response time of premptive resume priority queue mmm

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Computer Science Technical Reports Department of Computer Science

1-1-1981

The Response Times of Priority Classes underPreemptive Resume in M/M/m Queues

J. P. Buzen

A. Bondi

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Buzen, J. P. and Bondi, A., "The Response Times of Priority Classes under Preemptive Resume in M/M/m Queues" (1981).Computer Science Technical Reports. Paper 314.http://docs.lib.purdue.edu/cstech/314
http://docs.lib.purdue.edu/http://docs.lib.purdue.edu/cstechhttp://docs.lib.purdue.edu/comp_scihttp://docs.lib.purdue.edu/comp_scihttp://docs.lib.purdue.edu/cstechhttp://docs.lib.purdue.edu/


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The Response Times of Priority Classes underreemptive Resume in M/M/m Queues

J.P. ]luzrm

Des Systems. loc .. WalLhnm. ~ ' : u . s s l . \ c h u s c L l S "

I1.JJ, [Jonrii

I>cpnrlmcnL of Computer Sciences.Purdue Universily. West Ln[i.lycLLc, Inulana

CSlJ-TH No. 3U'I

MJSTRACTExpressions arc derived fol' the mco.l1 response l imes of each p r i o r ~

ity level in a mlllLi-scrvcr M/M/m queue opcraLing umler prccmpLivcresume scheduling. Exact rcsulLs arc obtained fo r cases where nil priol'ilics have th e same mean service Limes; approximate resulLs i.\l'Cobtained fo r Lhe morc gcncrnl CilSC where mean service Limes ow,)'ditTer. The rcsLl1ls hotd fol' i lny number of servers and illly numucr ofclasses. For c il ch priority level, i t is assumed l lwl arrivals arc jJoissonand scrviec Limes ilrc cxponcnLiatly dtslribulcd .

This work W;:lS s u . ? p o ~ t c d by NSF t::r;:lnl n'..lrr:bcr }.ICS7U.OI72!J.


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The Response Times of Priority Classes underPrccrnpLivc Resume in M/M/rll Q u c u c : ~ '

J.P. [ J u z ~ n

DGS Systems, lnc .. Waltham, 1;assnchuscLLs

Ali. fJand!.

DeparLment of Computer Sciences.Purdue Universily. West Lil(uyclLc, lnuiana

CSD-TH No. 00'7

Several authors have analyzed Lhe behavior of }'Uj'vi!l mutLiplc class queuesopcrnUng: under prccmpLivc resume priority. F or the two cli lss cusc, WhiLe and Clll'jS-Lic (1958) have produced gcncraLing funcLlons for the steady sLaLe queue l c n l ~ l h dislri-hutions . ."lurks (1973) has also produced genera Ling funclions [o r this proulclIl, uut ina form ";hich lends i lsel f to an eITicienL algorithm for compuling the joint slcj\(..ly stellequeue lenglh probubilitics.

The methods llsed in these papel's uTe noL easily generali2ed Lo the mulLiplc: scrvel'c[\!;;e, parliculo.rly when the classes have dilTel'cnL mean service limes. ]-Jowcvcr. whileil is not easy Lo exlract lhe joinl queue lengLh dislribulions fo r Lhis problem, i t i s possi -blc La ~ i \ Y c e ~ : p r e s s i o n s for cerlain pcrformance l11eaSUI'CS. 1"01' th e rIlll!liplu ~ c r v u rcase in whieh nil priority classes have lhc same mean service lime, I3rosh (lDG9) gives;:\11 ( ' ~ : r r c s s i o n [ or l he expecLed limc frolll arrival Lo inception of serVLce am i usL"d.llishesbounds for the expected rC!-iponsc limes and queu e le ne lh s of LIte dilTurunL CusLu!lLCr


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classes. For lower priority cuslomers, Lhe response time cannot be obtained directlyfrom Brosh's result as the service period will be subject Lo inLerruptions.

Other results on multiple serve rs and prioritic!:: have been published by Taylor andTempleton (19BO) and Abolnikov an'd Yasnogorotlsky (Hr/4). Theil- papers deal wiLh adiscipline in which priority is given Lo especially urgent job:; (c.e., in un 1l.1!luulunccscrviC'e) so Ion]; as th e number of busy servers exceeds i.\ sclllJl'c:>hohl. PrccllljlLion i::;noL used here und unanswereu urgent requests arc laslo

Obtaining the n:lsponsc limes of mulUplc server pl"ccl11plivc priority qucues WllCIlthe levels have diITcrent mean service limes has been ucscl'ibcu by Heyman (10'77) as ,-tparLicularly important unsolved problem. Hccenlly, MiLr


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derived. This algorithm which is based on Lhe exacL soluLion obLllincd in Sec Lion 8, isshown to yield good rcsulLs in a series of examples.

