V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

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1 R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon road, Clayon, Vcora 8, Ausrala e-mal: vyacheslav.abramov@sc.monash.edu.au Absrac In he recen paper [Abramov, RTA, 7), pp. 4-4], confdence nervals have been derved for symmerc large clen/server compuer newors wh clen servers, whch are subec o breadowns. The presen paper manly dscusses he case of asymmerc newor and provdes anoher represenaon of confdence nervals. Key words: Closed newors, Performance analyss, ormalzed queue-lengh process, Confdence nervals. Mahemacal Subec Classfcaon: 6K, 6K5. Inroducon Consder a large closed queung newor conanng a server saon nfne-server queung sysem) and sngle-server clen saons. The oal number of cusomers un s, where s assumed o be a large parameer. The deparure process from clen saons s assumed o be auonomous. For he defnon of queung sysems wh auonomous servce mechansm n he smples case of sngle arrvals and deparures see [], where here are references o oher papers relaed o ha subec. The arrval process from he server o he -h clen saon s denoed A, ). The servce me of each un n he server saon s exponenally dsrbued wh parameer λ. Therefore, he rae of arrval o clen saons depends on he number of uns n he server saon. If here s uns n he server saon n me, hen he rae of deparure of uns from he server n me s λ. There are clen saons n oal, and each 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. The probably dsrbuon of -h clen saon s G x). The clen saons are no necessarly dencal, and a un ransmed from he server saon chooses each one wh correspondng probably p. Ths probably p depends on confguraon of he sysem n a gven me momen, ha s on he number of avalable no falure) clen servers and here ndexes. In general such nd of dependence s very complcaed. However, n he case of he newor wh only wo clen saons hs dependence s smple. Ths smples case s us dscussed n he presen paper. The deparure nsans from -h clen saon,,,) are ξ,, ξ, + ξ,, ξ, + ξ, + ξ,, where each he sequence ξ,, ξ,, forms a srcly saonary and ergodc sequence of random varables s he seres parameer). The correspondng pon process assocaed wh deparures from he clen saon s denoed - -

2 R&RATA # Vol.) 8, March S ) Ι ξ, l, l and sasfes he condon lm S ) μ. The relaons beween parameers λ, p, μ,,,) and are assumed o be λp.) <, μ and for all,,,, λ.) >, μ a leas for one of s,,,,. In he sequel, asymmerc newors wll be dscussed for, and he relaons.) and.) wll be assumed for hs value of parameer. In he case of symmerc newor, where μ μ for all and p ) p l, where l s l he number of avalable clen saons n a gven me momen), condons.) and.) correspondngly are as follows: λ.) <, μ and λ.4) >. μ In hs case, condon.4) means ha afer one or oher breadown he enre clen saons λ become bolenec, and here s a value l max l : >. lμ The queue-lengh n he -h clen saon s defned as { Q s ) } ds ). Q ) A ) Ι > s In he case of large, he behavor of he queue-lengh process s as follows. When all of clen saons are avalable n me, mos of uns are concenraed a he server saon, and normalzed queue-lenghs q, ) Q ) / vansh as ncreases ndefnely. When afer one or anoher falure he clen saons are overloaded n me, hen q, ) converge n probably, - -

3 R&RATA # Vol.) 8, March as ncreases ndefnely, o some posve value. Then he queue-lenghs n clen saons ncrease more and more as ncreases. Then he sysem s assumed o be a rs f he oal number of uns n queues n clen saons ncreases he value α. Confdence nervals for symmerc large clen/server compuer newors have been suded n Abramov []. The movaon of hs problem, revew of he relaed leraure and echncal deals are gven n []. The presen paper manly dscusses confdence nervals for large asymmerc clen/server compuer newors and provdes new represenaons for confdence nervals. In he case of symmerc newor, a confdence nerval s characerzed by parameer α. More specfcally, for gven level of probably P, say P.95, here s value characerzng he guaraneed level of normalzed cumulaed queue, and a random) confdence nerval s assocaed wh hs value of. In oher words, along wh parameer α characerzng he sysem a rs we have anoher parameer, whch s closely assocaed whα and wh probably P. In he parcular case, he explc represenaon for has been esablshed n []. In he case of asymmerc newor such deermnsc parameer canno longer characerze confdence nervals. To see, consder a newor he parameers of whch are: λ, p p, μ 4, λp λp μ. Then, condon.) s fulflled, and <, <. μ 8 μ 4 λ λ Condon.) s fulflled as well, and <, >. In hs case one can expec he μ 4 μ suaon when he second server breadowns frs, and he cumulave normalzed queue-lengh process wll converge o zero as for all, and here s no observable parameer. Therefore, he newor can breadown unexpecedly whou any nformaon on s sae. In oher example, where he parameers of newor are: λ, p p, μ, μ, 4 we have he λp λp followng suaon. Condon.) s fulflled wh <, <. Condon.) s μ μ 4 λ 4 λ fulflled wh > and >. Therefore n he case when he frs clen saon μ μ breadowns frs, he lmng cumulave normalzed queue-lengh wll ncrease wh he rae, dfferen from ha would be n he case when he second clen saon breadowns frs. Therefore, by followng up he lmng cumulave normalzed queue-lengh one canno unquely characerze a confdence nerval as has been done n he case of symmerc newors. For hs reason we need n anoher represenaon for confdence nervals. The res of he paper s organzed as follows. In Secon we recall man equaons for lmng as ) cumulaed normalzed queue-lengh process from earler paper [], and derve slghly more general represenaon han n []. We hen derve an explc value for a confdence nerval, whch are closely relaed o he resul, obaned n []. In Secon, he resuls are derved for asymmerc newors n he case. We conclude he paper n Secon

