SU1 JouエTta］L oま 頁ath創natics （Po＝m●rly T富U Ha告h●matics） ▼011国匝e 26， 貰1国mber 1 （1990）， 101−1］LO
QUASI LOSS PROBABILITY．AND QUASI THROU．GHPUT
OF T正lE SYSTEM M／M／1／1∨→／M／1／1
Toji MAKINO
（Recei▼ed Harch 2， tg90；Revised Hay 7， 1990）ABSTRACT． We oonsider the two stage tandem queueing sys−
tems with量nite waiting room capacity． The interdeparture in． tervals， that is output，加m the first stage form the ilput to the seoond stage． In general， the interdeparture intervals．from the first stage are not mutually independent． Nevertheless， we compute the vahe of the loss probabi五ty for the s㏄ond stage assuming as if the output from the丘rst stage has i．i． d．． We caU the value．“quasi k）ss’ probability”． In the same sense， we com− pute the quasi throughput from the second stage． We guess the quasi values may be available as a good approximation to the true values． In a viewp・int・f ab・ve statement， we i皿vestigate these values of the loss probability and the throughput fbr thesystem．M／M／1／N→／M／1／1．
1980Matんematics subゴec￠classifi侃tions（19851∼四輌3輌oπ）． Pri−maエy 90B22；Secondary 60K25．
Keywords． Tandem queue， separability， Quasi loss probability， Quasi throughput． ． §1．introduction（cf．［6］，［7D In this paper we consider the tandem queueing system with two stages，M／M／1／1V→／M／1／1，
wh輌ch is de丘ned in「the f（）llowing way； The］［irst stage．is M／、M／1／」v， i．e．， the single server queueing system hav輌ng Poisson−arrival， exponential−service and fixed waiting room capacity 」V． Therefore if a customer（we say it“call，，）arrives’ to the firsレs輪ge when th・qu・u・・ize・i・N・，th・n． th・・u・t・聴・h・・ t・1・av・th・・y・t・m with・・t reCeiVing the ServiCe（we Call− it‘‘lost Cal，，）・101
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QU▲SI LOSS P且OB▲BILITY▲m｝QU▲SI rmOUGHPUT
The∫i耳terd｛epartUr（＞」耳tery司，・sρ．c｛ lled・・“output，，デfrorP the first stage queue M／M／1／IV． fb㎜．th（｝iエ｝te’arriyal ip￠rval・（that三is “inpUl，，）to the second queue， which is having ekponentia1 serVice and fixed waiting room capacity 1． Acustomer who丘nished the se‡vi（℃Ct the first stage is lost when the size of second queue is 1． It is wel known that the interdepa」rture intervals from the、M／G／1／O ar6血utually iidependent， ahd that the interdeparture interva1 from the system M／G／1／O has a same distribution as the random variable A十3， where． the rahd6血． variables」4 and 5mean inte士arrival time of the cuStb血6r and the Sefvice time in the cha皿ne1， respect輌vely」 Therefore， we cari dbtain the loss pr6bability at the second stage『of the system．M／G／1／0」ジ．／M／1／1 as． the loss probal）ility of the single server queue GI’／、M／1／1 with intefarrival time A十「S．、 ． ’ We caU．the ’propertY mentiOned 6bove“the 16ss ptobability of the second stage is separable”．（We abbrevi’ate the， property “10ss separability，，）． By’the way， We can See’that the l syStem with． the channel→M／1／O as the second s七age has the loss’separability regardless bf‘the dependency of the interdeparture ihterval L from the firSt stage． That is， we can calculate the loss probability fbr the second stage bY P（L＜s）， ■ where s meanS the service time pt ．the second stage・ So if we cons輌der the syste血’α／ハイ／1／」V『→／」∬／1／0，伍e五we can see the system has the property of loss separabilitY． H・WYρ・・th・・y・1・Iln．」叩4／1！N三μ／1／1．．h・・n・1・ng・・th・p・・p− erty of k）ss separability． Neverthdess， we compUte l the value Of the lossP・・b・biHty伽th・・ec・nd・t㎎e assuming th・t th・6・輌工加孟th・first
