Chapter 1. Analysis of a M/G/1/K Queue without Vacations

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Willis Wiggins
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1 Chatr nalysis of a M/G// Quu without Vaations W onsir th singl srvr finit aaity quu with Poisson arrivals an gnrally istriut srvi tims. Th M/G// systm may analys using an im Marov Chain aroah vry similar to th on follow in Stion 3.2. quu of this ty may a ttr rrsntation of a ral-lif systm. This is aus th infinit numr of uffrs imli y th M/G/ rally th M/G// mol of Stion 3.2 woul iffiult to satisfy in a ral systm t roaly as an aroimation. quu of this ty is illustrat in Fig.. -Waiting Positions rrivals Darturs Srvr Jos laving without srvi Figur. M/G// Quu For th M/M// ty of quu i.. onntially istriut intrarrival tims an srvi tims quilirium rsults ha n otain arlir in Stion 2.5. Not that following our usual notation rrsnts th

2 2 Chatr maimum of th total numr of os that an rsnt in th systm at any instant of tim i.. on o ing srv an - othrs waiting for srvi. Sin th quu is of finit aaity os arriving whn th systm is full i.. os in th systm ar lost an hav to lav th systm without gtting any srvi. Suh os ar also rfrr to as ing lo. Th roaility of os ing lost also rfrr to as th loing roaility P is as imortant a rforman aramtr for th finit aaity systm as its lay an throughut. Following our usual rati w fin th systm stat at tim t to th numr in th systm at that instant. s on for th M/G/ quu of Stion 3.2 w onsir th im Marov Chain of th systm stats ust aftr th artur instants of th os that lav th quu aftr otaining srvi. Not that th os whih gt lost aus of loing o not atually ntr th quu. Thir arturs without gtting srvi o not ontriut to th im tim instants onsir hr. Lt th avrag arrival rat of os from th Poisson arrival ross to th quu. Not that of ths arrivals only a fration -P will atually al to ntr th quu. Consir th im Marov Chain of systm stats at ths tim instants t i i23... whn th i th o arts from th systm aftr otaining srvi. t a tim instant t i th systm stat n i will th numr of os lft hin in th systm whn th i th o arts. Not that n i will rang twn an - sin th artur of th o annot lav th systm omltly full i.. with systm stat. Lt a i th numr of arrivals from th Poisson arrival ross in th i th srvi tim. Th quations for th orrsoning Marov Chain an thn writtn as n i min{ a min{ n i i } a i } for for n i n... i Not that th aroah follow in Stion 3.2 was to otain th gnrating funtion of th systm stat at quilirium using irtly th rssions for th im Marov Chain. For th M/G// quu it woul asir to irtly omut th quilirium stat roailitis...- at th artur instants of os from th quu. For this w will n transition roailitis of th im Marov Chain at quilirium. t quilirium ths ar fin to { i i P n n } - 2

3 . nalysis of a M/G// Quu without Vaations 3 Lt th roaility of o arrivals to th quu uring a srvi tim. whr th f of th srvi tim is givn as t with f t an Lala Transform L s. Using this w an fin as t t t t 3! t It may not that an also foun as th offiint of z in th ansion of L -z. This may rov y using of 3 to show that th gnrating funtion of th numr of arrivals in a srvi uration will L -z. This also follows from th ommnts ma at th n of Stion an th aroah us for otaining quations li 2.36 or 3.3. Th transition roaility for th two ass an... - will foun saratly using th valus of foun in 3. Th rssions for ths ar givn in 4 an 5 rstivly as on th osrvation that th final stat annot -. m 2 m 4 m m Using th transition roailitis of 4 an 5 th quilirium stat roailitis...- at th artur instants may alulat in th usual fashion y solving th - alan quations along with th normalisation onition. Ths quations will as follows Normalisation Conition 6

4 4 Chatr Th transition roailitis of 4 an 5 may now sustitut in 5 an 6. This givs a st of linar quations that may solv to gt th orrsoning stat roailitis. Not that only innnt quations ar n as thr ar only unnowns i to foun. This imlis that aart from th normalisation onition of 6 only -2 quations ar n from th - quations of 5. This st of - quations is summaris in ltrnativly on an solv first for th normalis varials / using an thn solv for using th normalisation onition to gt 9 Using this an th valus otain arlir for / on an thn otain th atual stat roailitis... - at th o artur instants. Consiring a systm at quilirium lt a... th roaility that a nwly arriving o irrstiv of whthr it finally oins th quu or not fins os waiting in th quu. For this systm lt... th roaility that th quu has os in it at an aritrarily hosn instant of tim. Using th PST rorty of Stion 2.5. w an thn laim that a... W an also fin a...- as th quilirium roaility of th systm stat as sn y an arrival whih os atually ntr th quu. as on th fat that th stat of th quu an hang y at most ± aus of ths arrivals an th arturs from it w an laim that

