New algorithms for Performance Measures Derivatives in the Erlang Multirate Loss Model including the Bandwidth Reservation Policy

Transcription

1 New lgorthms for Performce Mesures Dervtves the Erlg Multrte Loss Model cludg the dwdth Reservto Polcy I. Moscholos,*, J. Vrds, M. Logothets, A. oucouvls Dept. of Telecommuctos Scece d Techology, Uversty of Pelopoese, Trpols, reece WL, Dept. of Electrcl & omputer Egeerg, Uversty of Ptrs, Ptrs, reece ASTRAT We cosder sgle-l loss system wth fxed bdwdth cpcty ccommodtg servce-clsses of Posso trffc wth dfferet bdwdth-per-cll requremets. lls of ll servce-clsses compete for the vlble l bdwdth, uder two dfferet polces: the complete shrg S polcy d b the bdwdth/tru reservto R polcy. I the S polcy, cll s bloced d lost f d oly f the vlble l bdwdth s less th the cll s bdwdth requremet. Such sgle-l loss system hs bee lytclly descrbed by the Erlg Multrte Loss Model EMLM. O the other hd, the R polcy s used order to cheve cll blocg probblty P equlzto mog servce clsses, or, ltertvely, gurtee cert qulty of servce for ech servce-clss. Such sgle-l loss system hs bee lytclly descrbed by the EMLM uder the R polcy EMLM/R. I ths pper, we focus o the problem of determg, effcet lytcl wy, dervtves of blocg probbltes, throughput rtes d verge umber of clls the system, wth respect to offered trffc-lod, rrvl rte d servce rte of y servce-clss uder both the EMLM d EMLM/R models. We further show through lytcl formuls how dervtves c be used to successfully pproxmte the bove metoed performce mesures whe smll vrtos of offered trffc-lod re cosdered. eywords ll blocg, Dervtves, Posso, Loss model, Recursve formul, dwdth reservto.. INTRODUTION I teletrffc, the determto of dervtves of performce mesures such s blocg probbltes ot oly hs theoretcl vlue but lso s of gret mportce prctcl problems cludg: etwor optmzto terms of cpcty llocto or trffc shrg betwee dfferet servce-clsses [], [] d b forecstg of performce mesures e.g. pp.65 [3]. I [], for stce, the ll locg Probblty P dervtves wth respect to offered trffc-lod dervtves of the Erlg formul [3] cotrbute to the determto of the optmum prttog of trffc demd mog dfferet cells bse sttos so tht the totl bloced trffc s mmzed. Also, suppose tht we study, through dscrete evet smulto, loss system tht ccommodtes clls of dfferet servceclsses. Accordg to [3], f we wt to forecst the Ps but we roughly ow the offered trffc-lod we hve,,..., to smulte the system for the vlues ; ths pproch requres,,..., + smulto rus d cosderble executo tme. Altertvely, we c smulte the system for gve set d the c use P dervtves wth respect,,..., to offered trffc-lod order to forecst Ps the vcty of,,...,. The m of ths pper s to determe, coveet wy, dervtves of vrous performce mesures sgle-l multrte loss system whose lyss s bsed o the clsscl Erlg Mult-rte Loss Model EMLM. The EMLM s used to lyze the cll blocg behvor of sgle-l loss system tht ccommodtes servce-clsses wth dfferet bdwdth-per-cll requremets. lls of ech servceclss rrve to the system ccordg to Posso process d compete for the vlble l bdwdth uder the complete shrg S polcy. lls re bloced d lost oly f ther requred bdwdth s less th the vlble l bdwdth. Otherwse, they rem the system for geerlly dstrbuted servce tme [4]. I the EMLM, ccurte recursve formul ow s ufm-roberts recurso, R recurso hs bee seprtely proposed by ufm [4] d Roberts [5], by explotg the fct tht the stedy stte dstrbuto of the umber of clls the l hs product form soluto PFS [3]. The R recurso determes the l occupcy dstrbuto d smplfes the determto of P. Ths smplfcto resulted lrge mout of extesos of the EMLM d pplctos of the R recurso both wred e.g. [6]- [] d wreless etwors e.g. []- [5]. Amog other EMLM extesos, Roberts proposed [6] pproxmte recursve formul for clcultg P the EMLM uder the dwdth Reservto polcy EMLM/R. The R polcy s used whe P equlzto s requred mog servce-clsses, or ltertvely whe cert Qulty of Servce QoS should be cheved for ech servce-clss. Roberts formul s smlr but ot the sme wth the R recurso. Note tht, cotrry to the S polcy where the sttory probbltes hve PFS, the R polcy cot be