Stochastic Reliability Analysis of Two Identical Cold Standby Units with Geometric Failure & Repair Rates

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1 DOI: 0.545/mjis Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes NITIN BHARDWAJ AND BHUPENDER PARASHAR Received: Ocober 06, 06 Revised: December 0, 06 Acceped: January 4, 07 Publised online: Marc 05, 07 Te Auor(s) 06. Tis aricle is publised wi open access a Absrac Te presen paper is based on e analysis of an wo idenical uni sysem werein iniially one uni is in operaive sae and oer is in cold sandby and oal failure of e uni is via parial failure mode. Single repairman is always available wi e sysem for any repair of failed uni (parially/ compleely). Failure/repair ime is considered o be a Geomeric disribuion. Measures of sysem effeciveness ad also been calculaed for e sysem. Keyword: Geomeric disribuion, redundan sysem, discree random variable.. INTRODUCTION In some siuaions, discree failure disribuions is appropriae for lifeime model [,,4] e.g an discree disribuion is accurae for a equipmen operaing in pases and e number of pases observed prior o any failure. Discree failure case arises in several siuaions, e.g.: a) A device is moniored only once in any period of ime (e.g., ourly, daily, weekly, monly ec.), and e observaion is considered o be e number of ime periods compleed successfully, prior o device failure. b) Piece of equipmen performs in pases and e experimener successfully observes e number of pases, compleed prior o failure. Wen e observed values are assumed o be very large i.e. in ousands of pases, a coninuous disribuion is proved o be an adequae model for e discree random variable. However, for small observed values, e coninuous disribuion will no be describing a discree random variable. We see a many pracical siuaions of imporance are represened wi e elp of discree life ime models. Similarly, e repair ime may be considered as discree random variable as dividing e wole ime inerval ino Maemaical Journal of Inerdisciplinary Sciences Vol-5, No-, Marc 07 pp. 3 4

2 Bardwaj, N. Parasar, B. various small pars of ime. Discree ime models are considered by several researcers [, ]. Keeping in view ese concep of discree ime modeling, we analyze redundan sysem models [5] aving wo idenical unis wi geomeric failure and repair ime disribuion. In iniial sage, one uni is in operaive sae and second one as o be in cold sandby model. Te wo ypes of failure are oal failure and parial failure of e uni [3].. SYSTEM DESCRIPTION AND ASSUMPTION Sysem is analyzed under e following assumpions: (i) A sysem is consising of wo idenical unis i.e. operaive and sandby. Eac uni posses ree modes of saes: normal (N), parial failure (PF) and oal failure (F). Te sandby uni canno fail. (ii) Sysem is considered o be in failed sae, weer e cause of failure is parial or oal. (iii) Failures are self-announcing. (iv) Te failure and repair ime disribuion are independen aving geomeric disribuion wi parameer p and r respecively. (v) Te failure ime of all unis will be aken as independen random variable. (vi) Te sysem may go o parially failed or oally failed saes wi probabiliy b or a respecively. 3. NOTATIONS/SYMBOLS FOR STATES OF THE SYSTEMS a : b : Probabiliy of sysem going o failed sae. Probabiliy of sysem aving parially failed sae. N 0 : Uni aving operaive/normal mode. N s : Uni aving sandby/normal mode. F r /F wr : Uni aving failure mode and is repairing/waiing for repair. PF r /PF wr : Uni aving parial failure mode and is repairing/waiing for repair. Up Saes : S 0 (N 0, N s ), S (F r, N 0 ), S (PF r, N 0 ) Down Sae : S 3 (F r, F wr ), S 4 (PF r, PF wr ), S 5 (F r, PF wr ), S 6 (PF r, F wr ) 3

