Probabilistic Analysis of a Computer System with Inspection and Priority for Repair Activities of H/W over Replacement of S/W

Transcription

1 International Journal of Comuter Alications ( ) Volume 44 o. Aril Probabilistic Analysis of a Comuter ystem with Insection and Priority for Reair Activities of /W over Relacement of /W Jyoti Anand eartment of tatistics M..UniversityRohtak-4 aryana (India).C.Malik eartment of tatistics M..UniversityRohtak-4 aryana (India) ABTRACT The main aim of this aer is to carry out the robabilistic analysis of a comuter system of two identical units in which one is oerative and the other is in cold standby. In each unit h/w and s/w comonents fail indeendently and work together. A server visits the system immediately to insect the h/w comonents at their failure to see the feasibility of reair. If reair of the h/w is not feasible it is relaced by new one in the unit. owever only relacement of the s/w comonents is made by new one at their failure. Priority to the relacement and reair of the h/w comonents is given over the relacement of the s/w comonents. All the failure time distributions are assumed to be negative exonential while that of insection reair and relacement times are taken as arbitrary. ome reliability and economic measures of system effectiveness are evaluated using semi-markov rocess and regenerative oint technique. The grahs are drawn for a articular case to show the behavior of MTF availability and rofit of the system models. General Terms Reliability and Economic Measures Keywords Comuter ystem ardware and oftware Failures Feasibility of Reair Priority for Relacement Reair and Insection Probabilistic Analysis.. ITRUCTI In site of increasing develoment and availability of new comuter technologies a little work has been dedicated to the robabilistic analysis of a comuter system with indeendent failure of h/w and s/w comonents. And most of the research work in the subject of h/w and s/w reliability has been limited to consideration of either h/w subsystem alone or s/w subsystem alone. Friedman and Tran [] and Wilke et al. [] tried to establish a combined reliability model for the whole system introducing both h/w and s/w under the assumtion that h/w and s/w subsystems are indeendent to each other. Recently Malik and Anand et al.[34] have suggested some reliability models of a comuter system with indeendent h/w and s/w failures. In these models relacement of the comonents by new one is made in negligible time if insection reveals that reair of h/w comonents is not feasible. owever in aer [4] riority for the relacement at s/w comonent is also made by new one over reair and relacement activities of h/w failures. But the concet of riority to reair activities of the h/w over relacement of the s/w has not been studied so far by any researcher in the subject of reliability. In view of above the resent aer deals with the robabilistic analysis of a comuter system considering the concets of riority for the relacement and reair of the h/w comonents subject to insection over relacement of the s/w. For this a robabilistic model is develoed by taking two identical units of a comuter system. Initially one unit is oerative and other is ket as cold standby. Each unit has direct indeendent comlete failure from the normal mode. There is a single server who visits the system immediately to do insection. If reair of the defective h/w comonents is not feasible it is relaced by new one. owever only relacement of the s/w comonents is made by new one whenever they fail. The riority is given to relacement and reair of the h/w comonents subject to insection over relacement of the s/w comonents at their failure. The failure reair and insection time are taken as indeendent and uncorrelated random variables. The failure time of the unit follow negative exonential distributions while that of reair insection and relacement s/w and h/w are taken as arbitrary. To analyze the system robabilistically in detail exression for some reliability characteristics such as mean sojourn times mean time to system failure (MTF) availability busy eriod of the server due to h/w failure or due to s/w failure exected no. of relacement due to h/w failure or due to s/w failures & exected no. of visits by the server are derived by making use of semi-markov rocess and regenerative oint technique. The grahs are drawn for a articular case to show the behavior of MTF availability and rofit of the system models.. TATI E : The set of regenerative states cs : The unit is oerative and in normal mode : The unit is cold standby a/b : Probability that the system has hardware / / /q software failure : Constant hardware / software failure rate : Probability that reair of the unit due to hardware failure is not feasible / feasible FUr/FUR : The unit is failed due to hardware and is under reair /under reair continuously from revious state 3

