An Analysis of Queueing Model with a Multiple System Governed by a Quasi-Birth Death Process and PH Service and PH Repair

Transcription

1 Itertol Jourl of Scece d Reserch (IJSR) ISSN (Ole): Idex Copercus Vlue (3): 64 Impct Fctor (4): 56 A Alyss of Queueg Model wth Multple System Govered y Qus-Brth Deth Process d PH Servce d PH Repr M Re Sgy R, B Chdrsekr, Deprtmet of Mthemtcs, Scred Hert College(Autoomous), Trupttur , Vellore Dstrct, Tml Ndu, SId Astrct: I ths pper we study the multple systems of repr prolem d preset N-system wth oe workg d the other systems re repr, stdy or wtg repr Whe the workg system c fls t goes to repr d stteously stdy system ecomes the workg oe Stdy system stds to system whe the orgl system s o uder repr The systems re repr followg the rrvl order We let the tme etwee servce d repr tmes hve phse-type dstruto The workg system hs lfe tme govered y phse-type dstruto We show the process tht govers the system s qus-rth d deth process, we perform stedy stte lyss of ths model A umercl exmple s preseted d the expressos clculted through the pper re computtolly mplemeted usg the Mtl progrmme Keywords: Reprm prolem, Phse-type dstruto, Qus-Brth d deth process, Mtrx Geometrc Method, Codtol prolty of flure, Sttory dstruto Itroducto We cosder N-system d reprle stdy system wth sgle repr fclty There s oe workg server d the others re repr, stdy or wtg for reprg Oly the workg system c fl Upo flure, t goes to repr d s mmedtely replced y stdy server tht ecome the workg system We let tme etwee servce d repr tmes follows phse-type dstruto If t flure of system ll repr perso re usy the the fled system hs to wt utl repr perso s vlle Repr s s good s ew The queue dscple s frst come frst served The lfe tme of the workg system hs dstruto fucto F (t) The dstruto fucto of the repr tme s gve y G (t) Both of these rdom tmes re depedet We cosder multple system volvg PH-type dstruto d uder the ssumptos tht we estlshed for the system It s lmted the lterture o rellty for ths dstruto clss NeutsMF d MeeeKS [8] studed two ut system wth opertol d repr tmes follows phse-type dstruto cold stdy d wth two repr fcltes ChkrvrthyS [] studed smlr prolem from std pot of queue theory, cosderg the queue wth sgle repr fclty servce whe there re mches prllel wth opertol tmes expoetlly dstruted d whe the repr tme follows phse-type dstruto ChkrvrthyS d NtrR [] studed -system rdom evromet modelled y Mrkov process wth geertor y locks, smlr to the oes we ppled whe cosder phse-type dstruto We show tht the ove metoed multply N-system whch phse-type dstruto re volved s govered y qus-rth d deth process(qbd) These process exted the clsscl rth d deth processes to the vectorl cse, d the tr-dgol geertor of prmeters s susttuted y lock tr-dgol mtrx Severl uthors hve cosdered QBD processes from dfferet vew pot Neuts [7] troduced oe such process Volume 5 Issue Jury 6 wwwsret to queueg theory, showg tht the sttory prolty vector of QBD process wth fte sttes c e expressed terms of mtrx tht plys mportt role the theory(rte mtrx) NeutsMF, Perez-OcoR d Terres CstroI [9] studed reprle models wth opertg d reprs tmes govered y phse-type dstruto toucheg d RmswmV [4] troduced logrthmc reducto lgorthm for clcultg the rte mtrx Phsetype dstruto re recogzed s very good pproxmtos for most geerl dstruto ecoutered the rel lfe RmswmV d TylorAG [] studed the rte of opertors level depedet qus-rth d deth processes wth coutle umer of phses Thus works, d refereces there, complete the lyss of QBD processes t preset We hve ppled some of the geerl formule from [,3,4] to our system We cosder multple system volvg PH-type dstruto d uder the ssumpto tht we estlshed for the system For ths system, the sttory prolty vector, the vllty d the rte of occurrece of flures re clculte A umercl exmples re preseted d the expressos clculted through the pper re computtolly mplemeted usg the mtl progrmme Ths pper s rrged s follows: I secto, we lyss the PH-dstruto I Secto 3, we gves the model descrpto d equto coverg of the system I secto 4, preset qus-rth d deth(qbd) model formulto I secto 5, we ot the sttory prolty vector of the system sze I secto 6, preset queue legth d wtg tme dstruto of repr for the system I secto 7, d 8, we gve the vllty of the system d rte of occurrece of flure of system I secto 9, we gve the umercl exmples Cocluso re gve secto We summrze the followg deftos used the pper Pper ID: cesed Uder Cretve Commos Attruto CC BY

