Mathematical Modeling for Performance Analysis and Inference of k-out of-n Repairable System Integrating Human Error and System Failure

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1 Amerian Journal of Modeling and Optimization, 4, Vol., No., 6-4 Available online at Siene and Eduation Publihing DOI:.69/ajmo---3 Mathematial Modeling for Performane Analyi and Inferene of k-out of-n Repairable Sytem Integrating Human Error and Sytem Failure Vihwa Nath Maurya,,* Department of Mathemati, Shool of Siene & Tehnology, Univerity of Fiji, Saweni, Fiji Viion Intitute of Tehnology Aligarh, G.B. Tehnial Univerity, India *Correponding author: Reeived January, 4; Revied February, 4; Aepted February 7, 4 Abtrat Preent paper demontrate mathematial modeling and evaluation of performane meaure of k-out of-n repairable ytem in the mot influening ontraint of human error and ommon-aue failure have been taken into onideration. Firtly the mathematial modeling i developed for performane analyi of k-out of-n repairable ytem with tandby unit involving human and ommon-aue failure. Then, a ueful attempt ha been made to evaluate variou important performane meaure uh a availability of ytem, teady tate availability and mean time of ytem failure (MTSF, mean operational time(mot, expeted buy period (EBP and teady tate buy period et. Uing the plementary variable tehnique, Laplae tranform of variou tate probabilitie are explored. Moreover, a partiular ae when repair rate follow exponential ditribution ha alo been diued. In addition, numerial illutration ha alo been preented in order to enable a better mode for undertanding and teting the outome explored herein. Finally, table and graph for invetigated reult are diplayed for drawing ome ignifiant onluive obervation for teting their validity and oniteny. Keyword: Mathematial modeling, repairable ytem, human error, ommon-aue failure, teady tate availability, Laplae tranform, plementary variable tehnique, performane analyi Cite Thi Artile: Vihwa Nath Maurya, Mathematial Modeling for Performane Analyi and Inferene of k-out of-n Repairable Sytem Integrating Human Error and Sytem Failure. Amerian Journal of Modeling and Optimization, vol., no. (4: 6-4. doi:.69/ajmo Introdution Literature how that performane analyi of network with queue ha reeived oniderable attention by a large number of previou noteworthy reearher and it ha oied a prominent plae in Operational Reearh in reent year. Repairable ytem inorporating human error and ytem failure i a partiular type of queueing ytem. In erie ytem, the failure of one or more unit reult in the ytem failure. However, there exit ytem that are not onidered failed until at leat k unit or omponent have failed. Suh ytem are known a k-out of-n ytem. Example of uh ytem are: large airplane uually have three or four engine, but two engine may be the minimum number required to provide a afe journey. Similarly, in many power-generation ytem that have two or more generator, one generator may be uffiient to provide the power requirement. Performane analyi of wide range of queueing model in different frame work ha been onfined by everal noteworthy reearher, (e.g. [3,4,5,6,7,8,,3 and referene therein. Among them, reently Maurya [7,8 paid keen attention onerning with performane analyi of M X /(G,G / queueing