Lecure 4: Trunkng Theory nd Grde of Servce GOS 4.. Mn Problems nd Defnons n Trunkng nd GOS Mn Problems n Subscrber Servce: lmed rdo specrum # of chnnels; mny users. Prncple of Servce: Defnon: Serve user only on demnd llocon nd de-llocon of voce chnnels. Performnce nlyss s bsed on runkng heory.e., sochsc pproch. Grde of servce GOS bsed on prncples of sochsc runkng heory. GOS s mesure of he user's bly o ccess runked sysem durng buses hours. Purpose of runkng heory s: To deermne he requred cpcy nd lloce he proper number of chnnels n order o mee he GOS, where he probbly of GOS s deermned by lkelhood h cll s blocked; lkelhood of cll s experencng dely greer hn cern queung me. Common erms n Trunkng heory: Se-up Tme Blocked Cll Holdng Tme Trffc Inensy Lod GOS eques e,. There re severl bsc prmeers usully used n runkng heory: for user subscrber: Au - rffc nensy offered by ech user n Erlngs; - verge number of cll reques per un me; H - verge duron of cll.
b for sysem wh unspecfed number of chnnels: U - number of users n he sysem; A - ol offered rffc. c for chnnel for eqully dsrbued rffc mong he chnnels: C - number of chnnels. There re followng relons beween prmeers: for user A u H b for sysem A UA u c for chnnel Ac UAu / C All hese defnons nd prmeers llow us o obn probbly h cll s blocked, whch n lerure clled formul of Erlng B-model nd probbly h cll s experencng dely greer hn cern queung me, whch n lerure clled formul of Erlng C-model. We wll obn now hese formuls bsed on Queung nd Trunkng Theory. 4.2. Trunkng Theory Cellulr rdo sysems rely on runkng o ccommode lrge number of users n lmed rdo specrum. The concep of runkng llows lrge number of users o shre he relvely smll number of chnnels n cell by provdng ccess o ech user, on demnd, from pool of vlble chnnels. In runked rdo sysem, ech user s lloced chnnel on per cll bss, nd upon ermnon of he cll, he prevously occuped chnnel s mmedely reurned o he pool of vlble chnnels. Trunkng explos he sscl behvor of users so h fxed number of chnnels or crcus my ccommode lrge, rndom user communy. The elephone compny uses runkng heory o deermne he number of elephone crcus h need o be lloced for offce buldngs wh hundreds of elephones, nd hs sme prncple s used s desgnng cellulr rdo sysems. There s rde-off beween he number of vlble elephone crcus nd he lkelhood of prculr user fndng h no crcus re vlble durng he pek cllng me. As he number of phone lnes decreses, 2
becomes more lkely h ll crcus wll be busy for prculr user. In runked moble rdo sysem, when prculr user requess servce nd ll of he rdo chnnels s lredy n use, he user s blocked, or dened ccess o he sysem. In some sysems, queue my be used o hold he requesng users unl chnnel becomes vlble. To desgn runked rdo sysems h cn hndle specfc cpcy specfc grde of servce, s essenl o undersnd runkng heory nd queung heory. The fundmenls of runkng heory were developed by Erlng, Dnsh mhemcn who, n he le 9 h cenury, embrked on he sudy of how lrge populon could be ccommoded by lmed number of servers. Tody, he mesure of rffc nensy bers hs nme. One Erlng represens he moun of rffc nensy crred by chnnel h s compleely occuped.e. cll-hour per hour or cll-mnue per mnue. The grde of servce GOS s mesure of he bly of user o ccess runked sysem durng he buses hour. The busy hour s bsed upon cusomer demnd he buses hour durng week, monh, or yer. The busy hours for cellulr rdo sysems ypclly occur durng rush hours, beween 4 p.m. nd 6 p.m. on Thursdy or Frdy evenng. The grde of servce s benchmrk used o defne he desred performnce of prculr runked