5 . nalysis of a M/G// Quu without Vaations 5 a... Using P as th quilirium roaility that an arrival is lo aus th quu is full i.. in stat P w an s that a P a P... 2 Not that this may also onfirm y osrving that a P P a sin a an a Lt th man srvi tim of a o in th quu. Th traffi loa offr to th quu will thn givn y. Sin th avrag arrival rat of os atually ntring th quu also th avrag artur rat of os laving th quu is -P th atual traffi throughut of th quu will -P. Not that this imlis that th roaility of fining th quu mty at an aritrary tim will Using 2 for th as w an thn writ 3 P P Sin has n foun arlir using 7-9 w an us 3 to fin th loing roaility P or as P 4

6 6 Chatr Using th valus of otain using 7-9 an th rsults of 2 an 4 th quilirium stat istriution...- of th quu at aritrary tim instants may thn shown to... 5 Th quilirium stat istriution may now us in th usual fashion to fin th man numr N in th systm as N 6 Not that th fftiv arrival rat to th quu will givn y P 7 Using this an Littl's rsult th man total tim snt in systm y a o atually ntring th quu will N W 8 This may us to gt th man tim snt waiting in th quu W q as W W q whr is th man srvi tim. Th son momnt of th tim snt waiting in quu has also n otain in [Taagi2] an is givn y q w

7 . nalysis of a M/G// Quu without Vaations 7 whr 2 is th son momnt of th srvi tim. Proortionality Rlationshi twn th M/G/ an th M/G// Quus Consir th way 7 woul writtn for a M/G/ i..m/g// quu. Dnoting th orrsoning stat roailitis this quation for th M/G/ quu woul... 9 Not that for th M/G/ quu th stat roailitis at th artur instants arrival instants an at an aritrary tim instant woul th sam. This has n isuss arlir in Stion 3.2. Comaring th form of 7 an 9 w onlu that th stat roailitis of th M/G// quu an th stat roailitis of th M/G/ quu will roortional to ah othr for Using th normalisation onition th roortional rlation twn thm may shown to... 2 Not that 2 imlis that th quilirium stat roailitis at th artur instants of th M/G// quu may otain y a siml trunation an saling of th quilirium stat roailitis of th orrsoning M/G/ quu. This may us to fin for stats...- an thn 2 may us to fin th quilirium stat roailitis for...-. Finally th normalisation rsult may us to fin th loing roaility P. In orr to fin th quilirium stat roailitis of th M/G/ quu w an ithr irtly us 9 or invrt th gnrating funtion Pz otain in 3.4 of Stion

8 8 Chatr Pushout Oration of th M/G// Quu In our arlir srition for th M/G// quu w hav us th aroah that a nwly arriving o whih ss th quu full lavs without srvi. Not that th orr in whih os ar srv an FCFS LCFS or SIRO. s usual w may not that th squn in whih th os ar srv on thy ar in th quu will not afft th man rforman aramtrs i.. N N q W an W q. On an onsir an altrnat mtho for hanling th os that arriv whn th systm is full. In this a nwly arriving o is always at. If th quu is full whn th o arrivs it isars th on that has wait in th quu for th longst tim. Not that a o in srvi is nvr isar an is allow to ontinu its srvi until omltion. This stratgy is rfrr to as th ushout stratgy. Not that vn with th ushout stratgy on an still orat th quu following FCFS LCFS or SIRO srvi isilins. It may not that th ushout stratgy is a rasonal on to follow in systms whr a latr o/mssag/at arrival mas an arlir on runant in som way. For aml this may han in a systm hanling voi or vio ats whr on woul rfr to isar th olst at waiting in th quu rathr than th mor rnt arrival. Th man numr i.. N or N q in a M/G// quu following a ushout stratgy woul th sam as for on whr suh a stratgy is not ing follow. Th avrag artur rat of os an th atual throughut of th quu will also th sam in th two ass. tail rivation of th quu's rforman unr th ushout stratgy may foun in [Taagi2]. This rivation an th assoiat rsults ar somwhat iffiult. Howvr on an asily ommnt on th rlativ man rforman of th M/G// quu orat with an without th ushout stratgy. lying Littl's rsult to th M/G// quu with an without ushout stratgy w gt that 2 W q P Wq Wq P whr W q is th man waiting tim in th M/G// quu without ushout an W qp is th man tim snt in th quu rior to srvi y a o in th M/G// quu with ushout. Not that in th lattr as this waiting tim will inlu oth th os whih vntually gt srv an ons whih gt ush out an hn lav without srvi. Not that sin -P 2 imlis that

