The System Size Distribution for M/G/1 Queueing System under N-Policy with Startup/Closedown

Transcription

1 Busess o:436/b447 Pubshe Oe December (htt://wwwscrorg/oura/b) The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow Mgwu Lu Yoga Ma B Deg Schoo of Maagemet Chogqg Jaotog Uversty Chogqg Cha; Schoo of Maagemet a Ecoomcs Uversty of Eectroc Scece a Techoogy of Cha Chegu Cha E-ma: umgwu7@yahooc Receve Jue 4 th ; revse August 9 th ; accete Setember 7 th ABSTRACT Ths aer eveos a ew metho for cacuatg the system sze strbuto o two fferet M/G/ queueg system uer N-ocy wth geera startu/coseow Frsty the stochastc ecomosto roerty s use to erve the gf of the system sze strbuto By the Lebz formua of ervato we vestgate the atoa system sze strbuto a the we get the recurso exresso of system szes strbuto Fay severa exames are gve for ustratg the acato of the recurso exresso a sestvty aayss s aso erforme Keywors: Queue-Legth N-Pocy Stochastc Decomosto Recurso Exresso Lebz Formua Itroucto Ths aer resets a ew aroach for stuy M/G/ queueg system uer N-ocy wth geera startu/ coseow tmes N-ocy s a effectve mechasm for cuttg ow oeratg cost whch has bee wey ae moeg queue-e roucto system As soo as each servce s comete the server w be ture off for a raom coseow tme a a raom startu tme s eee before commecg hs servce whch refects more actua stuato Queueg system uer N-ocy frst cosere by Ya a Naor [] has bee extesvey exae by may authors Kea [] scusse N-ocy M/G/ queueg system wth server vacatos Bothaur et a [3] extee Baer s moe [4] wth exoeta startu tme to the geera startu tme Lee et a [5-7] vestgate the batch arrva N-ocy M/G/ queueg system wth a sge vacato a mute vacatos The gf (robabty geeratg fucto) of the system sze strbuto was erve whch show the famous stochastc ecomosto roerty roose by Fuhrma a Cooer [8] st came to exstece The cocet of coseow tme was trouce by Taag [9] Ke [] eveoe a M/G/ queueg system uer varetes a extesos of NT queueg systems wth breaows startu a coseow Ths wor was suorte by the New Cetury Exceet Taets Mstry of Eucato Suort Program uer Grat NCET-5-8 It s ffcut to get the aaytc steay-state system sze strbuto base o the gf uer the geera strbuto of servce tme Wag a hs co-authors [-4] erve the aaytc steay-state soutos of the N-ocy M/M/ queueg system the N-ocy M/H / the N-ocy M/E / a the M/H / resectvey Recety a maxmum etroy aroach was use to aayze the steay-state characterstcs of M/G/ queueg system [56] Base o ths metho Wag et a [7- ] eveoe the aroxmate robabty strbuto of the system sze for the N-ocy M/G/ queue wth varous cases Aso the maxmum etroy aroach cou be succeee ae to batch arrva queueg system [-3] Tag [4] eveoe a tota robabty ecomosto metho for ervg the recurso exresso of system sze strbuto equbrum Ths metho cou be use to scuss batch arrva cotuous-tme or screte-tme queueg system [56] whe the aayss roceure was very comcate Ths aer w reset a ew metho for aayzg the N-ocy queueg system Our metho ca erve the aaytca exressos of system sze strbuto whch s fferet from the maxmum etroy aroach It s to be ote that the aaytc system sze strbuto uer N-ocy s very ffcut to be erve by Tag s metho Our metho s much smer tha Tag s aayzg the N-ocy queueg system The arragemet of ths aer s as foows: Frst we Coyrght ScRes B

