Profit analysis of a computer system with preventive maintenance and priority subject to maximum operation and repair times

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1 ORIGINAL ARTICLE Proft analyss of a computer system wth preventve mantenance and prorty subect to maxmum operaton and repar tmes Ashsh Kumar Monka San Receved: 27 November 27 / Accepted: 3 February 28 Sprnger Internatonal Publshng AG part of Sprnger Nature 28 Abstract The am of the present study s to carry out the proft analyss of a computer system by consderng the concept of prorty to preventve mantenance over dfferent hardware and software repar actvtes For ths purpose three stochastc models have been developed under dfferent prorty polces A sngle repar faclty s avalable to perform all repar actvtes The hardware unt undergoes replacement after a pre-specfc maxmum repar tme The reparman performs the preventve mantenance of the whole system after a pre-fxed maxmum operaton tme All tme-dependent falure rates follow exponental dstrbuton whle the repar rates follow arbtrary dstrbuton Swtch devces repars and preventve mantenance are perfect All random varables are statstcally ndependent Sem-Markov process and regeneratve pont technque were used to derve the necessary system effectveness measure to evaluate the proft functon of all the three models To hghlght the mportance of the study graphs for proft dfference between models have been depcted for a partcular set of values of varous parameters wth respect to preventve mantenance Keywords Computer system Proft functon Sem-Markov process Regeneratve pont technque Prorty Introducton The outstandng progress n the feld of computer technology has resulted n the wdespread usage of computer applcatons n almost all academc medcal manufacturng busness and ndustral sectors Technology demands hghperformance hardware and hgh-qualty software for makng mprovements and breakthrough The sze and complexty of computer-ntensve systems has grown dramatcally durng the last few decades The demand for ntegrated hardware and software systems has ncreased rapdly than the ablty to desgn mplement test and mantan them When the requrements for and dependences on computers ncrease the possblty of ther falures also ncreases The mpact of these falures ranges from economc loss to human causaltes Therefore t becomes necessary to operate such systems wth hgh mportance and relablty Thus engneers and scentsts stress on the development of relable computer systems consderng varous operatonal and desgn polces Several B Ashsh Kumar ashshbarak22@gmalcom Department of Mathematcs and Statstcs Manpal Unversty Japur Japur Raasthan 337 Inda researchers have tme to tme tred to desgn the relablty models for computer systems under varous sets of assumptons By consderng the above facts n mnd n the present study an effort has been made to analyze the effect of prorty to preventve mantenance over dfferent hardware and software repars and upgradaton actvtes on the proft functon of computer systems For ths purpose three stochastc models have been developed usng the concepts of maxmum repar tme maxmum operaton tme preventve mantenance and prorty In the frst model no provson s made whle n the second and thrd models prorty to preventve mantenance over software upgradaton and h/w repar actvtes has been gven respectvely A sngle repar faclty s avalable wth a system whch performs all repar actvtes such as software upgradaton preventve mantenance hardware repar and hardware replacement The unt undergoes replacement after maxmum operaton tme and under preventve mantenance after maxmum operaton tme All tme-dependent falure rates follow exponental dstrbuton whle repar rates follow arbtrary dstrbuton Swtch repars and preventve mantenance are perfect All random varables are statstcally ndependent Sem-Markov process and regeneratve pont technque have been used to