2 Phse-Type dstruto Itertol Jourl of Scece d Reserch (IJSR) ISSN (Ole): Idex Copercus Vlue (3): 64 Impct Fctor (4): 56 Defto Cotuous Phse-type dstruto: et : N e Mrkov ch s defed wth stte X spce,,3,, m, m S, where the frst m sttes re trset d the lst stte s sorg, d trsto prolty mtrx T t where T s squre mtrx of dmeso m, t s colum vector d s row vector of dmeso m Sce Q s geertor mtrx, we hve T d t, S d T t Where s the colum vector of oes of the pproprte dmeso m The prolty dstruto of the tl stte s deoted wth the row vector, ) et ( m Z f ( : X m ) e the rdom vrle of the tme to the sorg stte m The dstruto of Z s clled phse-type dstruto wth represetto (, T) Note tht the kowledge of (, T) s suffcet sce t T, m The dmeso m of T s clled the order of the PH dstruto d the trset sttes {,,3,, m} re clled the phses The vector t cots the so-clled ext rtes The cumultve dstruto fucto of the PH dstruto Z : PH(, T) s gve y z FZ ( z) T for z,,, () d ts prolty desty fucto s fz () m fz ( z) e t for z,,3, Proof of equto (), tkg trsto mtrx fucto Qz Q ( z) e, we get ( Qz)! T t Z! e, e Tkg the tl dstruto m to ccout d usg expresso () for the trsto mtrx, we ot F ( z) e Z The servce tme of the systems re followg phse-type dstruto PH (, T) m mtrx The d T s m me servce tme s gve y T The repr tme dstruto s followg phse-type dstruto wth represetto (, S), where S S, S s d s the dmesol colum vector wth ll elemets () Volume 5 Issue Jury 6 wwwsret equl to The me repr tme s gve y S Upo the completo of repr, the system s s ew s oe 3 The Model Descrpto I ths secto, we frst descre the system model The we derve qus-rth d deth process of the system The symols d deote the kroecker product d the kroecker sum of two mtrces, respectvely Thus, f A s mtrx of order m m d f B s mtrx of order, the A B wll deote mtrx of order m th m whose (, ) lock mtrx s gve y B Pper ID: cesed Uder Cretve Commos Attruto CC BY For more detls o kroecker products, we refer to Bellm [3] Before we descre the Mrkov ch the reprm model, we represet revew of PH-dstruto The ssumpto of the system model re s follows Workg system c fls d goes to reprs wth the repr m dle d stdy ecomes the ew workg system The lock tht represets ths trsto s ( t ) of order ( m mk) Workg system fls d goes to the reprs queue(repr m usy) The order of the su-mtrx s mk mk t s ( t I) Whe repr s completed, the system goes to stdy d the repr ecomes dle The workg system does ot chge the phse The order of the su-mtrx s ( mk m) d t s ( I S) The reprm ecomes usy The order of the su-mtrx s ( mk m) d the mtrx s ( I S ) The trsto s govered ut the mtrx T The trsto ( > ) s covered y the mtrx S, whe there re chges mog the opertol or repr phses The lock s T S Tm Ik Im Sk order of the su-mtrx s mk mk The mtrx I, we deote detty mtrces of pproprte orders 4 Qus-Brth d Process Ths mtrx represets Qus-rth d deth process ws developed Neuts [7] to solve the sttory stte prolty for the vector stte Mrkov process wth repettve structure We develop the stedy-stte prolty The correspodg trsto rte mtrx t ths Mrkov ch hs the lock trdogol form Cosder the geertor mtrx Q s show elow

