Variance of Time to Recruitment for a Single Grade Manpower System using Order Statistics for Inter-decision Times and Wastages

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1 Vaance o e o Recuen o a Sne Gae Manowe Syse usn Oe Sascs o Ine-ecson es an Wasaes K. Eanovan, B. Ese Caa Asssan Poesso, Deaen o Maeacs, Rajah Seoj Govenen Coee Auonoous, hanjavu , a Nau, Ina. Asssan Poesso, Deaen o Maeacs, Bsho Hebe Coee Auonoous, uchaa , a Nau, Ina. Absac In s ae, a sne ae oanzaon s consee n whch successve e o esonne eas o wasae n e o o an hous ue o s ocy ecsons. A aeaca oe s consuce usn oe sascs o e oss o an hous an ne-ecson es base on shoc oe aoach w unvaae CM ocy o ecuen. he ean an vaance o e o ecuen s obane by assun secc sbuon o beaown esho. he nuence o noa aaees on e eoance easues ae sue nuecay w eevan nns an concusons ae esene. Keywos: Sne ae anowe syse, nvaae CM ocy, Oe sascs, Shoc oe, Vaance o e o ecuen. AMS Subjec Casscaon 00: Pay: 90B70, Seconay: 60H0, 60K05. INRODCION In ea e, once e e o esonne haens, e ecuen canno be nouce as s e ovewhen an cosy. Once e oa aoun o wasae cosses a cean esho eve, e oanzaon eaches an uneconoc saus whch oewse be cae e beaown on an ecuen s one a s on o e. Many oes have been scusse consen een ns o wasaes an een yes o sbuons o e oss o an hous, e esho an e ne-ecson es. Such oes cou be seen n [],[],[4],[5],[6],[7],[9] an [0]. In [8], a sne ae anowe syse w anaoy eonena esho o e oss o anowe, e auo sue syse chaacescs when ne-ecson es o an oe sascs an e oss o anowe os a seuence o neenen an encay sbue eonena ano vaabes. In [] an [] consen ee een sbuon o eshos, e auo has obane syse chaacescs when e ne-ecson es o an oe sascs an oss o an hous ae coeae eonena ano vaabes an vce vesa. he esen ae eens e esu when oe sascs ae o oss o an hous an ne-ecson es. MODEL DESCRIPION AND ANALYSIS FOR MODEL-I Conse an oanzaon an ecson a ano eochs 0, an a evey ecson eoch a ano nube o esons e o e oanzaon. hee s an assocae oss o an hous a eson es an s nea an cuuave. Le X be e oss o an hous wasae ue o e ecson eoch,,.... Assue a s ouaon s a seuence o neenen an encay sbue ano vaabes w sbuon G e an ensy uncon. sze 'n' seece o e ouaon. Le X, X, X,... X n be e oe sascs coesonn o s sae w esecve ensy uncons,.,.,.... X s e s w ean 0. Assue a X n be a sae o X. X X X n. Hee oe sascs saes w obaby ensy uncon X. an X n s e n oe sascs aes w obaby ensy uncon X X, X,... X, X n.. Noe a e ano vaabes n ae no neenen. Le,,,,... be e e beween ecsons w sbuon uncon ensy uncon.. Le an F. F an obaby be e sbuon obaby ensy uncon o. Le Y be a ano vaabe enon e esho o e oss o 559

2 an hous w cuuave sbuon uncon. H. an h. Le be a connuous obaby ensy uncon ano vaabe enon e e o ecuen n e oanzaon w obaby ensy uncon an sbuon uncon...,. an an L esecvey. Le. be e Laace ansos o.,. an. esecvey. Le V F F w F 0. Le E an V be e ean an vaance o e o ecuen. he unvaae CM ocy o ecuen eoye n s ae s as oows Recuen s one as an when e cuuave oss o an hous n e oanzaon ecees esho. MAIN RESLS he suvva uncon o s ven by P F F P X Y 0 By usn e aw o oa obaby an on scaon we e 0 G yh y P y Hee Y oows e eonena sbuon w aaee. F F G yh y P y P 0 0 F F 0 F 4 P he obaby sbuon uncon o s ven by L P an L Fo 5 an 6 we have he Laace anso o ensy uncon o s s s on scaon 8 s I s nown a, s s s * * s, E s E 0 s0 an V E E s s0 9 he obaby ensy uncon o G X I n,,... n X s ven by G G n n G en n G X I X n G en n G By hyoess * X n 4 e, we e n n n n!,, an X n... n Conse e ouaon X X n. 5 o neenen an encay sbue ne-ecson es w eonena cuuave sbuon F e an e coesonn ensy uncon. Assue a be a sae o sze ' ' seece o e ouaon. Le,,,... be e oe sascs coesonn o s sae w esecve ensy uncons.,.,.,..... Hee s e s oe sascs saes w obaby ensy uncon. an s e oe sascs aes w obaby ensy uncon.. Noe a e ano... vaabes an hence,,,..., ae no neenen. he obaby ensy uncon o e enea j j j j,,,... j oe sascs s ven by j j F F 6 560

