Analysis of a M/G/1/K Queue with Vacations Systems with Exhaustive Service, Multiple or Single Vacations

Transcription

1 Analyss of a M/G// uu wth aatons Systms wth Ehaustv Sv, Multpl o Sngl aatons W onsd h th fnt apaty M/G// uu wth th vaaton that th sv gos fo vaatons whn t s dl. Ths sv modl s fd to as on povdng haustv sv, as th sv annot go fo a vaaton untl all th obs psntly n th systm hav bn svd. Ths s th sv modl bng onsdd h as ths lads to a smpl analytal modl. Not that t s also possbl to hav a gatd sv modl wh th sv only svs thos ustoms that t fnds n th systm whn t fst stats sv followng ts vaaton. It thn lavs fo vaaton agan. Th vaaton modl tslf may b of dffnt typs. In Ston 4., w had onsdd a multpl vaaton modl. H a sv, on tunng fom a vaaton, gos fo anoth vaaton f t fnds th systm stll mpty. In ths as, th sv sums nomal sv f t fnds on o mo obs watng whn t tuns fom a vaaton. Not that multpl vaatons, on aft th oth, wll b possbl n ths modl. An altnatv vaaton modl, dsussd n Ston 4.2, assums that th sv gos fo only on vaaton whn th uu boms mpty. Evn f th uu s mpty whn t tuns fom th vaaton, t stays at th uu watng fo a ob to av. In th multpl vaaton modl, t would hav gon fo anoth vaaton n ths as. Oth vaatons of ths a also possbl suh as on wh th sv an go on multpl vaatons and sums sv only whn t fnds L o mo obs watng whn t tuns fom a vaaton. In th subsunt analyss, w fst onsd th multpl vaaton modl and thn loo at th as whn w allow only a sngl vaaton whnv th dpatu lavs bhnd an mpty systm.

2 2 Multpl aaton Modl ollowng ou usual notaton, w assum that th avals om fom a Posson poss wth at and that th sv tms a gnally dstbutd wth pdf bt, df Bt and has L B s as th Lapla Tansfom of th pdf. Lt th man sv tm b X µ -. As n Stons 4. and 4.2, w assum that a vaaton ntval has pdf f t, df t and has L s as th L.T. of th pdf. Lt th man vaaton ntval b. As n Stons 4. and 4.2, w also assum that th sv tms and vaaton tms a..d. andom vaabls whh a also ndpndnt of ah oth. o ths M/G// uu wth haustv sv and multpl vaatons, w onsd th analyss usng an mbddd Maov Chan appoah. o ths, th mbddd ponts a hosn to b at th tm nstants whn th a ob omplts sv o a vaaton has ndd. Ths hav bn llustatd n g. usng a typal plot of th sdual sv and vaaton tms wth th mbddd tm ponts mad wth shadd ls. t t: Rsdual Tm fo th Cuntly Ongong Sv o aaton Tm X : th sv tm : th vaaton tm X 2 X 2 X 3 X 4 X 2 X 2 X 3 Tm τ gu. Imbddd Tm Ponts fo th M/G// uu wth Multpl aatons and Ehaustv Sv Th systm stats at th mbddd ponts a psntd by both th numb n th systm watng and n-sv mmdatly aft th sltd tm nstant and th natu of th mbddd pont.. whth t s a sv omplton o a vaaton omplton. Th systm stat at th th mbddd pont s psntd by n, φ wh n numb of obs n th systm ust aft th th mbddd pont and φ f th th pont was a vaaton omplton f th th pont was a sv omplton Consdng th systm n ulbum, lt,,,..., b th pobablty of, and,,,...,- b th pobablty of,. Not that ust aft a sv omplton, th systm stat annot b whh s th

3 Analyss of a M/G// uu wth aatons 3 ason why s not dfnd. Lt f,,... b th pobablty of th bng obs n th systm ust aft a vaaton ntval. Ths wll b gvn by f t t f t dt,,...,! Lt α,,... b smlaly dfnd as th pobablty of avals n a sv tm. Ths wll b gvn by α t t b t dt,,..., 2!! Consdng th systm stat ust aft th mbddd ponts, th followng tanston uatons may thn b wttn.,,...,- 3 f f 4 α,...,-2 5 α - 6 Summng th pobablts of all possbl stats, w wll also gt 7 W nd to solv fo,,..., and,,,...,- usng 3-7 along wth th appopatly alulatd valus of f and α,,,...,. Ths s most onvnntly don by dfnng an ntmdat vaabl β,,,...,- as