3 Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes Figure.: Transiion Diagram 4. TRANSITION PROBABILITIES AND SOJOURN TIMES q Q 0 () ap q ( + ) Q 0 () bp q q ( + ) ( ) Q () arp ( + ) ( ) Q 3 () aps ( ) Q () brp ( + ) ( + ) ( ) Q 0 () rq ( ) Q 0 () rq + ( + ) ( ) Q 5 () bps ( ) Q () brp ( + ) ( + ) ( ) Q () arp ( + ) Q 4 () bps ( ) ( + ) Q 6 () aps ( ) ( + ) s Q 3 () Q 4 () Q 5 () Q 6 () r s ( + ) (. -.3) Te probabiliy of seady-sae ransiion from S i o S j is calculaed by using 33

4 Bardwaj, N. Parasar, B. and i is verified a P ij lim P 0 + P 0, P + P 0 + P + P 5 + P 3, P + P 0 + P + P 4 + P 6 P 3 P 4 P 5 P 6 Q ij 5. MEAN SOJOURN TIMES Le T i be assumed o be sojourn ime in sae S i (i 0,, ) and en, e mean sojourn ime of S i is calculaed by µ i PT ( > ) 0 i so a, µ 0 q µ µ µ 3 µ 4 µ 5 µ 6 s (.4-.6) By defining m ij in sae S i, wen sysem ransis ino sae S j i.e. m ij 0 q ij () we obained m 0 + m 0 qµ 0 m + m 0 + m + m 3 + m 5 () µ m + m 0 + m + m 4 + m 6 () µ m 3 m 5 m 4 m 6 (.7-.0) 6. MEAN TIME TO SYSTEM FAILURE Lebe e sysem probabiliy o work for a leas epocs, wen iniial i sars from sae R 0 () Z 0 () + q 0 () R + q 0 R R () Z () +q 0 R 0 +q R +q R R () Z () + q 0 R 0 + q R +q R Taking Geomeric Transform on bo sides, we ge R 0 *() N( ) D ( ) (..3) 34

5 D () [ q * ( )][ q * ( )] q * ( ) q * ( ) q * ( ) q * * ( ) [ q ( )] 0 0 * * * * * * q ( q ) ( q ) ( ) q ( q ) 0 ( q ) ( ) * * * 0 0 q ( ) q ( )( q ( )) * * * * * * * N () Z ( )[ q ( )][ q ( )] q ( ) q ( ) Z ( ) + q ( ) * * * * * * * * ( q ( )) Z( ) + q ( q ) ( Z ) ( q ) 0 0 ( q ) ( Z ) ( ) Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes + q * ( )( q * ( )) Z * ( ) 0 Ten, MTSF N /D (.4) were D (P ) (P ) P P P 0 P 0 (P ) P P 0 P 0 P 0 P 0 P P 0 P 0 (P ) N (µ 0 ) [P P (P ) (P )] + (µ + P 0 ) (P 0 (P ) + P 0 P ) + [(µ + P 0 ) P 0 (P ) + P P 0 ] 7. AVAILABILITY ANALYSIS Le A i () be e sysem probabiliy aving up sae a epoc '', wen iniial i sars from sae S i. () Z 0 () + q 0 A + q 0 A A () Z () + q 0 + q A + q 3 A 3 + q 5 A 5 + q A A () Z () + q 0 + q A + q A + q 4 A 4 + q 6 A 6 A 3 () q 3 A A 4 () q 4 A A 5 () q 5 A A 6 () q 6 A (.5-.3) By aking geomeric ransformaion, and solving above equaions we ge *() N ( ) D ( ) 35