3 Where T, Itertol Jourl of Scece d Reserch (IJSR) ISSN (Ole): Idex Copercus Vlue (3): 64 Impct Fctor (4): 56 t, I S, t I, (3) T S, I S The we hve the followg stlty codto for the QBD process 5 Sttory Prolty Vector We use [,,, ] It s kow from Neuts[7], 3 where ( (,), (,),, (, k)) to deote the sttory prolty vector, whch stsfes the mtrcl equto, suect to the ormlzto codto For,,, d (, k) s the prolty of fdg the system the stte (, k) s stedy stte Ths gves the followg equto (4) 3 Suect to (5) (6) I log wth the pot stuto, t my e show tht there exts costt mtrx such tht for,3, The su-vector ech other sce X re sd to e geometrclly relted to (7) d the rte mtrx c If the su-vectors d e foud the the remg su-vectors of the sttory dstruto my e formed usg equto (7), susttutg from equto (7), we ot for e,,3, A ( ) It s ow ppret tht, t s kow from Neuts [7] tht there exts mtrx whch s the mml o-egtve soluto to the mtrx qudrtc equto Volume 5 Issue Jury 6 wwwsret Gve the rte mtrx d locks,,,, d the system my e solved to ot d The computed soluto eeds to e ormlzed so tht the compoets of sum to I other words, we sst tht Thus Pper ID: cesed Uder Cretve Commos Attruto CC BY Ths mples the codto (8) (7) The egevlues of le sde the ut crcle, whch mes tht ( I ) s o-sgulr hece tht ( I ) Ths ele us to the ormlzg of the vectors d y computg (9) (8) N d cosequetly equto (4) d (5) c e expressed s [ [ I ] ] d dvdg the computg su-vector X d X y (9) (, ) ( ) (,) I () The frst colum of the cluded mtrx s removed to vod ler depedecy The removed colum s replced the ormlzto codto Equto () s solved for computg d 6 Stlty Codto The drft to hgher umered levels must e strctly less th the drft to lower levels let d The stedy stte prolty vector exts f oly f < Where X s the vrt prolty vector of the mtrx As s fte, the stedystte prolty of the ew CTMC c e esly computed The the drft to the ext hgher level c e descred s:, l) l ( d the drft to the ext lower level s:

4 l whch leds to (, l) Itertol Jourl of Scece d Reserch (IJSR) ISSN (Ole): Idex Copercus Vlue (3): 64 Impct Fctor (4): 56 For ths we c show the stlty codto for ths system to e s followg theorem Theorem The mrkov ch equto(3) s postve recurret ff < Proof It s kow from Neuts [7] tht f the mtrx s of fte the crtero for the mtrx equto(3) to e postve recurret s the me drft codto < The equto tht results from multplyg, ) y the ( frst colum of the correspodg to the ormlzg codto The ler equtos gve y equto () hve uque soluto tht lso stsfes the ormlzg codto If however s rreducle tht vector exts the ths result stll holds for the fte lock (see []) Therefore we hve ( )( t I) < ( )( I S ) ( t ) ( I) < ( I) ( S) ( ) < ( ) < Remrk: It c e show tht ( t) d ( t) Ths s tutve ecuse () s the sttory prolty of eg stte, whch s essetl the prolty of workg system The sme rgumet s true for servce d repr system It s kow from Neuts [7] tht t d I smlrly s d I 7 Queue egth d Wtg Tme dstruto of Repr A very mportt performce mesure of terest s the dstruto of the umer of workg system of rtrry tme, whch s oted from the sttory vector, where I secto 4 d Neuts [6], we c d ot the vector d ll formto out the queue legth For exmple, the me umer of workg system ( ) the repr fclty d ts mthemtcl expresso s ( ) [ I ] The me umer of system queue for repr s q J ( ) Volume 5 Issue Jury 6 wwwsret Pper ID: cesed Uder Cretve Commos Attruto CC BY J 8 Avllty We coclude the vllty d the rte of occurrece of flures, the me umer of o-workg system, d explct lgerc formule re gve All the mesures wll clculted the sttory regme The vllty s the frst mesure to e clculted the reprle system It s well kow tht the vllty s the prolty tht the system wll e servce Alytclly, the expresso of the vllty s exctly the ormlzto codto, tht s, A [ I ] Cosequetly, The o-vllty s gve y A A, whch s the prolty tht the system s workg 9 Rte of Occurrece of flure A flure c occurs whe the system occupes some of the sttes,,,, whe the system s stte, the flure occurs y sorpto the dstruto PH (, T) d reprm s dle The spce of repr must ot chge Thus, the rte of occurrece of flures s gve y the expresso t ( t ) t [ I ] ( t ) Multplyg o the rght the frst equto (3) determes relto etwee d, d the rte of occurrece of flure c e expressed terms of X s follows [ S ( I ) ( t )] () Numercl Exmples I ths secto we pply the clculto performed ove the prctcl cse The methods were mplemeted mtl progrmme We cosder the followg represets for the PH-dstruto of the servce tme d repr tme (,) d (,,) 3 T ; S The mtrx T dctes the workg system hs opertol tme tht udergoes successve decortg phses The order of T d S re, respectvely m d 3 Cosequetly, the mtrces, d re of order 6 6 ; d re of order 6 d 6 The vllty d the rte of occurrece of dfferet

