Scalability Analysis of a Model for GSM Mobile Network Design

Transcription

1 Sclblty Anlyss of Model for GSM Moble Network Desgn Rebecc F. Pnhero, Alee rbos de Agur, Plácdo Rogéro Pnhero, Álvro de Menezes S. Neto, Ruddy P. P. Cunh, Domngos Neto Grdute Progrm n Appled Informtcs, Unversty of Fortlez Av. Wshngton Sores 131, Sl J-30, Fortlez, CE, rzl, Abstrct - Ths work shows mthemtcl nd computtonl tool to desgn GSM (Globl System for Moble Communctons) Network, n the pont of vew of SC (se Stton Controllers) llocton nd dmensonng. It optmzes the totl trnsmsson cost nd SC cquston cost. It determnes how much SC re need, n wht stes they hs to be llocted, wht model ech one must hve to support the totl trffc demnd wthout wstng money wth ther cquston nd wht TS (se Trnscever Stton) must be lnked to wht SC for trnsmsson cost reducton. Its core s nteger progrmmng (IP) model s presented n Wolsey et l [8]. The pproch of dt generton to the model from the rel world s eplned too. In ths model, the SC nodes re llocted tkng ccount both fctors: Trnsmsson nd SC cquston costs. The trnsmsson cost nvolves dstnce nd cpcty of the E1 lnes. The lnks between TS nd SC re llocted, nd the ones between SC nd MSC re dmensoned n number of E1 lnes. The choce of the SC model tht hs the best cpcty to the totl trffc demnd gves fleblty for the moble network desgn comprng wth fed cpcty models. It s mportnt snce n rel cses, the SC equpments suppler gves confgurton optons from low cpcty nd prce, untl hgh cpcty wth good reltve cost. Ths model uses the trffc demnd n Erlngs nsted of number of voce chnnels. Ths pproch llows the lnks between SC nd MSC (Moble Swtchng Center) dmensonng usng the sttstc gn of telephony swtches. Otherwse, smple determnstc sum of voce chnnels would be very smplstc, but would oversze the lnks too. Other mportnt contrbuton n ths model s the fct tht t ddresses the new resources llocton technque of SC swtches tht rses ts cpcty. The trdtonl wy of resources llocton (processors, for nstnce) to the rdo chnnels ws determnstc nd fed. Thus, ts cpcty ws gven by totl number of voce chnnels (4096, for nstnce). Nowdys, the SC cn hndle pool of resources tht re llocted on-demnd. The cpcty rses nd s gven by ts totl trffc n Erlng. Key words: GSM moble network desgn, cellulr telephony, Integer Progrmmng (IP), Opertons Reserch. I. INTRODUCTION One GSM moble network s composed by mny knd of equpments. There re swtches clled MSC nd SC, HLR (Home Locton Regster) tht ct lke subscrber dtbses, SGSN (Servng GPRS Support Node) tht s the dt network swtch verson of the MSC, nd mny more. In ths work, we wll focus n the SS (se Stton Subsystem). The SS s the group of equpments tht goes from the SC to the moble phone sde. It s composed mnly by TS, SC, MSC nd trnsmsson network to lnk them ll. The TS rdtes the RF (Rdo Frequency) sgnl to the moble phones nd receve ts sgnl bck. Ths RF s rdted by ntenns n the top of towers or buldngs, cretng coverge res clled cells. The geogrphcl llocton of TS s guded by RF coverge nd trffc demnd. Fg. 1. Moble Network Desgn When the coverge s the gol, the RF engneerng s look for hgh lttudes nd free of obstcles stes to rech lrger dstnces. When the gol s trffc, hotspots re focused wth TS full equpped wth rdo chnnels n lmted nd controlled RF rdton. In n urbn re, the TS promty s lmted by nterference, snce there s lmted number of