lysed by the use of PFS. Ths s becuse oe-wy trstos pper the stte spce, whch destroy reversblty [7]. I [8], pproxmte lgorthm hs bee proposed whch gves better results th Roberts formul, compred to smulto results, whe P equlzto s requred. However, the preset pper we do ot dopt ths method becuse t s more complex th Roberts formul d my result worse P results compred to Roberts formul whe P equlzto s ot eeded. I the recet yers my studes e.g. [9]- [] focus o the potetl umercl stblty problems of the R recurso. Sce such problems c be overcome by ormlztos s clerly stted [9], here, we dopt the R recurso d the Roberts formul of [6], s sprgbord to the clculto of P dervtves wth respect to offered trffc-lod, rrvl rte d servce rte, the EMLM d EMLM/R, respectvely. The determto of P dervtves wth respect to offered trffc-lod, rrvl rte or servce rte, s sgfct multrte loss systems sce t ebles the study of the tercto betwee dfferet servce-clsses tht shre the sme system. I the cse of the EMLM, the lterture s lmted to refereces [0], []- [6]. I the cse of the EMLM/R, to the best of our owledge, o prevous wors exst, d therefore we propose here lgorthms for the determto of such dervtves. * orrespodg uthor: Eml: dm@uop.gr

2 I [],[3] t s proved tht the P dervtve of servceclss wth respect to offered trffc-lod of servce-clss equls the P dervtve of servce-clss wth respect to offered trffc-lod of servce-clss. Ths relto s ow s the recprocty relto [4] or s the elstcty result for product form loss etwors pp. 65 [3]. I [], lthough some results of P dervtves re extrcted, there s o lgorthm preseted ether [] or [3] o how these dervtves c be clculted. I [4], the dervtves of P, throughput rtes d verge umber of clls the system wth respect to the: offered trffc-lod, rrvl rte d 3 servce rte re proved, but g o lgorthm s gve o how such dervtves c be clculted. Such lgorthms re preseted [0], [4][5] d [6] oly for the cse of P dervtves wth respect to offered trffc-lod. The lgorthm of [0] s of the sme complexty wth the lgorthm proposed the preset pper, but less strghtforwrd. The lgorthm of [5] uses the R recurso s the bss for the clculto of P dervtves. However, t s more complex th our lgorthm, sce t requres the clculto of the dervtve of the l occupto dstrbuto wth respect to offered trffc-lod, whle our lgorthm does ot requre t. Flly, the lgorthm of [6] s less complex th ll lgorthms but requres the exstece of PFS. Ths c be prohbtve o PFS loss models such s the EMLM/R vestgted the preset pper d therefore t s ot dopted ths pper. I ths pper, we exted the wor of []- [4] d [6] by proposg lgorthms for the clculto of dervtves of P, throughput rtes d verge umber of clls the system wth respect to the: offered trffc-lod, rrvl rte d 3 servce rte, both for the EMLM d the EMLM/R. I the cse of the EMLM our bss s the R recurso, whle the cse of the EMLM/R our bss s Roberts formul. Hvg determed the dervtves, we c use them to pproxmte P, throughput rtes d verge umber of clls for smll vrtos of offered trffc-lod, rrvl rte or servce rte. I both the EMLM d the EMLM/R we propose formuls for these pproxmte clcultos. Note tht the lterture oly pproxmte P formul exsts, proposed [6] for the EMLM, tht uses the dervtves of P wth respect to offered trffc-lod. The remder of ths pper s s follows: I Secto we revew the EMLM d the EMLM/R. I Secto 3, we cosder dervtves the EMLM wth respect to offered trffc-lod. We propose lgorthms for the clculto of dervtves of P, verge umber of clls the system, throughput rtes d clculte pproxmte wy P, verge umber of clls the system d throughput rtes. I Secto 4 we propose the correspodg lgorthms for the EMLM/R. I Secto 5 we cosder the EMLM d the EMLM/R d propose lgorthms for the clculto of dervtves wth respect to rrvl rte d servce rte. I Secto 6 we preset umercl exmples whch show the vldty of our lgorthms, for smll vrtos of offered trffc-lod, the cse of dervtves wth respect to offered trffc-lod. We coclude Secto 7.. REVIEW OF THE EMLM AND THE EMLM/R. Revew of the EMLM osder l of cpcty bdwdth uts b.u. tht ccommodtes clls of dfferet servce-clsses. A cll of servce-clss =,, rrves the