sysem by specfyng desred lkelhood of user obnng chnnel ccess gven specfc number of chnnels vlble n he sysem. I s he wreless desgner s ob o esme he mxmum requred cpcy nd o lloce he proper number of chnnels n order o mee he GOS. GOS s ypclly gven s he lkelhood h cll s blocked, or he lkelhood of cll experencng dely greer hn cern queung me. Posson Formul nd egve Exponenl Lw. We consder elephone sysem exchnge, swch ec., nd we wn o nvesge he sscl feures of he npu process of ousde users. The mos fmous npu process s he Posson process.we consder me nervl of lengh, nd we denoe he r.v represenng he number of clls durng hs me nervl. Le us suppose h P, s he probbly h excly clls re rrve n me. We dvde hs me nervl no n sub-nervl of duron /n. In one nervl here s us one cll n n equl probbly, depends on he lengh of nervl cll. We suppose h one even, whch occurs n sub-nervl, s ndependen of oher sub-nervls. Then P, hs connuous dervon: dp, d Ech sub-nervl hs success f conns cll so: P, + o n n P, + o n n 3
ow we ke he probbly Pn of hvng clls n n sub-nervls: n n Pn o o n + n + n n By kng n o nfny we ge: P, e, 2.....! Ths s he Posson dsrbuon of ncomng clls. Th hs he followng chrcerscs: Men: E{ } Vrnce: vr{ } 4.2.. Loss Sysems Lfe Tme Dsrbuon. A cll s chrceres by he followng properes: Iner-rrvl me s he me nervl beween 2 consecuve clls. The holdng me s he me nervl beween begnnng nd ermnon of cll. The wng me s he nsn of rrvl of cll nd he nsn n whch s dmed o he sysem. They ll reffer o s lfeme. We denoe now L s he lfeme lengh. The CDF F s he probbly h lfe s no longer hn : And s PDF s: F F τ dτ F < fd pr{<l<+d} By developng he prolongon of lfe he cll exceedng +τ, we ge he condonl probbly h cll wll ermne durng,+d. Ths probbly s kd: kd d F F For holdng me replces k nd evenully we ge he densy funcon, whch s k s ds f k e 4
he exponenl dsrbuon.loss Sysems Below we descrbe he problem of loss sysems n cse of full vlbly. We hve o ke 3 specs whle nlyng hs problem: The npu nd ermnon process. b The sysem srucure. c The fe of unsuccessful cll. Le us ke he sysem, whch cn be presened, s schemclly shown n Fg. 4.: S groups of sources C groups of chnnels Fgure 4.: Full vlbly group In hs scheme ny source hs ccess o ny chnnels. Assumpons: All chnnels re fully vlble o clls from he sources. A cll, whch mees congeson, s dscrded mmedely. If no chnnel s vlble, source wll connue o demnd servce for me whch equls o holdng me. If durng hs me chnnel becomes vlble, source sees nd held for he remnng busy me. umber of busy chnnels n group equls uumber of busy members n s group of sources. The npu process s descrbed by PDF of ner-rrvl mes beween clls ncomng o C-group. Les Tr be he rndom vrble r.v. of me nervls beween 2 consecuve clls: r- o r. The r.v., Tr, r,2.., re ndependen nd hve he sme PDF u wh he.ג/ men vlue The npu process, Tr, s sonry wh respec o prmeer r < r <nfne, where s fne. The ermnon process, Lr, s descrbed s PDF of holdng-me of r-h cll. The r.v. Lr, r,2.., re ndependen nd hve he sme PDF f wh he men vlue /u. oe : Inpu nd ermnon processes do no nfluence one ech oher. The processes Lr nd Tr re ndependen. oe 2: All users hve he sme bly o produce clls. Sources re ndependen. 5
