e-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov

Transcription

1 June 7 e-ournal Relably: Theory& Applcaons No (Vol. CONFIDENCE INTERVALS ASSOCIATED WITH PERFORMANCE ANALYSIS OF SYMMETRIC LARGE CLOSED CLIENT/SERVER COMPUTER NETWORKS Absrac Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Clayon Campus, Buldng 8, Level 4, Wellngon road, Vcora 38, Ausrala. e-mal: vyacheslav.abramov@sc.monash.edu.au The paper sudes a closed queueng newor conanng a server saon and dencal clen saons. The clen saons are subec o breadowns, and a lfeme of each clen saon s assumed o be a random varable ndependen of all oher ones havng he probably dsrbuon G (x. The server saon s an nfne server queueng sysem, and clen saons are sngle server queueng sysems wh auonomous servce,.e. every clen saon serves cusomers (un only a random nsans generaed by srcly saonary and ergodc sequence of random varables. The oal number of uns n he newor s N. The servce mes of uns n he server saon are ndependen exponenally dsrbued wh parameer. The expeced mes beween deparures n clen saons are ( N μ. Afer a servce compleon n he server saon a un s ransmed o he h clen saon wh equal probably / l, where l s he number of currenly avalable (.e. no falure clen saons, and beng processed n he h clen saon he un reurns o server saon. The parameer N s assumed o be large. The am of hs paper s o sudy he behavour of bolenec queues and o fnd confdence nervals assocaed wh ncreasng a gven hgh level of queue proporonal o N n clen saons. Key words: Closed newors, Performance analyss, Normalzed queue-lengh process, Confdence nervals Mahemacal Subec Classfcaon: 6K3, 6K5. Inroducon Consder a large closed queueng newor conanng a server saon (nfne-server queueng sysem and dencal sngle-server clen saons. The oal number of cusomers (un s N, where N s assumed o be a large parameer. The deparure process from clen saons s assumed o be auonomous. Queueng sysems wh auonomous servce mechansm have been nroduced and orgnally suded by Borovov [6, 7]. The formal defnon of hese sysems n he smples case of sngle arrvals and deparures s as follows. Le A ( denoe an arrval pon process, le S( denoe a deparure pon process, and le Q( be a queue-lengh process, and all hese processes are sared a zero ( A( = S( = Q( =. Then he auonomous servce mechansm s defned by he equaon: 34

2 June 7 e-ournal Relably: Theory& Applcaons No (Vol. Q ( = A( Ι{ Q( s > } ds(. The queueng sysems wh auonomous servce mechansm have been suded n many papers (e.g. Abramov [, 3, 4], Frcer [8, 9], Gelenbe and Iasnogorods []. In he presen paper we sudy a closed clen/server newor (see Fgure. Fgure. An example of clen/server newor The arrval process from he server o he h clen saon s denoed A, N (. The servce me of each un n he server saon s exponenally dsrbued wh parameer. Therefore, he rae of arrval o clen saons depends on he number of uns n he server saon. If here s N uns n he server saon n me, hen he rae of deparure of cusomers from he server n me s N. There are clen saons n oal, and each of clen saon s a subec o breadown. The lfeme of each clen saon s a connuous random varable ndependen of lfemes of oher clen saons and has he probably dsrbuon G (x. All clen saons are assumed o be dencal, and a un ransmed from he server chooses each one wh equal probably. (For hs reason he newor s called symmerc. Therefore, f here are l avalable clen saons n me, hen he rae of arrval o each of hese clen saons s / l. The deparure nsans from he h clen saon ( =,,..., are ξ N,, ξ N, ξ N,, ξ N, ξ N, ξ N, 3,... where each sequence ξ N,, ξ N,.,... forms a srcly saonary and ergodc sequence of random varables ( N s he seres parameer. The correspondng pon process assocaed wh deparures from he clen saon s denoed N S N ( = Ι ξ = l= N, l, and sasfes he condon Ρ lm S, N μ ( = N =. 35