9 . nalysis of a M/G// Quu without Vaations 9 W P W 22 q P q an that thrfor W q P Wq 23 W may also not that for th M/G// quu with ushout th lay aramtr W qp atually onsists of two omonnts. On omonnt W qps is th man waiting tim in th quu as sn y os whih vntually o gt srv. Th othr omonnt W qpns orrsons to th tim snt waiting in th quu y os whih gt ush out aftr sning som tim waiting for srvi an lav without srvi. It may also not that P an -P ar th rstiv roailitis that a o is not vntually srv an that a o os gt srvi. Using this w gt that 24 W q P P Wq PS PW q PNS Sustituting 22 in 24 givs P W W P W 25 q q PS q PNS an that thrfor W W 26 q PS q Not that 26 las to th following imortant onlusion. For a M/G// quu with ushout th quuing lay sn y th os whih vntually gt srv will lss than what on woul osrv for a quu without ushout. Sin th systm throughut will th sam in oth ass th quu with ushout rovis a way of giving imrov srvi lowr lays to th os that atually o gt srvi. Th rar is rfrr to [Taagi2] for mor tail analysis of th M/G// quu orat with th ushout stratgy. n ltrnat Drivation for th Stat Proailitis at an ritrary Instant in a M/G// Quu It is ossil to rovi a mor irt aroah to fining th stat roailitis of th M/G// quu at an aritrary instant of tim for a quu

10 Chatr in quilirium. For this w first not that th man tim intrval twn sussiv im oints at th o artur instants will if th quu is mty at th arlir im oint if th quu is not mty at th arlir im oint Using th aov w otain th roaility that th quu is mty at an aritrarily hosn tim to 27 whih agrs with th rssion - otain arlir or as givn in 2 an 5 for. Now onsir th situation whr th aritrarily hosn tim instant falls within a srvi uration whr is th amount of srvi alray rovi. W onsir th as whr thr ar os in th systm for...-. Th f of may foun from rsiual lif argumnts to whr is th roaility that th aritrarily hosn tim instant will fall within a srvi tim. Consiring saratly th two ass whr th rvious artur lft th quu mty or lft th quu with... os in it w gt!! Lt th roaility that thr ar or mor arrivals uring a srvi tim whr

11 . nalysis of a M/G// Quu without Vaations ] [!! an 3 Th rivation of ths rsults is givn in th ni. Sustitution of 29 in 28 givs To simlify this furthr w n th following rsult whih may rov y rursion. 32 Sustituting this in 3 givs...- with th sam rssion as otain arlir in 2 an 5. W an us a similar aroah to fin th roaility i.. for whn th quu is foun to full at an aritrary tim instant. In this as w o n to ta into aount th situation that arrivals oming whn th systm is alray full will ni ntry into th quu an will lost. Following th sam argumnts as givn arlir w gt!! 33 Using 29 w an rwrit 33 as

12 2 Chatr 34 To simlify 34 furthr w n th rsult that 35 Not that this follows from summing ovr...- an using 3 to gt 36 Sustituting 35 in 34 w gt th sam rssion for as givn for th loing roaility P whih is th sam quantity in 4. ni: Consir 29 for i.. for. For this w n to rov that ] [!. Using intgration y arts w an show that.2 an that

13 . nalysis of a M/G// Quu without Vaations 3.3 Sustituting.3 in. w an show that 29 hols for. W an also show from.2 that [ ] [ ].4 an that [ ] ] [! ] [! ] [! ] [!.5 Sin w hav shown that 29 hols for w an now us inution to show th gnral rsult. Not that from th finition of w an also writ that!.6 ssuming that 29 hols for w an sustitut that rsult for in th RHS of.5 an us.5 to gt

14 4 Chatr RHS LHS!!! [ ] [ ]! W an thrfor laim that if 29 hols for thn it hols for. Sin w hav shown that it hols for w an laim that it hols for all 2... tually on an giv a hysial rasoning to ustify 29 without going through th tails of th roof givn aov. For this onsir arrivals to a siml M/G/ quu. Th ustifiation for as givn y 29 thn follows y onsiring an aritrary tim instant within a srvi tim. Lt this tim instant th tim at whih th th arrival in th on-going srvi tim ntrs th systm. For this lt th tim that has las from th ginning of th urrntly on-going srvi to this instant. Not that [-] will th roaility that th srvi tim will gratr than or qual to. Comining ths w gt that! [ ] from whih 29 follows. On w hav shown that 29 hols 3 follows irtly using [ ]! [ ] [ ]

MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c.

MA1506 Tutorial 2 Solutions. Question 1. (1a) 1 ) y x. e x. 1 exp (in general, Integrating factor is. ye dx. So ) (1b) e e. e c. MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