6 Bardwaj, N. Parasar, B. were N () D () ( * * * * * * * Z q q 0 q4 q q4 Z0 ( )[( ( ))( ( ))] ( )( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * * 3 * * * * * * q q Z0 q6 q q6 Z0 q3 q3 * * 4 * * * * 3 * ( q ( )). Z0 ( ) + q4 ( q ) 3( q ) 4 ( q ) 3( ) q( ) * * * * * * * * q q Z q6 q5 6 q5 + q ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( )) * * * * * * * * * q Z 3 q q q Z q0 q Z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + ( ) 3 * * * * * * * 3 * q0 q5 q5 Z q q0 Z q6 * * * * * * * * q q Z + q q Z q3 q3 ( ) ( ) ( ) ( )( ( )) ( ) ( ) ( ) * * q0 Z ( ) ( ) * * * * * * q q q q q 4 4 q ( ))( ( )) ( ) ( )( ( )) ( ) * * * * * * * q ( ) q 3 ( q ) ( q ) ( ) q ( q ) ( )( q ( )) * * * * * * * * q q q q 3 q q q q6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * * * * * * q q q q q0 q0 q4 ( ) ( ) ( )( ( )) ( ) ( ) ( ) * 3 * * * 4 * * * * 3 q4 ( ) q ( q ) 0( q ) 0 ( ) q5( q ) 0( q ) 0 ( q ) 4 ( ) * * * * * * * * * q q q q 4 q q q q q0 ( ) ( ). ( ) ( ) ( ) ( ) ( ) ( ) ( ) * * * * * ( q ( )) + q 4 3( q ) 0 ( q ) 3( q ) 0 ( ) were, Z 0 () q, Z () Z () (), Z 3 () Z 4 () Z 5 () Z 6 () s Hence, Z * i () µ i Te availabiliy of seady sae sysem is calculaed by lim Hence, using L Hospial Rule, we ge were () N () D () (.3) 36

7 and N () ( p p 4 ) (µ 0 (p ) + µ p 0 ) + (p + p 6 ) (µ p 0 µ 0 p ) p 5 p µ 0 p 3 (p ) µ 0 + µ p 0 (p + p 5 ) + µ p 0 (p p 3 ) + p 3 p 4 p 6 p 5 D () [µ (p p 4 p 0 p 0 ) + q µ 0 (p 0 (p + p 5 ) p 0 (p 4 p )) Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes + µ (p p 3 p 0 p 0 ) + sµ 3 [p 4 (p p 0 p 0 ) + p 5 (p 0 p 0 + (p +p 6 )) + p 3 (p p 4 p 0 p 0 ) + p 6 (p + p 0 p 0 )] Now, e sysem expeced upime a epoc is calculaed by µ up () x 0 (x) * *( ) so a µ ( ) A up 0 8. BUSY PERIOD ANALYSIS Le be e repair probabiliy i.e busy repair ime of any failed uni, wen iniial e sysem sars from. B 0 () q 0 B + q 0 B B () Z () + q 0 () + q B + q A + q 3 B 3 +q 5 A 5 B () Z () + q 0 B 0 + q B + q B + q 4 B 4 + q 6 B 6 B 3 () Z 3 () + q 3 B B 4 () Z 4 () + q 4 B B 5 () Z 5 () + q 5 B B 6 () Z 6 () + q 6 B (.33.39) 37

8 Bardwaj, N. Parasar, B. By aking geomeric ransformaion and solving above equaions we ge B 0 *() were, N3( ) D ( ) N q q q q q Z q q * * * * * * * * ( ) ( )( ( )) 3 ( ) ( ) ( ) ( ) ( ) ( ) * 3 * * * * 3 * * * * Z ( ) + q0( q ) ( q ) 4 ( Z ) 4 ( ) + q0( q ) ( q ) 6 ( Z ) 6 ( ) + * * * * 4 * * * * * q0( q ) 3( )( q ( )) Z3 ( ) q0( q ) 3( q ) 4 ( q ) 4 ( Z ) 3 ( ) + * * * 3 * * * * q0 q Z + q0 q6 q6 Z * * * + q0 q Z ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ( )) ( ) + ( ) ( )( ( )) ( ) + ( ) ( )( ( )) ( ) ( ) * * * * * * * * q0 q4 q Z4 q6 q0 q Z6 3 * q0 * * * 3 * * * * q3 q3 Z + q0 q3 q Z3 4 * * * * * + q0 q3 q6 q3 Z4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 4 * * * * * 4 * * * * * q0 q3 q6 q6 Z3 q0 q3 q6 q3 Z6 3 * + q0 * * * * * * * * q q Z + q q5 q6 q6 Z5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Probabiliy of a repair faciliy busy in repairing failed uni is: B 0 () lim Hence, using L Hospial Rule, we ge B 0 () N3() B 0 D () (.40) were N 3 () p 0 (p )+ µ (p 0 (p +p 6 ) p 0 p 4 )+ µ (p 0 p p 0 (p +p 3 )) + µ 3 {(p 6 + p 4 ) (p 0 p + p 0 (p )) + p 0 (p + p 6 ) (p 3 + p 5 )} 38