2 derve the necessary system effectveness measure to evaluate the proft functon of all the three measures To hghlght the mportance of the study graphs for proft and proft dfference between models have been depcted for a partcular set of values of varous parameters wth respect to preventve mantenance The manuscrpt s organzed nto the followng sectons The present secton s ntroductory n nature Secton 2 s concerned wth the lterature revew Notatons models descrpton and transton probabltes are gven n Sect 3 Development of recurrence relatons and analyss of results are gven n Sect 4 Secton 5 deals wth the proft analyss along wth the graphcal results Fnally the comparatve analyss of results have been appended n Sect 6 2 Lterature revew wth the concept of preventve mantenance But the effect of prorty to mantenance polces over repar actvtes has not been extensvely studed n detal and comparatvely for proft analyss of computer systems So n the present study an effort has been made n ths drecton and for comparatve analyss graphs of proft functon have been depcted The necessary data for proft analyss were collected wth the help of IT personnel of a prvate unversty 3 Notatons model state descrpton and transton probabltes 3 Notatons The followng notatons have been used n the present study: Fredman and Tran [] developed some relablty technques for combned hardware/software systems Welke et al [2] tred to develop relablty models for hardware/software systems La et al [3] establshed a stochastc model for avalablty analyss of dstrbuted software/hardware systems The work carred out by these authors has been lmted to the consderaton of ether hardware or software components Frst of all Malk and Anand [4] studed a computer system consdered as a sngle entty wth ndependent hardware and software falures Koutras and Plats [5] formulated a sem-markov performance model of a redundant system wth partal full and faled reuvenaton Jan et al [6] carred out the avalablty analyss of the software hardware system wth common cause shock falure spare and swtchng falure The concept of maxmum operaton and repar tme n the feld of computer systems relablty was ntroduced for the frst tme by Kumar et al [7] Kumar and Malk [8] developed a stochastc model for a computer system wth prorty to PM over software replacement Kumar and Malk [9] carred out the cost beneft analyss of a computer system wth maxmum operaton and repar tme In the paper prorty to software replacement over hardware repar actvtes has been gven Malk and Munday [] desgned a stochastc model of computer systems wth hardware redundancy Kumar et al [] carred out the performance analyss of a computer system wth mperfect fault detecton of hardware components Jan and Preet [2] evaluated the avalablty of software reuvenaton n an actve/standby cluster system Kumar and San [3] analyzed comparatvely varous relablty measures of a computer system under the concept of prorty for preventve mantenance over hardware repar Kumar et al [4] studed the proft functon of a computng machne wth prorty and s/w reuvenaton Kumar et al [5] probablstcally analyzed varous performance measures of a redundant system usng Webull falure and repar laws along N o /Cs a/b λ /λ 2 / α /β h(t)/g(t)/ m(t)/ f (t) Pm/WPm PM/WPM HFur/HFurp /HFwr HFUR/HFURP/ HFWR SFurp/SFwrp SFURP/ SFWRP /S K K = 7 LT/LST q (t)/q (t) q kr (t)/ Q kr (t) pdf/cdf Denotes operatve/cold standby unts Denotes probablty of hardware/software falure Constant rate of hardware falure/software falure/ maxmum operaton tme/maxmum repar tme Probablty densty functon of upgradaton tme of the software/repar/replacement of hardware/preventve mantenance of system Denotes unt under preventve mantenance/watng for preventve Mantenance Denotes unt contnuously under preventve mantenance/watng for preventve Mantenance from prevous state Denotes faled hardware unt under repar/under replacement/watng for repar