system ccordg to Posso process wth rrvl rte λ, requests b b.u. d f these b.u. re vlble t rems the system for expoetlly dstrbuted servce tme wth me. Otherwse the cll s bloced d lost. If we deote by j the occuped l b.u. j=0,, the the l occupcy dstrbuto,, s gve by the ccurte d recursve R recurso [4], [5]: for j = 0 = j - for j =,..., j 0 otherwse b b = where: α = λ s the offered trffc-lod of servce-clss clls. The P of servce-clss, deoted s, s clculted by the formul: where jb j0 s the ormlzto costt. Note tht f oly clls of oe servce-clss = exst the l wth bdwdth requremet b = b.u. d α=λ/μ the the clsscl Erlg- formul results [3]: 0! The proof of eq. s bsed o the fct tht the stedy stte dstrbuto of the umber of clls the l hs PFS. If we deote by the umber of clls of servce-clss the stedy stte d the correspodg vector =,,,,, the the stedy stte dstrbuto, P, s gve by [3]: P =! 3! where: s the ormlzto costt gve by Ω = Ω! 4 d Ω={:0 b } s the stte spce wth b=b, b,,b,, b d b = b. Note tht the exstece of the PFS of eq. 4 s bsed o the exstece of locl blce rte-up = rte-dow betwee djcet sttes see the stte spce Ω for two servce-clsses wth b =, b = d =8 b.u. Fg Fgure. Locl blce d Ω ={:0 b + b }= {:0 + 8} uder the S polcy Ω,

3 Due to the smplcty of the EMLM d the ccurte d recursve Ps clculto eqs. - t hs bee ppled dmesog wred d wreless multrte etwors. As smple pplcto of the EMLM the cse of wreless etwors, cosder the Iub terfce betwee Rdo se Stto d Rdo Networ otroller. Let three servceclsses shre the bdwdth, sy Mbps, the dowl drecto the upl drecto c lso be cosdered: Adptve Multrte servce-clss, rcut Swtched servceclss d Pcet Swtched servce-clss whose clls requre., 64 d 8 bps, respectvely. For presetto purposes d order to eep the exmple smple, we do ot cosder the vrous trsmsso overheds for ech servce-clss or y sgllg trffc. Assumg tht ll servce-clsses hve strct QoS requremets d lettg α, α, α 3 be the offered trffc-lod erl for ech servce-clss, the eqs. - c be used to provde the correspodg P. Hvg determed P we c clculte the verge umber of servce-clss clls the system,, d the rrvl rtes of servce-clss requests ccepted the system throughput, T, s follows: E 5 T 6 The reltoshp betwee eq.5 d eq.6 comes from the defto of = verge rrvl rte of clls ccepted the system*verge servce tme or = T /μ.. Revew of the EMLM/R If we pply the R polcy to the EMLM ccordg to Roberts [6], the the formul for the pproxmte clculto of tes the form: for j = 0 = D j b j -b for j =,..., j = 0 otherwse where: D j 7 b b for 0 for j t j t d t s the reserved bdwdth R prmeter for servceclss clls. Note tht eq.7 s recursve lthough the EMLM/R s o PFS model,.e. locl blce betwee sttes tht re djcet s destroyed see Fg. where two servce-clsses wth b =, b =, t=, t=0 re ccommodted l of =8 b.u. uder the R polcy. 4 3 Sttes whch locl blce does ot exst due to the pplcto of the R polcy., Fgure. Locl blce s destroyed whe the R polcy s ppled 8 The recursve feture s bsed o the ssumpto pproxmto tht clls of servce-clss do ot exst re eglgble sttes j > - t; ths s corported eq. 7 through eq. 8. I the exmple of Fg., locl blce s destroyed whe j = b + b = + > 7. The R polcy s used to tt P equlzto mog dfferet servce-clsses tht shre commo l. Ths s cheved by proper selecto of the R prmeters. If for exmple P equlzto s requred betwee clls of three servce-clsses wth bdwdth requremets b, b d b 3 b.u., respectvely, the the followg equto should hold: b + t = b + t = b 3 + t3. The P of servce-clss,, the cse of the EMLM/R s clculted by the followg formul: 9 j b t where j0 s the ormlzto costt. Note tht f t = 0 for ll =,, the the EMLM results. Hvg determed P we c clculte d T ccordg to eq. 5 d eq. 6, respectvely, whch the vlue of wll be provded by eq DERIVATIVES WITH RESPET TO OFFERED TRAFFI-LOAD THE EMLM ASE 3. Determto of P dervtves sed o eq. the dervtves of wth respect to offered trffc-lod of servce-clss re gve by [6]: or Sce, j b j b jb j jb... b Ω j Ω j 0 t s ppret tht order to determe the P dervtves of eq. we should clculte the quttes. y defto s gve by: P Ω 3 where: Ω Ω : b j! j s the stte spce where exctly j b.u. re occuped.