model with eond phae optional ervie and Bernoulli vaation hedule and ueeded to invetigate ignifiant performane meaure. It i relevant to mention here that mathematial modeling pay a key role for performane analyi of ytem repair problem like other mathematial tehnique. In thi diretion, ome previou reearher (e.g. [3,9,,6 and referene therein are worth mentioning. Although, reliability and/or performane analyi of ytem repair problem have been attempted by ome previou reearher, (e.g. [,4,5,6,7 but human error and ommon-aue failure have not been yet taken into onideration in their tudie. However, it ha been keenly oberved that human error and ommon-aue failure are very muh influening ontraint in performane and/or reliability analyi of ytem repair problem. Few earlier reearh worker onfined the reliability behavior of k-out of-n ytem taking into aount the ommon-aue failure, for example, Hughe (987 preented a new approah to ommon-aue failure in analyzing ytem repair problem and Who Kee Chang (987 attempted for reliability analyi of a repairable parallel ytem with tandby involving human failure and ommon-aue failure yet no attention ha been paid to the performane evaluation of k-out of-n repairable ytem due to human error and ytem failure ontraint. Jain et al ( tudied a k-out of-n ytem with

2 Amerian Journal of Modeling and Optimization 7 dependent failure and tandby port and Moutafa (997 alo tudied k-out of-n repairable ytem with dependent failure and imperfet overage. In thi paper, our main objetive i to invetigate performane meaure and valuable inferene of k-out of-n repairable ytem with tandby unit involving human and ommon-aue failure by uing mathematial modeling, partiularly to evaluate availability, teady tate availability, mean time of ytem failure (MTSF et. The plementary variable tehnique ha been ued here to evaluate Laplae tranform of variou tate probabilitie of the repair model. Numerial omputation and graph for availability and MTSF have alo been drawn. Following three partiular ae are alo developed: (i. The ae when there i repair due to hardware, human and ommon-aue failure (ii. The ae when there i repair due to hardware and human failure (iii. The ae with no repair. Hypothee of the Sytem Here, we onider k-out of-n repairable ytem with tandby unit involving human and ommon-aue failure. Following are ome aumption taken into our preent onideration for the repair model. (i. The ytem onit of n idential main unit and tandby unit (ii. Unit failure, human error and ommon-aue failure are ontant (iii. Repair rate from failed tate due to unit failure, human error and ommon-aue failure are generally ditributed (iv. Initially n unit are operating and unit are kept a old tandby (v. The entire ytem working if at leat k out of n unit or omponent are operating (vi. The ytem i aid to be in one of the failed if k+ unit have failed due to unit failure, human error and ommon-aue failure (vii. When any of the operating unit fail, it i replaed immediately by tandby unit (viii. If all the tandby are onumed, the ytem work a degraded ytem until k-unit work (ix. No repair will be undertaken until the ytem ha failed due to hardware and human error (i.e. until k+ unit have failed or due to ommon-aue failure (x. We aume that a repaired ytem i a good a new (xi. A perfet with i