Doth th e exact and lhe approximaLe solutions j ) ,!,C computationally :.;Lablc anelinvolve only Lhe direct evaluaLion of simple algebraic expressions. These ,'esulls arcapplicable Lo the study of compute r sys tems wilh multiple processors whose jobs ur csubject Lo preemptive resume priority scheduling.

1. Nolillion

Consider u preemptive resume .\-IlM/m queueing syslem with cuslomers "i. and N(p) to denot e the ovel"all Llvc!'Cl.Ge orthe mean response limes of th e p highest priorities. Thus,

7'n(p)=L>""ii' p=1,2, ... ,r .i =Il 'A{pj=L;A,. p=i,2, .. "T ,

i : : ,Dy LitLle's Law,

, . ,, J"')'"(8)

- p 1.,[1H ( p } = L . : ~ ' -'-, p=1.2, ... ,r.' i=l i \ ~ )

To enSlU'C the exisLenee of finite wiliLing times for the p highest prlor ity classes(CombhCl.tTI [1955]), also a::;sllme LllCl.t the Lotul trafTie intensity salis[jes

l'P{pj=L;(A,/ml ' ;)


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of i), simple F'eFS !vI/M/m queue with arl"ival rale 11. and service rate u. Dy Klcinrock(1 Q'iG)

where1R ~ - +u

( J J ~ m (p)"(1-1')

ilnu

p=rL/ t fLu) l 'o(p)(mp)'"P'" (p ::; - - - -__m!( l -p ) " )v(G)

_ m ~ i ! . : . L ~ (mp)'fl _1',(1')-[ 2., " - " ' (1 - ) J Ii=O 1. . 7lL PP",(p) is Lhe probability of ha.ving Tn or more cuslomers

('I)in an 1U;\Um queue ":ilhtraGic intensity p. white lJo(P) is Lhe probability of the queue b c [ n ~ empty,

2. Two Priority Clas::;cs wilh J ~ q u a l NeUll ~ c r v i c c TimesFor t he ' ip cdal case where r=2 and /-LI ;:: J . L ~ ::; J-L. expressions fo r Lhe r c " p o n s ~

Limes of the intlividual c la sses can be easily derived using llle following Lwo inLuiUn:U:OSUll1pLiOIlS. which arc pl"ovctl in th e appendices.

11ss'lLTnplian AThe ITJeill1 response Lime of the highest prior ity jobs. R I . I:; cqu,J,! Lo I.he mean

response Lime or an j\U.\(j/nl queue wilh arl'ivnl rale ,\\ illld lllC'a.l1 st:l"viee J"ale f.t ulHlcrFe]"::.> sl:!leduling. Thus, lllC 10\'1 priorily jobs may lJe disre11urdcd when c O ! l l p l l L i I l ~ H),lilt: response Lime of l ll c h igh priorily jobs. In general. Im'oer pdoril) ' e u s l u m c r ~ ; have110 impaeL on lilt: menn response Lime of higher priority clislolllers. The proof is givenin :\ppendix A.

A!>'S"lL11lplian UThe u\l'erilge response Lime tukcn over LlU cuslomers, N : ; ~ } , is equal lo lh e mean

response time of un j \ l ! ~ U r n qu(;ue wilh arrival I"ule '\1+>-2 ,md mean sCI"vice rule 1.1.under reFS seheuuling. Thus. while pl'ccrnpLivc resume schedUling witl afIccl lh e meanresponse Lime of lhe individual clusscs, lilc average response time of lllc combined


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classes will be the same as under To'CFS scheduling. The proof is g iven in Appenuix D.NoLe Lhat AssumpLion U depends sLrongly a l l the condition Lhal J.L(::=J.L2;;j.J. and lhuL

ull service Limes are memoryless (exponential). In Lhts ca::ie changing lhe ser ....ie e dis-cipline from FCJi'S Lo preemptive resume leuves t he l lepa rLur c process al th e servercxacUy the sallle.