4 4 β,,...,- 8 W an thn gt β,,,...,- usng th followng uson β β β f β α,..., α 2 9 Substtutng 4 nto 7 and smplfyng gvs f β and usng β f β Usng th valu of obtand fom, w an fnd,,..., usng 3 and 4. Usng ths valus of and th valus of β obtand al, w an now fnd,,,..., usng β,,...,- Th pobablts,,..., and,,,...,- may now b usd to gt som of th pfoman paamts of th systm. W an s that f th stat s th, o, at any mbddd pont, thn th tm to th nt mbddd pont would ospond to a vaaton. Th pobablty of ths would b whh would thn also b th pobablty that a sngl vaaton omplton would follow an abtaly sltd mbddd pont whh ould b th a vaaton omplton o a sv omplton. It would thn also follow that - - would b th pobablty that a ob sv omplton mbddd pont would follow an abtaly sltd mbddd pont. It may also b notd that by summng 3 and 4 fo all,,.., w gt

5 Analyss of a M/G// uu wth aatons 5 2 Lt b dfnd as th ad load,.. th pobablty that th sv s busy at an abtay tm. Not that f w loo at all th ntvals btwn sussv mbddd ponts ov a long tm duaton say T, thn w an asly onlud that lm T vaaton tms n T X X sv tms n T sv tms n T 3 wh th offd load s dfnd as usual to b X 4 Usng 3 and 4, th blong pobablty P B may b found usng PB to gt P B 5 Sn a faton P B of th avals wll b blod and wll not b allowd to nt th uu, th thoughput at γ of th systm wll b gvn by γ P 6 B Anoth usful uantty that may b obtand fom th abov analyss s th man tm btwn sussv mbddd ponts of th abov analyss whn th systm s n ulbum. It follows fom th dfntons that ths wll b X 7

6 6 It follows fom 3 that thfo 8 X It should b notd that th analyss gvn abov lads only to th uu sz dstbutons at th mbddd ponts ospondng to th a sv omplton o a vaaton omplton. Atually, ths a th dstbutons ust aft th mbddd pont. To analys th systm mo ompltly, w nd to atually fnd th ospondng dstbutons at an abtay nstant of tm as thn w an us that to fnd th man uu lngth and oth latd paamts at an abtay nstant. Sval mthods fo dong ths a gvn n [Taag2]. On of ths mthods s gvn nt. Consd th pobablts,,,..., and R,,...,- dfnd at abtay tm nstants as follows Ths ospond to th pobablts and dfnd al at th mbddd tm ponts. P{ obs n systm, sv untly n a vaaton},,..., R P{ obs n systm, sv untly svng a ob},...,- W dfn as th pobablty that o mo obs av to th systm dung a vaaton tm. Ths may b valuatd usng to b f t! t t t! [ t ] dt t f t dt 9 wh t s th df ospondng to th pdf f t of a vaaton ntval. It may also b notd fom th dfntons that Thfo f f E{numb of avals n a vaaton ntval} 2

7 Analyss of a M/G// uu wth aatons 7 W smlaly dfn A as th pobablty that o mo obs av dung a sv duaton. Usng α of 2 as th pobablty of ob avals n a sv tm, w gt th followng sults whh a smla to thos gvn n 9 and 2. [ ] dt t B t dt t b t A t t t α!! 2 X A 22 wh s th load offd to th systm whh sults n th ad load, <. In od to fnd, onsd an abtay tm nstant that falls wthn a vaaton suh that th a avals n th tm ntval say btwn th stat of that vaaton ntval and th tm nstant sltd. Not that th pobablty of sltng a vaaton ntval would b -. Th pobablty of avals n th tm ntval would b gvn by th Posson dstbuton,..!. Th pdf of th tm ntval tslf would b gvn by sdual lf typ agumnts to b. Usng ths, w an wt d d!,...,! 23 Usng 9, w an smplfy ths to gt

8 8,..., 24 uth smplfaton of 24 s possbl by notng that by usng 3 and 4, w an wt,..., 25 Substtutng ths n 24 and usng 8 and 2 lads to th fnal psson fo as,..., 26 To fnd R, w smlaly onsd an abtay tm nstant that falls wthn a sv tm suh that th a avals n th tm ntval say btwn th stat of that sv ntval and th sltd tm nstant. Not that th pobablty of sltng a sv ntval would b. Gvn that th sv ntval wll hav to stat wth a non-mpty uu, th pobablty of t statng wth obs n th systm wll b fo,...,- and wll b fo. Th pobablty of m avals omng n th m ntval wll b. Th pdf of th tm ntval tslf would b m! B gvn by sdual lf typ agumnts to b. Usng ths, w gt X