ued to with-on the tandby unit and with-over time i negligible 3. Notation and Deription of the Sytem In order to analyze our preent repair model in the diretion a indiated earlier, we ue underlying notation; P i,j (t Probability that the ytem i in tate (i, j at time t. State (i, j i the tate of the ytem when i unit failed due to hardware failure and j unit due to human error, i, j,,,, k+. State (, i the initial tate at t and tate (i, k+ i, i,,., k + are the failed tate of the ytem. State i when the ytem ha failed due to ommon-aue failure. λ ontant hardware failure rate of a unit. h ontant human error rate of a unit. λ ommon-aue failure rate Λ n λ, H n h ρ repair rate when the ytem i in tate (i, j ρ repair rate when the ytem i in tate P i,j (y,t Probability denity funtion that the failed ytem i in tate (i, j and ha an elaped repair time of y at time t; j k + i, i,,,.k + P (y,t Probability denity funtion that the failed ytem i in tate and ha an elaped repair time of y at time t.. Laplae tranform variable, F ( Laplae tranform of the probability denity funtion f(. 4. Evaluation of State Probabilitie of the Sytem In view of our aumption and notation a in etion -3, the et of governing differential equation for the model an be expreed a following; d ( +Λ+ H + λ P,,( t dt k+ i ρ( y Pik, + i( y, t dy (4. + ρ ( y P ( y, t dy d ( +Λ+ H + λ Pi, j( t Λ Pi, j( t + HPi, j ( t (4. dt i, j,,,... k, i, j k d d ( + + ρ ( y Pik, + i( yt, (4.3 dy dt d d ( + + ρ ( y P ( yt, (4.4 dy dt 4.. Boundary Condition P (, t Λ P ( t + HP ( t; i,,,... k+ (4.5 ik, + i i, j i, j k k i λ i j i, j P (, t P ( t (4.6 k+ k+ i i j Pi, j ( t + P ( yt, ( Initial Condition,, (, i, j(,,,,,... + (4.8 P t P t i j k 5. Solution of the Mathematial Model By taking the Laplae tranform of equation (4.- (4.7, one may get a et of following integral and differential equation: ( +Λ+ H + λ P ( (5. ( y P ( y, dy + ( y P ( y, dy,, k i + ρ i, k+ i ρ

3 8 Amerian Journal of Modeling and Optimization L i k i L il, i i Λ, P ( ( H P (/ A, i,,,... L, L,,... k (5. d ( + + ρ ( y Pik, + i( y, (5.3 dy d ( + + ρ ( y P ( y, (5.4 dy 5.. Boundary Condition Pik, + i(, Λ Pi, k+ i( + HPik, i(; i,,,... k+ k k i λ i j i, j P (, P ( (5.5 (5.6 k+ k+ i i j Pi, j ( t + P ( yt, (5.7 By integrating equation (5.3, (5.4, we get: Pik, + i( y, Pik, + i(, exp( y ρ( y dy (5.8 P ( y, P (, exp( y ρ ( y dy (5.9 Again integrating equation (5.8, (5.9, uing equation (5.5-(5.7, we get k+ i k+ i k Pik, + i( [ F (( i Λ H / A; i,,,... k + k L L i k i L P ( λ [ G ( L i ( i Λ H / A (5. (5. Uing equation (5.5-(5.7, we have following equation alo from equation (5.8-(5.9; ρ ( yp ik, + i( y, Pik, + i(, F ( k+ i k+ i k F (( i Λ H / A ρ ( yp ( y, i,,,... k + (5. P (, G ( (5.3 k L L i k+ i L G ( L i ( i Λ H / A Latly by ubtituting equation (5., (5. in equation (5., we an eaily get P ( {( +Λ+ H + λ,, k L L i k + i i Λ, L L i k+ i i Λ k+ i, K [ λ G ( ( H P (/ A [ F ( ( H P ( / A } The Laplae tranform of the reliability of the ytem and teady tate availability of the ytem are: k k i k k i P ( P (, P ( lim P ( i, j i, j i j i j 6. Partiular Cae When repair rate follow exponential ditribution. Setting ρ ρ F (, G ( + ρ + ρ (6. The Laplae tranform of the probabilitie of the ytem are: (6. k L L i k i L PiL, i( L i ( i Λ H / A P H P A k+ k+ i k+ i k ik, + i( i ( i Λ,(/ (6.3 + ρ λ k L L i k+ i L P ( L i ( i Λ H / A ρ,, + (6.4 P ( {( +Λ+ H + λ 6.. Cae I L i k+ i k L ( i Λ H L L i A (6.5 λρ [ + ρ ρ k+ k+ i k+ i K [ i ( i Λ H P,( / A } + ρ Conider the ae when n,, k, ρ, ρ are the repair rate when ytem i in tate (i, j, and in tate repetively. In thi ae we have: Λ λ, H h, A + ( λ+ h + λ Moreover, from equation (6.