2 364 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow eveo a ew aroach for erve the recurso exresso of system sze strbuto equbrum Seco some seca cases are resete ths aer for our ew metho Fay umerca exames are gve for ustratg the accurate cacuato of the system sze strbuto a the sestvty aayss s vestgate Moe Descrto a Assumto I ths aer we scuss two fferet s M/G/ queueg system uer N-ocy wth startu/coseow The moes are escrbe etae as foows: Assumto of the moe ) Customers arrve accorg to a Posso rocess wth rate Arrvg customers form a sge watg e a are serve accorg to the orer of hs arrvas The server ca rocess oy oe customer at a tme ) The servce tme rove by a sge server s a eeet a etcay strbute raom varabe ( G) wth a geera strbuto fucto Gt 3) Wheever the system becomes emty the sever shuts ow wth a raom coseow tme ( D ) Whe the umber of arrvas the queue reaches a reeterme thresho N the server mmeatey reactvates a erforms a startu tme wth raom egth ( U ) before startg servce Whe the server cometes hs startu he starts servg the watg customers ut the system becomes emty 4) If a customer arrves urg a coseow tme the server s mmeatey starte wthout oeratg N-ocy a wthout a startu tme Assumto of the moe The frst three assumtos are the same as those moe However the fourth assumto ow s that eve f a customer arrves urg a coseow tme the server st be shut ow As soo as the coseow s over the server mmeatey reactvates Before startg servce the server st ee a raom startu tme ( U ) however wthout satsfyg the cotos of N-ocy I may teratures [9] they suose that the server w resume shuttg ow a offers servce whe a customer arrves urg a coseow tme However some rea-wor the server (mache) restarts ut the coseow s fshe for rotectg the mache whe a customer arrves urg the coseow tme 3 Premary Formua for the System Sze Dstrbuto for M/G/ Queueg System Before scussg our moes et us reca some resuts the orary M/G/ queueg system For the sae of coveece we efe severa tems We ca a tme terva whe the server s worg a busy ero A tme terva whe the server s uavaabe (such as startu/ coseow or e) a e ero Let be the robabty strbuto of the system sze equbrum From Tag [4] we have () ( ) () g t t ( )! e G t t t t e G t g! t g e G t a f We eote by the robabty strbuto of system sze urg the busy ero at statoary Note that EG s traffc testy a t shou to be suose to be ess tha uty whch esures the system reachg at equbrum The gf of the queue sze of the M/G/ queueg system s gve by z z gz g zz 4 Aayss of Moe 4 The gf of the System Sze For ths system the server w start u ut the umber of customer arrva reach N urg the e ero The gf of the system sze at the begg of the busy s gve I ths secto we erve the gf of the system sze strbuto foowg the smar argumets as [] Moe s as a exteso of the M/G/ queueg system gve Secto 3 At the begg of the busy ero for Moe ca be escrbe by Case : o customers arrve whe the server s shut- tg ow whch occurs wth robabty D s LST of U Case : Some customers arrve whe the server s shuttg ow whch occurs wth robabty D I ths case the servce s starte mmeatey wthout a startu tme a wthout N-ocy There s oy oe customer at the begg of the busy ero By the stochastc ecomosto resuts [8] the gf of the umber of customers fou at the begg of the busy ero s gve by N by z U z U s N U z (3) z D z U z D z (4) eotes the gf of the umber of customers arrve urg the startu tme Foowg the resut of Meh a Temeto [7] the stochastc e- Coyrght ScRes B

3 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow 365 comosto of the gf of umber of customers the N ocy M/G/ queueg system wth startu/coseow st comes to exstece We obta z z z (5) z D N E U D s the execte umber of customers that arrve urg the e ero us startu tme 4 Atoa System Sze Dstrbuto From Equato (5) we see that the statoary system sze s sum of two raom varabes oe of whch s the system sze of the geera M/G/queue The other s the atoa system sze strbuto ue to N-ocy a startu/coseow The gf of atoa system sze strbuto s gve by z a z (6) z Foowg the efto of gf the robabty of atoa system sze strbuto ca be wrtte by ca be wrtte by z a a z (7)! Notce that whe N we have ( ( z) )) z D ( ) (8) N A whe N we have z z D N z U z N t N z t D! e U t (9)! Substtute (8) a (9) to (7) we get the atoa system sze strbuto a a D () N () a m N t t m m! D e U t N N () 43 System Sze Dstrbuto I ths secto we w erve the statoary-state system sze strbuto The cha rue of fferetate s use as the ma too for cacuato Obvousy the robabty of the umber of customers the queue whch equas to zero s gve by z z (3) Whe N by the Lebz formua from (5) the robabty of the umber of customers s the queue s aso gve by z z! z z z (4)! s gve by Formuas () a () We e- ote ( z) by ( z ) usg the rue of cha of z ervato aga cotog o N thus we have z z z! z z! D!! D (5) Substtute (5) to (4) a combg () we get the system sze strbuto D D D N EU D N (6) Whe N f N that s N we have Coyrght ScRes B

