Cost-Benefit Analysis of a Computer System with Priority to S/W... over H/W Repair Activities Subject to Maximum Operation and Repair...

Cost-Benefit Analysis of a Computer System with Priority to S/W... over H/W Repair Activities Subject to Maximum Operation and Repair...

Journal of Scientific & Industrial Research
Vol. 73, October 2014, pp. 653-655
Cost-Benefit Analysis of a Computer System with Priority to S/W Replacement
over H/W Repair Activities Subject to Maximum Operation and Repair Times
A Kumar* and S C Malik
Department of Statistics, M.D. University, Rohtak-124001, Haryana (India)
Received 24 August 2012; revised 28 December 2013; accepted 12 June 2014
The emphasis of the present study is on the evaluation of reliability measures of a computer system with independent
hardware (h/w) and software (s/w) failures. A single repair facility is provided immediately for conducting repair activities
of h/w and s/w. Preventive maintenance of the system is done after a maximum operation time. Replacement of the h/w by
new one is made in case server fails to complete its repair in a given maximum time. However, only replacement of the s/w
is done by new one giving some replacement time. Priority is given to s/w replacement over h/w repair activities. The failure
time distribution of the h/w and s/w follows negative exponential while the distributions of preventive maintenance, repair
and replacement time are taken as arbitrary with different probability density functions. Illustration for a particular case is
given to show the graphical behaviour of some important reliability measures.
Keywords: Computer System, Maximum Operation and Repair Times, Preventive Maintenance and Cost- Benefit Analysis.
Introduction
The use of computer systems has been increasing
day by day in most of the critical and sensitive
areas like aircraft, automobiles and home appliances.
And, impact of their failures may be costly and
dangerous to the society. It is, therefore, of great
importance to handle such systems with very care and
high reliability. The reliability of these systems can
be improved by adopting some suitable techniques
including redundancy and other cost-effective repair
policies. The technique of redundancy has not been
used much more in case of computer systems.
However, Friedman and Tran (1992) tried to develop
a combined reliability model for the whole system
in which hardware and software components
work together without considering the concept of
redundancy. Recently, Malik and Kumar [2011, 2012]
investigated reliability models for a computer system
with preventive maintenance and repair subject
to maximum operation and repair times. In view
of the above observations and facts in mind, the
concentration of the present paper is on the evaluation
of some important reliability measures of a computer
system with cold standby redundancy. A reliability
model is developed considering independent h/w and
s/w failures. A single repair facility is provided who
conducts preventive maintenance after maximum
——————
*Author for correspondence
Email: sc_malik@rediffmail.com
operation time and also makes replacement of the
h/w by new one in case its repair is not possible
by him in a given maximum time. Priority is given
to s/w replacement over h/w repair activities.
All random variables are statistically independent.
The distributions of failure time and rate by which
system undergoes for preventive maintenance are
taken as negatively exponential. The distributions
of h/w repair, s/w replacement and h/w replacement
are assumed as arbitrary with different probability
density functions. The system passes through the
following transition states: S0(N0, Cs), S1(N0, Pm),
S2(N0,HFur),S3(N0,SFurp),S4(N0,HFurp),S5(HFUR,W
Pm),S6(HFwr,PM),S5(SFURP,HFwr),S8(PM,SFwrp),
S9(SFURP,WPm),S10(SFURP,SFwrp),S11(HFwr,SFur
p),S12(HFUR,HFwr),S13(PM,Wpm),S14(HFurp,HFWR
),S15(HFurp,Wpm),S16(HFURP,Wpm),S17(HFwrp,SFu
rp) and S18(HFURP,HFwr). On the basis of these
transition states following reliability measures are
obtained using semi-Markov process and regenerative
point technique:
Mean Time to System Failure (MTSF)
Let i(t) be the cdf of first passage time from the
regenerative state i to a failed state. Regarding the
failed state as absorbing state, we have the following
recursive relations for
i (t) as i t    Qi, j t   j t    Qi,k t 
j
k
...(1)
654
J SCI IND RES VOL 73 OCTOBER 2014
Where j is an un-failed regenerative state to which
the given regenerative state i can transit and k
is a failed state to which the state i can transit
directly. The MTSF can be obtained by taking
LST of above relation (1) which is given by
~
1  0 ( s)
s o
s
MTSF = lim
Steady State Availability
Let Ai(t) be the probability that the system is in upstate at instant 't' given that the system entered
regenerative state i at t = 0. The recursive relations for
Ai (t) are given as
Ai  t   M i  t    qi(,nj)  t  A j  t 
... (2)
j
Where j is any successive regenerative state to which
the regenerative state i can transit through n
transitions. Mi(t) is the probability that the system is
up initially in state Si  E is up at time t without
visiting to any other regenerative state, we have
by taking LT of above relations (2) and solving
for A0* ( s) , the steady state availability is given by
A0 ()  lim sA0* ( s).