We defne: Tr - r.v., represenng he me nervl beween he rrvls of r- o r clls. Lr r.v., represenng he ermnon process; Lr s he holdng me of he rh cll. Y r.v., represenng he number of busy sources n he S-group of sources. Then, he probbly of hs even s I well-known h P, PrY,,, 2,,. Z - number of busy chnnels, for whch he probbly equls: nd P, Q, PrZ,.. 2.,. Defnons of Loss: Q, Tme Congeson S s he probbly h me, ll devces n group re engged. In hs cse cll rrves o devce s los. Cll congeson H s he condonl probbly h nsn he group s blocked when cll rrves. In our dervons, we wll del wh 2 ypes of loss sysems: Loss Clls Cler L: A cll rrves o sysem whle ll chnnels re busy s clered nd he source does no ge servce s used for Erlng B model. Loss Clls Held Lh: A cll rrves o sysem whle ll chnnels re busy s wng unl one chnnels becomes free s used for Erlng C model. The Tme Congeson S: A. In cse of los clls cler L: where s he number of chnnels. B. In cse of los cll held Lh: S Pr{Z} Q, S Pr{ Y } Q, 6
The cll congeson H: Le Pb s he condonl probbly h cll rrves o group when he group s blocked, h s, he possbly h cll s orgned by source when C- group s n vrous ses, n prculr n. Le P s he probbly h cll rrves o C-group. The probbly h nervl d cll rrves o C- group s blocked s PbS. Thus he cll congeson s: Pb H S P A cse of Poson rrvl process: In cse of Poson rrvl process he probbly of new cll s ndependen of he se of he group so Pb P. So we ge he heorem: Theorem: For loss sysems wh Poson npu, he me nd cll congeson re equl. Trffc nlyss of sysem we wll sr wh mn defnons: Trffc Offered {A} s he expeced number of clls durng he verge holdng me The rffc offered o chnnel by sources. If number of clls durng nervl s, srng me, s s, -s nd he holdng me CDF s Fs, hen: A, s sdfs Trffc Offered per source s he verge number of ncomng clls los or no durng me nervl correspondng o he verge holdng me. c where c s cllng re per source, u s he verge holdng me. 7
Trffc Crred s he verge number of occuped devces n group, n gven nsn of me. Trffc Los s he verge number of blocked sources. Trffc Congeson H Trffc_ Los Trffc_ Offered Cse : Los Cll Clered L. In hs cse s mpossble o more hn sources o be busy smulneously. All sources hve he sme npu process X nd ccess o ll chnnels.we my noe h no ll sources h reques for servce wll ge snce he upper lm of smulneously served sources s. In L sysem Y Z so we ge: nd Q, P, for P, for > E{Y} E{Z} The rffc los s: Trffc Los A E{ Z} The rffc congeson s: In equlbrum H H: H Trffc_ Los Trffc_ Offered A E{Z} +AH A EZ { } A Menng: The rffc offered by sources s he sum of rffc crred by chnnels nd he rffc los from source o chnnels. Cse 2: Los cll held Lh. 8
There my be more hn busy sources smulneously, up o sources re served. If C- group s blocked so S-group s vrully no blocked so he process Y nd Z re no equl. We hve o defne he relons beween Y nd Z: EZ { P, The npu process X yeld he rffc offered A verge number of orgned clls. So: EY { } A The rffc crred by C-group s E{Z}. The verge blockge of sources of he rffc los s: Trffc_Los E{ Y } E{ Z } The Trffc Congeson s herefore:. H Trffc_ Los Trffc_ Offered EY { EZ { } P, EY { } A + 9
4.2.2. Brh nd Deh Process Ths secon s n overvew of he Brh nd Deh B&D process descrbed by Feller n 939, whch cully s specl cse of he Mrkov process of 2 nd ype. We show he bsc feure of he mehod usng Kolmogorov dfferenl equon whch s derved from he Chpmn Kolmogorov equon. We consder sysem wh he followng chrcerscs ndom Vrble of process Sources Y Chnnels Z Probbly funcon P, Q, We wll dsnc beween he los cll cler sysem L o he los cll held Lh. We consder sysem n se, were here re sources fne of nfne. So here re + ses of. We ssume lso h he flucuons of ses n me consue dscree Mrkov process wh connuous me prmeer, h chrceres n s rnson probbly: PrY Ys Pr, s;,, s h s, he probbly of sysem n se nsn, gven h me s he sysem ws n se. The Chpmn-Kolmogorov Equon: The rnson probbly s expressed