Applied Mathematical Sciences, Vol. 7, 2013, no. 17, HIKARI Ltd, Optional Re-service. Monita Baruah

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1 le Mathematcal Sceces ol. 7 3 o HK Lt M X G G acato ueue wth alkg a Otoal e-sevce Mota auah College of Scece Uvesty of aha Kgom of aha mauah@uo.eu.h motah@gmal.com coesog autho Kalash C. Maa College of fomato Techology hla Uvesty Kgom of aha kmaa@ahla.eu.h kcmaa@yahoo.com Tllal la uel usess School uel Uvesty Ute Kgom Tllal.la@uel.ac.uk stact The eset ae ams at stuyg a queug moel wth two stage heteogeeous sevce whee custome aval s atches a has a sgle seve ovg sevce two stages oe afte the othe successo. sevce s sa to e comlete whe oth sevces ae oe. lso the seve s assume to go fo vacato afte comleto of the two stages of sevce wth oalty o may cotue sevg the et custome wth oalty -. ato we have clue alkg whch eflects custome s matet ehavo whee a avg atch eces ot to o the queue fo some easos. We also ae otoal e-sevce whee a custome may eque e-sevce fo ay of the sevces o eat fom the system oce sevce s comlete. We eve the steay state queue se stuto a scuss some secal cases. Keywos: ueue se atch val Two-Stage Heteogeeous Sevces alkg Otoal e-sevce

2 838 M. auah K. C. Maa a T. la toucto ueues wth seve vacatos have gae coseale motace the aea of eseach ecet yeas. t has emege as a motat aea of stuy a eseaches has ee elog ffeet cocets of queug moels wth vacatos. Most of the lteatue queug wth atch aval a vacato has ee stue y emet authos lke Lee et al 9]Takag 8] Shathkuma J.G ] Chouhuy G. & Maa K.C.6] Kelso a Sev ] to quote a few. mao cotuto stues o two stage atch aval queug system s see y authos lke Chouhuy G a Maa K.C. 7]. the last few yeas we see stues o queues wth alkg a eegg gag sgfcat motace. ueug system wth matet customes s a famla heomeo we come acoss may eal lfe stuatos. Customes may e scouage to o a queue ue to log watg tme o sevce tmes o fo othe costats a may leave the queue wthout og. Ths s alkg. We see alcatos of queue wth alkg emegecy sevces hostals ealg seous atets commucato systems oucto a vetoy system a may moe. queue wth alkg was tally stue y Haght 5]. Sce the etesve amout of wok has ee oe o queug systems elate to matet customes. ueues wth alkg has ee stue y authos lke ltma. a Yechal U. 4] cke et al ] Chouhuy. a Meh ] Zhag et al 8] the last few yeas. Most of the authos metoe hee have stue sgle seve moels. e-sevce s also a eal lfe heomeo whee customes ecevg some k of sevce may ee to eeat o ema e-sevce fo the sevce take. atets mght ema e-sevce whle vstg a octo s clc maufactug uts a usty whle secto may sot some efects a as a esult may eque e-sevce a may eal lfe alcatos. e-sevce was tally stue y Maa 5].ecetly the cocet of e-sevce has ee stue y authos lke Ta a Ke 6] Jayakuma a umugaatha 7]. Though we see few woks o e-sevce t s a motat asect of queug theoy a ca e stue moe etesvely. Hee we stuy the steay state ehavo of a ulk queue system havg two-stage geeal heteogeeous sevce alog wth alkg a e-sevce wth seve X vacatos. We may thus eote ou moel as M G G s L O. We efe the mathematcal moel ue the assumtos gve elow.

3 M X G G vacato queue 839. The Mathematcal Moel Customes ave atches followg a comou osso ocess wth ate of aval. Let a t 3. e the fst oe oalty that customes atches of se ave at the system at a shot teval of tme t] whee a a a The seve oves two stages of heteogeeous sevces oe afte the othe. aval atch shall fst eceve the sevce offee y seve followe y the sevce offee y seco seve efe as the fst stage FS a seco stage SS sevce esectvely. The sevce scle s assume to e o a fst come fst seve ass FCFS. We assume that the sevce tme S of the th stage sevce follows a geeal oalty stuto wth stuto fucto s s eg the oalty esty fucto a as the th momet of the sevce tme S Let μ e the cotoal oalty of stage sevce ug the eo t] gve elase tme s such that μ s a s μ e μ Oce the seco stage sevce SS of a ut s comlete the seve s assume to take vacato wth oalty o may cotue to offe sevce wth oalty -. s soo as the vacato eo of the seve s ove he os the system to cotue sevce of the watg customes. We assume the vacato tme to e a aom vaale followg geeal oalty law wth stuto fucto gve y W v a esty fucto y wv a v s the th momet.

