ISSN: [Yamuni* et al., 5(9): September, 2016] Impact Factor: 4.116

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1 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 JESRT NTERNTONL JOURNL OF ENGNEERNG SCENCES & RESERCH TECHNOLOGY BULK RRL RETRL UEUE WTH FLUCTUTNG OES OF SERCE ETE FEEBCK SERER CTON N ORBTL SERCH. Yau K. Kua a. K. Uaya Chaa 3 offesso a Hea eatet of atheatcs ashlga Uesty Cobatoe a O:.58/eoo.554 BSTRCT Ths ae eals wth sgle see etal queue wth fluctuatg oes of sece. See oes oes of sece wth ffeet sece ates. t the aal eoch of a batch f the see s le the oe custoe etes fo sece a othes jo the obt. Othewse all the custoes jo the obt. fte coleto of sece the usatsfe custoes ot fo e-sece. t the coleto eoch of each sece the see taes a sgle Beoull acato. fte acato coleto the see ay seach fo custoe the obt. The suleetay aables coesog to etal te sece te a acato te ae cooate to etee the queue se stbuto. The ea ube of custoes the obt the ea ube of custoes the syste a syste obabltes ae obtae. Fally the stochastc ecoosto law s efe. The aalytcal esults ae alate wth the hel of uecal llustatos. KEYWORS: Retal ueue Fluctuatg oes eate Feebac acato a Obtal Seach. NTROUCTON ueueg systes wth eeate attets ae chaactee by the heoeo that a custoe fg all the sees busy uo aal s oblge to leae the sece aea a eeat the equest fo sece afte soe ao te. Retal queues hae bee wely use to oel ay obles telehoe swtchg systes telecoucato etwos a coutes coetg to ga sece fo a cetal ocesso. The ecet bblogahes o etal queues ca be fou Fal (99) Fal a Teleto (997) a talejo (999 ). Re-sece s a eal lfe heoeo whee custoes eceg soe of sece ay ee to eeat o ea e-sece fo the sece tae. Re-sece was tally stue by aa et al. (4). Recetly the cocet of e-sece has bee stue by Taj a Ke (8) Kua a uugaatha () a Bauah et al. (3). the etal setu each sece s ecee a followe by the see s le te because of the goace of the status of the see a obtal custoes by each othe. See s le te s euce by the toucto of seach of obtal custoes eately afte a sece coleto. The seach fo custoes eately o teato of a sece was fst touce the classcal queue by Neuts a Raalhoto (984). Obtal seach afte sece coleto hae bee estgate by talejo et al. () Kshaoothy et al. (5) Sutha a Uaya Chaa () a Rajaua et al. (5). ths ae we aalyse Bul aal etal queue wth fluctuatg oes of sece eate feebac (esece) see acato a obtal seach. OEL ESCRTON Cose a sgle see etal queueg syste whch custoes ae batches accog to a coou osso ocess wth ate. The batch se Y s a ao aable wth stbuto fucto (Y=) = C =... a obablty geeatg fucto C() hag fst two oets a. The see oes heteogeeous oes of sece a the obablty of og oe sece s ( ). f a ag batch fs the see fee oe of the custoes the batch begs ay oe of the oes a the htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [76]

2 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 est jo the obt. te-etal tes hae a abtay stbuto fucto () esty fucto a() Lalace Steltje s tasfo (s) a cotoal coleto ate η() = a() [ ()]. The sece te of oe (= ) follows a geeal stbuto wth stbuto fucto B () esty fucto b () Lalace Steltje s tasfo B () = b () [ B ()]. (s) th factoal oets a cotoal coleto ate t the coleto of each sece the see taes a sgle acato wth obablty. The acato te s geeally stbute wth stbuto fucto () esty fucto () Lalace Steltje s tasfo (s) o wats fo the et custoe wth coleetay obablty th factoal oets a cotoal coleto ate γ () = () [ ()]. fte coleto of oe sece the custoe ay ot fo the sae sece wth obablty o leae the syste wth ts coleetay obablty (- ). ths case t s assue that the custoes ae allowe to eeat the sece oly oce. fte the coleto of acato f the obt s o-ety the see seaches fo the custoes the obt wth obablty o eas le wth obablty. STEY STTE STRBUTON Let N (t) eote the ube of custoes the obt at te t a C(t) eote the state of the see efe as f the see s le f the see s busy oe sece C(t) = f the see s busy oe e sece f the see s o acato The state of the syste at te t ca be escbe by the ao ocess X t;t C t N(t) ξ (t) ξ (t) ξ (t) ξ 3 (t); t }. f C(t)= the ξ (t) eesets the elase etal te f C(t)=( ) ξ (t) eesets the elase sece te f C(t)=+ξ (t) eesets the elase e-sece te a f C(t)=+ ξ 3 (t) eesets the elase acato te. efe the followg obablty estes (t) = {C(t) = N(t) = } ( t) = {C(t) = N(t) = < + } ( t) = {C(t) = N(t) = ξ ξ ( t) = {C(t) = + N(t) = ( t) = {C(t) = + N(t) = t t < + } =. ξ t < + } =. < + } ξ t Let () () () a () be the steay state obabltes of (t) ( t) ( t) ( t) a ( t) whee =. The syste of equlbu equatos goeg the oel s ge below γ () η () htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [76]

3 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [76]... c (3)... c (4) (5) C γ wth bouay cotos ) (6 γ.... γ η C (7) (8)... ; γ C η C... (9) ) ( The oalg coto s () efe the obablty geeatg fuctos