stage has i．i． d．． We cal the value“quasi loss probaility，，． In the section 2， we wil compare the quasi value with the tru（｝vah坦of loSs probability In genera1， we will guess the quasi loss probability may be available as a good apProxilnation’to the−k）ss probability」t． Lastly we investigate thg thrQughput from the second stage of the sys− tem M／」M「／1／」V→／M／1／1．“ T’h6 value eati biE ealculated by using the loss probal）ility for second stage． So we calculate the throughput using the quasi loss pmbabilit苫We c泣l the value｛‘qu∼msi th迦gh画t”． In the section 3， we will． compare ・the，quasi・value with the true value qf thfoughput． ．、 §2．Qudsi．16ss之pfbbability二〇f M／、M／1／」v→／M／1／1 】‘ ＼ Lー ヒ， ． ． ． i ’』 、 ㎡ Let us consider the system M（λ）／、M・（μ）／1／N．・→／M（μ）／1／1， where theT，．．．HAKINO
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p紅a1鞭tersλ，Earβarriv尋r『de tO the fiエst stage．麺d．the seryice rate at the ・h皿・・IS侮st・；han・・1 aPd・ec。nd也・頑），・e・p・cti・ely・ At first， we calculate the quasi lOss prObability午， which is the loSs probab盗ty at the second stage．When・ψe interdβpartUre inte㎜1s丘om the 丘rst stage are assumθd mutua∬y independent． So we conSider the output distribution from the first stage M／」M／1／1V． N・wl・t pS−）（・一・，2，3，…，蜘＋・）b・th。，teady，t。t。 p，。b。bility that the state is n just beforg the departure from the first stage， where the state means the number of custolners in the first Stage． Let P be the transition matrix between the’states just befbre the de・ parture from the first stage．P＝
1 2 3 …
． ・ ・N
2V十1
1ゐ0克1鳶2
ゐ1V＿1 α」V2
ゐ0ゐ1ん2
克1V＿1 α」V 3 ：0 ゐo先1・
動V＿2 α」V＿1 ・．N 「O．0 0
克1 α2」V十1
0 0 0
ゐ0 α1 The element kr of P describes the probability that r customers wil arrive during a serVice， and it is given by kr−f。°°9：A sg，！：．X（！Ax）’…一・・d・一（、＋4」）r＋、・whereρ＝λ／μandαπ＝Σ三πゐr．
We can obtain the value P（一）一（P’（i−）pS’）…縞）， using the equation P（一）・P−P（ヨ． E・p㏄i・lly・we・can g・t・th・val…f bS・一）・品姐・w・．、・ ・1−）一、＋，＋，・≒…＋，N・ （・） N・t・・．Lgl・P・（n−・，．・，2，…，N＋・）’b・・h・…a4・・ta・e・i・b・bili・蜘・・e πmeans the number of custom但s in the systeln．．＄o We Cap． also get th・valu・・f pS’） ti・i・9・th・・el・ti・n ・1−）一．、、竺，。1104
QU▲SI LOSS PROB▲B工LITY▲ND QU▲S工 maOUGHPUT Then we can describe the m．g．￡（moment generating function）Mu（θ）of the interdeparture interval U from the system M（λ）／」M（μ）／1／」V aぶfbllows． M。（θ）−MA＋s（θ）・pS−）＋M、（θ）・｛・−pl−）｝，‘ （2） where MA＋s（θ）a皿d Ms（θ）are the m．g．￡6f、4十S and S， respectively In this case， we have． MA＋・（θ）一、1、・，！θ
M・（θ）一，1−b・ Thus we have the expression M・（・）一（、1、）・（。：、）・、＋，＋，，≒…＋，N＋（μμ一θ）・1鵠ギ1芸；N・ （・）
Now we consider the loss probability q fbr the single server system GI／M／1／1． In order to calculate the value of g we putP（A＞Sl）＝α
P（A＞Sl十S2）＝b，
︶︶
45
︵︵
where、A represents the interarrival interval of the customers， and Si，52 are the serVice times of customers． Then we can get the value of ・q by expressi皿（6）．・−i語・ （・）
Next， let us calculateもhe、quasi 1（）ss probability at the second stage of the system M（λ）／M（μ）／1／N→／M（μ）／1／1． Here we assume the output L丘om the single server queueing system M（λ）／M（μ）／1／」Vis a sequence of i．泌． random variable， and m．g．￡of L is given by the eXpression（3）・We putP（L＞S1）＝αo，
P（L＞Sl十S2）＝bo．
︶︶
7・只︶︵︵
r．目▲KINO