4 Assumg tht the vlues of P re ormlzed the c be expressed by: P 4! Ω Ω j j I Fg.3 we grphclly represet Ω d j Ω the cse of two servce-clsses whose clls requre b d b b.u., respectvely, d compete for the vlble l bdwdth uder the S polcy. Fgure 3. Stte spces Ω d uder the S polcy sed o eq. 4, we determe j Ω s:!!! Ω j Ω j whch leds to the followg equto: j b for j=b,b +,, 5 where the vlues of j-b hve lredy bee clculted through the R recurso eq.. Equto 5 s grphclly terpreted wth the d of Fg. 3. The dervtve of wth respect to α leds to shft of the stte spce from Ω to Ω. Hvg determed the vlues of s follows: j j-b we clculte... b... 0 sed o eq. 5, eq. 6 c be rewrtte s: 6 b... b b b b b=j j=0,, Ω j Ω b= where the lst equto holds from the P defto of eq. d the vlues of s hve lredy bee determed by the R recurso eq.. Note tht the result s ot ew gve tht smlr formul bsed o P d ot o s hs bee proved [], [3]. b sed o eq. 5 d eq. 7, eq. tes the form: b b... b b 8 whch gves the dervtves of wth respect to offered trffclod of servce-clss bsed o the vlues of s d Ps lredy provded by the R recurso. Note tht the vldty of eq. 8 c be exmed by the recprocty relto of [],[3] whereby =. Furthermore we ote tht eq. 8 ppers lso [6]. The dfferece wth our pproch s tht [6] s re ot obted ccordg to the R recurso but bsed o lgorthm tht successvely covolves the dvdul =,, servce-clsses. 3. Approxmte P determto Hvg determed ccordg to eq. 8 we dopt the followg pproxmte P formul [6]:,,...,,,...,... 9 for smll vrtos eth er postve or egtve of offered trffc Δα. Note tht the vlues of,,..., hve lredy bee determed by eq.. Accordg to [6], eq. 9 c geerl, d ot lmted to the EMLM, solve some bsc problems, cludg: Dscrete Evet Smulto Suppose we wt to forecst P system, but hve rough de of the offered trffc-lod,,...,. I tht cse we should smulte the system for the vlues,,..., whch requres + smulto rus d cosderble executo tme. Altertvely, we c smulte the system for,,..., d the use eq. 9 to pproxmte Ps the vcty of,,...,. Approxmte clculto If we c lytclly determe Ps for gve set,,.. the we my use eq. 9 to pproxmte., Ps for,,...,. 3.3 Determto of d T dervtves sed o eq. 5 the dervtves of wth respect to offered trffc-lod re gve by: I smlr wy d bsed o eq. 6 we c determe the dervtves of T wth respect to offered trffc-lod s follows: for for 0

5 T where both eq. 0 d the vlues of by eq. 8. for for re gve 3.4 Approxmte determto of d T Hvg determed E d T ccordg to eq. 0 d eq., respectvely, we propose the followg pproxmte formul for the clculto of, T :,,...,,,...,... T,,..., T T T T,,...,... 3 for smll vrtos ether postve or egtve of offered trffc-lod Δα. Note tht the vlues of E d,,..., T,,..., hve lredy bee determed by eq. 5 d eq. 6, respectvely. To summrze our lgorthm for the pproxmte P, d T clculto the EMLM we preset the followg steps: For gve set of clculte s,,..., ccordg to the R recurso eq.. Determe Ps ccordg to eq.. 3 lculte for ll servce-clsses ccordg to eq lculte E d T ccordg to eq. 0 d eq., respectvely. 5 Determe pproxmte Ps for smll vrtos ether postve or egtve of offered trffc-lod Δα. ccordg to eq Determe pproxmte d T for smll vrtos ether postve or egtve of offered trffc-lod Δα. ccordg to eq. d eq. 3, respectvely. 4. DERIVATIVES WITH RESPET TO OFFERED TRAFFI-LOAD THE EMLM/R ASE 4. Determto of P, d T dervtves sed o eq. 9 the dervtves of wth respect to offered trffc-lod of servce-clss re gve by: j b t 4 or jb t jb t 5 Although the EMLM/R s o PFS model we me the ssumpto tht eq. 5 holds the cse of the R polcy. Note however tht the vlues of j-b eq. 5 should ow be clculted by the Roberts formul eq. 7. Furthermore, followg the lyss of secto 3. we c use eq. 7 for the determto of b... b b... 0 j : b 0 b where the vlues of s re determed by eq Note tht the cse of the EMLM/R the result s ot vld due to the defto of tht tes to ccout the R prmeter t. sed o eq. 5 d eq. 6, eq. 5 tes the form: b b... b b t b j0 7 efore we proceed to the pproxmte P clculto we meto tht the troducto of the R polcy the EMLM, destroys the recprocty relto tht holds the EMLM. Therefore,. I order to clculte the dervtves of d T we c use eq. 0 d eq., respectvely, where the vlues of re gve by eq Approxmte P, d T determto Hvg determed ccordg to eq.7 we dopt eq. 9 for the pproxmte P clculto whe smll vrtos ether postve or egtve of offered trffc-lod Δα re cosdered. Furthermore, hvg clculted E d T ccordg to eq. 0 d eq., respectvely, we propose eq. d eq. 3 for the pproxmte clculto of, T. To summrze our lgorthm for the pproxmte P, d T clculto the EMLM/R, we preset the followg steps:

6 For gve set of,,..., clculte s ccordg to the Roberts formul eq. 7. Determe Ps ccordg to eq lculte for ll servce-clsses ccordg to eq lculte E d ccordg to eq. 0 T d eq., respectvely. 5 Determe pproxmte Ps for smll vrtos ether postve or egtve of offered trffc-lod Δα. ccordg to eq Determe pproxmte d T for smll vrtos ether postve or egtve of offered trffc-lod Δα. ccordg to eq. d eq. 3, respectvely. 5. DERIVATIVES WITH RESPET TO ARRIVAL RATE AND SERVIE RATE Followg the lyss of Sectos 3 d 4 we c determe the P, d T dervtves wth respect to rrvl rte d servce rte. Sce the lyss s strghtforwrd we preset here oly the correspodg formuls wthout gog to further detl. 5.. Arrvl rte cse ppled to both the EMLM, EMLM/R I the cse of the EMLM, the dervtves of wth respect to rrvl rte of servce clss re gve by the formul: λ 8 b b... b b whle the cse of the EMLM/R by the formul: λ b b... b b t α b j0 9 Note tht the vlues of s eq. 8 d eq. 9 re ot the sme. Those of eq. 8 re clculted by the R recurso eq. whle those of eq. 9 re gve by the Roberts formul eq. 7. I both the EMLM d EMLM/R the dervtves of, T wth respect to rrvl rte re gve by: for E 30 E for T E for for Hvg determed, E d λ λ propose the followg pproxmte formuls for: P clculto λ,,...,,,...,... b lculto of the verge umber of clls the system T 3 we 3,,..., 33,,...,... c lculto of throughput T,,..., T T T T,,..., Servce rte cse ppled to both the EMLM, EMLM/R I the cse of the EMLM, the dervtves of wth respect to servce rte of servce-clss re gve by the formul: μ 35 b b... b b whle the cse of the EMLM/R by the formul: μ b b... b b t b j0 36 Note tht the vlues of s eq. 35 d eq. 36 re ot the sme. Those of eq. 35 re clculted by the R recurso eq. whle those of eq. 36 re gve by Roberts formul eq. 7. I both the EMLM d EMLM/R the dervtves of, T wth respect to servce rte re gve by: for 37 for

7 T for for Hvg determed, E d μ μ propose the followg pproxmte formuls for: P clculto μ,,...,,,...,... b lculto of the verge umber of clls the system T 38 we 39,,..., E E E 40,,...,... c lculto of throughput T,,..., T T T T,,..., NUMERIAL RESULTS-EVALUATION I ths secto, we exme through pplcto exmple the vldty of the proposed lgorthms of Sectos 3 d 4. We preset results oly the cse of P d verge umber of clls the system wth respect to the offered trffc-lod. The ccurcy of the results proves to be qute stsfctory for smll vrtos of offered trffc-lod. Sce smlr ccurcy ppers the cse of dervtves wth respect to rrvl rte d servce rte for: P, verge umber of clls the system d 3 throughput rtes we do ot preset such results ths secto. We cosder sgle l of cpcty = 00 b.u. tht ccommodtes clls of two servce-clsses wth the followg trffc chrcterstcs: st servce-clss: α = erl, b = b.u d servce-clss: α = erl, b = 6 b.u. lls compete for the vlble l bdwdth uder: the S polcy EMLM cse d the R polcy EMLM/R cse. I the EMLM/R cse, we choose the followg R prmeters: st set: t=8 d t=0 o P equlzto s cheved d set: t=5 d t=0 P equlzto s cheved, sce b + t = b + t Ths exmple hs bee preseted the cse of the S polcy [7] pp. 39, fg. 3 d lter [] pp.0, Fg. 3 order to show the osclltos tht occur the P of the st servceclss clls, whe α creses. Such osclltos re chrcterstc of the S polcy, especlly whe two servceclsses re cosdered d the dfferece the bdwdth requremet s hgh. A depth lyss of ths pheomeo c be foud [7]. I Fgs. 4-9 we preset the EMLM cse. I Fgs. 4 d 5 we preset the exct P bsed o eq. of the st d d servceclss clls, respectvely, whe α creses from to 40 erl steps of erl. The osclltos tht occur the P of the st servce-clss clls c be tutvely expled through Fg. 6 whch presets the l of =00 b.u. tht ccommodtes clls of both servce-clsses. For presetto purposes oly, we hve llustrted the bdwdth requremet of the d servce-clss clls 6 b.u s cotguous bloc of bdwdth uts. I the S polcy, however, there s o such requremet of cotguous bdwdth uts. osder ow stt where cll of the st servce-clss rrves the system, whle 4 clls of the st servce-clss d 5 clls of the d servce-clss re servced; therefore, t ths stt, there re 6 vlble b.u the l depcted by whte smll squres Fg. 6. I tht cse, the