by relon s known s Chpmn-Kolmogorov equon. I mens he followng: Le us consder 3 momens: s, nd +h, where s < < +h. A he me s he sysem s n se. A me he sysem s n se v. The probbly, h here re busy sources me +h, when here re busy sources nsn s, s sum of probbles of rnsons from busy sources no busy sources occurrng hrough nermedry suon v, nsn. The mn propery of hs equon s h he fuure, of nsn +h depends only on he presen nd no on he ps he rnson from s o s ndependen of s. Therefore, we cn wre he rnson probbly P, s;,+h s P;,+h. So Ps,;, + h Psv,;, Pv,;, + h v Trnson Prob. h sysem n se nsn +h, gven h ws n me Trnson Prob. of sysem n se v nsn, gven h ws n me Condonl Prob. h f sysem n se v nsn, wll chnge no durng h
We cn llusre s Fg. 4.2: T s T sv,v2,... Fg. 4.2 T+h s In order o nlye B&D sysem we hve o ke he followng ssumpons: The sysem chnges hrough rnsons from one se, o s neghbor se - or +. If he sysem s n se nsn, hen he condonl probbly h he followng me wll ump o se + s: Pr{ Y + + Y } + o Ths s he brh rnson cused by orgnng cll by source. If he sysem s n se < nsn, hen he condonl probbly h he followng me wll ump o se + s Pr{ Y + Y } + o Ths s he deh rnson by cll ermnon of one cll by source. Here: o s he probbly h more hen one rnson occur durng. s clled Brh coeffcen. s clled Deh coeffcen. From bove he followng conclusons cn be sed: The probbly of rnson s proporonl o. The chnce of occurrence of brh s : / +. The chnce of occurrence of deh s : / +. B&D equons of se The Chpmn-Kolmogorov equon cn be wren n order o show he rnson probbly o be s se s follows :
P ;, + P ;, + P; +, + + P;, + o Trnscon from - o Trnscon from + o o Trnscon occurred More hen one Trnscon occured A common ype of represenon hs process s by Fg. 4.3: - - - - + + dp;, P ;, + P ;, d lm P ;, ; P ;,, Equon bove s he forwrd Kolmogorov dfference- dfferenl equon h lso known s equons of se. For <, <: The nl condons: The boundry condons for nd : 2
dp ;, P ;. + P ;, d dp ;, P ;. + P ;, d The Congeson Funcons In Ths chper we denoe some defnons of sysems feures, n vew of congeson. Usng he rnscons probbly P ;, we obn he probbly h new cll wll rrve: The bsolue probbly cn be defned s: Λ dp;, Λ P, nd Λ d pr{ Y + d Y } whch s clled he Cllng e for sysem comprsng sources, whch s he probbly of orgnng new cll by source n d. The vergng of he cllng re for long me clled he Averge Cllng e C For he ergodc process : lm Λ P C C remns he sme for L nd Lh sysems. The bsolue probbly cn be defned s: Γ P, nd Γ d pr{ Y + d Y } whch s clled he Termnon e for sysem comprsng sources, whch s he probbly of orgnng new cll by source n d. The lm vlue of ermnon re over long me s he Averge Termnon e. 3
Congeson funcons n L nd Lh sysems: The cllng nd ermnon res Λ nd Γ re vld for L nd Lh sysems. Bu for L sysems he brh coeffcens vnsh whle >-: sysems. ג dffer beween L nd Lh Λ Q, ; C Q The probbly h free chnnel wll become busy durng d s clled he Cllng e for Chnnel. The verge cllng re s herefore: Λo pr{ Z + d Z d Q, Co Q C C Q Congeson n Lh sysem: The Tme Congeson Funcon S n Lh sysem As ws defned before: S P The Cll Congeson H n Lh sysem: As menoned before, he Cll Congeson s he probbly h cll rrved o sysem wll fnd les n se : we ge: P pry { Y + d Y P P, Λ J P, P, 4
nd n equlbrum condons: Congeson n L sysem: The Tme Congeson Funcon S n L sysem cn be defned s before: S P P C Q The Tme Congeson Funcon H n L sysem: When new cll rrves, ll chnnels re engged: nd ג for >. so we ge: nd n equlbrum: Trffc properes usng congeson funcons: P pry { Y + d Y Pr Y + d Y nd Y d Q, P Q, Λ Trffc Crred By Group Of Sources s he verge number of busy sources: or {Lh sysem : E{ Y } For L sysem, snce : P J, P E{ Z } Q J, Q 5
