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Navneet Malik , Pardeep Goel , Surender Garg , Nidhi Choudhary , Vijay Goyal : Behavioral Analysis of Single Module under Software and Hardware Failure using Regenerative Point Graphical Technique

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ISSN (Online):2347-1697 INTERNATIONAL JOURNAL OF INFORMATIVE AND FUTURISTIC RESEARCH ( IJIFR) Volume-1, Issue-3, November 2013 Research Area: Mathematics, Page No. : 56-65

Behavioral Analysis of Single Module under Software and Hardware

Failure using Regenerative Point Graphical Technique

Navneet Malik 1 Pardeep Goel

2 Surender Garg

3 Nidhi Choudhary

4 Vijay Goyal

5

1. Research scholar, Mewar University Rajasthan

2. Associate Professor, Department of Mathematics, M.M.(P.G.) College, Fatehabad

3. Asst.Prof. Department of Mathematics, M.M.(P.G.) College, Fatehabad

4. Asst. Prof. Department of Mathematics, AIMT, Karnal

5. Asst.Prof. Department of Mathematics, M.M.(P.G.) College, Fatehabad

Abstract

This paper discusses the Behavioral and Availability Analysis of Single Module System, in

which module can work in full working and reduced working capacity. This is single module

model which can work in reduced state which means the software is having old features and

need perfective maintenance to reach the full capacity working state. So two type of failure

are there: Partially failed and completely failed. This model consists of single module ‘A’ it

can work in reduced capacity. The module ‘A’ can fail partially and hence can be in upstate,

partially failed state (reduced state) or totally failed state which means software is not

working due to module failure .The module failure can be due to S/W or H/W. The software

system is discussed for steady state conditions. Regenerative Point Graphical Technique

(RPGT) is used to measure the performance of the system.

Keywords: Reliability, Availability, Base-State, RPGT, MTSF, Software Failure, Hardware Failure,

Probability, Module.

1 Introduction

Reliability, Availability and Maintenance have become more significant in recent years due to

the large number of competitors in services, growing needs and overall operating costs. Performance

of software and hardware depends on reliability and availability used in processes, operating

environment, maintenance actions, as well as efficiency, and technical expertise of operators. When

reliability issues are low, actions are needed to improve them by reducing the failure rate or increasing

the repair rate for the components or whole sub systems. In the present scenario of competitive market

to cut down the production cost of delivery performance of software by end user require continuous

and long term use of the software to meet the ever increasing demand at low cost. Availability of

application software can be enhanced by maintenance and inspection. Software may not be working

with full functionality at particular instant of time, this state of affairs is dealt with by considering the

fuzziness measure of the state which makes the problem more realistic. It also makes possible to

identify when the software is to be declared in failed state which helps the management to take

decision whether to replace or enhance the existing system. The system is discussed for steady state

conditions. Using the Regenerative Point Graphical Technique (RPGT) the following system

characteristics have been evaluated to study the system performance.

i. Mean Time To Software Failure (MTSF).

ii. Total fraction of time for which the software is available.

iii. The busy period of the repairman doing any given job.

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iv. The number of the repairman’s visits.

Tables are there to represent the behavior of the MTSF and Steady state Availability of the software

for a particular case.

2 Assumptions and Symbolizations

Assumptions

1) The software contains a single module which can work in reduced state after partial failure

but cannot work in completely failed state.

2) The module ‘A’ can work partially and hence can be in upstate and this state is considered

to be perfective maintenance category to make the system perfect according to time

requirement of user. The software can work with reduced capacity in a partially failed

state.

3) Single repair facility is there for completely failed state and reduced capacity state of a

module.

4) Repairs are perfect i.e. the repair facility never does any damage to the software.

5) A repaired module works like a new-one.

6) The software is down if module is in completely failed state.

7) When the software system is in failed state due to H/W failure it can not fail further due to

S/W failure.

8) The software is discussed for steady state conditions.

9) The distribution of failure time and repair time are exponential and general.

10) The failure time and repair time are independent for S/W and H/W failure.