5 Itertol Jourl of Scece d Reserch (IJSR) ISSN (Ole): Idex Copercus Vlue (3): 64 Impct Fctor (4): 56 flures, for the system wth dfferet umer of workg system The vllty decreses slghtly whe the umer of workg system creses d stlzes of roud the 97 % The dfferet rte of occurrece of flures chge slowly wth '' The su-vector of the sttory prolty vector re gve elow, where the vlue 4 re susttute y stersk, (35,6) (49,85,3,3,4,) (,4,3,,,) The performce mesures clculted ths model deped o the su-vectors (, ), they c e explctly determed, d they re The umer of queued d wtg for repr s 776 q The me totl expected the repr fclty s 365 The smll vlue re expected for the umer of system repr, sce the me tme of the workg system s susttlly loger th the expected tme to complete repr Cocluso I ths pper, we hve comprtve lyss for repr prolem where oth the systems d the reprm c fl The repr tmes of the systems d the servce tmes of the reprm re modelled usg dfferet PH-dstruto We clude recet d fster computtol methods for prctcl pplctos of QBD processes Clculte the sttory prolty vector d to determe pproprte from for the tertve umercl clculte of the locks of the mtrx some prtculr smple cses, s the model Erlg/ph/ Our results c e treted s performce evluto tool for the cocered system whch my e suted to my cogesto stutos rsg my prctcl pplctos ecoutered computer d commucto systems, dstruto d servce sectors, producto d mufcturg system, etc, Refereces [] Chkrvrthy S, Rellty Alyss of Prllel System wth expoetl fe Tme d Phse-Type reprs, Trsp Sc, OR Spekrum, 5; pp5-3(983) [] Chkrvrthy S d Ntr R, A Study of -ut System Opertol Rdom Evromet, Eur JOperRes, 66; pp3-36(999) [3] Bellm R, Itroducto to mtrx lyss, McGrw Hll, New York(96) [4] toucheg d RmswmV, Algorthmc Reducto for Qus-Brth d Deth Process, J ApplPro, 3;pp65-674(993) [5] touche G d Rmswm V, Itroducto to mtrx methods Stochstc Modellg, Phldelph: ASA- Volume 5 Issue Jury 6 wwwsret SIAM Seres o Sttstcs d Appled Prolty (999) [6] NeutsMF, Prolty Dstrutos of phse-type, I: er Amcorum Professor Emrtus HFlor Deprtmet of Mthemtcs, Uversty of ouv, Belgum, pp83-6(978) [7] Neuts MF, Mtrx Gometrc Soluto Stochstc Models: A Algorthmc Approch, Joh Hopks Uvrsty Press, Bltmore(98) [8] Neuts MF d Meee KS, O the Use of Phse-Type Dstruto Rellty Modellg wth two Compoets OR Spekrum, ; pp7-34(986) [9] Neuts MF, Perez-Oco R d Terres Cstro I, Reprle Models wth Opertg d Repr Tmes Govered y Phse-Type Dstruto Advces Appled Prolty, 3; pp () [] Rmswm V d Tylor AG, Some Propertes of the Rte Opertors evel Depedet Qus-Brth d Deth Processes wth Coutle Numer of Phses, Stochstc Models,(); pp43-64(996) [] Svzl BD d Wg KH, Dffuso pproxmto to the G / G/ R mche repr prolem wth wrm stdy spres, Nv Res ogstcs, 37; pp753-77(99) [] Tweede R, Opertor-Geometrc Sttory Dstruto Mrkov Che wth Applctos to Queueg Models, Adv ApplPro,4(); pp368-39(98) [3] Wg K, Proft lyss of the mche repr prolem wth cold stdys d two modes of flure, Mcroelectro Rel,34; pp635-64(994) [4] Wg K d ee H, Cost lyss of the coldstdy M / M/ R mche repr prolem wth multple modes of flure, Mcroelectro Rel,38; pp435-44(998) [5] Wrtehorst P, N Prllel queueg systems wth server rekdows d repr, Eur J Oper Res, 8; pp3-3(995) [6] v der Duy SFA d Wrtehorst P, Two mche repr model wth vrle repr rte, Nv Res ogstcs, 4; pp495-53(993) Pper ID: cesed Uder Cretve Commos Attruto CC BY