2 RF chnnels nd they re repeted on nd on. The TS stes re llocted n trngulr grd pttern, where t s possble. Ths llocton s due to the three cells of ech TS tht re formed by the coverge of the tree groups on ntenns, dsposed wth 10º ngles between then. Once TS llocton s fnshed ts tme to geogrphclly llocte the SC. SC re smll telephony swtches tht control the TS. Its gol s to crete n ddtonl level n the network herrchy nd ncrese the effcency, bsed on the sttstcl gn. It s n eclusvty of GSM system. An IS- 136 nd CDMA fmly hsn t ths equpment. Its lnks wth TS re E1 lnes tht holds voce chnnels slots confgured determnstclly n one-to-one bss wth TS's rdo chnnels slots. It s clled Abs nterfce. On the other hnd, SC's trunks wth MSC re E1 lnes dmensoned by the totl trffc from ll of ts TS. It s clled A nterfce. These trunks re smlr to trunks between two MSC or other telephony swtches. The voce chnnels n these cses re sezed sttstclly nd t vres wth the hours. All clls must pss through the MSC, even when both subscrbers re close, n the sme SC coverge. The Erlng formul cultes the blockng probblty (or congeston, or Grde of Servce GoS) to gven number of resources (voce chnnel, normlly) nd trffc offered. Ech of the three vrbles n ths formul cn be culted from the two others to ech stuton. We cn culte the percentle of clls tht re lost wth the number of voce chnnels vlble n some equpment nd the mesured trffc. We cn culte how much chnnels would be necessry to flow ths trffc f the GoS ws the desred (%, for nstnce). Ths s used to djust the network. We cn lso culte how much trffc cn we flow wth gven number of chnnels nd the desred GoS. The Erlng formul s shown below: e b = Σ n = n n! 0! (1) Where n (1) eb s the probblty of blockng, lso known s GoS, n s the number of resources (voce chnnels n ths cse) nd s the mount of trffc offered n Erlngs. Some SC hs determnstc wy of resources llocton. When new rdo chnnel s nstlled n TS, the requred resources (processors, for nstnce) re bound wth ths new rdo chnnel n fed wy. Ths resources re compromsed wth the rdo chnnel despte t s n cll or s dle. Thus, the SC hs fed mml cpcty n number of rdo chnnels. For nstnce, 4096 rdo voce chnnels (slots). Some more modern SC uses pool of resources tht re bound to the rdo voce chnnel on demnd, when cll s mde. Ths feture rses the SC cpcty. Now the mmum SC cpcty n ths stuton cn't be determned by ts number of rdo chnnels, but by ts trffc cpcty n Erlngs. For nstnce, the 4096 rdo voce chnnel SC would be trnsformed n 4058 Erlngs (t % GoS) SC, wth vrtully unlmted number of rdo voce chnnels. Therefore, there re determnstc chnnels n E1 lnes from TS to SC. These lnes wste trnsmsson resources. And there re sttstcl chnnel n E1 lnes from SC to MSC. These lnes re effcent. The more SC we dstrbute the less trnsmsson costs, snce ths equpment reduces the dstnces of TS to SC lnks tht wste trnsmsson lnes. On the other hnd, the SC hs ts cquston cost. The blnce between these two costs s reched wth the optml geogrphcl llocton of the SC, ssocted wth ts correct choce of model tht mples n ts cpcty nd cost. A typcl GSM network hs hundred or thousnd TS nd tens or hundreds of SC. The humn cpcty of desgnng effcent networks s very lmted nd the costs re hgh. The use of computtonl tools cn reduce these costs rdclly. Ths s ths work's trget. II. THE MATHEMATICAL PROGRAMMING MODEL Ths s the proposed Integer Progrmmng model [8]. T = t1,t TS nodes;,t3,, tm,b3,, bn,w3,, wo,c1,c,, cp = b1,b SC nodes; W = w 1,w SC models; C = c... 0 Lnk cpctes; j Decson vrbles for lnk llocton between TS node nd SC node j;