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III this system， we have U＝λ・（1−9｛一））（1−ql−））， （13） wh・re 9S−），φ一）m・…th・1・ss pr・圃hty楓h。 first、t。g。 and・th。、ec。nd stage， respectively． On the other hand， （14）． ぎ we calculate the quasi throughput U by eXpression ∂一λ・（1 一 qS’））（1−7）． （14） Thus we have：−1誇）・． （・5）
ぎ The ratio ofσ／U in case of N＝1，2and 5楓he system MIM／1／N→ ／M／1／1have also shown in Table 1∼3． プ From Tables， we can recognize the value of U／σis neaエly equal to 1．0． Table 1． Tr…al・・91−）皿輌・・温。。7。f th。1。ss p，。b。bihty。t th。，ec。nd stage and the ratio of throughput fbr the system M／ルf／1／1→／M／1／1 ρ 一q
γ 岬ﾐ／σ 0．1 0．0063 0．0082 1，001 0．2 0．0215 0．0270 1，005 0．3 0．0412 0．0502 1，009 0．4 0．0625 0．0745 1，012 0．5 0．0839 0．0980 1，015 0．6 0．1044 0．1199 1，017 0．7 0．1235 0．1397 1，Ol8 0．8 0．1411 0．1575 1，Ol9 0．9 0．1570 0．1734LO19
1．0 0．1714 0．1875 1，019 1．5 0．2246 0．2377 1，017 2．0 0．2564 0．2667LO13
2．5 0．2764 0．2845 1，011 3．0 ⑩．2897 0．2961 1，009106
QU▲SI LOSS PROB▲BエLエTヱご▲ND qU▲S工 THReUGHPUT We calculate the quasi loss probability 7 by using ao，bo instead ofα， b in the expression（6）， that is， we have ．・−1等云竺）；。・． （・） The probabi五ty ao is calculated as fbllows． αo＝1．−ML（一μ） 1 1． ＝ラ｛1＋（・＋，）（・＋，＋，・＋．．．＋，N）｝・ （1°）
whereρ＝λ／μ． And we get d 妬＝・・一μ・｛評・（θ）1・一一・｝’÷、（、＋，）・（、＋芸号・＋．．．＋，N）・ （・・）
Substitutillg（10）’and（11）’ intO（9）， we can get the value of 7． O・th・・th・・h・nd， w・g・t th・1・ss p・・b・bility ql−）a・・f・11・w・． I」etρらゴ（i＝．0，1，2，・一，1V十1：」＝0，1，2）be the s七eady sta七e probability， where i andゴmean the number of customers in the first stage and the second sfage， respectively：Solving．the bala nce eq［ua七iOns concetningρら」， we can ・bt・i・th・1・ss p・Ob・bilityφ一）by』th・’exP，essi・n（12）． −Lρ1，2＋P2，2＋…＋PN，2＋PN＋1，2ql−、一（PO，O十PO，1十PO，2）・ （・2）
T・bl・1∼3・h・杣・；・IU・S’・f gl−）・nd・7 i・。a・e。f N＝1，2。nd 5・fth・・y・t・m MIM／li／N→／M／1／1． IFr・m th・Tab1・・we c・n・ee th・ fbllowing facts．． The quasi ValUe§慨are not’identical with the true values ql−）． Th・・ef・・e・th・・y・tem h・…。t th， P・6Pe・ty・f・ep・，ability． H。w。ve，， 7my b・avail・b1・a・ag・・a・PP・・xim・ti・n’・t・qll）． Fig・・e 1・1…h・w・th・iv・1・rr・f・1−）ψ嗣一〇，・，・・nd・・． §3．Quasi throughput of．M／M／．1／」V 4・IM／1／1 Let us consider the throughputσ丘om the system M（λ）／M（μ）／1／ハr→／M（μ）／1／1． The throughput means the rate of depa rtures du血g unit time．T？盟▲K工NO
107
T・bl・2．・Tr・・valu・ql→and・q…ival。。7。f th。1。ss p。。b。bihty。t・th。。㏄。。d stage and the ratio of throughput for the system、M／M／1／2→／M／1／1 ρ 一 X27
’ﾐ／σ 0．1 0．0086 0．0089 1，000 0．2 0．0299 0．0311 1，001 0．3 0．0574 0．0604 1，003 0．4 0．0870 0．0919 1，005 0．5 0．1159 0．1225 1，007 0．6 0．1428 0．1506 1，009 0．7 0．1669 0．1755 1，010 0．8 0．1881 0．1971 1，011 0．9 0．2066 0．2156 1，011 1．0 0．2226 0．2314 1，011 1．5 0．2744 0．2811 1，009 2．0 0．2994 0．3039 1，006 2．5 0．3123 0．3154 1，004 3．0 0．3195 0．3216 1，003 T・bl・3． T・u・val・・ql−）and・qu・・i・・1・・7。f th。1。ss p。。b。bility。t・th。，㏄。nd stage and the ratio of throughput for the system M／M／1／5→／M／1／1 ρ 一 X2 γ ’ﾐ／σ 0．1 0．0090 0．0090 1，000 0．2 0．0322 0．0322 1，000 0．3 0．0645 0．0646 1，00σ 0．4 0．1016 0．1019 1，000 0．5 0．1398 0．1405 1，000 0．6 0．1763 0．1775 1，001 0．7 0．2091 0．2108 1，002 0．8 0．2370 0．2391 1，002 0．9 0．2598 0．2521 1．003馳 1．0 0．2777 0．2800 1，003 1．5 0．3198 0．3210 1，001 2．0 0．3295 0．3300 1，000 2．5 0．3320 0．3322 1，000 3．0 0．3328． 0．3329 1，000一 108 → ．β2工的8［ 茄 α 0 3 コ 0 5 り⇔ ロ 0 0 つ一 ト 0 5 ー ロ 0 0 1 ひ 0 5 0 0 0 QU▲SI LOSS PROBMBIL工T．Y’▲M）．qUASI pmOUGHPUT 1 ・＝ el−）（f・r N＝。。） 1．，｝（f°「 ’V＝2） 92・ 1。）｝（f・r N ＝1） 92 （一・） 7＝92 （f・r・N＝O） 0．5 1．0 1．5 2．0 215 3．0 ρ→