4 366 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow z z D U! (7) * D U D U D D N EUD From the (5) (7) a combg () we obta the system sze strbuto N N m N t t U e U t m! m m N t t U e U t m! m The recurso exressos of system sze strbuto are gve by (3) (6) a (8) whch ca be use for cacuatg the robabty of system sze equbrum Remar 4 If the startu tme equas to zero whch mes EU a U U moe ca be reuce to the M/G/ queueg system wth eaye N-ocy I ths case these resuts coce wth those of Tag s system [8] The aroach scusse ths aer s much easer tha the tota robabty ecomosto metho for the system sze strbuto use [8] 5 Aayss of Moe 5 The gf of the System Sze I ths secto we coser aother N-ocy for M/G/ queueg system wth startu/coseow I ths system the oy fferece from the moe s that f some customers arrve urg the coseow ero the server w st be shut ow orer to rotectg the server After the coseow the server mmeatey commeces a startu wth a raom tme The server w offer servce wthout oeratg N-ocy ut the startu s fshe For ths system from the smar argumets as for moe the gf for the umber of customers fou the system at the begg of the busy ero s gve by ( ) ( ) N ( ) z D z U z D z ( D ( )) D ( z) U ( z) (9) eotes the gf of the umber of customers arrve urg the coseow tme A the execte umber of customers at the begg of busy ero s as foows D N E U D ED EU () N (8) From (9) a () we have the strbuto for the system sze at a arbtrary tme z z z z () 5 System Sze Dstrbuto I ths secto we erve the system sze strbuto Smary from (9) () we have the robabty that the system s emty D D U ( ( )) z () ( )! () As Secto 4 whe N the robabty of the system sze equas to s gve by z (3) we eote get z z by z z z D D U D N From (9) we t t e t Dt e t Ut Thus we get z (4) D D U z!! D (5)! Tae (5) to (3) we get D D U Coyrght ScRes B

5 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow 367! D! D D D U N (6) Whe N we aayze the foowg two cases ) If N that s N we have z z U D!! (7) D! ) If N whch mes N we get! z z D! (8) Substtute (7) a (8) to (3) we have z N ( ( )) z () ( )! N! z z U D D! N U D! D N! D N (9) We get the recurso exressos of the system sze strbuto for the moe whch are show by () (6) a (9) Remar 5 Suose that we et PU whch cates EU a U I ths case our moe ca be smfe to M/G/ queueg system wth geera coseow tme It s fferet from Tag s system [8] because the server w be shut ow eve f some customer arrves urg the coseow tme We get the system sze strbuto as foows: D D D N D E D D N D E D (3) D D D! D D! D N D E D D N D (3)! N D! D N! N (3) t e t Dt Remar 5 If we et the coseow tme equa to zero whch mes ED a D I ths case our moe a ca escrbe the N-ocy M/G/ queueg system wth geera startu tme The system sze strbuto ca be smfe to the foowg exressos: (33) N E U N EU N (34) N N N E U U U N (35) Coyrght ScRes B

6 368 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow 6 Numerca Exame To ustrate the mcato of cacuatg the system sze strbuto we coser the moe of a M/G/ queueg system uer N-ocy wth geera startu/ coseow whch base o the moe scusse Secto 4 Here we assume that the servce tme foows the egatve exoeta wth mea EG The startu tme s 3-stage Erag strbuto wth mea EU a the coseow tme s 4-stage Erag 3 strbuto wth mea ED 4 The umerca resuts are ustrate Tabe Observg the Tabe t s cear that as creases ) the robabty of the system beg emty creases whe other system arameters are costat; ) whe N the robabty of system sze beg creases; 3) whe N N the robabty of system sze beg ecreases The execte system sze (EL) aso ecreases as creases Deserve to be metoe s that the mea system sze s ot eough for the system esg a cotro As we ca see the execte system sze s o more tha 3 whe the sum of robabty of system sze exceeg the mea system sze resete the ast s about 3% whch ca ot be egecte 7 Cocusos I ths aer we erve the recurso exressos of system sze strbuto for two fferet N-ocy M/G/ queueg systems wth geera startu/coseow We frst utze the ervato of the cha rue combg wth the famous stochastc ecomosto resuts for eveog the robabty strbutos of system sze the system We reset severa exames to ustrate the memet of cacuatg the system sze strbuto from these recurso exressos a vestgate the effects for fferet system arameters We get the system sze strbuto characterstc whch s mortat ractce The metho ths aer ca be use N- ocy batch arrva queueg system Tabe The strbuto of the system sze ( N 5 ) P P P P P P P P P P P P P P P P P 6 3 P 7 3 P 8 P 9 EL Sum Coyrght ScRes B