-(6.5, one may get : h P,( [ + ( h+ λ + λ P,( λ,( [ + ( h+ λ + λ P 4h P,( ( + ρ[ + ( h+ λ + λ 8λh P,( P, ( ( + ρ[ + ( h+ λ + λ P 4,( λ,( ( + ρ[ + ( h+ λ + λ P λ( + 4( h+ λ + λ P ( ( + ρ[ + ( h+ λ + λ (6.. (6.. (6..3 (6..4 (6..5 (6..6 ( + ρ( + ρ( + ( λ+ h + λ (6..7 C ( C ( ( + ρ( + ρ( + ( h+ λ + λ λρ ( + ρ( + 4( h+ λ + λ 4 ρλ ( + h ( + ρ Therefore the Laplae tranform of probability of the ytem i in -tate i:

4 Amerian Journal of Modeling and Optimization 9 P ( P ( + P ( + P (,,, ( + ρ( + ρ( + 4( λ+ h + λ P ( 3 ( + k + k+ k3 k ρ + ρ_{ } + λ + 4( h+ λ k ρ( ρ λ + (( h+ λ + λ + ( ρ + ρ(( h + λ + λ k3 ρ( ρ λ(4h+ 4 λ+ λ + ( ρ + ρ(( h+ λ + λ 4 ρ( h+ λ ( P,( + P, ( + P,( + P ( ( P ( + Pi, k+ i ( (6..8 ( + 4( h+ + ( + + 4( h+ ( + C ( λ λ λ ρ λ ρ Taking invere Laplae tranform of equation (6..8, one may get: rt rt rt 3 rt (6..9 P ( t ke ke ke ke k4 ρρ(4( λ + h + λ / ( rr 3 r (( r+ ρ( r+ ρ ( r+ 4( λ+ h k5 ( r( r r( r r3 (( r + ρ( r + ρ( r + 4( λ+ h + λ k6 ( r( r r( r r3 (( r3 + ρ( r3 + ρ( r3 + 4( λ+ h + λ k7 ( r3( r3 r( r3 r and rr,, r3 are the root of the equation k + k + k 6... Mean Operational Time (MOT From equation (6..9 one may get: µ ( t o P ( t dt k4 rt k 5 rt k 6 rt k 3 7 rt ( e + ( e + ( e + ( e 4 r r r3 r4 ( Buy Period B ( P ( + Pi, k+ i ( ( + 4( h+ + ( + + 4( h+ ( + 3 ( + k + k+ k3 (6.. λ λ λ ρ λ ρ Taking invere Laplae tranform of equation (6.., it i fairly eay to get buy period: rt rt rt 3 rt Bt ( ke ke 5 + ke 6 + ke 7 (6.. k4 ( λρ (4( h+ λ + λ + 4 ρ( h+ λ /( rr 3 r ( λ( r+ 4( h+ λ + λ( r+ ρ + 4( h+ λ ( r+ ρ k5 ( r( r r( r r3 ( λ( r + 4( h+ λ + λ( r + ρ + 4( h+ λ ( r + ρ k6 ( r( r r( r r3 k7 ( λ( r3 + 4( h+ λ + λ( r3 + ρ + 4( h+ λ ( r3 + ρ Expeted Buy Period (EBP From equation (6.. one may get the expeted buy period a following; µ ( t B( t dt B o 4 rt k 5 rt k 6 rt k 3 7 rt 4 k ( e + ( e + ( e + ( e r r r r 3 4 ( Steady State Availability of the Sytem P lim P ( ρρ (4( λ + h + λ / C ( (6..4 : C ( (( h+ λ + λ ( ρ + ρ + ρρ(( h+ λ + λ λρ (4( h+ λ + λ 4 ρ( h+ λ λρρ Steady State Buy Period B( ( λ( + 4( h+ λ + λ( + ρ + 4( h+ λ ( + ρ / C ( Mean Time of Sytem Failure (MTSF (6..5 MTSF lim P ( ρρ (4( λ + h + λ / C( (6..6 : C( ρρ(( λ + h + λ λλρρ(4( λ + h + λ 4 ρρ ( λ + h 6.. Cae II We onider the ae when n,, k, ρ are the repair our when ytem i in tate (i, j. In thi ae, we have: Λ λ, H h, A + ( λ+ h + λ and from equation (6.-(6.5, one may get h P,( [ + ( h+ λ + λ P,( λ,( [ + ( h+ λ + λ P 4h P,( ( + ρ[ + ( h+ λ + λ 8λh P,( P, ( ( + ρ[ + ( h+ λ + λ (6.. (6.. (6..3 (6..4