3 June 7 e-ournal Relably: Theory& Applcaons No (Vol. Then, he relaons beween parameers, μ and are assumed o be (. <, μ and (. >. μ Condon (. means ha all of he clen saons are nally non-bolenec,.e. he servce rae s greaer han arrval rae. Condon (. means ha afer one or oher breadown all of he clen saons become bolenec. Denoe l = max l : >. lμ he maxmum number of avalable clen saons when he clen saons all are bolenec. Then for all l l he res l clen saons wll be bolenec as well. The queue-lengh process n he h clen saon s defned as ( Q ( s ds (, Q N ( = A N ( Ι N > N s where A, N ( s he arrval process o he clen saon. Le α < be a gven posve number. We say ha he newor s a rs f he oal number of uns n clen saons exceeds he value αn. Assumng ha a he nal me momen all of he uns are n he server saon, he am of hs paper s o fnd a confdence nerval [, θ such ha wh gven hgh probably P (say P =. 95 he newor wll no be a rs durng ha me nerval [, θ. For newors wh an arbrary number of clen saons hs problem s hard, because he behavour of bolenec queues s very complcaed (see nex secon for explc resul. Therefore n he presen paper we sudy hs problem for he case of newor wh wo clen saons only. A large closed clen/server queueng newor wh bolenecs has been suded n many papers. The bolenec analyss of Marovan newors has been provded by Kogan and Lpser []. Abramov [,, 3] has exended he resuls of [] o non-marovan newors. Specfcally, [] has suded he varan of newor wh auonomous servce mechansm n clen saons. The resuls of [] have been hen exended o newors wh wo ypes of node and mulple un classes n [4]. However, he conrbuon of he aforemenoned papers s purely heorecal. All of hem provde he bolenec analyss for he parcular case of one bolenec saon and under he assumpon ha a he nal me momen all of he uns are concenraed a he server saon. 36

4 June 7 e-ournal Relably: Theory& Applcaons No (Vol. The dealed bolenec analyss of he newor ncludng all cases relaed o bolenec saons as well as nal condons has been recenly done n Abramov [5]. The resuls of [5] are promsng for he soluon of many appled problems. Specfcally, he analyss of [5] s devoed o closed clen/server newors n sem-marov envronmen requrng he sudy of hese newors under mos general assumpons. The asympoc soluon of he problem of he presen paper, as N ncreases ndefnely, s based on he sudy of [5]. I s worh nong ha relably of compuer sysems hemselves has been suded n many papers. We refer he boo of Xe, Da and Poh [], where a reader can fnd he dealed nformaon relaed o hs subec. The confdence nervals ha are suded n he presen paper are relaed o relably of nformaon, whch heavly depends on relably of he newor. The paper s movaed by sgnfcan praccal problems n elecommuncaon sysems. Suppor and exchange of nformaon s very expansve and ofen ncreases he relaed coss of he equpmen self. On he oher hand, relable suppor of nformaon s dervave from hgh relably of equpmen and drecly depends on ha relably. A specal crcle of praccal problems s relaed o suppor of large daabases. Then uns" are assocaed wh uns of nformaon (record, and clen saons" are assocaed wh users of a daabase. Falng saon" can be assocaed wh absence of connecon or very low rae of exchange. Low exchange n ceran saons can resul n bolenec of enre newor leadng o desrucon of a daabase. The paper s organzed as follows. In Secon we recall some of he resuls of [5] whch are necessary for our purpose and hen adap hem o he case of symmerc newor consdered here. In Secon 3 we derve he dsrbuon of he normalzed queue-lengh processes n avalable clen saons. In Secon 4 we esablsh resuls for confdence nervals n he parcular case of wo clen saons. In Secon 5 we gve a smple numercal example. In Secon 6 we conclude he paper.. Bolenec clen saons In hs secon we recall some resuls of bolenec analyss of [5] correspondng o he cases consdered n he presen paper. We sar from he elemenary case of l equvalen bolenec saons exacly,.e. he case ha a he nal me momen = here are l bolenec saons s dscussed. For smplcy, assume ha all of hese l saons are absoluely relable, and a he nal me momen = here are ( β N uns n he server saon, < β, and he res βn are dsrbued beween l clen saons. So, because he newor s symmerc, he assumpon ha here are approxmaely βn / l uns n each clen saon n me =, accordng o he law of large numbers, s reasonable. The assumpon ha he clen saons are bolenec means ha ( β > lμ. The resul on asympoc behavour of normalzed queue-lengh n clen saons follows from Proposon 5.3 of [5] whch relaed o an asymmerc newor wh bolenec saons and arbrary nal queue-lengh. Recall hs resul. Lemma.. Assume ha all clen saons are nally bolenec, and he nal queue-lenghs n clen saons are asympocally equvalen o Nβ, Nβ,..., Nβ correspondngly ( β β... β, as N. Then, he normalzed queue-lengh n he h clen saon n lm as N s deermned as 37