cll s ccepted the system whch ow hs 5 vlble b.u. The, f cll of the d servce-clss rrves the system t wll be bloced d lost, levg the 5 vlble b.u. for ew clls of the st servce-clss. I such cse, crese α wll ot led to P crese. As α cotues to crese, P of the st servce-clss clls wll crese utl other bloc of 6 vlble b.u. becomes vlble to ew clls of the st servceclss. Such osclltos P show tht tteto s eeded whe dmesog system uder the S polcy, especlly whe two or three servce-clsses re cosdered d oe of them requres much more bdwdth th the others. Ths s becuse there mght be cse where the P of servce-clss wth low bdwdth requremet s cceptble for specfc trffc level d ucceptble whe ths trffc level decreses. For exmple Fg. 4 the P of the st servce-clss clls s doubled whe α decreses from 7 to 4 erl. I Fg. 7 we preset the P dervtves,. A smlr fgure hs bee preseted [] pp. 0, Fg. 3. y exmg Fg. 4 d Fg. 7 t s ppret tht the P of the frst servce-clss s lmost phse wth. Such observto s helpful whe we wt to predct the P behvour of servce-clss whe trffc chges d we lredy ow the form of P dervtves of the other servce-clss wth respect to trffc. I Fg. 8 we focus o cert vlue of α, α =0 erl d show both the exct d pproxmte P results bsed o eq. 9 of the st servce-clss clls, whe Δα vres from - to erl steps of erl.e. α vres from 8 to erl. Accordg to Fg. 8, we see tht eq. 9 performs qute well eve re where P osclltos occur. It s obvous tht sce eq. 9 s ler fucto of Δα t cot cpture the oscllto pheomeo. The correspodg pproxmte P for the d servce-clss, preseted Fg. 9, re lmost detcl to the exct P sce P osclltos do ot occur for ths servce-clss. I Fgs. 0-5 we cosder the EMLM/R cse d the st set of R prmeters. I Fg. 0 we preset the P of the st servceclss clls whle Fg. we preset the P of the d servce-clls. ompred to Fgs. 4-5 where the S polcy s cosdered, we see tht the pplcto of the R polcy mproves the P of the d servce-clss but substtlly deterortes the P of the st servce-clss. Furthermore, Fg. 0 shows g the oscllto effect but compred to Fg. 4 s smoother. Ths s expected sce t = 8 b.u. re reserved for clls of the d servce-clss. If we cosder g the exmple of Fg. 6, d pply the R polcy wth t = 8 b.u. the ew clls of the st servce-clss c ow use oly 6 t = 8 b.u out of the 6 vlble b.u. I Fg. we focus, g, o cert vlue of α, α =0 erl d preset P results from Roberts formul eq. 7 d the proposed P formul eq. 9 of the st servce-clss clls, whe Δα vres s Fg. 8. Accordg to Fg., eq. 9 performs qute well the re of P osclltos. The correspodg pproxmte P for the d servce-clss, preseted Fg. 3, re lmost detcl to the P gve by Roberts formul. I Fgs. 4-5 we preset the vlues of d, respectvely, d compre the results of eq. 5 wth the proposed formul of eq.. Ag we focus o cert vlue of α, α =0 erl d let Δα vry from - to erl steps of erl. The comprso shows tht the proposed formul of eq. gves lmost detcl results to eq. 5.

8 I Fg. 6 we study the EMLM/R cse d the d set of R prmeters. Sce b + t = b + t = 6 b.u. o osclltos occur the equlzed P results. We cosder g α =0 erl, d preset the pproxmte equlzed P results bsed o eq. 9 d the correspodg P results of Roberts formul eq. 7 whe Δα vres s prevously. Accordg to Fg. 6 the proposed lgorthm for the P clculto the EMLM/R cse performs qute well. P st servce-clss P d servce-clss Osclltos Osclltos α Fgure 4. P of st servce-clss clls exct vlues α Fgure 5. P of d servce-clss clls exct vlues P dervtves wth respect to offered trffc-lod st servce-clss d servce-clss / / α Fgure 7. P dervtves of both servce-clss clls wth respect to α P st servce-clss Exct P Approxmte P α Fgure 8. P of st servce-clss clls exct vs pproxmte vlues Exct P Approxmte P L of cpcty =00 b.u. b = b.u. b = 6 b.u. st Servce-clss clls Offered trffc d Servce-clss clls Trffc Loss loced d lost tme =00 Free dwdth Ut rred trffc omplete Shrg Polcy P d servce-clss Fgure 6. Two servce-clsses ccommodted l uder the S polcy α Fgure 9. P of d servce-clss clls exct vs pproxmte vlues

9 P st servce-clss Osclltos Osclltos α Fgure 0. P of st servce-clss clls R polcy, t=8, t=0 P d servce-clss α Fgure. P of d servce-clss clls R polcy, t=8, t=0 P st servce-clss α Roberts formul Proposed lgorthm Fgure. P of st servce-clss clls Roberts vs proposed vlues P d servce-clss Roberts formul Proposed lgorthm α Fgure 3. P of d servce-clss clls Roberts vs proposed vlues Avr. Num. of st servce-clss clls Equto 5 Proposed equto α Fgure 4. Averge umber of st servce-clss clls wth respect to trffc R cse, t=8, t=0 Avr. Num. of d servce-clss clls α Equto 5 Proposed equto Fgure 5. Averge umber of d servce-clss clls wth respect to trffc R cse, t=8, t=0