3 Case : he obaby ensy uncons o an by F F an ae ven 7 I en s s e By hyoess 8, e obaby ensy uncon o e s oe sascs s ven by s 9 s Fo 9, 0, 5 an 9 we e E an 0 E sn 0 an n e vaance o e o ecuen o Case o oe I s obane. Case : Suose a. In s case s s he obaby ensy uncon o e ven by s =! s λ + s λ +.s λ Subsun n 9 we e s s s s Now, s s0 oe sascs s E an E 7 8 Subsun 7 an 8 n e vaance o e o ecuen o Case o oe I s obane. MODEL II Hee, Y oows e eene eonena sbuon w aaee θ w shae aaee. In s case, F P F he obaby sbuon uncon o s ven by 0 he Laace anso o ensy uncon o s s s s s s Case : Poceen as Case o oe I, usn euaons 0,, 5, 9 an we e E 4 E 4 an sn an 4 n, e vaance o e o ecuen o Case o oe II s obane. an s s s0 Fo 9, 0, 5,, 4, 5 an 6 we e 6 Case : Poceen as Case o oe I, usn euaons 0, 5,,, 5, 6 an we e E an 5 56

4 E 6 Subsun 5 an 6 n, e vaance o e o ecuen o Case o oe II s obane. Moe III Hee, Y oows e SCBZ oey w aaee θ, θ an θ. In s case, F P 7 an w. he obaby sbuon uncon o s ven by F L 8 9 he Laace anso o ensy uncon o s s s s 40 Case : Poceen as Case o oe I, usn euaons 0, 5, 9 an 40 we e E 4 an E 4 Euaons, 4 an 4 ve e ean e o ecuen o Case o oe III. Case : Poceen as Case o oe I, usn euaons 0, 5,,, 5, 6 an 40 we e E an 4 E 44 Subsun 4 an 45 n, e ean e o ecuen o Case o oe III s obane. NMERICAL ILLSRAION he nuence o noa aaees on e eoance easues naey ean an vaance o e e o ecuen s sue nuecay. he aaees 0. 0., 0.,.4, 0 an e sae szes an 5 n ae e.

5 abe. Eec o on e eoance easues E an V Moe I Moe II Moe III G G G G G G X X X X X X E V E V E V E V E V E V abe. Eec o on e eoance easues E an V Moe I Moe II Moe III G G G G G G X X X X X X E V E V E V E V E V E V FINDINGS a b Fo abe, nceases, e aveae oss o an hou eceases an hence e ean an vaance o e o ecuen ncease when e oss o an hou ocess os s oe as we as n oe sascs. Fo abe, nceases, e aveae ne ecson es ecease an hence e ean an vaance o e o ecuen ecease when e CONCLSION ne-ecson e ocess os s oe as we as oe sascs. Fo e nueca usaon we concue a oe I s eeabe an oes II an III. In oe I, e o ecuen s osone when oss o an hou ocess an e ocess o ne-ecson es os s oe sascs. 56

6 REFERENCES [] D.J. Baoew, Sochasc oes o soca ocesses, n Eon, John Wey & sons, New Yo, 97. [] D.J. Baoew, an A.F. Fobes, Sasca echnues o anowe annn, John Wey & sons, Chchese,979. [] K. Eanovan, B. Ese Caa, an A. Snvasan, Vaance o e o ecuen o a sne ae anowe syse w coeae ne-ecson es an wasaes as oe sascs, Annas o Manaeen Scence, 5, 49-57, 07. [4] R.C. Gno an K.. Masha, Manowe annn Moes, No Hoan, New Yo, 977. [5] J. Guan, Dsbuon o e au o e aec ean o coeae ano vaabes, Ann. Ma. Sascs,6, 94-00, 995. [6] S. Kan an H.M. ayo, A s couse n sochasc ocesses, Acaec Pess, New Yo, San Fancsco, Lonon, 975. [7] A. Muayan, A. Suaan, A Sochasc oe o esaon o eece e o ecuen une coeae wasae, a Sc. Phys. Sc., M, 4-48, 009. [8] A. Muayan, A. Suaan an R. Sayaooh, A Sochasc oe base on oe sascs o esaon o eece e o ecuen, Aca Cenca Inca, 5, , 009. [9] Sayaoo R an Eanovan R. A Shoc oe aoach o eene e eece e o ecuen, Jouna o Decson an Maeaca Scences, -, 67-68, 998. [0] R. Sayaoo an S. Paasaay, On e eece e o ecuen when esho sbuon has SCBZ oey, Inenaona Jouna o Manaeen an Syses, 9, -40, 00. [] J. Shaan, K. Eanovan an A. Snvasan, Eece e o ecuen n sne ae anowe syse une coeae wasae, Inenaona Jouna o Innovave Scences, Enneen an echnooy, 7, 79-85,

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