5 Amerian Journal of Modeling and Optimization P 4,( λ,( ( + ρ[ + ( h+ λ + λ P λ( + 4( h+ λ + λ P ( [ + ( h+ λ + λ ( + ρ( + 4( λ+ h + λ ( + ρ( + ( λ+ h + λ 4 ρλ ( + h (6..5 (6..6 (6..7 Therefore the Laplae tranform of probability of the ytem i in -tate i: P ( + P,( + P, ( ( + ρ( + 4( λ+ h + λ P ( 3 + k + k+ k3 k ρ + 4( h+ λ + λ k (( h+ λ + λ + ρ(4( h+ λ + λ k3 λρ (4( h+ λ + λ ( P,( + P, ( + P,( + P ( 4 h ( + λ + λ ( + ρ( λ+ h ( ( + ρ( + ( h+ λ + λ (6..8 Taking invere Laplae tranform of equation (6..8, one may get after a little implifiation: rt rt rt (6..9 P ( t k ke ke ke k4 (( r+ ρ( r+ 4( λ+ h + λ / (( r r( r r3 k5 (( r + ρ( r + 4( λ+ h + λ / (( r r( r r3 k6 (( r3 + ρ( r3 + 4( λ+ h + λ / (( r3 r( r3 r and rr,, r 3 are the root of the equation 3 + k + k+ k3 P λ_{ }( + ρ( + 4( h+ λ + λ_{ } + 4( λ+ h down (( + ρ( + ( λ+ h + λ 4 ρλ ( + h 6... Expeted Operational Time From equation (6..9, one may get the expeted operational time a below; µ ( t o P ( t dt k5 rt k 6 rt k (6.. 7 rt k ( e + ( e + ( e r r r Buy Period B ( P ( ik, + i 4( h + λ ( + ρ( + ( λ+ h + λ 4 ρλ ( + h {} (6.. k ρ + λ + 4( h+ λ k (( h+ λ + λ(( h+ λ + ρ + λ k λρ(4h+ 4 λ+ λ 3 Taking invere Laplae tranform of equation (6.. and after a little implifiation, one may get: rt rt rt Bt ( ke ke 5 + ke 6 (6.. k4 4( h+ λ / ( r r( r r3 k5 4( h+ λ / ( r r( r r3 k6 4( h+ λ / ( r3 r( r3 r Expeted Buy Period From equation (6.., one may get µ ( t B( t dt B o k4 rt k 5 rt k ( rt ( e + ( e + ( e 3 r r r Steady State Availability of the Sytem P lim P ( ρ(4( λ+ h + λ (6..4 (( h+ λ + λ + ρ(( h+ λ + λ Steady State Buy Period (4( h + λ B( (6..5 (( h+ λ + λ + ρ(( h+ λ + λ Mean Time of Sytem Failure (MTSF MTSF lim P ( 6.3. Cae III ( ρ(4( λ+ h + λ (6..6 ( ρ(( λ+ h + λ 4 ρλ ( + h Conider the ae when n,, k, and no repair, we have: Λ λ, H h, A + ( λ+ h + λ And from equation (6.-(6.5, one may get h P,( [ + ( h+ λ + λ P,( λ,( [ + ( h+ λ + λ P 4h P,( [ + ( h+ λ + λ 8λh P,( P, ( [ + ( h+ λ + λ P 4,( λ,( [ + ( h+ λ + λ P (6.3. (6.3. (6.3.3 (6.3.4 (6.3.5

6 Amerian Journal of Modeling and Optimization λ(4( h + λ + λ P ( [ + ( h+ λ + λ ( + ( λ+ h + λ (6.3.6 (6.3.7 Therefore the Laplae tranform of probability of the ytem i in tate i: P ( P ( + P ( + P (,,, ( + 4( λ+ h + λ P ( 3 + k + k k 4( h+ λ + λ k (( h+ λ + λ ( P,( + P, ( + P,( + P ( λ(4( h+ λ + λ + 4( λ+ h ( ( + ( h+ λ + λ (6.3.8 Taking invere Laplae tranform of equation (6..8 one may get: rt rt (6.3.9 P ( t ke ke k3 (4( λ+ h + λ / ( r r k4 (4( λ+ h + λ / ( r r and rr,, are the root of the equation 6.3..Mean Operational Time (MOT µ From equation (6.3.9, one may get: + k + k k3 ( ( ( rt k 4 r ( t t o P t dt e + e (6.3. r r Steady State Availability of the Sytem (4( λ+ h + λ P lim P ( (( h + λ + λ Mean Time of Sytem Failure (MTSF (4( λ+ h + λ MTSF lim P ( (( λ+ h + λ 7. Numerial Illutration (6.3. (6.3. In thi etion, our endeavor i to tet the validity of variou performane meaure explored herein by way of numerial illutration. Setting λ., h., λ., ρ.8, ρ.8, one an ompute reult ae wie a following; 7.. Cae I when n,, k, ρ, ρ are the repair rate when ytem i in tate (i, j, and in tate repetively 7... Availability Analyi P ( t 4.97 exp(.737 t +.93exp( t 6.8 exp( t Expeted Operational Time µ ( t.8446 t.33687(exp(.737 t 3.6(exp( t (exp( t Buy Period (7.. (7.. Bt ( (exp( t.564(exp( t ( (exp(.737 t Expeted Buy Period µ ( t.43t (exp( t +.54 (exp( t ( (exp(.737 t 7.. Cae II when n,, k, ρ are the repair our when ytem i in tate (i, j 7... Availability Analyi P (exp(.744 t exp( t exp(.976 t (7.. 3 ( t µ (exp( t.346(exp(.743 t ( (exp(.977 t 7... Buy Period Bt ( (6.868t exp( t 4.99 (exp(.744 t ( (exp(.976 t Expeted Buy Period µ 7.3. Cae III ( t exp( t (exp(.744 t 8.77(exp(.976 t when n,, k, and no repair (7..4