Denotes faled hardware unt contnuously under repar/under replacement/watng for repar from precedng state Denotes faled software unt under upgradaton/watng for upgradaton Denotes faled software unt contnuously under upgradaton/watng for upgradaton from prevous state Laplace convoluton/laplace Steltes convoluton Revenue generated by system per unt up tme Expendture per unt tme durng reparman s engaged n dong preventve mantenance hardware repar software upgradaton hardware replacement per unt hardware replacement software upgradaton and vsts by server Laplace transformaton/laplace Steltes transformaton pdf /cdf of passage tme from regeneratve state to a regeneratve state or to a faled state wthout vstng any other regeneratve state n ( t] pdf/cdf of drect transton tme from regeneratve state to a regeneratve state or to a faled state vstng state k r once n ( t] Probablty densty functon/cumulatve densty functon

3 32 Model state descrpton In ths subsecton three stochastc models have been developed for a two-unt cold standby system by consderng the computer system as a sngle unt The computer system comprses hardware and software components whch fal ndependently The unt undergoes preventve mantenance after a maxmum operaton tme and faled hardware s replaced by the new one after a maxmum repar tme Intally one unt s operatve and other s taken as a cold standby The state descrpton of all models s as follows: Model I total states Model II total states Model III total states S =(N o Cs) S =(N o Cs) S =(N o Cs) S =(N o Pm) S =(N o Pm) S =(N o Pm) S 2 =(N o HFur) S 2 =(N o HFur) S 2 =(N o HFur) S 3 =(N o SFurp) S 3 =(N o SFurp) S 3 =(N o SFurp) S 4 =(N o HFurp) S 4 =(N o HFurp) S 4 =(N o HFurp) S 5 = (HFURWpm) S 5 = (HFURWpm) S 5 = (HFURWpm) S 6 = (HFwr PM) S 6 = (HFwr PM) S 6 = (HFwr PM) S 7 = (SFURP S 7 = (SFURP S 7 = (SFURP S 8 = (PM SFwrp) S 8 = (PM SFwrp) S 8 = (PM SFwrp) S 9 = (SFURP WPm) S 9 = (SFwrp Pm) S 9 = (SFURP WPm) S = (SFURP SFwrp) S = (SFURP SFwrp) S = (SFURP SFwrp) S = (HFURSFwrp) S = (HFURSFwrp) S = (HFURSFwrp) S 2 = (HFUR S 2 = (HFUR S 2 = (HFUR S 3 = (WPm PM) S 3 = (WPm PM) S 3 = (WPm PM) S 4 = (SFWRPHFurp) S 4 = (SFWRPHFurp) S 4 = (SFWRPHFurp) S 5 = (HFurp HFWR) S 5 = (HFurp HFWR) S 5 = (HFurp HFWR) S 6 = S 6 = S 6 = (HFWRPPm) (HFurpWPM) (HFurpWPM) S 7 = (HFURP Wpm) S 7 = (HFURP Wpm) S 7 = (HFURP SFwrp) S 8 = (HFURP SFwrp) S 8 = (HFURP SFwrp) S 8 = (HFURP S 9 = (HFURP S 9 = (HFURP Operatve and regeneratve states Operatve and regeneratve states Operatve and regeneratve states S =(N o Cs) S =(N o Cs) S =(N o Cs) S =(N o Pm) S =(N o Pm) S =(N o Pm) S 2 =(N o HFur) S 2 =(N o HFur) S 2 =(N o HFur) S 3 =(N o SFurp) S 3 =(N o SFurp) S 3 =(N o SFurp) S 4 =(N o HFurp) S 4 =(N o HFurp) S 6 = (HFWRPPm) S 9 = (SFwrp Pm) S 4 =(N o HFurp) S 5 = (HFURWpm) 33 Transton probabltes and mean soourn tmes By consderng smple probablstc arguments the below mentoned process gven n Eq () yelds the transton probabltes at all stages of model I ε = = α = α e α t e t e t e st dt e (+ +α +s)t dt α + s () Takng the lmt s neq() we get α p = lm + s α = The transton probabltes and mean soourn tmes for model II and model III are obtaned by applyng the same procedure For Model I α p = p 2 = p 3 = p = f ( ) p 6 = [ f ( )] = p 26 p 8 = [ f ( )] = p 38 p 2 = g ( ) α p 3 = [ f ( )] = p 3 p 3 = h ( ) β p 24 = [ g ( + β +β )] p 4 = m ( ) α p 25 = [ g ( + β +β )] p 5 = g (β ) p 56 = g (β )