2 International Journal of Comuter Alications ( ) Volume 44 o. Aril FUi/FUI : The unit is failed due to hardware and is under insection/ under insection continuously from revious state FWi/FWI : The unit is failed due to hardware and is waiting for insection/ waiting for insection continuously from revious state FUR/FURP : The unit is failed due to the software and is under relacement/under relacement continuously from revious state FWR/FWRP : The unit is failed due to the hardware and is waiting for relacement/waiting for relacement continuously from revious state h(t) / (t) : df / cdf of insection time of unit due to hardware failure f(t) / F(t) : df / df of relacement time of the / : ymbol for Lalace-tieltjes convolution/lalace convolution ~ / * : ymbol for Lalace teiltjes Transform ' (desh) (LT) / Lalace Transform (LT) : Used to reresent alternative result The following are the ossible transition states of the system: The following are the ossible transition states of the system: = ( cs) = ( FUi) = ( FUR) 3 = ( FUr) 4 = (FURP FWR) 5 = (FUI FWi) 6 = (FUR FWi) 7 = (FUr FWi) 8 = (FUI FWR) 9 = (FUr FWRP) = (FUR FWR) The state 3 are regenerative states while the states 4 are non-regenerative as shown in figure 3. RELIABILITY IICE 3. Transition Probabilities and Mean ojourn Times g(t) / G(t) q ij / Q ij (t) software : df / cdf of reair time of the unit due to hardware failure : df / cdf of assage time from regenerative state i to a regenerative state j or to a failed state j without imle robabilistic considerations yield the following exressions for the non-zero elements Q ( ) q ( t) dt ij ij ij by taking all distributions exonential t i.e. h(t) = e f(t) = e t and g(t) = e t : visiting any other regenerative state in ( t] q ij.kr /Q ij.kr : df/cdf of direct transition time from regenerative state i to a regenerative state j or to a failed state j visiting state k r once in ( t] m ij : Contribution to mean sojourn time ( i ) in state i when system transits directly to state j so that i m and j ij = a = b q 3 a b a b a b a a 5.5 a b a b qa b.57.8 a b a b qb.89 a b a b b a 4 a b a b b a.4 36 a b a b m ij = tdq ( t) q () ij * ' ij 4

3 International Journal of Comuter Alications ( ) Volume 44 o. Aril b a a b a b 3. b a b It can be easily verified that.9 q () 3. Reliability and Mean Time to ystem Failure (Mtsf) Let i (t) be the cdf of first assage time from regenerative state i to a failed state. Regarding the failed state as absorbing state we have the following recursive relations for i (t): + = = = = + 9= = = +.9 = () The mean sojourn times ( i ) is the state i are a b a b 3 a b a b (3) Also m m m m3 m5 m8 m m4 m m3 m36 m3 3 m m9 (4) And m ( m m.5 m.57 m.8 m.89 say) m m m say ( ).4. m m m (say) (5) t For h(t) = e f(t) = e t and g(t) = qa + ( al + bl ) a + ( al + bl ) qq m = q a ( al + bl + q ) 3 e t we have (6) Where j is an un-failed regenerative state to which the given regenerative stat I can transit and k is a failed state to which the state I can transit directly. Taking LT of above relation (7) and solving for f () s We have R*(s) = - f ( s) (8) s The reliability of the system model can be obtained by taking Lalace inverse transform of (8). The mean time to system failure (MTF) is given by ~ MTF = ( s) lim = so s = 3 3 = teady tate Availability Let A i (t) be the robability that the system is in u-state at instant t given that the system entered regenerative state i at t =. The recursive relations for A i (t) are given as (7) (9) () Where j is any successive regenerative state to which the regenerative stste I can transit through n (natural number) transitions. M i (t) is the robability that the system is u initially in state i any other regenerative state we have E is u at time t without visiting to M t e () a b t abt M ( ) ( ) t e t 5