s 0
Busy Period Analysis for Server
Let
BiP (t ) BiR (t ) BiS (t ) and BiHRp (t ) be the
probabilities that the server is busy in Preventive
maintenance of the system, repairing the unit due
to hardware failure, replacement of the software
and hardware components at an instant ‘t’ given that
the system entered state i at t = 0. The recursive
relations for BiP (t ) BiR (t ) BiS (t ) and BiHRp (t ) are
formulated in same manner as in steady state
availability. By using Laplace transformation
on
recurrence
relations
and
solving
for Bi*P ( s ) Bi*R ( s ) Bi*S ( s) and Bi*HRp ( s) , the
time for which server is busy due to repair and
replacements respectively is given by
B0H  lim sB0*H ( s) =
s 0
N 3H
, B0S  lim sB0*S ( s)
s 0
D2
N 3S
NR
, B0R  lim sB0*R ( s)  3
=
D2
D2
s 0
N3HRp
HRp
*
HRp
And B
 lim sB0
(s) 
0
D2
s 0
Expected Number of Replacements and Visits by
the Server
Let RiH (t ) and RiS (t ) be the expected number
of replacements of the hardware and software
components by the server. Let Ni (t ) be the expected
number of visits by the server in (0, t] given that
the system entered the regenerative state i at t = 0.
The recursive relations for RiH (t ) RiS (t ) and Ni (t )
,
are formulated in same manner as in steady
state availability. By using Laplace transformation
on
recurrence
relations
and
solving for
Fig.1—Profit Vs. Preventive Maintenance Rate (α)
KUMAR & MALIK : COST-BENEFIT ANALYSIS OF A COMPUTER SYSTEM
RiH (t ) , RiS (t ) and Ni (t ) the time for which server is
busy due to repair and replacements respectively is
given by
R0H ()  lim sR0H ( s) =
s 0
R0S ()  lim sR0S ( s) =
s 0
N 4H
and
D2
N 4S
D2
Cost- Benefit Analysis
The profit incurred to the system model in steady
state can be obtained as
Conclusion
Considering exponential distribution for all
random variables, the study reveals that a computer
system in which chances of h/w failure are high can
be made more profitable by using cold standby
redundancy, by conducting preventive maintenance
and by giving priority to s/w replacement over h/w
repair activities.
References
1
2
P  K0 A0  K B  K 2 B  K3 B
P
1 0
HRp
4 0
K B
R
0
S
0
 K5 R0H  K6 R0S  K7 N0 … (3)
K0 = Revenue per unit up-time of the system and
Ki = Cost per unit time for which server is busy due
to various repair activities.
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3
Friedman M A & Tran P, Reliability Techniques for
Combined Hardware/Software Systems, Proc Annual
Reliability and Maintability Symp(1992) pp.290-293.
Malik S C & Kumar A, Profit Analysis of a Computer
System with Priority to Software Replacement over
Hardware Repair Subject to Maximum Operation and Repair
Times, Int J Engine Sci Technol, 3, No. 10(2011) pp.
7452- 7468.
Malik S C & Kumar A, Stochastic Modeling of a Computer
System with Priority to PM over S/W Replacement Subject
to Maximum Operation and Repair Times, Int J Comput
Applicat, 43 (3)(2012), pp. 27-34.