4 p 2 = [ g ( + β +β )] p 62 = f () p 72 = h () p 22 = [ g ( + β +β )] p 83 = f () p 9 = h () p 37 = [ h ( )] = p 327 p 3 = h () p 3 = g (β ) α p 39 = [ h ( )] = p 39 p 4 = g (β ) p 22 = g (β ) p 3 = [ h ( )] = p 33 p 25 = g (β ) p 3 = f () α p 47 = [ m ( )] = p 47 p 43 = m () p 52 = m () p 48 = [ m ( )] = p 438 p 6 = m () p 7 = m () p 49 = [ m ( )] = p 429 p 83 = m () p 92 = m () α p 25 = [ g ( + β +β )]g (β ) α p 256 = [ g ( + β +β )][ g (β )] p 23 = [ g ( + β +β )][g (β )] p 234 = [ g ( + β +β )][ g (β )] p 222 = [ g ( + β +β )]g (β ) p 2225 = [ g ( + β +β )][ g (β )] (2) μ = μ = μ 2 = μ 3 = μ 4 = The mean soourn tmes (μ ) s the state S are + α + θ + β + β + γ (3) μ 2 = β + γ(θ + β) ( + + θ + α + β) + ( + + α){ θ 2 (θ + β) 2 + γθ( + + θ + α + β) +β(θ + β)( + + θ + α +β)( + + α + β) β(θ + β)γ θ( + + θ + α + β) +(θ + β)γ ( + + α + 2β)( + + α + β)} (θ+β) 2 ( + +θ+α + β)( + + α + β) 4 Development and analyss of recurrence relatons 4 Avalablty analyss By smple probablstc arguments sem-markov process and regeneratve pont technque the recurrence relaton for systems avalablty are as follows: For Models I II and III: A (t) = M (t) + q (n) (t) A (t); denotes the regeneratve states (4) where A (t) s the probablty that the system s n up-state at nstant t gven that the system entered the regeneratve state S at t = and M (t) s the probablty of remanng n up-state at the regeneratve state wthout vstng any other state For Models I II and III: M (t) = e (+ +α )t M (t) = e (+ +α )t F(t) M 2 (t) = e (+ +α +β )t G(t) M 3 (t) = e (+ +α )t H(t) M 4 (t) = e (+ +α )t M(t) (5)

5 Usng Laplace transformaton on Eqs (4) and (5) and solvng for A (s) the steady state avalablty s gven by A ( ) = lm s sa (s) 42 Busy perod analyss By smple probablstc arguments sem-markov process and regeneratve pont technque the recurrence relaton for reparman s busy perod are as follows: For Models I II and III Usng Laplace transformaton on equaton (6) &(7) and solvng for B P (s)b R (s)b S (s)andb HRp (s)thetmefor whch the server s busy due to preventve mantenance h/w repar and h/w and s/w upgradatons respectvely s gven by: B H = lm s sb H (s) = N 3 H B S = lm s sb S (s) = N S 3 B p (t) = W (t) + B R (t) = W (t) + B S (t) = W (t) + B HRp (t) = W (t) + q (n) (t) B p (t) ; q (n) (t) B R (t) ; q (n) (t) B S (t) ; q (n) HRp (t) B (t) B R = lm s sb R (s) = N 3 R and B HRp = lm sb HRp s (s) = N HRp 3 43 Expected number of hardware repars software upgradaton and vsts of reparman denote the regeneratve states (6) where B P (t) B R (t) B S(t)andBHRp (t) are the busy perod probabltes due to preventve mantenance hardware repar software upgradaton and hardware replacement at an nstant t gven that the system entered state S at t = and W (t) s the probablty of remanng busy due to any repar actvty The expresson of W (t) for I & III model was derved usng the same procedure as that used n model II For Model II: W = e (+ +α )t F(t) +(α e (+ +α )t )F(t) +( e (+ +α )t )F(t) +( e (+ +α )t )F(t) W 9 = F(t) W 2 = e (+ +α +β )t G(t) +(α e (+ +α +β )t )G(t) +( e (+ +α +β )t )G(t) +( e (+ +α +β )t )G(t) W 3 = e (+ +α )t H(t) +( e (+ +α )t )H(t) +( e (+ +α )t )H(t) W 4 = e (+ +α )t M(t) +(α e (+ +α )t )M(t) +( e (+ +α )t )M(t) +( e (+ +α )t )M(t) (7) By smple probablstc arguments sem-markov process and regeneratve pont technque the recurrence relaton for the expected number of hardware repars software upgradaton and vsts of reparman are as follows: For Models I II and III: R H (t) = ] Q (n) [δ (t) + R H (t) ; R S (t) = [ Q (n) (t) δ + R S ]; (t) N (t) = Q (n) (t) [ δ + N (t) ] ; denotes the regeneratve states (8) where R H (t) R S (t)and N (t) s the expected number of hardware repar software upgradaton and vsts by reparman at an nstant { t gven that the system entered state S at ; f s regeneratve state t = and δ = ; otherwse Usng LST n Eq () and solvng for R H (s) R S (s) and Ñ (s) We obtan the expected number of hardware repars software upgradatons and vsts by the server usng the followng expressons: ( ) = lm s R H (s); s R S ( ) = lm s R S (s); s N ( ) = lm s Ñ (s) s R H