10 Equlzed P Roberts formul Proposed lgorthm α Fgure 6. Equlzed P Roberts vs proposed vlues 7. ONLUSION We propose lgorthms for the pproxmte clculto of P, verge umber of clls the system d throughput rtes the cse of the EMLM uder two dfferet bdwdth llocto polces: the S polcy d the R polcy. The lgorthms re bsed o the clculto of the correspodg performce mesures dervtves wth respect to: offered trffc-lod, rrvl rte d servce rte. Such clcultos eble the study of the tercto betwee dfferet servce-clsses tht shre the sme system. The proposed lgorthms provde good ccurcy, for smll vrtos of offered trffc-lod, rrvl or servce rte, eve the worst cse sceros.e. res of trffc where P osclltos occur. REFERENES [] S. Lu-Rmírez, M. Torl, M. Ferádez-Nvrro, V. Wlle, Optml trffc Shrg ERAN, Wreless Persol ommuctos 009. [] F. elly, Routg d pcty llocto Networs wth Tru Reservto, Mthemtcs of Opertos Reserch, Vol. 5, No. 4, 990, pp [3]. Ross, Multservce Loss Models for rodbd Telecommucto Networs, Sprger, U, 995, ISN: [4] J. ufm, locg shred resource evromet, IEEE Trsctos o ommuctos, Vol. 9, No. 0, 98, pp [5] J. Roberts, A servce system wth heterogeeous user requremets, Proc. of Performce of Dt ommuctos systems d ther Applctos,. Pujolle ed.. North Holld, Amsterdm, 98, pp [6] A. reeberg, R. Srt, omputtol Techques for Accurte Performce Evluto of Multrte, Multhop ommucto Networs, IEEE/AM Trsctos o Networg, Vol. 5, No. 997, pp [7] M. Logothets,. os, Pth dwdth Mgemet for Lrge Scle Telecom Networs, IEIE Trsctos o ommuctos, Vol. E83-, No. 9, 000, pp [8] H. Shegye, Y. Wu, F. Sul, S. Hu, oordto-bsed optmsto of pth bdwdth llocto for lrge-scle telecommucto etwors, omputer ommuctos, Vol. 7, No., 004, pp [9] I. Moscholos, M. Logothets,. os, ll-burst blocg of ON-OFF trffc sources wth retrls uder the complete shrg polcy, Performce Evluto, Vol. 59, Issue 4, 005, pp [0] E. Nordström, J. rlström, A ew rewrd model for MDP stte ggregto wth pplcto to A d Routg, Europe Trsctos o Telecommuctos, Vol. 6, No. 6, 005, pp [] M. lbows, M. Sts, P. Zwerzyows, ommucto Networs Modellg of vrtul-crcut swtchg odes wth multcst coectos, Europe Trsctos o Telecommuctos, Vol. 0, No., 009, pp [] P. Fzes, S. Imre, M. Tele, Modelg d Alyss of rodbd ellulr Networs wth Multmed oectos, Telecommucto systems, Vol. 9, No. 3-4, 00, pp [3] D. Stehle, A. Mäder, A Alytc Approxmto of the Upl pcty UMTS Networ wth Heterogeeous Trffc, Proc. 8 th Itertol Teletrffc ogress IT, J. hrzs, R. Lehert, P. Tr- eds, erl, ermy, 003, pp [4] A. Mäder, D. Stehle, Alytc Modelg of the WDMA Dowl pcty Mult-Servce Evromets, Proc. IT Speclst Semr o Performce Evluto of Wreless d Moble Systems, Atwerp, elgum, 004, pp [5] M. lbows, M. Sts, A. Wsews, P. Zwerzyows, locg Probblty lculto for ellulr Systems wth WDMA Rdo Iterfce Servcg PT d PT Multrte Trffc, IEIE Trsctos o ommuctos, Vol. E9-, No. 4, 009, pp [6] J. Roberts, Teletrffc models for the Telecom Itegrted Servces Networ, Proc. of 0 th Itertol Teletrffc ogress, Modrel, d, 983. [7] F. elly, Reversblty d Stochstc Networs, Wley Seres Probblty d Mthemtcl Sttstcs, Wley, New Yor, 979, ISN: [8] M. Sts, M. lbows, A smple pproxmto of the l model wth reservto by oe-dmesol Mrov ch, Performce Evluto, Vol. 4, Issues -3, 000, pp [9] S. erezer, A. rzess, A effcet stble recurso to compute multservce blocg probbltes, Performce Evluto, Vol. 43, Issues -3, 00, pp [0] A. Nlsso, M. Perry, A. ersht, V. Iverse, O mult-rte Erlg- computtos, Proc. of 6 th Itertol Teletrffc ogress, P. ey, P. Smth eds, Scotld, 999, pp [] A. Nlsso, M. Perry, Multrte blocg probbltes: umerclly stble computtos, Proc. of 5 th Itertol Teletrffc ogress, Wshgto D., USA, 997, pp [] J. Vrtmo, Prtl shrg ccess cotrol polcy swtchg two dfferet trffcs tegrted etwor, OST 4 Techcl Documet 068, 986. [3] J. Vrtmo, Recprocty of locg Probbltes Multservce Loss Systems, IEEE Trsctos o ommuctos, Vol. 36, No. 0, 988, pp [4] S. Jord, P. Vry, Throughput multple servce multple resource commuctos etwors, IEEE Trsctos o ommuctos, Vol. 39, No. 8, 99, pp. 6-. [5]. Aswul, J. rr, Error lyss of multservce sgle-l system studes usg ler pproxmto model, Proc. of IEEE Itertol oferece o ommuctos, Vol., ore, 005, pp [6] V. Iverse, S. Stepov, Dervtves of locg Probbltes for Mult-servce Loss Systems d ther Applctos, Lecture Notes omputer Scece, Sprger, 007; pp [7] S. Johso, A performce lyss of tegrted commuctos systems, rtsh Telecom Techol. Jourl, Vol. 3, No. 4, 985, pp ogrphes Ios D. Moscholos ws bor Athes, reece, 976. He receved the Dpl.