9 9. PROFIT FUNCTION ANALYSIS Te profi expeced in seady-sae is P C 0 C B 0 (.4) were C 0 / C : per uni up/down ime revenue/expendiure by/on e sysem. Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes 0. PARTICULAR CASE P 0 a, P 0 b, P P arp q P 0 P 0 rq, P P On fixing following numerical values as:, P 3 P 6 brp, P 5 P 4 P 0., r 0.5, a 0.45, b Te values of differen sysem effeciveness measures are as: MTSF 5.33 ime unis Availabiliy ( ) Busy period of analysis of repairman (B 0 ) Grapical Represenaion: aps bps Fig..3 reflecs e beavior repair rae I is clear a, e increases wi increasing repair rae. Fig..4 reflecs e paern of e profi failure rae for disinc repair rae Profi decreases wi increasing failure rae and is iger for bigger values of repair rae Following observaions from e grap are: (i) For r 0.5, P > or or < 0 according as p < or or > So, sysem is profiable only if, failure rae is less an (ii) For r 0.75, P > or or < 0 according as p < or or > Tus e sysem is no profiable wen > Hence e companies using suc sysems can be suggesed o purcase only ose sysem wic do no ave failure raes greaer an ose discussed in poins (i) o (iii) above in is paricular case. 39

10 Bardwaj, N. Parasar, B. Fig..5 reflecs e paern of e profi w.r. repair rae for disinc failure rae Profi increases wi increasing repair rae and is lower for iger values of failure rae Figures.: reflecs e beavior of failure rae I is clear a, e decreases wi increasing failure rae. Figure.3 40

11 Socasic Reliabiliy Analysis of Two Idenical Cold Sandby Unis wi Geomeric Failure & Repair Raes Figure.4 Figure.5 4

12 Bardwaj, N. Parasar, B. CONCLUSIONS Te analysis discussed above sows a e MTSF and e expeced up ime of used sysem decreases wi increasing e values of rae of parial and oal failures. For e profi of e sysem, e analysis saed various cu of poins of e revenue per uni up ime and cos per uni repair of e failed uni o enance e profi of e sysem. REFERENCES [] Savlia and Bollinger, On discree azard funcions IEEE Transacions on Reliabiliy, Vol. R-3, (December 98), No. 5. [] Padge and Spurrier, On discree failure models IEEE Transacions on Reliabiliy, Vol. R , No. 3. [3] B. Parasar, G. Taneja, Reliabiliy and Profi Evaluaion of a PLC Ho Sandby Sysem Based on Maser-Slave Concep and Two Types of Repair Faciliy, IEEE Transacions on Reliabiliy,Vol. 56, Sepember 007, No. 3, [ ]. [4] Lance Fiondella, Liudong Xing, Discree and coninuous reliabiliy models for sysems wi idenically disribued correlaed componens, Reliabiliy Engineering & Sysem Safey, Volume 33, January 05, Pages 0. [5] Niin Bardwaj, J. Bai, B.Parasar, Nares Sarma, Socasic Analysis of Sandby Sysem Wi Differen failure. Inernaional Journal of Applied Engineering Researc, ISSN , Vol. 0, (05), No. 35 Researc India Publicaions. 4