(t) + i4l.67(0 + i4l.67(5,7)"(0 + ^4l.8,ll (t)]©BH (t)
BH(t) = q9i(t)©Bf(t)
Where, Wf (t) = [e-(%&i+(&2+^t + (aXle-(ax!+(x2+^)£©^e-^£©s(t)©l) + (ax1e-(ax!+bx2+^)t©l)]H(t) Taking LT of above equations and solving for #H (s), then busy period of server due to h/w repair is given by
BH(«0 = limsBH'(s) = B&, where Ns = (l- p^o - PosPso)^* (0) and D+ = (Pl4^4 + Vl + Pl9^9)(l - Ps,io - PosPso) + Pio[Vo(l - Ps,io) + №Pos]
ii)Software Up-gradation
Bs0 (t) = qoi(t)©BG(t)+qos(t)©Bi(t) BG(t) = qio(t)©BsS(t)+[qii,2(t) + qn.2(5,7)# (t)]©BS(t) + qi4(t)©B^(t) + qi9(t)©B9s(t) BsS(t) = Wss(t) + qso^BG^+qsi.i+^Bitt) + fe.ioW©^)
B^(t) = [q4l(t) + 94l.67(0 + 94l.67(5,7)"(t) + ^4l.8,ll (t)]©Bl(t) Bi(t) = W9s(t) + q9i(t)©Bi(t)
Where, Wss(t) = [e-(a&i+(&2)t + (axle-(ax!+bx2)t©l) + (bx2e-(ax!'bx2)t©l)]U(t) and W9s = U(t) Taking LT of above equations and solving for (5) (same as 4.3), busy period of server due to s/w up-gradation is given by
Boi(™) = limsBiM = tp where N4 = wf(0)posPio + W9S'(0)pi9(l - ps,^ - PosPso) and
s^o C2
D+ = (Pl4^4 +Vl + Pl9^9)(l - Ps,io - PosPso) + Pio[Vo(l - Ps,io) + ^Pos]
f) Expected Number of Server Treatment
Let Tf (t) be the expected number of repairs of the unit by the server in (0, t] such that the system entered regenerative statei at t = 0. The recursive relation forTf (t) are given as:
TlR(t) = Ilj Q$(t)®[8j +T? (t)] (4)
Where j is any regenerative state to which the given regenerative state i transits and 8- = l if j is the regenerative state where the server does job afresh, otherwise, 8- = 0. Thus, the following
R.K. Yadav, N. Nandal, S.C. Malik
STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM RT&A, No 4 (71)
WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY Volume 17, December 2022
equations are obtained by using (4) as:
UO = Q0i(t)©Ti(t) + Q03(t)®T3(t)
Ti(t) _ Qw(t)®T0(t) + [Q ll.+ it) + Qll.2(5,7)n (0]©7\(i:) + Qi4 T3(t) = Q30(t)®T0(t) +Q 3l.l2 (t)®Tl(t) + Q 33.l0 T4(t) _ [Q4l(t) + Q4!.67(t) + Q4l .67(5,7)n (t) + Q4l.8,ll(t)]®[1 + Tl(t)]
T9(t) _ Q9l(t)®Tl(t)
Taking LST of above relation and solving for T0*(s) (same as 4.3). The expected no. of the server treatments is given by
T0(m) _ ^^^ _ ^ where Ns = Wl - P3,w - P03P30)(Pl2P25 + Pl4) and
D+ _ (Pl4fr4 + Pw^i1 - P3,l0 - P03P30) + Pw^oi1 - P3,w) + Po3]
IV. Profit Analysis
The profit function in time 't' of the computer system is given by P (t) = Expected revenue in (0, t] - expected total cost in (0, t]
In steady state, the profit of the computer system model can be obtained by the following formula: P _ ZM«>) - Z+BFW - Z3BG(<x>) - Z4T0QX) (5)
V. Particular Cases
Let us assume h(t) _ ae at, u(t) _ fie ft and s(t) _ ye yt then reliability measures are determined as follows:
_ a _ __k __bx"
Pn _ mrmmr >pi+ _ TZ~TZTT^TZ>'pI4 _ .,,.„>p-5
ax1 + bx2 + K+a ax1+bx2 + K+a 4 ax1 + bx2 + K+a ax1+bx2 + K'a K _ 1 _ bx2 _ y _ l
P+6 n+a' ^30 nY-.+bY^'I' P3,10 nr-A-bY-A-ft' ^4l nY-.+bY^'v' 0
K+a ax1 + bx2+f ^ ax1+bx2+f ax1+bx2+y' 0 ax1+bx2'
_ l _ l _ l , _ ya+axi(n+y)
frl — , — , W4 — , frl — ,
ax1 + bx2 + K+a axi+bx2+f axi+bx2+y ya(ax1+bx2 + K+a)
"-ia^S't^- urS*m\ u?H*m\ - (K+a)(a+y)(a+axi) + Kyaxi
¿3 =i= W (0) = W5 (0)' W? (0) = .