Symbolizations

Pr/pf Probability/transition probability factor Qi,j(t) Probability density function (p.d.f.) of the first passage time from a regenerative

state i to a regenerative state j or to a failed state j without visiting any other

regenerative state in (0,t].

r-th directed simple path from i-state to j-state; r takes positive integral values

for different paths from i-state to j-state

Ri(t) Reliability of the system at time t, given that the system entered the un-failed

regenerative state i at t=0.

Ai(t) Probability that the system is available in up-state at time t, given that the

software system entered regenerative state i at t=0.

Bi(t) :probability that the repairman is busy doing a particular job at time t, given that

the system entered regenerative state i at t=0.

Vi(t) The expected number of repairman visits for a given job in (0,t], given that the

software system entered regenerative state i at t=0.

Wi(t) probability that the repairman is busy doing a particular job at time t without

transiting to any other regenerative state ‘i’ through one or more non-

regenerative states, given that the system entered the regenerative state ‘i’ at t=0.

mean sojourn time spent in state i, before visiting any other states;

.

the total un-conditional time spent before transiting to any other regenerative

states, given that the system entered regenerative state ‘i’ at t=0.

expected waiting time spent while doing a given job, given that the system

entered regenerative state ‘i’ at t=0;

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Fuzziness measure of the j-state and assumed to be 1.

Constant failure rate of the unit ‘a’ to a partially failed state/ from partially failed

state to a totally failed state.

Constant failure rate for hardware failure.

Probability density function/cumulative distribution function of the repair-time of

the module ‘a’ from the partially failed state due to h/w failure.


the module ‘a’ from the completely failed state due to s/w failure.


the hardware failure.

A/ a Software in full capacity working/ partially failed state/completely failed state.

A(h) Software failure due to h/w failure from full capacity working state.

(h) Software failure due to h/w failure from reduced capacity working state.

3 Evaluation of parameters of the system

The key parameters of the software system are calculated by finding the ‘base-state’ and

applying RPGT. The MTSF is determined w.r.t. the initial state ‘0’ and the other parameters are

obtained by using base-state.

3.1 Determination of base-state

From the transition diagram, all the paths (P0) from one regenerative state to the other reachable states

are determined and shown in Table- 2. The Primary, Secondary, Tertiary circuits at all vertices are

shown in Table- 3.

Table 1: Paths from State ‘i’ to the Reachable State ‘j’:P0

i j = 0 j = 1 j = 2 j = 3 j = 4

0 {0,1,0}

{0,2,0}

{0,2,3,0}

{0,1} {0,2} {0,2,3} {0,2,4}

1 {1,0}

{1,0,1} {1,0,2} {1,0,2,3} {1,0,2,4}

2 {2,0}

{2,3,0}

{2,0,1}

{2,3,0,1}

{2,0,2}

{2,3,0,2}{2,4,2}

{2,3} {2,4}

3 {3,0} {3,0,1} {3,0,2} {3,0,2,3} {3,0,2,4}

4 {4,2,0}

{4,2,3,0}

{4,2,0,1}

{4,2,3,0,1}

{4,2} {4,2,3} {4,2,4}

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Table 2: Primary, Secondary, Tertiary circuits at a Vertex

Vertex

I

Primary Circuits Secondary Circuits Tertiary Circuits

0 {0,1,0}

{0,2,0}

{0,2,3,0}

NIL

{2,4,2}

{2,4,2}

{1,4,1}

NIL

NIL

NIL

1 {1,0,1}

{0,2,0}

{0,2,3,0}

{2,4,2}

{2,4,2}

2 {2,0,2}

{2,4,2}

{2,3,0,2}

{0,1,0}

NIL

{1,4,1}

NIL

NIL

NIL 3 {3,0,2,3} {0,1,0}

{0,2,0}

{2,4,2}

NIL

{2,4,2}

NIL

4 {4,2,4} {2,0,2}

{2,3,0,2}

{0,1,0}

NIL

As there are three simple circuits associated each of the vertices 0 & 2.So, any of these can be

the base-state of the system. We choose the vertex ‘0’ as a base-state.