5 June 7 e-ournal Relably: Theory& Applcaons No (Vol. (. q ( [ ( ] ( ( = β β μ r s ds, = ( ( = μ (. r( = exp (, ( = = where q ( denoes he normalzed queue-lengh process n he h clen saon n lm,.e. q ( s he lm n probably of Q, N ( / N as N ncreases ndefnely. In he noaon of hs lemma ( N denoes he nsananeous rae of uns o he h clen saon n me =, and μ N denoes he servce rae n he h clen saon. In our parcular case he number of nodes s l, he nsananeous rae of uns o each clen saon s ( β N / l and he servce rae s μn and all q ( are equal,.e. q ( g( for all =,,..., l. Therefore n our case from hs Lemma. we have he followng saemen. Proposon 3.. We have: β ( ( (.3 ( ( β β g = β ( μ r s ds, l l l where μl ( β (.4 r( = ( e. ( β 3. Lmng normalzed cumulave queue-lengh process In hs secon we sudy he lmng (as N normalzed cumulave queue-lengh process n clen saon. The lmng normalzed cumulave queue-lengh process s denoed q(. A he nal me = here are avalable clen saons. Le,,..., be he momens of her breadown,.... The above momens of breadown are assocaed wh he behavour of he medependen newor whch can be consdered as a newor n sem-marov envronmen. Therefore one can apply Theorem 5.4 of [5]. The random me nerval [, ] s he lfeme of he enre sysem. Therefore q( s o be consdered durng he aforemenoned random nerval [, ]. Recall ha l = max( l : /( lμ >. Therefore, accordng o Theorem 5.4 of [5] we oban ha n he random nerval [,, q ( =. Nex, n he random nerval [, he equaon for q ( s 38

6 June 7 e-ournal Relably: Theory& Applcaons No (Vol. (3. q( = ( lμ( r( ds, where r ( s gven by (.4. Equaon (3. follows from (. and (. as follows. Seng β =, l = and replacng wh from (. we oban: l g( ( = μ ( r( ds. l l Hence, ang no accoun ha q ( = l g( we arrve a (3.. In he nex nerval [, l >, we have he followng equaon:, q( [ ( ] q (3. g( = [ q( ]{ ( μ l l [ q( ] r( ds}. l Therefore, n he me nerval [, (3.3 q q( [ q( ]{([ q( ] μ( l ( ( = [ q( ] r( ds}. In an arbrary me nerval [,, = l, l,...,, we have: q = q( [ q( ]{([ q( ] μ( ( ( [ q( ] r( ds}. In he las endpon we se q ( =. 4. Confdence nervals The formulae for he lmng normalzed cumulaed queue-lengh process are complcaed. Therefore n hs secon we oban confdence nervals for he parcular case of wo clen saons. In hs case only smple represenaon (3. s used, whch n he case of wo clen saons loos as follows: 39