where
f 3 v ' 5 v j? l a(ax1+bx2 + K+a)(n+a)(a+yy
? fia(axl + bx2 + y) + ya + fiaxl(^ + y) fiya(axl + bx2 +y)
MTSF _ ¿0(M) = ^ ' B?(™) _ B*M _ ^' T0(n) _
Oi D2 0 ^ ; D2 0\ j D2 0v J D2
^ _ {(axi + bx2 + K+a)(axi + bx2+y)—yn}{axi+2bx2+f}+axi(axi+bx2 + K+y)iaxi+bx2+f)
l (axi+bx2 + K+a)(axi+bx2+y)(axi+bx2)(axi+bx2+f)
{(axi+bx2 + K+a)(axi + bx2+y)—yK}{(axi+bx2)(axi + bx2+f) — bx2f} —axia(axi+bx2+y)(axi+bx2+f)
Dl _
(axi+bx2 + K+a)(axi+bx2+y)(axi + bx2)(axi+bx2+f) (axi+bx2+K+y)+a(axi+bx2+y)
D2 _
n2 _ ■
(axi + bx2 + K+a)(axi+bx2+y)(axi+bx2) axi(axi + bx2+f)[(axi+bx2+y){afy+faxi(K+y)+afK+aybx2}+ayK+fnaxi(K+y)] +a2y(axi+bx2+y)[f(axi+bx2+f)+bx2(axi+bx2)} afy(axi+bx2 + K+a)(axi+bx2+y)(axi+bx2)(axi+bx2+f) ^ _ axi{(n+a)(y+a)(a+axi)+ynaxi} 3 a(axi+bx2 + K+a)(axi+bx2)(n+a)(y+a) bx2(a+axi)
N4 _
f(axi+bx2 + K+a)(axi+bx2) ^.axl(a + axl +
N6 _-
(ax-L + bx2 + ^ + a)(axl + bx2)(^ + a)
VI. Graphical Presentation
The graphical representation of MTCSF, availability and profit function has been shown in figures 2, 3 and 4 respectively to check their behavior with respect to the values of the parameters associated with failure and repair rates. From Figure 2, it is observed that the MTCSF of the system
decreases when failure rate of hardware and software is increased from 0.01 to 0.1. Also, MTCSF increases with an increase in hardware repair rate, software up-gradation rate and treatment rate
Figure 2: MTCSF Vs Hardware Failure Rate (Xi)
From Figure 3, it is clearly seen that the availability of the system decreases rapidly with increase of failure rate of hardware and software. Also, availability of the system increases with an increase
Figure 3: Availability Vs Hardware Failure Rate (Xi)
From Figure 4, it is observed that the profit decreases when failure rate of the hardware and software increases. Also, the profit of the system is increases with an increase of hardware repair rate, software up-gradation rate and treatment rate of the server.
7000 -i > x2=.005, ц= 001, ci=2, ß = 5.
6995 - Y=10, a=.6, b = .4
6990 - ■ x2=.007
S 6985 - «•^K * M=.004
<«—I О £ 6980 6975 _ ) с et=4
6970 - —t—ß = 8
6965 - ™ —•— Y=15
6960
1 1 1 1 1 1 1 1 1 O.Ol 0.02 0.03 0.04 0 05 0.06 0.07 0.08 0.09 Hardware Faillir eRate (xl) 1 01 —+—a=.4, b=.6
Figure 4: Profit Vs Hardware Failure Rate (Xi)
VII. Conclusion
The present study mainly focuses on MTCSF, availability and profit analysis of a computer system with unit wise redundancy and failure of service facility. The preference is given to the software up-gradation over hardware repair. The graphical behavior of some important measures such as MTCSF, availability and profit has been observed w.r.t. hardware failure rate (xi) and for the fixed values of server failure rate, repair rates of components and server's treatment rate as shown in the respective figures (Fig.2, Fig.3 and Fig.4). From these figures, it is concluded that MTCSF (Fig.2), availability (Fig.3) and profit (Fig.4) decrease with increase in hardware failure rate (xi) & software failure rate (x2) and increase with increase of hardware repair rate (a) and software up-gradation rate ([) and treatment rate (Y) of server. It is also examined that the provision of priority to software up-gradation of one unit over the hardware repair of other unit can only be helpful in increasing the profit of the system model provided the software up-gradation rate is increased.
VIII. Illustration
Suppose the department office has two computers for furnishing day to day assigned jobs. The official starts the jobs initially at a single computer (unit) and the other computer system is kept as spare in order to makes its use as and when required at any type of problems which occur in the initial operative computer system. The computer can have problems in both hardware and software like damage of RAM, defects in CPU and short-circuit in the monitor as the hardware problems while software can fail to follow the instructions due to malware in the system and failure of drivers. In that situation it becomes necessary to take the help of another computer system in order to complete the assigned jobs in time. In order to secure the data from any kind of malware attack the priority to up-grade the software is required instead of repair of any type of hardware faults. On the other hand, it is not necessary that the service facility can be made available immediately to rectify the faults and in that case we can consider the failure of the service facility. On the basis of the experience and practices the present study is illustrated on a computer system by considering the ideas of unit wise redundancy, priority to software up-gradation and failure of the service facility. The reliability characteristics such as MTCSF, availability and profit have been obtained by taking the hypothetical values for the parameters as:
Here, X" = 0.04,x2 = 0.007, p. = 0.00l,a =2, [ =5, Y = l0,a = 0.6 and b = 0.4 , Z" = 7000, Z2 = l000,Zs = 800,Z4 = 500.
We have
MTCSF = 3046.58l, Availability = 0.999848 and Profit = 6986.86
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STOCHASTIC ANALYSIS OF A COLD STANDBY COMPUTER SYSTEM RT&A, No 4 (71)
WITH UP-GRADATION PRIORITY AND FAILURE OF SERVICE FACILITY Volume 17, December 2022
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