-Eg. degree Electrcl & omputer Egeerg from the Uversty of Ptrs, Ptrs, reece, 999, the M.Sc. degree Spcecrft Techology & Stellte ommuctos from the Uversty ollege Lodo, U, 000 d the Ph.D. degree Electrcl & omputer Egeerg from the Uversty of Ptrs, 005. From 005 to 009 he ws Reserch Assocte t the Wre ommuctos Lbortory, Dept. of Electrcl & omputer Egeerg, Uversty of Ptrs. urretly, he s Lecturer the Dept. of Telecommuctos Scece d Techology, Uversty of Pelopoese, reece. Hs reserch terests clude smulto d performce lyss of commucto etwors. He hs publshed over 65 ppers tertol jourls/ cofereces. He s member of the Techcl hmber of reece TEE. Joh S.Vrds ws bor Alexdr, reece, 979. He receved hs Dpl.-Eg. degree Electrcl & omputer Egeerg from the Democrtus Uversty of Thrce, reece,

11 004. Sce 005 he s Ph.D studet t the Wre ommuctos Lbortory, Deprtmet of Electrcl d omputer Egeerg, Uversty of Ptrs, reece. Hs reserch terests clude teletrffc egeerg optcl d wreless etwors. He s member of the Optcl Socety of Amerc OSA d the Techcl hmber of reece TEE. Mchel D. Logothets ws bor Stees, Adros, reece, 959. He receved hs Dpl.-Eg. degree d Doctorte Electrcl Egeerg, both from the Uversty of Ptrs, Ptrs, reece, 98 d 990, respectvely. From 98 to 990, he ws Techg d Reserch Assstt t the Lbortory of Wre ommuctos, Uversty of Ptrs, d prtcpted my tol d EU reserch progrmmes, delg wth telecommucto etwors, s well s wth offce utomto. From 99 to 99 he ws Reserch Assocte NTT s Telecommucto Networs Lbortores, Toyo, Jp. Afterwrds, he ws Lecturer the Dept. of Electrcl & omputer Egeerg of the Uversty of Ptrs, d recetly he hs bee elected Professor the sme Deprtmet. Hs reserch terests clude teletrffc theory d egeerg, trffc/etwor cotrol, smulto d performce optmzto of telecommuctos etwors. He hs publshed over 0 coferece/jourl ppers d hs over 0 thrd-prt cttos. He hs publshed teletrffc boo ree. He hs orgsed the 5th Itertol oferece o ommuctos Systems, Networs d Dgtl Sgl Processg, SNDSP 006. He served/s servg o the Techcl Progrm ommttee of severl tertol cofereces whle he orgzes d chrs severl techcl sessos. He hs become uest Edtor three jourls, whle prtcptes the Edtorl ord of jourls. He s member of the IARIA Fellow, IEEE Seor, IEIE, ETRI, FITE d the Techcl hmber of reece TEE. Athoy. oucouvls receved the.sc. degree Electrcl d Electroc Egeerg from Newcstle upo Tye Uversty, Newcstle, U.., 978, the M.Sc. d D.I.. degrees ommuctos Egeerg from Imperl ollege, Uversty of Lodo, Lodo, U.., 979, d the Ph.D. degree fber optcs from Imperl ollege, 98. Subsequetly, he joed the E Hrst Reserch eter d becme roup Leder d Dvsol hef Scetst worg o fber-optc compoets, mesuremets, d sesors utl 987, whe he joed Hewlett Pcrd Lbortores HP s Project Mger. At HP, he wored the res of optcl commucto systems, optcl etwors, d strumetto, utl 994, whe he joed ouremouth Uversty, ouremouth, U.. I 996, he becme Professor Multmed ommuctos, d 999 he becme Drector of the Mcroelectrocs d Multmed reserch eter. urretly, he s the Hed of Deprtmet of Telecommuctos Scece d Techology, Uversty of Pelopoese, reece. Hs curret reserch terests sp the felds of wreless commuctos, optcl fber commuctos d compoets, multmed commuctos, d hum-computer terfces where he hs publshed over 00 ppers. He hs cotrbuted to the formto of IrDA s dustry stdrd d he s ow member of the IrDA Archtectures oucl cotrbutg o ew IrDA stdrds. Dr. oucouvls s Fellow of the Royl Socety for the ecourgemet of Arts, Mufcturers d ommerce, Fellow of Isttute of Electrcl Egeers IEE d, 00, becme Fellow of IEEE, for cotrbutos to optcl fber compoets d optcl wreless commuctos. He s Member of the New Yor Acdemy of Sceces d AM. He s Edtor of the IEEE Wreless ommuctos Mgze, IEEE Trsctos o Wreless Networs, Assocte Edtorl Member for the Wreless ommuctos d Moble omputg Jourl d Vce hrm of the IEEE U&RI ommuctos hpter. He s member of Techcl ommttees umerous cofereces.