4 84 M. auah K. C. Maa a T. la Hee we assume that φ e the cotoal oalty of a vacato eo ug the teval ] gve that elase tme s so that a thus W φ 3 W v w v φ ve φ 4 ato we assume that customes avg fo sevce may ecome matet y osevg the queue se o the seve eg usy may alk efuse to o the system. Hee we assume that s the oalty that avg atch os the system at the tme whe seve s usy a s the oalty that a avg atch os the system ug the eo whe seve s o vacato. lso we assume that as soo as seco stage sevce s comlete the custome has the choce to leave the system o o the system aga fo e-sevce f ecessay. custome may eeat the sevce the two stages wth oalty o may leave the system wth oalty. 3. eftos a Notatos We assume that steay state ests a efe oalty that thee ae customes the system clug Thus oe custome tye sevce a elase sevce tme s s the coesog steay state oalty esectve of elase tme. oalty that thee ae customes the system clug oe custome who s eeatg sevce. a elase sevce tme s.

5 G G M X vacato queue 84 ccogly s the coesog steay state oalty esectve of elase sevce tme. oalty that thee ae customes the queue a seve s o vacato a elase vacato tme s. s the coesog steay state oalty esectve of elase vacato tme. Steay state oalty of the seve s le as the seve takes vacato The oalty Geeatg Fuctos ae efe as: 5 ; 6 ; ; 7 a ; 4. quatos goveg the system Ue ths moel we costuct the ffeetal equatos as ; a μ 8 a ; μ 9

6 84 M. auah K. C. Maa a T. la { μ } a ; { } μ a ; { φ } a ; { φ } 3 μ φ 4 whee occug the equatos 8- gve aove. The aove ffeetal equatos ow have to e solve suect to the followg ouay cotos: a μ μ φ μ μ 6 μ 7 μ 8 μ 9 5

7 M X G G vacato queue Steay State ueue stuto Multlyg equatos 8- y summg ove sutale values of a smlfyg we get { μ } { μ } { μ } { μ } 3 { φ } 4 tegatg equatos 9-3 etwee lmts to we ota e μ t t ] 5 e μ t t ] 6 6 e μ t t ] 7 e μ t t ] 8 φ t ] e t 9 quatos 5-9 hol fo all >

8 844 M. auah K. C. Maa a T. la We et multly the ouay cotos y sutale owes of a takg summato ove all ossle values of a usg 4 we get afte smlfcato φ μ μ 3 μ μ 3 μ 3 μ 33 μ 34 ga multlyg equato 5 y μ a tegatg y ats w..t. etwee the lmts to we ota μ 35 oceeg smlaly fo equatos 5-8 we ota μ 36 μ 37 μ 38 W φ 39

9 G G M X vacato queue 845 Now utlg elatos equatos 3-34 we get W ] 43 ] 44 Whee e s the Lalace Tasfom of fo a W e W s the Lalace Tasfom of W. Susttutg the values fom a 44 4 we get ] ] ] } ] { ] W W

10 846 M. auah K. C. Maa a T. la fte smlfyg we get ] W 45 ga susttutg fom we ota ] } { W 46 thus ] ] W 47

11 G G M X vacato queue 847 ] } { W 48 ] } { ] W 49 We futhe tegate equatos 5-9 w..t a afte utlg elato ota ] 5 Smlaly ] ] } } ] { { 5

12 848 M. auah K. C. Maa a T. la ] ] ] 5 ] ] ] 53 ] ] ]} { ] ] W 54 whee W ] ] ] ] ]} { ] ] ]} { 55 Let q e the oalty geeatg fucto esectve of the tye of sevce eg ove y the seve such that we have N q 56 To f we use the omalg coto Sce the. H. S of 55 s etemate of the eoeo fom at alyg L Hotal s ule we ota

13 G G M X vacato queue 849 ] lm q q 57 Whee s the mea se of avg customes s the mea of sevce tme of fst stage sevce s the mea sevce tme of seco stage sevce a s the mea of vacato tme a W. Now usg the elato q we ota ] 58 a ρ gves ] ρ < 59 s the stalty coto ue whch steay state ests. Now let s e the steay state queue se stuto. Usg the aove elatos we ota

14 85 M. auah K. C. Maa a T. la ] ] ]} { ]} ]{ ]} ] { } { ] ] ] a W S 6 a s gve y equato 55 Thus elato 6 gves the steay state queue se of a O L G G M s X queug system. 6. Mea ueue Se a Mea Watg Tme ueue Hee we eve the mea queue se of ths O L G G M s X queue. Let L eote the mea queue se at aom eoch the q L 6 Usg L Hotal s ule twce as the. H.S of q s etemate of the fom lm N N L 6 whee mes a oule mes eote the fst a seco evatves at Smlfyg 56 we get ] N

15 G G M X vacato queue 85 ] ] N 3 ` whee ae the seco momets of fst stage a seco stage esectvely s the seco momet of vacato tme a s the seco factoal momet of se of avg atch of customes a W has ee otae 58 Thus we ca eve the mea queue se of the system y usg the elato L S s at.

