International Journal of Scientific & Engineering Research, Volume 3, Issue 10, October ISSN

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1 Ieraoal Joural of cefc & Egeerg Research, Volue, Issue 0, Ocober-0 The eady-ae oluo Of eral hael Wh Feedback Ad Reegg oeced Wh o-eral Queug Processes Wh Reegg Ad Balkg ayabr gh* ad Dr a gh** *Assoc Prof RKD (PG )ollege, Kahal-607(Haryaa), Ida sayabrsghehla@galco ** Prof of aheacs(red), Dep of aheacs & ascs,, Haryaa Agrculural Uversy, Hsar 5004 (Haryaa), Ida Absrac - Ths Paper cosders he seady-sae behavor of seral queug processes wh feedback ad reegg coeced wh o-seral queug chaels havg reegg ad balkg whch () seral queug process wh feedback ad reegg are coeced wh o-seral chael wh reegg ad balkg () A cusoer ay o ay chael fro ousde ad leave he syse a ay sage afer geg servce () Feedback s pered fro each chael o s prevous chael seral chaels (v) The pae cusoer ay leave he servce facly seral ad o-seral chael afer a wa of cera e ad balkg has bee corporaed o-seral chaels oly (v) The Ipu process deped upo queue sze oseral chaels ad Posso arrvals ad expoeal servce es are followed (v) The queue dscple s rado seleco for servce (v) Wag space s fe Expressos for ea queue leghs have bee derved Key Words : eady-ae, wag space, seral, o-seral, rado seleco, Posso arrvals, expoeal servce, feedback, balkg, reegg, argal probables ad ea queue legh ITRODUTIO o Bre [6], Jackso [] ad Hu [4] suded he probles of seral queues he seady sae wh Posso assupos I hese sudes, s assued ha he u us go hrough each servce chael whou leavg he syse Barrer [] obaed he seady-sae soluo of a sgle chael queug odel havg Posso pu, expoeal holdg e, rado seleco roducg reegg Fch [] suded sple queues wh cusoers a rado for servce a a uber of servce saos seres wh feedback gh [8] suded he proble of seral queues roducg he cocep of reegg Pua, gh ad Ashok [7] foud he seady-sae soluo of seral queug processes where feedback s o pered Forulao of he odel The syse cosss of he seral queues Q,,,, ad o-seral chaels wh respecve servers,,,, ad,,, usoers deadg dffere ypes of servce arrve fro ousde he syse Posso srea wh paraeers,,,, ad a Q,,,,,,, ad bu he sgh of log queue a ay dscourage he fresh cusoer fro og ad ay decde o o eer he servce chael a The he Posso pu rae a (,,, ) Q would be where s he queue sze IJER 0 hp://wwwserorg of Q,,,, Furher, he pae cusoer og ay servce chael Q,,,, ad ay leave he queue whou geg servce afer wa of cera e ervce e dsrbuos for servers,,,, ad,,, are uually depede egave expoeal dsrbuo wh paraeers,,, ad,,, respecvely Afer he copleo of servce a, he cusoer e- p or os he ex her leaves he syse wh probably chael wh probably q or o back he prevous chael wh probably r such ha p q r,,, ad afer he copleo of servce a he cusoer eher leaves he syse wh probably o back he prevous chael wh probably r or o ay queue Q,,, wh probably q,,, such ha p q r p or I s beg eoed here ha r 0 for as here s o prevous chael of he frs chael The applcaos of such odels are of coo occurrece For exaple, cosder he adsrao of a parcular dsrc a parcular sae a he level of dsrc head quarer cossg of Block Develope offcer, Tehsldar, ub- Dvsoal agsrae, Dsrc agsrae ec These offcers correspod o he servers of seral chaels Educao De-