4 International Journal of Comuter Alications ( ) Volume 44 o. Aril abt M ( ) ( ) t e F t abt (b) ue to relacement of the software () M ( ) ( ) 3 t e G t Taking LT of above relations () and solving for the steady state availability is given by A * () s Let B i (t) be the robability that the server is busy due to relacement of the software at an instant t given that the system entered the regenerative state i at t =. We have the * A( ) lim sa( s) following recursive relations for B i (t):() s = [ + 3 ( ) ] + ( 33) and +[ + 3 ( ) ] = [ + 3 ( ) ] + ( ) 3 + [ ( ) ) ] ( 3.4 Busy Period Analysis for erver (a) ue to ardware Failure Let B i (t) be the robability that the server is busy in reairing the unit due to hardware failure at an instant t given that the system entered state i at t =. The recursive relations for B i (t) are as follows: (3) W i (t) be the robability that the server is busy in state i due to relacement of the software u to time t without making any transition to any other regenerative state or returning to the same via one or more non-regenerative states and so W ( t) e ( a b ) t ( a b ) t F( t) ( b e ) F( t ) (4) Taking LT of above relations () and (3) and solving for * B () s and * B () s the time for which server is busy due to reair and relacements resectively is given by = 3 B lim sb ( s) * s And * B lim sb ( s) = 3 s () (3) (5) W i (t) be the robability that the server is busy in state i due to hardware failure uto time t without making any transition to any other regenerative state or returning to the same via one or more non-regenerative states and so a b t a b W ( t) e ( t) ae qh(t) G( t) ( b e ( a b ) t ( a b ) t qh( t) ) G( t) ( b e qh( t) ) F( t ) a b t a b W3 ( t) e G( t) ae G( t) b e G t a b t ( ) 3 ( W () 3W 3 ()) ( ( 3 3. ) W () 3 ( ( 3 3. ) W () and is already mentioned. 3.5 Exected umber f Relacements f The Units W (a) ue to ardware Failure ( t) ( ) () (4) t Let R i (t) be the exected number of relacements of the failed hardware comonents by the server in ( t] given that 6

5 International Journal of Comuter Alications ( ) Volume 44 o. Aril the system entered the regenerative state i at t =. The Taking LT of relation (9) and solving for () s recursive relations for R. The i (t) are given as exected number of visit er unit time by the server are given by (6) (8) ( ) lim s ( s) = 5 () s Where j is any regenerative state to which the given regenerative state I transits and δ j = if j is the regenerative state the server does job a fresh otherwise δ j =. (b) ue to oftware Failure Let R i (t) be the exected number of relacements of the failed software by the server in ( t] given that the system entered the regenerative state i at t =. The recursive relations for R i (t) are given as Where j is any regenerative state to which the given regenerative state I transits and δ j = if j is the regenerative state the server does job a fres Taking LT of relations (6) and (7). And solving for R () s and R () s. The exected numbers of relacements er unit time to the hardware and software failures are resective of given by R ( ) lim sr ( s) = 4 s 5 = [ + 3 ( ) ] and is already secified. 4. PRFIT AALYI The rofit incurred to the system model in steady state can be obtained as (7) (9) P K A K B K B K R K R K K = Revenue er unit u-time of the system K = Cost er unit time for which server is busy due to hardware failure K = Cost er unit time for which server is busy due to () R ( ) lim sr ( s) = 4 And s software failure (8) () K 3 = Cost er unit relacement of the failed hardware comonent 4 = ( )+ ( ( 3 3. ) ) 4 =( +.4 )+ [ + 3 ( ) ] and is already mentioned. 3.6 Exected umber of Visits by The erver Let i (t) be the exected number of visits by the server in ( t] given that the system entered the regenerative state i at t =. The recursive relations for i (t) are given as (9) Where j is any regenerative state to which the given regenerative state I transits and δ j = if j is the regenerative state the server does job afresh otherwise δ j =. K 4 = Cost er unit relacement of the failed software K 5 = Cost er unit visit by the server and A B B R R are already defined. 5. PARTICULAR CAE uose g(t) = a e - at h(t) = q e - q t f(t) = We can obtain the following results MTF (T ) = Availability (A ) = Busy eriod due to hardware failure 3 B Busy eriod due to software failure 3 B qe - qt 7