16 85 M. auah K. C. Maa a T. la lteatvely we ca f L S L ρ Futhe we ca ota the aveage watg tme the queue a system y the elatos L L W a W S 6. atcula cases:. M X G G acato ueue wth alkg a o e-sevce Hee we cose a queug system wth two stage heteogeeous sevce vacato a alkg ut wthout e-sevce thus takg ou ma esults we ota s ρ W ] W ] { } ] { } ] N N { } ] { } { } ] { } ] { } ] { } 63 Thus equato 63 gves the steay state queue se of a M X G G vacato a alkg. wth. M X G G acato ueue wth o e-sevce a o alkg

17 M X G G vacato queue 853 Hee we cose the aove elato 59 a queug system offeg o e-sevce a avg customes oes ot alk the ou GF euces to ρ W ] W ] S ] { } ] N N { } ] ] { } ] { } ] { } 64 The elato 63 gves the steay state queue se of a wthout alkg a e-sevce. X M G G s 3. M X G G wth o vacato o alkg a o e-sevce We take.e. we cose a queug system whee the seve oes ot go fo vacato the avg atch oes ot alk a the system oes ot offe e-sevce the ou GF 6 ecomes ρ S { } { }] N { } { }] N 65

18 854 M. auah K. C. Maa a T. la ] ] ] The elato 65 gves the steay state queue se of a M X G G ueue. 4. M X G ueug system wth sgle seve ato to the aove case 3 f we assume that thee s o seco stage sevce we acheve the esult y otae y Gave 3] as ρ N S N ] 66 The aove elato66 gves the steay state queue se of a M X G queue. efeeces.. Chouhuy. Meh alkg a eegg Multseve Makova ueug Systems. teatoal Joual of Mathematcs Oeatoal eseach C.J. cke J... Gafaa Some queug olems wth alkg a eegg. Oeatos eseach Gave mee makov cha aalyss of a watg le ocess cotuous tme.. Mathematcal Statstcs ltmau. Yechal alyss of customes matece queue wth seve vacatos ueug Systems F.. Haght ueug wth alkg ometka

19 M X G G vacato queue G. Chouhuy K. C Maa Steay State alyss of a M X G G queue wth estcte msslty a Mofe eoull Seve acatos. Joual of oalty a Statstcal Scece G. Chouhuy K. C. Maa Two-Stage atch val ueug System wth aom Set u Tme ue Mofe eoull Scheule acato. evsta vestgaco Oeacoal ol. 6 No H. Takag Tme-eeet aalyss of a MG acato Moels wth haustve sevce. ueug Systems H. W. Lee S. S. Lee J. O. ak a K. C. Chae alyss of M X G ueue wth N- olcy a Multle acatos. Joual of le oalty J. Kelso L.. Sev yamcs of the MG vacato moel. Oeato eseach J. G. Shathkuma O Stochastc ecomosto the MG Tye ueues wth Geeale acatos. Oeatos eseach J.. taleo G. Chouhuy Steay state alyss of a MG ueue wth eeate ttemts a Two- hase Sevce. ualty Techology of uattatve Maagemet. : K. C. Maa MG ueue wth seco otoal sevce. ueueg Systems K. C. Maa O a Sgle seve ueue wth Two- Stage Heteogeeous Sevce a eoull Scheule Seve acatos. gyta Statstcal Joual K. C. Maa.. l-nasse.. l Mas O M X G G ueue wth Otoal e-sevce. le Mathematcs a Comutato.lseve

20 856 M. auah K. C. Maa a T. la 6. L. Ta & J.C. Ke Hysteetc ulk ueue wth a Choce of a Sevce a Otoal e-sevce. ualtatve Techology of uattatve Maagemet S. Jeyakuma. umugaatha No-Makova ulk queue wth Multle acatos o equest fo e-sevce. ualty Techology of uattatve Maagemet. ol 8 No Y. Zhag. Yue W. Yue alyss of a MMN ueue wth alkg eegg a Seve acatos. teatoal Symosum o O a ts lcatos.5 eceve: Octoe