Uy AssumpLion A, thc value Df 11'1111.:\)' \)e: obLaint:d by r . : \ a l u ' - \ l l l l i ~ CqW.tLiUII (.:) "filIIU;;AI unt! 'u::=J.L. SllnUur])', ASSUll lpl ion U implies Lhat h ' : ~ ) nlil}' IJr;- obLuincci uy c\"


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To apply equatioll (11), it is necessary to know the va lues of Itr,p). The rcHawingussumplion can be used Lo derive these quantities. Like Assumplions A ilnd 13. thisilssumplion is InLuitively appealing. IL is proYed rigorously in Appendix C.

Ass'u.mplia"n CThe mean response Lime of the aggregale uf th e firsL]J priority c l a ~ s c ; , J((p). i

ullllallu Lhe rnoun response Lime of an ! I ' ~ / M / m queue with a r r i ' l ~ 1 ! !'


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ily level: however, the cxacl values of R(y) ilfC not readi ly available, since t \ssllmption Cno longer holtls. Therefore. approximate express ions fo r Rfp) will be derived [or 1) =2,3, .... :r . SubstituLion of t he se express ton:; into equat ion (11) \\"ill then yield approximale values of Up.

Some addilional noLi.\Uon is required La facililaLe discussion of this prohlem. Lcl,u(p) denole the mean sCI'vice raLe. weighted by arrival rule, of th e p l1ighc::L priorityclilsses.

1 ~ J.l(p)= jJ t...J Ai) ' A-I II. ~ = I.....J ) t-'"Ji-o:l

_ Ar1)mp(p )

Let R(d,!i.(p)")':PJ,m) denote Lhe mean rcspono;c lime of Lhe }J h i g l l C ~ L classes in an~ ' j / : \ - I / m queue opcl'uLing uHuer uisclplinc d, w!lcrc l i ( ; !) ulld .1,:,,) un , p-vccLun; of ~ ; c : ! ' v i t : l Jo.nu urriv


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(1" )

(: 0)

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meun service times of the individual classes. rr th e JLi'S m'o approximately equal. 7Jshould iJc very close Lo uniLy since l:onvcrLing from PH ] Lo ]i'Cf'S will only have a smat!eCTect on Lhe departure process. Conversely, if the l . l L ' ~ arc VC1"Y dissimilar, converlingfrom pm Lo FeF3 could have a substantial eCTecL on Lhe ucpartul"C process and 1} couhldiCTer signHlcilnLly from unily. I-lenec, any IlIOtlincalion uf Lhe sy:sLcm which r n ~ s e r v c : . : ;Lhe fuLias between the /-Li. t :::; und also IJrcscrves Lhe lrillTic intensiLy shaull! nuL -llIed Llwvalue of 1J significantly.

One such modification is Lo repl ace t he m servers by iJ. sinJlc SerVOl " TTL limes a.'j[ilSL. This preserves Lhe ralios between Lhe fJ.i '5, while lcilvinb Lhe Lnl.fTic inlcnsilics l lwsame, ilnc1 yields Lhe follOWing approximation:

II' ( l JR f ,l!:.(p),.A.(1.),tn) ..., II' (PRJ ,m.IL(p ) ~ i ! ) ' ~ II' (Ji"CrS .1!..(p) ,-Ali') ,m) II' (reFS, Tn 1l.;1') )"(p) ,1)

Hearri.lJlging lo oblain ilpproXimilliotls Lo lh e ljw.\nlilies need.ed in equo.tion ( ~ 1 ) .J:'(FCFS .J! '/J),A'IJ),m);Y (P J(J ,l1(il )'..\-,;.0)' Tn ):::.J JY (PJi'l .TnJJ.(p ).N"p), 1) I" (1"[-',i'


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analyzing: th e appropriate NIlJJp 1m ilnd }..! !lIp /1 queues respectively, but this approachis numericully complex. Instead. consider Lhe facLors thaL inOucncc {.

Nole thaL Lhe In serveI' quell? in Lhe nUllleraLor and Lhe single server queue in LhedenomlnaLor have Lhe same arrival process parameters .A0J). Also, Lhe service comple.lion processes a rc idcnLical whenever ul! servers in Lhe 'l it. ::;crvcl' queue an:: adivc . UllLhe olhcl' !1;:md, uS Lhe number of Ll.cLivc servers in the Tn SCr\."Cl' queue d C C 1 ' C d ~ C S frulJlIn Lo 1. lh e service {;umplcliull processes il l Lhe two queue::; uccomc incn;Qsiligly UlJ-~ i l l 1 i l a r .