2 Ieraoal Joural of cefc & Egeerg Research, Volue, Issue 0, Ocober-0 pare, Healh Depare, Irrgao Depare ec coeced wh he las server of seral queue correspod o oseral chaels The people ee he offcers of he dsrc coeco wh her probles I s also a coo pracce ha offcers call he cusoers (people) for hearg radoly The seor offcer ay sed ay cusoer o hs uor f soe forao regardg he cusoer s proble s lackg Furher Dsrc agsrae ay sed he cusoers o dffere depares such as Educao, Healh, Irrgao ec f here probles are relaed o such depares The cusoer afer seeg log queues before ay o-seral servce chael ay decde o o eer he queue I geerally happes ha perso becoes pae afer og he queue ad ay leave he chael whou geg servce Forulao of Equaos: Defe: P,,,,,,,,, ; probably ha a e here are = he cusoers (whch ay leave he syse afer servce or o he ex phase or o back he prevous chael or reege) wag before,,,,, ; cusoers (whch ay balk or reege) wag before he severs,,, We defe he operaors T, T, T,, T, o ac upo he vecors or,,, as fallows,,, T,,,,, T,,,,,, T,,,,,,,, T,,,,,,, Followg he procedure gve by Kelly [5], we wre he dfferece dffereal equaos as d P, ; d d P T, ; p P T, ; P, ; q P T,,, ; P, T ; d P, T ; For 0,,,,, 0,,, where x f x 0 0 f x 0 ; ad P, ; 0 f ay of he argues egave 4 eady ae Equaos We wre he followg eady sae equaos of he queug odel by equag he e-dervaes o zero he equao () d P, P T, q P T,, p P T, r PT,, q P,,, T,, () P T d P T For 0,,,,, 0,,, ; 5 eady-ae oluos The soluos of he eady-ae equaos () ca be verfed o be q P T,, ; r PT,, ; IJER 0 hp://wwwserorg

3 Ieraoal Joural of cefc & Egeerg Research, Volue, Issue 0, Ocober-0 r P, P0,0 q r q 4r q r q q q! d! d q! d 0,,,, (4), 0,,, Where r q r q 4r q r (4) q olvg hese (4) -equaos for wh he help of deeras, we ge q q q q q q q q q q q q q q r where q r oug hs way Where q r (4) (44) r q r r4 4 q q r q r r q q ce s obaed, we ca ge by pug he value of 4, he las equao of ad by pug he values of he las bu oe equao of 4 coug IJER 0 hp://wwwserorg

4 Ieraoal Joural of cefc & Egeerg Research, Volue, Issue 0, Ocober-0 4 hs way, we shall oba, 4,,,, ad, Thus, we wre 4 as uder P, P 0,0 q q q P0,0 e! d! d! d q Where,,,, for 0,,,,,,,, 45 d We oba P 0,0 fro he oralzg codos, 0,0 P P q q! d! d (47) q q! d! d We oba P 0,0 fro (46) ad (47) as Thus P, s copleely deered P, (46) 0, 0 ad wh he resrco ha raffc esy of each servce chael of he syse s less ha uy, ad d are he reegg raes a whch cusoers reege afer a wa of e cusoers he servce chael T0 e T0,,, e e T0 ad d,,, T0 e T 0 wheever here are Q ad Q ad Here s eoed ha he cusoers leave he syse a cosa rae as log as here s a le, provded ha he cusoers are served he ordered whch hey arrve Pug,,, ad d d,,, he seady-sae soluo (4)we ge 6 eady-ae argal Probables e P be he seady-sae argal probably ha here are us he queue before he frs server Ths s deered as P, P,, 0 0! d! 0,0 P,, 0 0 q q q! d! d Thus P larly P q d IJER 0 hp://wwwserorg

5 Ieraoal Joural of cefc & Egeerg Research, Volue, Issue 0, Ocober P Furher, le P, P, P,, P be he seadysae argal probables ha here are,,,, cusoers wag before server (,,,, ) respecvely P, P 0,, 0 0,0 P 0,, 0!!! P! e larly P ; 0! e P! e ; 0 ; 0 ea queue legh before he server s deered as larly ;,,, Hece ea queue legh of he syse s k k Refereces [] Barrer, DY (955): A wag le proble characerzed by pae cusoers ad dffere clerk Jourals of Operaos Research ocey of Aerca Vol : 60 [] Fch, PD (959): yclc queues Ops Res Quar, Vol 9, o [] Jackso, RR P(954) : Queug syse wh phase ype servce opera Res Quar 09-0 [4] Hu, G (955) : equeal arrays of wag les opps Res Vol 4, o 6 [5] Kelly, FP (979): Reversbly ad sochasc eworks Wley, ew York [6] O Bre, GG (954) : The soluo of soe queug probles Joc Id Appl ah Vol, -4 [7] Pua; gh; ad Ashok (0): The eady-ae soluo of seral queug processes wh reegg due o log queue ad urge essage Appled scece perodcal Vol XII, o [8] gh (984): eady-sae behavour of seral queug processes wh pae cusoers ah operaos forsch, Usas er asvol 5o, ea Queue egh ea queue legh before he server s deered by 0 0 P larly IJER 0 hp://wwwserorg