9 Analyss of a M/G// uu wth aatons 9 d X B d X B R!,...,! 27 Usng 2 and X, w an smplfy ths to obtan A A R,..., 28 W an futh show usng 2-6 that,..., A 29 Ths s most onvnntly don by fst showng that 29 holds fo and thn usng mathmatal nduton to pov t fo a gnal valu of, assumng that t holds fo -. Now usng 29 and 2 along wth th nomalsaton ondton of 7, w an pov that A 3 Smplfyng 28 futh usng 29 and 3 and usng 8 and 22, w gt th fnal psson fo R as

10 R,..., 3 Usng th vlus of,..., and,...,- pobablts ust aft th mbddd Maov ponts whh w obtand al, w an now us 26 and 3 to alulat th pobablts,..., and R,..., at any abtaly hos tm nstant. o a systm n ulbum, w dfn p,,..., as th pobablty of th systm bng n stat at an abtaly hosn tm nstant. Usng th valus of and R obtand abov n 26 and 3, w an alulat th stat pobablts p,,..., as R p R p p,..., 32 Ths pobablts may now b usd to fnd th usual uung paamts as dfnd n Chapt. o ampl, th man numb N n th systm and th man tm W spnt n th systm by a ob whh s not blod wll b gvn by N 33 B P N W 34 It s also vdnt that th pobablty that th sv s busy o dl.. on a vaaton wll b gvn by P{sv s busy} R p 35

11 Analyss of a M/G// uu wth aatons P{sv s on vaaton} p 36 Sngl aaton Modl In th sngl vaaton modl, th sv stll gos fo vaaton whnv th systm boms mpty. Howv, unl th multpl vaaton as dsbd al, on t oms ba fom ths vaaton, t dos not go fo anoth vaaton vn f t fnds th uu mpty on ts tun. It gos fo ts nt vaaton only aft th systm boms mpty on agan followng a "sv busy" ntval. If ths modl s bng onsdd, th stuaton dptd n g. wll hang and loo as shown n g. 2. t t: Rsdual Tm fo th Cuntly Ongong Sv o aaton Tm X : th sv tm : th vaaton tm X X 2 X 3 X 4 X X 2 X 3 Tm τ gu 2. Imbddd Tm Ponts fo th M/G// uu wth Sngl aaton and Ehaustv Sv Th sngl vaaton modl wth haustv sv may b analysd n ssntally th sam fashon as th multpl vaaton modl onsdd al. W follow an mbddd Maov Chan appoah wth th mbddd ponts sltd as th ponts ospondng to th dpatu nstants of obs that hav fnshd sv o th nd of ompltd vaatons. Ths mbddd ponts a th ons llustatd n g. 2 usng shadd ls. Th systm stats at th mbddd ponts a psntd by both th numb n th systm watng and n-sv mmdatly aft th sltd tm nstant and th natu of th mbddd pont.. whth t osponds to a sv omplton o a vaaton omplton. W an wt a Maov Chan fo th systm stats, dnotd n ths fashon, btwn th mbddd

12 2 ponts. Th systm stat at th th mbddd pont s psntd by n, φ wh n numb of obs n th systm ust aft th th mbddd pont and φ f th th pont was a vaaton omplton f th th pont was a sv omplton Consdng th systm n ulbum, lt,,,..., b th pobablty of, and,,,...,- b th pobablty of,. Ths dfntons a th sam as th ons usd fo th multpl vaaton modl. Th dffn fo th sngl vaaton modl s th stuaton whn th vaaton omplts but th systm s stll mpty. In that as, th nt mbddd pont s th on ospondng to th dpatu of th ob that s th fst to av aft ths vaaton omplton vnt. W us h th sam notaton fo haatsng th Posson aval poss at, th sv tm dstbuton bt, Bt, L B t and X and th vaaton tm dstbuton f t, t, L t and as usd al fo th multpl vaaton modl. Lt f and α b as dfnd n and 2, sptvly. Consdng th systm stats ust aft th mbddd ponts, th followng uatons may b wttn latng th tanstons fom on mbddd pont to th nt.,,...,- 37 f f 38 α α,...,-2 39 α α - 4 Summng all th stat pobablts at th mbddd ponts, w wll also gt 4