3 y Decson vrbles for choosng the cpcty c of E1 ( Mbps) lnes between SC l nd MSC; z lw Decson vrbles for SC l model w choce. ct j Lnk cost between TS nd SC j nodes n n nlyss tme perod; cm Lnk cost of cpcty c between SC l nodes nd MSC n n nlyss tme perod; cb w SC model w cquston cost, consderng n nlyss tme perod; TS trffc demnd n Erlngs; f c Lnk cpcty c n Erlngs; e w Erlngs. A. Objectve Functon SC model w trffc cpcty n The objectve functon (1) mnmzes totl cost of lnks between TS nd SC, plus cost of E1 lnes between SC nodes nd MSC, plus totl cost of SC's cquston. 1 mnmze T j. Restrctons ct j j + l c C cm y + d k W cb z In (), ech TS must be connected to one nd only one SC: j j = 1, In (3), the y dmensonng s mde. It llows ll trffc from TS ssgned to one SC to flow over ts lnks: 3 T l c C In (4), the SC dmensonng s mde ccordngly to the gven models nd the totl trffc demnd. 4 T 5 j j k W e k f c z y jk,, T j l k 0,1, T j dk y 0,1 l c C 6 z 0,1 l k W 7 lw III. MODEL APPLICATION Ths model hs some ssues n rel pptons tht must be observed. The set of TS nodes T s known prevously becuse ts desgn s mde by RF engneers s the frst step. Its geogrphcl locton s determned by coverge nd trffc. Its trffc demnd s known prevously too by mesure of other moble network (old one tht s beng reped, or by other technology such s TDMA (Tme Dvson Multple Access) or CDMA (Code Dvson Multple Access)) or by estmton bsed on verge subscrber trffc nd subscrber forecst bsed on populton. The set of SC nodes cn be generted bsed on ll vble stes possbltes. The stes tht wll hve TS re good cnddtes, snce ts spce wll be lredy vlble by rentl or buy. Other compny buldngs cn be dded to the set. The set represents the possbltes, not the ctul SC lloctons. The more optons ths set hs, the better the llocton of the needed SC wll be. The W set contns the possble models of SC. Normlly SC mnufcturer offers dfferent models. Ech one hs ts cpcty n Erlng (n our model) nd prce. The C set s tble of trffc cpctes for n nteger number of E1 lnes. Ech E1 lne hs number of tme-slots llocted for voce from the 31 vlble. Other tme-slots re used for sgnlng nd dt lnks. Thus, the frst E1 lne my hve dfferent number of voce tme-slots thn the second E1 lne, nd so on. Ech voce tme-slot crres 4 compressed voce chnnels. The elements of the C set re culted by the reverse Erlng formul, tkng the number of voce chnnels nd the desgn GoS s ncomng dt nd the trffc s outgong dt. The frst element of C set s 0 E1 lnes, tht hs 0 Erlng. The second element of C set s 1 E1 lne nd hs the trffc culted for 4 tmes the number of tme-slots llocted for voce n ths E1 lne. The thrd element of C set s E1 lnes nd hs the trffc culted for 4 tmes the number of tme-slots llocted for voce n ll E1 lnes, nd so on. The sze of the C set s determned by the mml cpcty of the lrger SC model. The lnk costs ct nd cb n gven perod of nlyss must be determned by the trnsmsson nture. If the trnsmsson network belongs to the moble compny, ts cost cn be determned by dstnce bnds or lnerly plus n equpment fed cost. If moble compny contrcts