Since Lhe number of acliVl: servers is prinlcu'ily a [uncUon uf Lhe ovct'ull Lr


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where Lhe values of Tj; fo r k =1,2.... ,T u fC given in equation (17). The approximatevalues of Hi (i=::2) may then be derived dirccLly [rom equation (11). HI may beevaluated as though Lhe other classes did not exist, using equation (4) with a=>..[ andU=J.LI_

This approximate melhod of obtaining response Limes may be- implcrllcnlcll cheaply. ror thl: C


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applicable Lo an arbi trury number of customer clusscs. lL aLLempts Lo


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fr'or f , n denoles th e number of customers. ilnd m deno te s the number n[ serversilvilililblc to th e n cusLomers.

Also, lel

j = 1.2,....TP('!.1:.) denoLe::; Lhe probability that Lhe syslem is in sL[ltc 11:.

With this nolaLion, the steady slaLe equations arc given by, ,

(A(r)+f (2; nj ,T n )p.)?(n)= L 9 (n, -1,,( P('n -!i.i)i:;; I i= I ,+J.J.L f ( ~ + l . r n - n ( i _ , ) P ( n + . f u )

1:;;'(A;)

To obtain L11C marginal disLribution of n l' SUnt (1\1) oyer ;'Ill:!l such Lha.t n I::::'/.; fo r a l l l ~ ,Venne

IT};= lJ(nl,'lt:,:, ... ,71.1')"l"'''Then, we bave, on Lhe lefL hund sitl12 of (AI),

VI,(r)+! (L n" ,m )/-L)P(".!.!:.):,:'i = I=A(r) L: Ft!J:.}+J-LJ(k,m) 2: P('1!:.)n,=k n,=k,+p. L I O : ' > ~ i , 7 r L - k ) P ( n )

n l =},; (=2

,-M 2: [J O ~ n i , m - k ) P ( I ! : ) ]

Il l =;; ~ = On th e r igh t hand side We hil\'C

, ,L Ly(ni-1)AiP(n-.!:!..i)+j.L L ~ J ( n ( + l , m - n ( i _ I ) g ( m . - n ( i _ I ) P ( n + . . ! 1 : ) (1\3)1I 1=ki ' " TL1=loi=1

=>"19 (k -1 );rJ.: - , +(A(r}-A')"l: +p,f (k +1,Tn)Y (m )111; +I,+f.l L L!(n i +l,711-n(i-I))u(m-n(i_l)P(!!:+..u..,;)"l=!:i=:.!

Because th e sel of I!: oyer which \"Ie n rc summing is infInite, ilntl bccause lerms '-iilh


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=0 (i2.:2) make no contribution to lhe rates of Oow between states. the last term in(A3) is equal to Lhe lo.sl term in (A2). It therefore follows that

NoLe that (I "+f (k ,m )1',)1f. =y (k -1 )AI1f. -I +f (k + I ,m ),L1f. . . , k;"O(A4-) is Lhe steady state equation for an M/11Jm system(M)

with FCFSdiscipline.arrival fille AI .and service rale /-t. regardless of how many clnsscs there arc.

Thus, Lhe bighcsl priodLy customers will have Lhe queue length dislr lbullon of ~ 1 l 1M/,\Vl1l :>y:;Lclil with arrival raLe Al amI service rL.l.lc J.L1 I ' c g a r d l c s ~ uf LIlt: olher c 1 a ~ : : ; c sof customers.

Hence. the response time of th e highest p rior ity class may be computed uSdescribed in J\ssumption A.

Appendix U: The ::ilcady Slale D i ~ l r i l J u l i o n of Lhe flccrcgaLc ( ; l a s ~Let PI.:. be Lhe probabUiLy that there arc k cusLomers in Lhe sysLcm in LoLal. k :::

0.1,2 .... ncgardlcss or llw combinulion or cUsLomCI' clUS:i Lype::;, Lhc uggl'cgtLLe eusLu-mer completion ("elLe of Lhc sysLem will be kf.l. if k ~ amI mf.l. oLherwisc. l"uI"Lhennore,i f Lh,nc Ut-e k cusLolllers in Lhe sysLcm, Lhc number w;lI be increased Lo k + 1 aL ruLeA I + ' \ ~ + ... +Ar;;;;A(T),sincc lite arrival procc::;::; of ull classes or cllsLolllcrs is Poisson.TbercCorc. Lhe sLeudy sLuLe clluuliollS of Lhe aggregaLe sysLcm musL uc given uy