4 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 ; ; ;... a ultlyg equatos () to () by a sug oe fo all ossble alues of we obta the followg esults η () c... c... c γ (5) γ (6) η C (7)... γ B h (8) B h B h (9) whee h c Solg the atal ffeetal equatos () to (5) we get e () e B c e B c c e Usg equatos (7) (8) (9) () () a (3) equato (6) a slfyg we get CT h h T whee (3) (4) () () (3) (4) htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [763]

5 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 T B h B h h T C T h Usg equato (4) the equato (7) becoes c... (5) setg equato (5) equato (8) a (9) we get B h c... (6) C T (7) Substtutg the eessos of ( ) ( ) ( ) a ( ) equatos () () () a (3) we get the followg esults C T h h T e (8) = h C e B... (9) B hc e B h h... C T e (3) (3) The atal obablty geeatg fucto of the obt se whe the see s le s ge by () = ( ) CT h h T (3) The atal obablty geeatg fucto of the obt se whe the see s busy s ge by B() = [ () + ()] = B h B h... (33) The atal obablty geeatg fucto of the obt se whe the see s o acato s ge by = T h (34) obablty geeatg fucto of the ube of custoes the obt s ge by q = (35) obablty geeatg fucto of the ube of custoes the syste s ge by htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [764]

6 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [765] s = T (36) ERFORNCE ESURES obablty that the see s le s ge by = l () (37) whee obablty that the see s busy s ge by B = l Z B() (38) obablty that the see s o acato s ge by = l () (39) Usg oalg coto s obtae as (4) ea ube of custoes the obt L q ue steay state coto s ge by L q = l q() = N N (4) whee N() a () ae the Nueato a eoato of q (). N The ea ube of custoes the syste s ge by

7 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 L s = l s() = L q + B (4) STOCHSTC ECOOSTON Theoe: The ube of custoes the syste (L s ) ue steay state ca be eesse as the su of two eeet ao aables oe of whch s the ea ube of custoes (L) the classcal batch aal queueg syste wth fluctuatg oes of sece eate feebac a obtal seach a othe s the ea ube of custoes the obt (L ) ge that the see s le o o acato. oof: The obablty geeatg fucto π () of the syste se the classcal batch aal queue wth fluctuatg oes of sece eate feebac a obtal seach s ge by π T T (43) The obablty geeatg fucto acato s ge by ψ ψ T of the ube of custoes the obt whe the syste s le o o (44) Fo equatos (36)(43) a (44) we see that s () = π(). ψ() (45) ffeetatg (45) wth esect to a tag lt as we get L s = L + L NUERCL RESULTS ssue that the etal te sece te a acato te follow eoetal stbuto wth aaetes η a γ. esectely. Nuecal aalyss ae cae out wth stablty coto by settg =.3 η = 5 µ = µ = 3 =.4 =.4 =.7 =.3 =.5 = γ = 7 =.6. The fluece of aaetes ( ) ( ) a (η) o the efoace easues + a L s ae slaye Fg.4. to 4.3. Fo the fgues t s obsee that obablty + that the see s le eceases fo ceasg alues of a ceases fo µ a eeet of η a ea ube of custoes the syste L s ceases fo ceasg alues of a a eceases fo µ η a htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [766]

8 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS L s Ls u.4.4 µ u l l. µ.7.8 Fg.4. Effect of ( ) o + a Ls L s Ls t t...8 Fg.4. Effect of ( ) o + a Ls L s Ls th. 3 th. 3 η η Fg.4.3 Effect of ( η ) o + a Ls CONCLUSON sgle see bul aal etal queueg syste wth fluctuatg oes of sece eate feebac see acato a seach of custoes fo the obt s stue. Elct esults fo the obablty geeatg fuctos a othe steay state syste efoace easues ae ee. Nuecal aalyss s efoe. htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [767]

9 SSN: [Yau et al. 5(9): Setebe 6] act Facto: 4.6 C alue: 3. COEN: JESS7 REFERENCES [] J.R. talejo Classcal Bblogahy of Reseach o Retal ueues: ogess (999) To [] J.R. talejo.c. Joshua a. Kshaoothy /G/ Retal ueue wth Obtal Seach by the See aces Stochastc oellg Notable ublcatos c. NJ () [3] J.R. talejo ccessble Bblogahy o Retal ueues: ogess -9 atheatcal a Coute oellg () [4]. Bauah K.C. aa a T. Elab [] /(G G )/ acato ueue wth Balg a Otoal Re-sece le atheatcal Sceces (3) [5] G.. Fal Suey of Retal ueues ueueg Systes (99) [6] G.. Fal a J.G.C. Teleto Retal ueues Loo Chaa a Hall (997). [7]. Kshaoothy T.G. eea a.c. Joshua /G/ Retal ueue wth No-esstet Custoes a Obtal Seach Stochastc alyss a lcatos (5) [8] J.S. Kua a R. uugaatha No-aoa Bul ueue wth ultle acatos o Request fo Re-sece ualty Techology of uattate aageet () [9] K.C. aa. l-nassa ja a beel-ae l-as O [] /(G G )/ ueue wth Otoal Re-Sece le ath Coutato (4) [].F. Neuts a.f. Raalhoto Sece oel whch the See s Reque to Seach fo Custoes Joual of le obablty (984) []. Rajaua K. ha.. Chaaseaa a.c. Saaaaaja alyss of a X /G/ Feebac Retal ueue wth Two hase Sece Beoull acato elaye Rea a Obtal Seach aces hyscs Theoes a lcatos (5) []. Sutha a K. UayaChaa Retal queueg syste wth Statg Falue Sgle acato a Obtal Seach teatoal Joual of Coute lcato () [3] L. Taj a J.C. Ke Hysteetc Bul ueue wth a Choce of a Sece a Otoal Re-sece ualtate Techology of uattate aageet (8) htt: // teatoal Joual of Egeeg Sceces & Reseach Techology [768]