4 R&RATA # Vol.) 8, March. The case of symmerc newor Lmng as cumulaed normalzed queue-lengh process s denoed q). Le λ l max l : >, le,,..., be he momens of breadown of clen saons, lμ.... Then q ) for all l, and n any arbrary me nerval [, + ), l l +,...,, we have he equaon, q ) q ) + [ q )]{[ q )] λ μ )) ) where [ q )] λ r s ) ds}, μ ) [ q )] λ r ) [ q )] λ In he las endpon we se q ). In he case of he confdence me nerval s he sum of wo nervals. The frs nerval s [, ). The second one s [, θ ), where he endpon θ s defned as follows. In [, ) for ) q we have he equaon: q ) λ μ) ) λ r s ) ds, where μ r λ λ To fnd θ we have he followng equaons: { q ) } [ )], { q ) < } [ )][ )], where Gx) denoes lfeme dsrbuon of each of dencal clen saons and s such he value of under whch λ μ) λ r ds. The value of can be found from he relaon - 5 -

5 R&RATA # Vol.) 8, March [ )][ [ )] d )] d If he correspondng value of s no greaer han α, hen he value θ of he nerval [, θ ) should be aen θ +. Oherwse, f > α, he value θ should be aen θ +. α The above resul has been obaned n []. Le us now exend hs resul for a more general suaon of an arbrary number of clen saons under he specal seng assumng ha l. Ths means ha q ) n he random nerval [, ), and q ) > n he nerval, ). In hs case we have he followng relaonshps: { q ) } [ )] [ G )], { q ) < } [ )] [ )] [ G )], where he value of can be found from he relaon [ )] [ )] [ G )] [ )] [ G )] d d Agan, f he correspondng value of s no greaer han α, hen he value θ of he nerval [, θ ) should be aen θ +. Oherwse, f > α, he value θ should be aen θ + α. The above consrucon gves us a random confdence nerval [, θ ) correspondng o he level of probably no smaller han P. We now fnd a deermnsc confdence nerval correspondng o he level of probably no smaller han P. Tha deermnsc confdence nerval wll be a guaraneed nerval, and he probably ha he sysem wll be avalable s no smaller han P. We have { θ > } { + > } { > } [ )]. P Therefore, he desred deermnsc nerval s [, z + ], where he value z s gven from he condon [ G z)] Then he consrucon of deermnsc nerval n he case s as follows

6 R&RATA # Vol.) 8, March Accordng o he aforemenoned relaons we fnd he value of nerval. If he correspondng value of s no greaer han α, hen we accep hs nerval and se Τ :. Oherwse, we se Τ : α, where he value α s deermned from he relaon μ λ μ) λ r ds α, and r λ λ We fnd he value from he relaon [ G )] The confdence nerval s hen aen as [, +T].. The case of asymmerc newor The case of asymmerc newor s smlar o ha of symmerc newor. I s based on he formula for he oal probably. Specfcally, n he case we are o sudy he cases as ) he frs clen saon breadowns frs and ) he second clen saon breadowns frs. Usng he noaon, G x) { χ x}, we have { χ χ } followng wo values and, such ha { q ) < χ χ } [ G )][ G )], [ G x)] dg x). ex, we have he { q ) < χ χ } [ G )][ G )], and he correspondng values of and are found from he relaonshps [ G [ G [ G )][ G )][ G [ G )][ G )][ G )] d )] d )] d )] d P, where n each case, f or s greaer han α, hen he correspondng value s replaced by α, and hen he correspondng value of or s o be replaced by α as well. Smlarly o he case of symmerc newor n hs case we have he followng. We fnd he value of nervals and. If he correspondng value of or s no greaer han α, hen we accep hs nerval and se Τ : or Τ : Oherwse, we se Τ : α or Τ : α, where he value α s deermned from he relaon μ λ μ) λ r ds α, and r λ λ - 7 -

7 R&RATA # Vol.) 8, March Usng he formula for he oal expecaon we fnd Τ Τ [ G x)] dg x) + Τ G x) dg x). We fnd value from he relaon [ G )][ G )] The confdence nerval s hen aen as [, +T]. 4. Concludng remar In he presen paper we esablshed confdence nervals for large closed clen/server compuer newors wh wo clen saons. Unle he earler resul esablshed n [] for symmerc newor, he confdence nervals are deermnsc. The advanage of he resul of [] s ha one can udge abou he qualy of sysem from he nformaon on he sysem sae. However, approach of [] s no longer avalable for asymmerc sysems. The advanage of he resuls of he presen paper s ha hey provde confdence nervals for boh symmerc and asymmerc newors ha gve us he enre lfeme of daa sysem n he newor wh probably no smaller han P. Acnowledgemen The auhor hans he suppor of he Ausralan Research Offce gran # DP778). Reference [] V.M.Abramov, 7. Confdence nervals assocaed wh performance analyss of symmerc large closed clen/server compuer newors. Relably: Theory and Applcaons,, Issue,