4 trnsmsson lnes from other compny, the costs must be culted bsed on the busness rules. Quntty dscounts cn be ppled, for nstnce. The model cn be dpted to work wth SC tht hs mmum number of rdo chnnels cpcty, nsted of mmum trffc cpcty. IV. COMPUTATIONAL RESULTS Smultons were mde wth mny network szes. The bgger network szes tht could be solved n resonble tme hs bout 50 stes. The dfferent generted dt cused bg dfferences n the solvng tme. For nstnce: The smller tme for 50 stes wth 301 nteger vrbles nd 150 restrctons ws 4 seconds nd 4 hundredths of second, whle other dt mde the solver to spent more thn 30 mnutes to solve. The dt ws generted usng the followng ssumptons. The trnsmsson cost ws lner n functon of the lnk dstnce. The locl mrket ppromted cost where used. The cost of more thn one E1 lne n the sme lnk ws lner too. The TS nd MSC ste geogrphcl loctons where generted rndomly. To ech TS ste, SC ste cnddte ws generted. The trffc of the TS ws generted rndomly from 0 to 80 Erlngs tht s the ppromted vlue tht TS cn hndle wth n E1 lne. The C set ws generted wth 41 vlues, from 0 E1 lnes untl 40 E1 lnes. For ech cpcty, the correspondent trffc ws culted ccordngly to the eposed n the model ppton sesson (3). Three SC models where used n ths smultons: A smll model wth 51 Erlngs of cpcty, medum model wth 048 Erlngs of cpcty nd lrge model wth 4096 Erlngs of cpcty. Ech one hd n cquston prce comptble to the locl mrket relty. OPL ntegrted modelng envronment nd Cple 10.0 solver lbrry [9] from Ilog Inc. were used n the smultons. Its ense hs unlmted vrbles nd restrctons, nd full set of optonl lgorthms. To ths model, the rnch nd ound solver ws selected utomtclly. It rn n 64 bts Intel Core Qud processor wth.4 GHz clock nd 4 G of RAM memory. Despte the fct tht 50 stes s very smll comprng to the hundreds or even thousnd stes of the rel moble networks, the smultons shown the correctness of the model. Vryng the costs, more or less SC were llocted. Its model ws correctly chosen ccordngly to the totl trffc demnded by the TS llocted to ech SC. The dstnces were mnmzed ndrectly becuse of the lner cost by klometer. The trunk between SC nd MSC ws szed to flow the totl trffc demnd of the SC, nd ts dstnce to MSC ws toke ccount, snce the mount of E1 lnes ws greter thn one. 0 nstnces were used wth 5, 10, 15, 0, 5, 30, 35, nd 50 stes, wth rndomly generted dt. The results re shown n tble 1. TALE 1 CHARACTERIZATION OF THE INSTANCES USED IN THE EXPERIMENTS T S Vr. Const. Densty Avg. Tme V. SCALAILITY ANALYSIS Std. Devt ,7% 50,0 1, ,95% 40,0 8, ,43% 33,0 8, ,57% 853,5 86, ,01% 3561,5 371, ,60% 19689,0 87, ,9% 4687,5 4890, ,05% ,1 8063, ,86% 36303, , ,70% 7574, ,35 Due to the wde rnge of rndom generted vlues the problem nstnces hve very hgh complety vrton. Thus, there were problem nstnces wth 40 TS tht could not be solved wthn resonble tme threshold. Some tmes the solver crshed becuse of memory k. ut, for the sme reson, there re be problems nstnces lrger thn 50 TS tht cn be solved n tme ntervl even smller thn some prtculr nstnces of 40 TS. The proposed model here s n Integer Progrmmng one. Ths clss of model requres n lgorthm lke rnch-nd-bound, rnchnd-cut or others. Ths sort of lgorthms hve n eponentl complety. Ths fct lmts the lrger nstnce sze tht cn be hndled.