7 Amerian Journal of Modeling and Optimization Availability Analyi P ( t. exp{.6 t} +.6t exp{.6 t} (7.3. µ ( t 3.58( exp{.6 t } t exp{.6 t } (7.3. Setting t,,,..., in equation (7.., (7.. and (7.3., one get Table. Variation of availability w.r.t. time in three ae i hown in Figure. t Table. Availability of the ytem for three ae at time t Availability of the ytem with ρ and ρ Availability of the ytem with ρ Availability of the ytem without repair ² ².444 ².46 ² t, in equation (7.., (7.. Setting,,,... and (7.3., one get Table. Variation of expeted operational time w.r.t. time in three ae i hown in Figure. Table. Expeted operational time of the ytem for three ae at time t Expeted Expeted Expeted operational time of operational time operational time t the ytem with ρ of the ytem with of the ytem and ρ ρ without repair Setting h,.,.4..., in equation (6..4, (6..4, one get Table 3. Variation of teady tate availability for different value of human error in three ae i hown in Figure 3. Table 3. Relationhip between teady tate availability and human error for three ae h Steady tate Availability of the ytem with ρ and ρ Steady tate Availability of the ytem with ρ Setting h,.,.4..., in equation (6..6, (6..6 and (6.3., one get Table 4. Variation of MTSF for different value of human error in three ae i hown in Figure 4. MTSF of the ytem with ρ and ρ, MTSF of the ytem with ρ Table 4. Relationhip between MTSF and human error for three ae MTSF of the MTSF of the MTSF of the h ytem with ρ and ytem without ytem with ρ repair ρ Figure. Variation of availability w.r.t. time in the three ae Figure. Variation expeted operational time of ytem w.r.t. time in the three ae

8 Amerian Journal of Modeling and Optimization 3 Figure 3. Variation of teady tate availability for different value of human error for ae II &III Figure 4. Variation of MTSF for different value of human error for ae II & III 8. Senitivity Analyi and Conluive Obervation Table ompute availability of the ytem at any time. Figure how availability of the ytem dereae in the interval (, t for the three ae. By omparing the availability with repet to time for three ae with and without repair graphially, it i oberved that: The availability of ytem dereae with repet to time. We onlude that the ytem availability with ρ and ρ i greater than the ytem with ρ, and availability with ρ i greater than without repair. Table ompute expeted operational time for the ytem at any time. Figure how expeted operational time inreae in the interval (, t for the three ae. By omparing the expeted operational time with repet to time t for three ae with and without repair graphially, we oberve that: The ytem expeted operational time inreae with repet to time. And it i alo notieable that the ytem expeted operational time with ρ and ρ i greater than the ytem with ρ and with ρ repair i greater than the ytem without repair. Table 3 ompute the relationhip between teady tate availability and human error for three ae. Figure 3 how variation of teady tate availability with repet to human failure for the two ae of repair. By omparing the teady tate availability with repet to human failure for the ytem with ρ and ρ graphially, it i remarked that: The human failure rate h inreae, however the teady tate availability of the ytem dereae at ontant λ., λ., ρ.8, ρ.8. The teady tate availability