In order o fnd he expresson for Y - From he Chpmn-Kolmogorov dfferenl equon we obn: d d E{ Y } P ;, P ;, Where E{Y } defned s: E{ Y } P ;, Snce he process s ergodc, he lm of E{Y} s consn, ndependen of. If he nl dsrbuon of se s known, he vergng of ll nl ses yelds for Lh sysems: nd for L sysems we ge: d d E{ Z } Λo Γ Γ C for Lh nd Γ Co for L umber of clls n Averge Holdng Tme: Suppose ll clls hve he sme dsrbuon of holdng me, Fs, wh men vlue T/u. If he nl dsrbuon s known so n he nervl -s, he number of ncomng clls : s Λ τ dτ The verge number of ncomng clls durng he verge holdng me s: [ F τ ] d + E { Y } F c A Λ τ τ In equlbrum A s consn so he verge rffc offered for Lh s: A C T And for L sysem s: 6
E{ Z } Co T 4.2.3. The Soluon Of B&D Equon In hs secon we derve he soluon of he rnson probbly Pr;, for boh cses of L sysem s well s Lh. The fnl gol s o formule he lm P for ech cse under some ssumpons regrdng he brh nd deh coeffcens. In hs chper we summre he soluon of he B&D equon whou rckng ll he wy o he soluon s n [] pp 32-48. We ke some ssumpons bou he brh coeffcens: B: The probbly of new cll n,+d s proporonl o he number of dle sources:,,,..., fne Ths s he mos common cse. B2: The probbly of new cll n, +d s ndependen of he number of busy sources.,,,..., n fne Ths cse of hgh number of users h ends o nfny. B3: The probbly of new cll n, +d s ndependen of he number of busy sources, unless ll chnnels re busy.,,,...,, Ths s vrn of B2 for L sysems. fne B4: The probbly of new cll n, +d proporonl o he number of busy sources excep for nd :,,2,...,, fne or no B5: The brh coeffcen s lner funcon of he number of busy sources: 7
α + β,,,..., fne or no Ths cse ncludes ll he oher cses. We ke he followng ssumpons for he deh coeffcens: D: The probbly of cll ermnon n, +d s proporonl o he number of busy sources.,,,..., fne or no D2: The probbly of cll ermnon n, +d s ndependen of he number of busy sources.,,,..., fne or no D3: The probbly of cll ermnon n, +d s proporonl o he number of dle sources.,,,..., fne ; We consder holdng me h hve n.e.d nd he coeffcen he verge holdng me T. T s equl o he nverse of Theorem : f / s he verge duron of ech holdng me, nd f he holdng mes re ndependen, hen he only form of deh coeffcen s D. The prncple of he B&D equon In order o solve he B&D equon, we use he generng funcon g.f h defned s: And nl condons: ψ, P ;, Z, Z < ψ ψ Z, Z, 8
9, Z ψ We follow he followng seps: Ge he rnsformed prl dfferenl equon, so we ge he expresson for By he nverse rnsform, ge he requred P;,. By lmng hs expresson, we ge he lm dsrbuon P. Usng he expresson of me congeson for [L] or [Lh] we compue S. Usng he expresson of cll congeson for [L] or [Lh] we compue H. For ech cse we show he dfferenl equon, he soluon P;,, P, S nd H. Soluon of B&D equon for [Lh] sysem B-D cse Lh where : And he explc soluon s: k k k k k k k P +, ; By lmng P we ge P: P + + We defne : + So, usng he expressons of S nd H nd usng he verge cllng re we ge: H S ; +,,, Z Z ψ ψ ψ [ ] [ ], Z Z ψ e P ;, + + + e P ;, + + + +
2 B2-D cse Lh The equon: The soluon: We ge from hs expresson he Poson Dsrbuon for he probbly h excly sourses re busy n equlbrum condon: The verge cllng re : And he congeson funcons: B3-D cse Lh If we ssume C, so we ge SH,,, Z ψ ψ ψ, e Z e e ψ,,...! e u P A Y E } { lm A A H S e! P!!
2 Soluon of B&D equon for [L] sysem B-D cse L The congeson funcon s known more s he Erlng-B formul. C H S!!!! C A Q J!!,,2... S H A A J J!!