Table 3: Primary, Secondary, Tertiary Circuits w.r.t. the Simple Paths (Base-State ‘0’)

Vertex j : (P0)

(P1) (P2) (P3)

1 :{0,1}

NIL

Nil Nil

2 :{0,2}

{2,4,2}

Nil Nil

3 :{0,2,3}

{2,4,2} Nil

Nil

4 :{0,2,4}

{2,4,2}

Nil

Nil

3.2 Transition probabilities and the mean sojourn times

Transition Probabilities:

qi,j(t) : probability density function (p.d.f.) of the first passage time from a regenerative

state i to a regenerative state j or to a failed state j without visiting any other

regenerative state in (0,t].

pi,j : steady state transition probability from a regenerative state i to a regenerative state

j without visiting any other regenerative state. pi,j = ; where denotes Laplace

transformation.

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Table 4: Transition Probabilities

(0)

=

=

=

=

=

Where

=1

= =1

It can be easily verified that;

=1; =1; =1; =1;

;

Mean Sojourn Times:

Ri(t) : reliability of the software time t, given that the software in regenerative state i.

:mean sojourn time spent in state i, before visiting any other states;

.

Table 5: Mean Sojourn Times

Ri(t) .

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3.3 Evaluation of Parameters

The mean time to software system failure and all the parameters of the system (under steady

state conditions) are evaluated, by applying Regenerative Point Graphical Technique (RPGT) and

using ‘0’ as the base-state of the system as under:

The transition probability factors for all the states reachable from the base state ‘0’ are:

= 1

=

=

=

Where

(a). MTSF( ): From Fig.1, the regenerative un-failed states to which the system can transit(initial

state ‘0’), before entering any failed state are: i = 0,2. For ‘ ’ = ‘0’, MTSF is given by

MTSF=

=[(0,0) +(0,2) ] [1– ] = N D

Where, = (0,2,0) =

N=[(0,0) +(0,2) ]= + = +

D = [1 – ]= 1-

(b). Availability of the software system: From Fig. 1, the regenerative states, at which the software

system is available are: j = 0,2 and the regenerative states are i = 0 to 4. For ‘ ’ = ‘0’, the total span of

time for which the system is available is given by:

=

=

=

=[ + ]

=

Where,

=[ + ]

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=

(c). Busy period of the Repairman: From Fig. 1, the regenerative states where repairman is busy

while doing repairs are: j = 1,3,4; the regenerative states are: i = 0 to 4.For ‘ ’ = ‘0’, the total span of

time for which the repairman is busy is

=

=

=

=[ + ]

=

Where, =[ + ]

=

(d). Expected number of Repairman’s visits: From Fig. 1, the regenerative states where the

repairman visits (afresh)for repairs of the system are: j = 1,3 and the regenerative states are: i = 0 to 4.

For ‘ ’ = ‘0’, the expected number of server’s visits per unit time is given by

=

=

=

=[ ]+

=

=

=

4. Particular Case

Let us take;

= , =

Assuming = =w

= , = , =1, = ,

= - *

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= - * , =1, =1

= , = , = , = , =

By using these results, we get the following:

MTSF( )= + =( + )/ 1-

Availability( )=

Availability=[ + ]

5. Analytical Discussion

The following tables and conclusions are obtained for:

= 0.01; = 0.002;

5.1 MTSF vs. Repair Rate:

The MTSF of the system is calculated for different values of the Failure Rate ( ) by taking

= 0.003, 0.005, 0.007, 0.008, 0.009 and 0.01 and for different values of the Repair Rate w by

taking w = 0.80, 0.85, 0.90, 0.95 and 1.0. The data so obtained are shown in Table 7.

Table 6: The behavior of the MTSF (T0) vs. the Repair Rate (w) of the Software modules

(w=0.80)

(w=0.85)

(w=0.90)

(w=0.95)

(w=1)

0.003 992.61083 993.03944 993.42105 993.76299 994.07114

0.005 987.77506 988.47926 989.10675 989.66942 990.17681

0.007 983.00970 983.98169 984.84848 985.62628 986.32818

0.009 978.31325 979.54545 980.64516 981.63265 982.52427

0.01 975.99039 977.34994 978.56377 979.65412 980.63891

Table 7 shows the behavior of the MTSF(T0) vs. the Repair Rate(w) of the Software modules

of the System for different values of the Failure Rate( ). It is observed that MTSF increases with

increase in the values of the Repair Rate (w). Further it can be observed from the table that values of

MTSF (T0) decreases with the increase in the values of Failure Rate( ).