7 The System Sze Dstrbuto for M/G/ Queueg System uer N-Pocy wth Startu/Coseow 369 REFERENCES [] M Ya a P Naor Queueg Systems wth a Removabe Servce Stato Oeratoa Research Quartery Vo 4 No [] O Kea The Thresho Pocy the M/G/ Queue wth Server Vacatos Nava Research Logstcs Vo 36 No [3] A Borthaur J Meh a R Goha Posso Iut Queueg Systems wth Startu Tme a uer Cotro Oeratg Pocy Comuters a Oeratos Research Vo 4 No [4] K R Baer A Note o Oeratg Poces for the Queue M/M/ wth Exoeta Startu INFOR Vo No [5] H W Lee S S Lee J O Par a K C Chae Aayss of M X /G/ Queue wth N Pocy a Mute Vacatos Joura of Ae Probabty Vo 3 No [6] S S Lee H W Lee S H Yoo a K C Chae Batch Arrva Queue wth N Pocy a Sge Vacato Comuters a Oeratos Research Vo No [7] H S Lee a M M Srvasa Cotro Poces for the M X /G/ queueg System Maagemet Scece Vo 35 No [8] S Fuhrma a R B Cooer Stochastc Decomostos the M/G/ Queue wth Geeraze Vacatos Oeratos Research Vo 33 No [9] H Taag Vacato a Prorty Systems Part I I: Queueg Aayss: A Fouato of Performace Evauato Vo I Amsteram 99 [] J-C Ke O M/G/ System uer NT Poces wth Breaows Startu a Coseow Ae Mathematca Moeg Vo 3 No [] K H Wag Otma Cotro of a Removabe a No- Reabe Server a M/M/ Queueg System wth Exoeta Startu Tme Mathematca Methos of Oeratos Research Vo 58 No [] K-H Wag a K-L Ye Otma Cotro of a M/ E / Queueg System wth a Removabe Server Mathematca Methos of Oeratos Research Vo 57 No [3] K H Wag Otma Cotro of a M/E / Queueg System wth Removabe Servce Stato Subect to Breaows Joura of the Oeratoa Research Socety Vo 48 No [4] K H Wag K-W Chag a B D Svaza Otma Cotro of a Removabe a No-Reabe Server a Ifte a a Fte M/H / Queueg System Ae Mathematca Moeg Vo 3 No [5] J E Shore Iformato Theoretc Aroxmatos for M/G/ a G/G/ Queueg Systems Acta Iformato Vo 7 No [6] M A E-Affe a D D Kouvatsos A Maxmum Etroy Aayss of the M/G/ a G/M/ Queueg Systems at Equbrum Acta Iformato Vo 9 No [7] K-H Wag S-L Shuag a W L Pear Maxmum Etroy Aayss to the N Pocy M/G/ Queueg System wth a Removabe Server Ae Mathematca Moeg Vo 6 No 5-6 [8] K-H Wag L-P Wag J-C Ke a G Che Comaratve Aayss for the N Pocy M/G/ Queueg System wth a Removabe a Ureabe Server Mathematca Methos of Oeratos Research Vo 6 No [9] K-H Wag T-Y Wag a W L Pear Maxmum Etroy Aayss to the N Pocy M/G/ Queueg System wth Server Breaows a Geera Startu Tmes Ae Mathematcs a Comutato Vo 65 No [] K-H Wag a K-B Huag A Maxmum Etroy Aroach for the < N>-Pocy M/G/ Queue wth a Removabe a Ureabe Server Ae Mathematca Moeg Vo 33 No [] K-H Wag M-C Cha a J-C Ke Maxmum Etroy Aayss of the M x /M/ Queueg System wth Mute Vacatos a Server Breaows Comuters & Iustra Egeerg Vo 5 No 7 9- [] J-C Ke a C-H L Maxmum Etroy Soutos for Batch Arrva Queue wth a U-Reabe Server a Deayg Vacatos Ae Mathematcs a Comutato Vo 83 No [3] J-C Ke a C-H L Maxmum Etroy Aroach for Batch-Arrva Queue uer N Pocy wth a U-Reabe Server a Sge Vacato Joura of Comutatoa a Ae Mathematcs Vo No 8-5 [4] Y H Tag The Traset Souto for M/G/ Queue wth Server Vacatos Acta Math Sceta Vo 7(B) No [5] Y H Tag a X W Tag The Queue-Legth Dstrbuto for M x /G/ Queue wth Sge Server Vacato Acta Mathematca Sceta Vo (B) No [6] YH Tag X Yu a S J Huag Dscrete-Tme Geo x /G/ Queue wth Ureabe Server a Mute Aatve Deaye Vacatos Joura of Comutatoa a Ae Mathematcs Vo No [7] J Meh a J G C Temeto A Posso Iut Queue uer N-Pocy a wth a Geera Start u Tme Comuters a Oeratos Research Vo 9 No [8] Y H Tag The Traset a Equbrum Dstrbutos of the Queue-Legth for M/G/ Queue wth Deaye N-Pocy System Egeerg - Theory & Practce Vo 7 No ( Chese) Coyrght ScRes B