(A[r) +J (k .m )/J.)Pk =g (Ie -1 )'\[r)J'k _I +J (k +1,m )/J.}J" q (lJ:)wherc J and 9 uxe ucfincll us Lnl\ppcnuix A. J-Jem;c, Lhc uggrcgaLc llueuc lengLh disLri.bUlion is Lhe sume us Lhut oC an M/;\-l!m sysLem wiLh arrivul ruLe A(T) uno 5crviec l'uLe /.t.

/ippcndix C: The Sleady ::Rale Vislriblion of lhe Tolal of Ule p l I i I ~ h e s l Priorily CIil:;::;Cll::;Lomers in Lhe System

For Assumplion C, it is suIIieicnt to show thaL Lhe combinalion of clusses 1,2, ... ,pwill lw.vc Lhe sumc qUCllc IcngLh disLl"ibuliol1 as un ) ,UM/m qucue wiLh arrivul rule ,\(J!)und service r u l l ~ ,u ior p = 1,2.... ,T .

Le t ~ } be Lhe sLcudy sLaLc probalJHiLy that Ulere ure k jou:..o of elusscs 1,2, ... ,p [nLhe syslem. Then


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" " L P(",)Tl{p)"k

The SLCiltly stale cquulions fo r the w,; 's arc obLained by summing (A 1) 0\"121' n!l1!,such Lhill L:ni=k.1' " I

Con::;ider the left hullll side of ( A ~ ) f irst . The cocrTiciclll of F(!.l:.) isr ;';-A:r )+ J ( L"lLj ,m )J-L= ,\:.Jll+ ('\(r) -,\0.' ))+ / ( 2..: Hi ,m J/.L+J ( : n., ,m )f-L

;= J ~ J i:.:;) I_ITherefure, UPO!l : : iUl l l lnaLion, the len hum! .::;ide uf l : \ ~ ) bccot,lcS

,( ' \ ~ } " i - J ( k , 1 r L ) f 1 . ) W ) ; + ( A : r ) - > " : , ; ; ) ) : : . . ' } : + ~ f ( 7l.,m-k).w')(.!...'.)

T1(}.J'=!: i =p ! lFor the u rr ival lc l' ll lS on t he r ig hl hUIltl side. we hil \ 'C

' l " )'.

1("')," y

" ,= L,\:.J};_I+ L L; v(n,-:),\JJ(!2:-E.J (CJ)T1v,):::/;i=1 "U.)"ki=Pl-l

Xow, Lhe lasL lCrJn in this expression LS equal Lo,L L y(n,}AiP(n)

"(J,)=I; i=l '+Jsince the oule!' sum is infiniLe i lnd [] (71) i ~ un indicnto!' [unction Lal':ll1L: Lhe '-illue 0 01'Thi:; reduces Lo

,L '\ ;S P ( l l : ) : = ( A ( r ) - A ~ , , ) ) ( J k (C-,)'=1' I I "(;J)" I;'i'l1l:s, Lhe: lC1Tn in (\r)- '\(:.) on th e lefl b and Slue; ill (e2) is bulnllced by ull equal lerl1lon th e r i g h l l l o . ~ l ~ t . I . ~ i u c , umllhc) ' eililecl.

The eodTicil: /J l uf 11011 th e ri21ll huntl ~ i u u i ~ p ,-L L.r (n( + :i.,1IL -n: ( - l ) )P( l i + ~ J + L L; f (7l; + 10m -n(i-I))PCn.+!iJ0":1.1':=;"=1 1,(.,)=ki=p '-1

)J ,.= L J( ' i /L. ,m)P(l l : )+ I ( 2..; - n 1 + : . U I . - / ~ ) P { ! ! J" (p ) " / : H l= 1 "(.0)=1; \=" I I,=J(.i.:+l,m);":.l ;f l+ L; J( L 'I l l ,m-/.:)J)("!.d

: ' t ~ ) = . l ; i=iJ I Jsincc th e sum over ,dl"ll- sllch lhul "IL;p):=k is infinile, us be[orc.