of ytem with repair ρ only i greater than the ytem availability with ρ and ρ (i.e., ytem inlude ommon-aue failure. The availability of the ytem in the teady-tate i equal to zero. Table 4 ompute variation of mean time of ytem failure (MTSF with repet to human failure. Figure (4 diplay the variation of MTSF with repet to human failure for the eond and third ae. By omparing the MTSF with repet to human failure for the ytem with ρ and without repair graphially, we oberve that: The mean time of ytem failure (MTSF with ρ and ρ MTSF with ρ (ontant. However, in ae of MTSF without repairing, a for a the value of human failure rate h inreae at ontant λ., λ., ρ.8, ρ.8, the MTTF of the ytem dereae. Finally with paing above remark, we have ueeded to invetigate ignifiant performane meaure of k-out of-n repairable ytem involving human and ommonaue failure. It i highly expeted that our preent ontribution for performane analyi and inferene of k- out of-n repairable ytem involving human and ommonaue failure will be ueful for reearher, tatitiian, mathematiian and management profeional for their future reearh and development in thi diretion, e.g. very reently Maurya (3 propoed a omputational approah to ot and profit analyi of k-out of-n repairable ytem integrating human error and ytem failure ontraint. Aknowledgement The preent paper i an integral part of the pot-dotoral diertation entitled Performane Analyi And Inferene of Mixed Poion Queueing Model publihed in Sholar Pre Publihing, Germany. Prof. V.N. Maurya; author of the paper would like to expre hi heartiet gratitude to hi D.S. advior Prof. (Dr. R.B. Mira. Ex- Vie Chanellor, Dr. Ram Manohar Avadh Univerity Faizabad, India; Hon ble Vie Chanellor, Univerity of Fiji, Saweni and Dean, Shool of Siene & Tehnology, Univerity of Fiji for their enouragement and inere port to provide all eential failitie. Referene [ Hughe, R.P., 987. A new approah to ommon-aue failure. Reliability Engineering and Sytem Safety, Vol. 7, pp [ Jain Madhu, Sharma G.C. and Alok Kumar,. K-out of-n ytem with dependent failure and tandby port. JKAU: Engineering Siene, Vol. 4, No., pp [3 Maurya Avadheh Kumar and Maurya V.N., 3. A novel algorithm for optimum balaning energy onumption LEACH protool uing numerial imulation tehnique, International Journal of Eletroni Communiation and Eletrial Engineering, Algeria, Vol. 3, No. 4, pp. -9. [4 Maurya Avadheh Kumar, Maurya V.N. and Arora D. Kaur, 3. Linear regreion and overage rate performane analyi for optimization of reeived ignal trength in antenna beam-tilt ellular mobile environment, International Journal of Eletroni Communiation and Eletrial Engineering, Algeria, Vol. 3, No. 7, pp. -4. [5 Maurya V.N.,. On buy period of an interdependent M/M/:(;GD queueing model with bivariate Poion proe and ontrollable arrival rate, IEEE Tranation, pp [6 Maurya V.N.,. Determination of expeted buy period in fater and lower arrival rate of an interdependent M/M/:(; GD queueing model with ontrollable arrival rate, International Journal of Engineering Reearh and Tehnology, Engineering Siene & Reearh Sport Aademy (ESRSA Publiation, Vadodara, India, Vol., No. 5, pp. -5. [7 Maurya V. N., 3 a. Senitivity analyi on ignifiant performane meaure of bulk arrival retrial queueing

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