5 Actul networks often hve hundred of TS tht s fr beyond the rnge of ths ect method. Agur nd Pnhero [13] used Lngo solver lbrry nd t ws not ble to hndle problem nstnces lrger thn 40 TS. The dopton of Cple [9] epnded ths boundry to 50 TS, but t remns too smll. A men squres non-lner regresson of the verge tmes ws mde to determne the observed symptotc complety functon. It s shown on equton 8 nd fg.. y = 0,851e 0,44 (8) In computtonl results, sze nd performnce smulton ws descrbed. The sclblty nlyss ws mde nd some conclusons were descrbed. We cn see the model by tself cn't be used to the rel networks becuse of ts sze. Smulton wth rel networks cn't show the optmzton potentl becuse smll networks re well desgned by humn ntuton nd hve smll costs too. Some methodology must be ppled to etend the sze of the problems to cheve hundred or thousnd TS stes. Thus, the optmzton gn wll be very effectve , , , , , , , ,00 0,00 Averge Tme (cs) y = 0,851e 0, verge tme (cs) Epon. (verge tme (cs)) Fg.. Averge tme versus nstnce sze The key to brek ths lmtton nd turn bg network desgns fesble s to use ppromte pproches. Some methodologes lke Lgrngen relton n Smple Subgrdent, undle Methods nd Spce Dltton Methods ( Shor et l [6, 7] ) cn be used. Rgolon et l [3] shows tht the use of ths tool n the frst model etends the sze of the moble network to be desgned. A frmework tht hybrdzes ect method nd met-heurstcs hs presented good results n epndng these boundres n other clsses of problems. Nepomuceno, Pnhero nd Coelho [11] used ths frmework to solve contner lodng problems. In the sme problem ctegory, Pnhero nd Coelho [1] presented vrton of the mplementton to work wth cuttng problems. VII. CONCLUSION Ths work gve soluton to network desgn problem of moble GSM opertors cpturng ts essence n mthemtcl model. In ntroducton some telecommunctons bckground ws gven to help understndng the model. Then, the model ws presented nd eplned. After the model presentton, we showed the model ppton tht eplns how to lnk techncl detls of the rel world wth the model's generted dt. ILIOGRAPHICAL REFERENCES [1] Kubt, P e Smth, J. McGregor. A mult-perod network desgn problem for cellulr telecommuncton systems. Europen Journl of Opertonl Reserch, 134: , 001. [] Kubt, P., Smth, J. McGregor e Yum. C. Desgn of celullr networks wth dversty nd cpcty constrnts. IEEE Trnsctons on Reblty, 49: , 000. [3] Rgolon, A. A., Pnhero, P. R., Mcmbr, E. M., Ferrer, L. O. R. A. Appromte Algorthms n Moble Telephone Network Projects. Interntonl Conference on Telecommunctons nd Networkng, rdgeport, Sprnger Verlg, 005, v. XV, p [4] Rodrgues, S. I. M. Relção Lgrngen e subgrdentes com dltção de espço pdos um problem de grnde porte. RJ, [6] Shor, N. Z.Utlzton of the operton of spce dltton n the mnmzton of conve functons. Cybernetcs, 1:7-15, [7] Shor, N. Z. Zhurbenko, N. G. A mnmzton method usng the operton of etenson of the spce n the drecton of the dfference of two successve grdents. Cybernetcs, 7(3): , [8] Wolsey, L. A. Integer progrmmng. John Wley & Sons, [9] ILOG. ILOG CPLEX 10.0 User's Mnul, Jnury 006. [10] Shrge, L., Optmzton Modelng wth Lngo. Lndo Systems Inc., [11] N. V. Nepomuceno, P. R. Pnhero, A. L. V. Coelho. Tcklng the Contner Lodng Problem: A Hybrd Approch sed on Integer Lner Progrmmng nd Genetc Algorthms. Lecture Notes n Computer Scence, v. 4446, p , 007. [1] N. V. Nepomuceno, P. R. Pnhero, A. L. V. Coelho. A Hybrd Optmzton Frmework for Cuttng nd Pckng Problems: Cse Study on Constrned D Non-gullotne Cuttng. In: C. Cott nd J. vn Hemert. (Org.). Recent Advnces n Evolutonry Computton for Combntorl Optmzton. erln / Hedelberg: Sprnger-Verlg, 008, v. 153, p [13] A.. de Agur nd P. R. Pnhero. A Model for GSM Moble Network Desgn, chpter Innovtve Algorthms nd Technques n Automton, Industrl Eletroncs nd Telecomunctons, pges Sprnger Netherlnds, Dordrecht, September 007.