(C:.i)


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The l"(P)[J (k -1):J1; -I +fl.! (k + l .m )"-'/; + I '

highest prioriLy classes combinetl havc the same(eG)

queue length disLriuu-lion as un }.:/;\I/m sysLem with arl"ivul raLe AllJ ) and se rv ic e I"ClLe ,u regiJ.nJle;;s of UJ(":I : ~ l \ " r ; r oriority ,da,;sI..:Oi. uS required. This is sulTidcnt Lu pn.'\c t!l"l . ' \ s ~ U t i l p ~ i O l l C IS

COfl;;Cl[u(:Illly, aile In ay quickly obtain the result de:.ocl'ibetl in l\ppendlx U, lucre!:"by seLting p ::;./. in equation (CG). Also, the queue lenglh d l ~ l n l J l I L i o n of th e h i [ ~ h l : : . ; tpriority c la ss may be o!.>lilLned immediulely by selLing p : : ; ~ .I ~ e r c r c n c c : : ;

A J U ~ : i : ; ( Q \ " L..\:. A:-i!.l YAS:,\UGO:WDSi\iY, H,.\,;., "f'. C l i l ~ ; ~ , of Q l l c u c i n L ~ P1'obll..:;ll:; wiLI\ i'riori-~ j C S \'.'hen Lherc ilre UrgenL Oedel's," Hng. C U / H ~ ' I " n . 12(1) pp . G ~ > ? 2 (lU-r


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T/\YLOR. LV.S. ,\ND T E ~ I P L ! ' ; T O N , J.G.C., "Wailing Time in 0. 11ulLiscrvcr CutofT PriorityQueue, and iLs ApplicaLion to an Urbal l Ambulance Serv ice," Opns. Res. 2U pr .I1GO-1HlO (1900).WfIITr:, J-I. AXi) C : i - J R i S l ' l i ~ , L. . "Queueing with Pl 'ccmrLivc Priorities '.I'ilh TJn;ollkdo'.'(ll,'Upm;. Hes. Gpp. 'jI9-Wj (10:.iD).


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Table 1: Comparison of Approximale and Exact Response Times for N ServersOne Class with Heavy TralTie. One Class with Light Trame

D.OG1.000

0.1l2.000

L.--,-__. +- ----:C-c_1"_"-c"-cI:- - f C_1"_S_".2_' IIArrival Rates N"1.GOO N"O,050!Service Rates,i! Trame lnlensiLy

0.000O.OU:)U.uuo

He l. E:IT.

3.4-121.0002.9001.000

I N I Priorily Ordering 'Exac t Approx. He!. r;I'I'. I Bxad (\pPI'O;{.I . :r--2l , 2 1.309 1.309 0.000 10.(05 10.091 0.022 iI ' !L:!-_-r=J2 1.901 1.90t 0.002 __._.0___8__1_.0_0_3.__0_._00_0_1i ,;. 1 2 D.O?3 0.073 o.ooo! :.i.GUO G.09a 00:30 Ii 2 1 i . l12 1.110 0.003 J--' . 'OOO 1.000 0:000 Ir--.--- ------.---- I -------- "r, 12 0.'116 0.716 O.OOO! ~ - . O ( J 1.2:32 O.Oc;.1.,! 2 1 O.OIl? 0.1l0D 0.003 I 1.000 l.000 0.000 i~ : i - ---1-,--2-----+-0,--.:--01----:2--0.-0 4.2:---0.-0:---00l - --- ;202: - I , ,1 O.?OO 0700 0.003 1.000I-- i !, 10" : 1 2 I0.002 0.002 o.ooota.noI 2 1 a.Gog a.Goa O.UOO ~ . O O U-----' - - - - - - - -_ ._-

.. The exacL r c ~ p o l 1 S c l i m c ~ in lbis row and the n ex t dWC1' [ru!lL th e c u r r c ~ p o f l l . H l l : . . :ones in },;ilruni and J'\ing (19U1). We have been inrurmec.! by Or. J\iu)J U!cIL Lhenumbers ~ h o w n here arc th e correeL o n . c ~ .


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Table IJ: Comparison of Approximate and f:xacL Hesponse Times for N Sf rvers

l.UOOI:\

Both Classes with Moderate 'l'rufTic

0.:)00elas::; 2

2.0uU/:\

,,I[--._------I If- -11---,..--,--t\rrival Rales 0./':30,: Sc'!'vicc RuLes r

a.oou

O.OOU

U.QuO

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