6 International Journal of Comuter Alications ( ) Volume 44 o. Aril Exected number of relacements at hardware failure 4 R Exected number of relacements at software failure 4 R Exected number of visits by the server 5 (4) a b a b a b b a b aq a b R a b a b a a b b a b -a a b a b R b R a b a b a b a a q a b q q a b R a b a a a b a a b a a b b 3 q a a b a b R a a b a b a b q q a b qa b a b a b a b a b a b a b a b a b a b a b a a b b 3 4 a a b a a b b 4 a b a b a q a b a a b 5 R 6. CCLUI In the resent study the numerical results considering a articular case are obtained to carry out the rofit analysis of a comuter system by giving the riority to reair activities of h/w comonents over relacement of s/w comonents. Using these results the grahs for mean time to system failure (MTF) availability and rofit are drawn with resect to h/w failure rate (λ ) for fixed values of other arameters as shown resectively in figures nd 3 rd and 4 th. From these figures it is concluded that MTF deceases with the increase of h/w and s/w failures rates. owever MTF goes on increasing as reair rate (α) relacement rate (θ) of the unit at s/w failure and relacement rate (θ ) of the unit at h/w failure increase. The results obtained for availability and rofit indicate that the value of these measures decrease with increase of h/w and s/w failure rate (λ ) and (λ ) resectively. But their values increase if reair rate (α) and relacement rates (θ) and (θ ) increase. Thus it is concluded that the concet of riority given to the relacement and reair of h/w comonents over relacement of s/w comonents is not much economically beneficial as comare to the system in which no such riority is given. 8

7 International Journal of Comuter Alications ( ) Volume 44 o. Aril tate Transition iagram h(t) FUi cs q h(t) aλ 3 FUr FUR FWR bλ f (t) aλ h(t) bλ aλ FUR FWi 6 bλ FUR FUI FWi 5 FUr FWI FURP FWR h(t) 8 bλ FUI FWR q h(t) f (t) f (t) 4 aλ 7 FUR FWRP h(t) FUi FWR q h(t) h(t) q h(t) 9 FUr FWRP Fig. U-state Failed state Regenerative oint 9

8 PRFIT AVAILABILITY MTF International Journal of Comuter Alications ( ) Volume 44 o. Aril 4 GRAP BETWEE MTF A FAILURE RATE a=.7b=.3λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.3q=.7α=3.5 a=.7b=.3λ=.5θ=3 θ==.3q=.7α=.5 a=.7b=.3λ=.θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.7q=.3α=.5 a=.3b=.7λ=.5θ= θ==.3q=.7α= FAILURE RATE (λ) Fig. GRAP BETWEE FAILURE RATE A AVAILABILITY a=.3b=.7λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.7q=.3α=.5 a=.7b=.3λ=.5θ= θ==.3q=.7α=3.5 a=.7b=.3λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ=3 θ==.3q=.7α=3.5 a=.7b=.3λ=.5θ= θ==.3q=.7α= FAILURE RATE (λ) Fig.3 GRAP BETWEE FAILURE RATE A PRFIT a=.3b=.7λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ=3 θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.3q=.7α=3.5 a=.7b=.3λ=.5θ= θ==.3q=.7α= FAILURE RATE (λ) Fig 4. a=.7b=.3λ=.5θ= θ==.3q=.7α=.5 a=.7b=.3λ=.θ= θ==.3q=.7α=.5 a=.7b=.3λ=.5θ= θ==.7q=.3α=.5

9 International Journal of Comuter Alications ( ) Volume 44 o. Aril 7. REFERECE [] Friedman M.A. and Tran P. 99: Reliability Techniques for Combined ardware / oftware ystems. Proceedings of Annual Reliability and Maintainability ymosium [] Welke.R.; Johnson B.W. and Aylar J.. 995: Reliability Modeling of ardware oftware ystems. IEEE Transactions on Reliability Vol. 44(3) [3] Malik.C. and Anand Jyoti : Reliability And Economic Analysis of a Comuter ystem With Indeendent /W and /W Failures. Bulletin of Pure and Alied ciences (BPA) Vol.9E(o.).4-53 [4] Malik.C. and Anand Jyoti : Reliability Modeling of a Comuter ystem With Priority for Relacement at oftware Failure over Reair Activities at /W Failure. International Journal of tatistics and ystem (IJ) I Vol. 6(3) [5] Malik. C. and Ashish Kumar. Profit Analysis of a Comuter ystem with Priority to oftware Relacement over ardware Reair ubject to Maximum eration and Reair Times International Journal of Engineering cience & Technology Vol.3 o [6] Lai C..; Xie M.; Poh K.L.; ai Y.. and Yang P. : A model for availability analysis of distributed software / hardware systems Information and oftware Technology Vol [7] Malik.C. 8: Reliability modeling and rofit analysis of a single-unit system with insection by a server who aears and disaears randomly Journal of Pure and Alied Mathematika ciences Vol. LXVII (- )