13 Analyss of a M/G// uu wth aatons 3 Solvng 37-4, w an obtan th ulbum pobablts fo,,,..., and,,,...,- fo ths M/G// uu wth sngl vaatons and haustv sv. Summng 37 ov,,...,- along wth 38, w obtan that 42 Ths would ospond to th pobablty that a sv omplton nstant lavs th uu mpty,.. t s th stat of a vaaton. Th tm ntval btwn sussv mbddd ponts would b on of th followng th possblts. a sv tm wth man X wth pobablty - - b vaaton tm wth man wth pobablty sv tm and nt-aval tm wth man X - wth pobablty. fnng as bfo to b th man tm ntval btwn sussv mbddd ponts at ulbum, w gt X X X 43 Consdng th man tm ntval btwn sussv mbddd ponts, w an s that th ntval atual osponds to a ob bng svd only wth pobablty -. Thfo, th ad load wll b gvn by X 44 ospondng to an offd load of an thn b found as X. Th blong pobablty P B

14 4 P B 45 It should b notd that th stat pobablts,,,..., and,,,...,- a only vald fo th systm at th mbddd ponts ospondng to th a sv omplton o a vaaton omplton. Howv, w an us thm along wth som addtonal analyss along th sam lns as bfo to gt th pobablts p,,..., fo obs n th systm watng and n sv at an abtay nstant of tm. Consdng th pobablty of th numb n th systm at an abtay nstant of tm, w an fnd th lmtng valus p and p by a smpl agumnt. Consdng th ntval btwn sussv mbddd ponts, th pobablty p wll b gvn by th ato of th man tm spnt dl n th man ntval. Ths gvs p 46 Usng 43 and 44, ths may b smplfd to gt p 47 whh would b th ptd sult fo th faton of tm th sv wll b dl n th systm. Smlaly, a ob wll b blod and dnd nty n th uu only whn th systm s full,.. n stat. Thfo, p P B 48 Consd th pobablts,,,..., and R,,...,- dfnd at abtay tm nstants as follows Ths ospond to th pobablts and dfnd al at th mbddd tm ponts. P{ obs n systm, sv untly n a vaaton},,..., R P{ obs n systm, sv not on vaaton},...,- Not that whn th sv s not on vaaton, t would th b busy svng a ob o t would b watng dl fo th fst ob to av followng a vaaton whh was ompltd wthout any ob aval. Ths s ally th

15 Analyss of a M/G// uu wth aatons 5 dffn btwn th dfnton of R h and that gvn fo th multpl vaaton modl al. In od to fnd, onsd an abtay tm nstant that falls wthn a vaaton ntval suh that th a avals n th tm ntval say btwn th stat of that vaaton ntval and th sltd tm nstant. Th pobablty of fallng n a vaaton ntval would b. Th pobablty of avals n th tm ntval would b gvn by th Posson dstbuton,..!. Th pdf of th tm ntval tslf would b gvn by sdual lf typ agumnts to b. Usng ths, w an wt!,...,! d d 49 W dfn as bfo to b th pobablty that o mo obs av to th systm n a vaaton ntval. Usng 9 and 2, w an thn smplfy 49 to gt,..., 5 To smplfy ths futh, w not fom 37 and 38, that,..., f 5 and that

16 6 f 52 Substtutng 5 and 52 n 5, w gt ou fnal psson fo as,..., 53 To fnd R, w smlaly onsd an abtay tm nstant that falls wthn a tm ntval wh th sv s not on a vaaton. Not that fndng R s patulaly staghtfowad as ths wll ospond to th faton of tm, wthn th ntval btwn sussv mbddd ponts, whn th sv s dl although t s not on a vaaton. Ths wll thfo b R 54 Not that w pt p to b ual to R. Ths may b vfd usng 54, 53 fo and 42. Th podu fo fndng R fo th oth valus of,..,..., s smla to ou al appoah and lad to th sult R,..., 55 Th pobablty p of fndng obs n th systm at an abtay nstant of tm fo,..., may thn b found as R p,..., 56 Usng ths and 54, w gt ou fnal sult fo th stat pobablty at an abtay nstant as

17 Analyss of a M/G// uu wth aatons 7 p,..., 57 Th man numb n th systm N at an abtay nstant of tm may b found usng 57 to b N 58 Th man tm W spnt n systm by a ob whh atually dos nt th systm.. s not blod and dnd nty may b obtand by applyng Lttl's sult to 58 to gt W N 59 P B W also gt that P{sv s on vaaton} P{sv s busy svng a ob} X P{sv s not on vaaton but s dl} P{sv s dl} p