7 June 7 e-ournal Relably: Theory& Applcaons No (Vol. (4. q( = ( μ( r( ds, where μ (4. r( = ( e. The confdence nerval s srucured from wo nervals. The frs one s [,, where he lmng normalzed cumulave queue-lengh s zero. The second nerval s [, θ ], where θ s a pon where q ( θ α. Equaons (4. and \(4. are defned for <, where s a random breadown pon of he second clen saon. * Le θ be a pon where ( * * q θ = α. The pon θ s a random pon dependng on. However, * under he assumpon ha one or oher clen saon s acve, he lengh of he nerval [,θ ] s fxed and unquely defned from (4. and (4.. Le us derve probably dsrbuon of he process q (. Clearly, ha he probably ha q( = concdes wh he probably ha he lengh of he nerval [, s greaer han. Therefore, (4.3 Ρ ( q( = = [ G( ]. Nex, (4.4 Ρ ( q < = [ G( ][ G( ], ( where s such he value of under whch (4.5 ( μ r( ds =. Equaons \(4.3 and (4.4 are easly obaned by sandard argumens of probably heory. Then he probably ha he lmng normalzed cumulaed queue wll reach he value before absorbng a s [ G( ][ G( ] d. [ G( ] d The problem s o fnd he value α such ha 4

8 June 7 e-ournal Relably: Theory& Applcaons No (Vol. [ G( ][ G( ] d (4.6 P. [ G( ] d I s wren he nequaly raher hen equaly because he exac equaly can be reached for > α, whle for all α here mus be nequaly ( Numercal example x Consder he followng example. Le = 4, μ = 3, α =., P =.95, G( x = e. From (4.6 we have: e ( e e 4( d d = e. Soluon of he equaon e =. 95 yelds = From (4.5 we oban: = e d = e d =.5.5e = Ths value of s less han α =., and herefore hs value = s he requred value for a confdence nerval. 6. Concludng remars We found confdence nervals assocaed wh ncreasng a gven hgh level. The confdence nervals ha esablshed n he presen paper are random. They are obaned n erms of he parameer, whch s he value of lmng cumulave normalzed queue-lengh under whch he probably ha he sysem wll be acve s no smaller han a gven value P. Thus, he sraegy s o observe he cumulave queue-lengh process unl he oal number of uns n clen saons reaches he value N. As soon as he oal number of uns exceeds hs value here s no enough confdence ha he sysem and/or nformaon wll be avalable. Acnowledgemen The research was suppored by Ausralan Research Councl, gran # DP

9 June 7 e-ournal Relably: Theory& Applcaons No (Vol. References [] V. M. Abramov,. A large closed queueng newor wh auonomous servce and bolenec. Queueng Sysems, 35, [] V. M. Abramov,. Some resuls for large closed queueng newors wh and whou bolenec: Up- and down-crossngs approach. Queueng Sysems, 38, [3] V. M. Abramov, 4. A large closed queueng newor conanng wo ypes of node and mulple cusomers classes: One bolenec saon. Queueng Sysems, 48, [4] V. M. Abramov, 6. The effecve bandwdh problem revsed. arxv: mah [5] V. M. Abramov, 7. Large closed queueng newors n sem-marov envronmen and her applcaon. arxv: mah. 64. [6] A. A. Borovov, 976. Sochasc Processes n Queueng Theory. Sprnger-Verlag, Berln. [7] A. A. Borovov, 984. Asympoc Mehods n Queueng Theory. John Wley, New Yor. [8] C. Frcer, 986. Eude d'une fle GI/G/ á servce auonomé (avec vacances du serveur. Advances n Appled Probably, 8, [9] C. Frcer, 987. Noe sur un modele de fle GI/G/ á servce auonomé (avec vacances du serveur. Advances n Appled Probably, 9, [] E. Gelenbe and R. Iasnogorods, 98. A queue wh server of walng ype (auonomous servce. Ann. Ins. H. Poncare, 6, [] Ya. Kogan and R. Sh. Lpser, 993. Lm non-saonary behavour of large closed queueng newors wh bolenecs. Queueng Sysems, 4, [] M. Xe, Y. S. Da and K. L. Poh, 4. Compuer Sysems Relably: Models and Analyss. Kluwer, Dordrech. 4