6 Fg Proft dfference of model I and model II vs preventve mantenance rate α -2 Iran Journal of Computer Scence Proft Dfference 2 3 Vs Preven ve 4 Mantenance 5 6 Rate 7(α) 8 9 α=5β=β=5γ=8λ=7λ2=5θ=2a=3b=7 Proft Dfference α=β=β=5γ=8λ=7λ2=5θ=2a=7b=3 α=5β=β=5γ=8λ=7λ2=5θ=3a=7b= α=5β=β=5γ=8λ=7λ2=5θ=2a=7b=3 Preven ve Mantenance Rate(α) α=5β=β=γ=8λ=7λ2=5θ=2a=7b=3 Fg 2 Proft dfference of model I and model III vs preventve mantenance rate α Proft Dfference Proft Dfferences Vs Preven ve Mantenance Rate (α) α=β=β=5γ=8λ=7λ2=5θ=2a=7b=3 α=5β=β=5γ=8λ=7λ2=5θ=3a=7b=3 α=5β=β=5γ=8λ=7λ2=5θ=2a=3b= α=5β=β=5γ=8λ=7λ2=5θ=2a=7b=3 α=5β=β=γ=8λ=7λ2=5θ=2a=7b=3 Preven ve Mantenance Rate (α) 5 Proft functons The proft ganed by the system models has been obtaned by usng the formula: upgradaton and hardware repar actvtes s more economcal to use as compared to the system where no provson of prorty s made P = K A K B P K 2B R K 3 B S K 4B HRp K 5R H K 6R S K 7N (9) 6 Concluson The graphs of proft dfference of all the models wth respect to preventve mantenance rate (α) for fxed values of other parameters are drawn for a partcular case g(t) = θe θt h(t) = βe βt f (t) = αe αt andm(t) = γ e γ t as shown n Fgs and 2 The fgures reveal that the proft ncreases wth the ncrease of PM rate (α) and hardware repar rate (θ) However the value of these measures ncreases wth the ncrease of maxmum operaton tme (α ) Agan f we ncrease the value of the maxmum constant rate of repar tme (β ) the value of MTSF avalablty and proft also ncrease From Fgs and 2 t s concluded that model II and model III are more proftable than model I Hence the concept of prorty to preventve mantenance over s/w References Fredman MA Tran P: Relablty technques for combned hardware/software systems In: Relablty and Mantanablty Symposum 992 Proceedngs Annual (pp ) IEEE (992) 2 Welke SR Johnson BW Aylor JH: Relablty modelng of hardware/software systems IEEE Trans Relab 44(3) (995) 3 La CD Xe M Poh KL Da YS Yang P: A model for avalablty analyss of dstrbuted software/hardware systems Inf Softw Technol (22) 4 Malk SC Anand J: Relablty and economc analyss of a computer system wth ndependent hardware and software falures Bull Pure Appl Sc 4 53 (2) Vol 29 E (Math & Stat) 5 Koutras VP Plats AN: Sem-Markov performance modellng of a redundant system wth partal full and faled reuvenaton Int J Crt Comput Based Syst ( 3) (2) 6 Jan M Agrawal SC Preet C: Avalablty analyss of software hardware system wth common cause shock falure spare and swtchng falure Int J Int Acad Phys Sc 4() 3 (2) 7 Kumar A Malk SC Barak MS: Relablty modelng of a computer system wth ndependent H/W and S/W falures subect

7 to maxmum operaton and repar tmes Int J Math Arch 3(7) (22) 8 Kumar A Malk SC: Stochastc modelng of a computer system wth prorty to PM over S/W replacement subect to maxmum operaton and repar tmes Int J Comput Appl 43(3) (22) 9 Kumar A Malk SC: Cost-beneft analyss of a computer system wth prorty to S/W replacement over H/W repar actvtes subect to maxmum operaton and repar tmes J Sc Ind Res 73() (24) Malk SC Munday VJ: Stochastc modellng of a computer system wth hardware redundancy Int J Comput Appl 89(7) 26 3 (24) Kumar A San M Malk SC: Performance analyss of a computer system wth mperfect fault detecton of hardware Proc Comput Sc (25) 2 Jan M Preet C: Avalablty analyss of software reuvenaton n actve/ standby cluster system Int J Ind Syst Eng 9() (25) 3 Ashsh K Monka S: Comparson of varous relablty measures of a computer system wth the provson of prorty In: Afzalpulkar N Srvastava V Sngh G Bhatnagar D (eds) Proceedngs of the Internatonal Conference on Recent Cognzance n Wreless Communcaton and Image Processng Sprnger New Delh (26) 4 Kumar A San M Srvastava DK: Proft analyss of a computng machne wth prorty and s/w reuvenaton In: Mshra D Nayak M Josh A (eds) Informaton and Communcaton Technology for Sustanable Development Lecture Notes n Networks and Systems vol 9 Sprnger Sngapore (27) 5 Kumar I Kumar A San M Dev K: Probablstc analyss of performance measures of redundant systems under webull falure and repar laws In: Vshwakarma H Akashe S (eds) Computng and Network Sustanablty Lecture Notes n Networks and Systems vol 2 Sprnger Sngapore (27)