5.2 Availability ( ) vs. Repair Rate (w)

The Availability of the software system is calculated for different values of the Failure Rate

( ) by taking = 0.005, 0.006, 0.007, 0.008, 0.009 and 0.01 and for different values of the repair

rate (w)by taking w = 0.80, 0.85, 0.90, 0.95 and 1.0. The data so obtained are shown in Table 8.

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Table 8: Shows the behavior of the Availability (A0) vs. the Repair Rate (w) of the Unit of the System

Table 8 shows the behavior of the Availability (A0) vs. the Repair Rate (w) of the Unit of the

System for different values of the Failure Rate ( ). It is observed that Availability of software system

increases with increase in the values of the Repair Rate (w).

Further it can be observed from the table that values of Availability (A0) shows the expected

trend for different values of Failure Rates. Availability decreases with the increase in the values of

Failure Rate ( ).

6 Conclusion

From the Tables, we see that, Availability of the Software System is increasing, as the Repair

Rate (w) increases .The study can be extended for two or more modules software system. In future,

parameters can be evaluated, when Repair rate and Failure rate are variable. Results obtained can be

used for cost and benefit analysis. Results can be applied to find the Waiting Time of software and

Number of time repairman will be required. Any state can be taken as the Base-state to evaluate the

various parameters. Study can also be extended for time dependent cases also.

7 References

[1]. Cao, Jinhva and Wu, Yanhong (1989). Reliability Analysis of A Two-unit Cold Standby System With

Replaceable Repair facility. Microelectronics, Reliab., Vol. 29(02), pp. 145-150

[2]. Chander, S., and Bansal, R.K., Profit analysis of a single-unit Reliability models with repair at different

failure modes; Proc. Of International Conference on Reliability and SafetyEngineering; Dec,2005, pp 577-

588.

[3]. Malik, S.C., Chand, P.,& Singh, J., Reliability And Profit Evaluation of an Operating System with

Different Repair Strategy Subject to Degradation; JMASS, Vol.4, No. 1, June, 2008, 127-140.

[4]. Gupta, V.K., Singh, J., & Vanita; The New Concept of a Base-State in the Reliability Analysis; proce.

Of National Conference;JMASS, Vol. 6, No. 1; ISSN-0975-5454 Dec.,2010.

[5]. Das, P., Effect of Switch-Over Devices on Reliability of a Standby complex system; Nav. Res. Log.

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[6]. Fukuta, J. and Kodama, M., Missin Reliability for a Redundant Repairable system with two dissimilar

units; IEEE Trans. On Reliability, R-23, 1974.

[7]. Osaki, S., A two-unit standby redundant system with imperfect switch-over, IEEE Trans. Rel, R-21, 20-

24, 1972.

(w=0.80)

(w=0.85)

(w=0.90)

(w=0.95)

(w=1)

0.005 0.9987399938 0.998814658 0.9988809697 0.998940256 0.998993577

0.006 0.9987377062 0.9988126288 0.9988791574 0.9989386275 0.9989921059

0.007 0.9987354268 0.9988106088 0.9988773508 0.998937004 0.998990639

0.008 0.9987331561 0.9988085912 0.9988755504 0.9989353857 0.9989891765

0.009 0.9987308939 0.9988065828 0.9988737557 0.9989337723 0.9989877183

0.01 0.9987286395 0.9988045817 0.998871967 0.9989321642 0.9989862645

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[8].Chung, W.K., Reliability Analysis of repairable and non- repairable system with common cause failure;

29, 545-547, 1989

[9]. Gupta, V.K., Kumar, Kuldeep; Singh, J.& Goel, Pardeep;, Profit Analysis of A Single Unit Operating

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[10]. Jindal, G., Goel, P., Gupta, V.K., Singh, J.; Availability and Behavioral Analysis of a Single Unit

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Issues and Challenges in Higher Education; 4-6, Nov, 2011.

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Behavioral Analysis of Single Module under Software and ... 1 a1a.pdf · Module under Software and Hardware Failure using Regenerative Point Graphical Technique ... M.M.(P.G.) College,