Energy saving in fixed wireless broadband networks

Transcription

1 Energy savng n fxed wreless broadband networks Davd Coudert, Napoleao Nepomuceno, Issam Tahr To cte ths verson: Davd Coudert, Napoleao Nepomuceno, Issam Tahr. Energy savng n fxed wreless broadband networks. Internatonale Network Optmzaton Conference (INOC), Jun, Hamburg, Germany. Sprnger, 67, pp ,, Lecture Notes n Computer Scence. <.7/ _53>. <nra > HAL Id: nra Submtted on Apr HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 Energy savng n fxed wreless broadband networks Davd Coudert Mascotte, INRIA, I3S, CNRS, Unversté de Nce Sopha Sopha Antpols, France davd.coudert@nra.fr Napoleão Nepomuceno Insttut for Matematk og Datalog Syddansk Unverstet, Campusvej 55, DK-53 Odense napoleao@mada.sdu.dk Issam Tahr Mascotte, INRIA, I3S, CNRS, Unversté de Nce Sopha Sopha Antpols, France ssam.tahr@nra.fr Aprl, Abstract In ths paper, we present a mathematcal formulaton for savng energy n fxed broadband wreless networks by selectvely turnng off dle communcaton devces n low-demand scenaros. Ths problem reles on a fxed-charge capactated network desgn (FCCND), whch s very hard to optmze. We then propose heurstc algorthms to produce feasble solutons n a short tme. Introducton Fxed broadband wreless communcatons s a sector of the communcaton ndustry that holds great promse for delverng prvate hgh-speed data connectons []. Such network comprses remote locatons, each of them served by a rado base staton (RBS), connected by means of hghcapacty mcrowave rado lnks. A bdrectonal lnk connectng two RBSs requres a dedcated par of outdoor unts (ODUs), each one drectly coupled to a hgh-drectonal antenna. Commonly, n ths context, the network s bult n a robust fashon to guarantee fault protecton and to support the extremely bursty traffc behavors. As a drawback, snce ODUs consume substantal power whenever the lnk s up, t brngs forth mportant energy waste to provde extra resources whch could be used only n crtcal stuatons. Therefore, the traffc fluctuaton over the tme offers an opportunty to energy savngs by handlng traffc effcently and turnng off devces used to keep mcrowave rado lnks whose capactes are underused. In ths work, we consder the problem of decdng both the network s confguraton and flows that mnmze the total energy expendture. Partcularly, by confguraton, we mean the choce of whch communcaton devces we need to keep on to successfully meet the traffc requrements. Ths work has been partally supported by project APRF RAISOM (PACA & FEDER), ANR DIMAGREEN, and Vllum Kann Rasmussen foundaton.

3 Ths problem reles on a fxed-charge capactated network desgn (FCCND), whch s very hard to optmze [6]. Among others, [4] and [7] tackled smlar problems on dfferent networks. We present an exact formulaton for ths problem and propose heurstcs that may be employed to produce good feasble solutons n a short tme. Problem modelng and lnear formulaton The network topology s modeled by a dgraph H = (V, E) where every node v V represents a base staton and every arc vw E represents a rado lnk. Every lnk has a capacty c vw and can be ether actve (whle consumng energy) or not. Traffc demands are defned by D pars (s d, t d ), wth s d, t d V and by an average volume per demand h d. We assume that H s symmetrc snce n the type of studed networks, rado lnks are usually symmetrc. Ths mples that for every node v V, the enterng neghborhood s the same as the leavng neghborhood,.e. δ + (v) = δ (v) = δ(v). Also the cost of an actve lnk s constant and equal to CL (as shown n [5]). Another assumpton that we consder n ths work, and whch s not always true, s the possblty of routng the traffc of the same demand d through dfferent paths from s d to t d (mult-routng). The problem can be formulated as a mxed nteger program (MIP). We defne two types of varables: to represent the state of lnk vw we consder a bnary decson varable, beng equal to f the lnk s actve and otherwse. Snce symmetrc lnks must be n the same state, and n order to reduce the total number of bnary varables, we use a sngle varable u vw wth v < w (assumng some orderng of the nodes) for the par of symmetrc lnks vw and wv. We also employ a varable x d vw to ndcate the volume fracton of the demand d whch s routed through the lnk vw. In the MIP formulaton, Eq. () s the objectve functon, Eq. () are the capacty constrants on the lnks, and Eq. (3) are the flow conservaton constrants. mn s.t. vw E CL u vw () D x d vw c vw u vw vw E () d= w δ(v) x d wv w δ(v) h d, f v = s d, x d vw = h d, f v = t d,, otherwse v V, d =,..., D (3) x d vw [, h d ] vw E, d =,..., D (4) u vw {, } vw E (5) 3 Hybrd algorthm The model cted n the prevous secton s a mxed nteger lnear program. Even though t can be handled by a solver lke CPLEX, ths may take a very long tme on relatvely large networks (contanng more than a hundred nodes). The number of varables and constrants can be huge and not even ft n memory. For such networks, we bult a hybrd soluton by combnng a heurstc based on smulated annealng and a lnear program wth real varables (Mult Commodty Flow - MCF). The former would be the master and on every teraton t chooses the lnks to turn on/off. The latter, whch s the slave, wll only fnd out whether there s a feasble soluton wth ths confguraton or not (see Fgure ). Actually the lnear program

4 Intal Soluton f e a s b l t y Generator of MCF Instances (Metaheurstc Smulated Annealng) Hybrd Algorthm Flter of Feasble Instances (Exact Method LP Solver) n s t a n c e Fnal Soluton Fgure : The framework of the hybrd algorthm wll have the same formulaton except that the lnk state varables u vw wll now be constant. Therefore there wll be no more nteger varables n the program, whch makes t faster to solve. At each teraton of the smulated annealng process, a new network confguraton s generated from the current soluton by swtchng the state (Up:Down or Down:Up) of a couple of symmetrc rado lnks. Ths couple s randomly chosen. Then, we apply the flter to check f ths new confguraton leads to a feasble soluton. In the affrmatve case, we attrbute the total power utlzaton as energy score value of ths soluton. Otherwse, we dscard ths nfeasble soluton. We then follow the orgnal descrpton of the smulated annealng process [9], where the algorthm replaces the current soluton by the new soluton wth a probablty that depends on the dfference between the correspondng energy score values and a global parameter T (called the temperature). 4 Sparse cuts based algorthm As an alternatve approach, we propose a heurstc to construct a good confguraton. It s also hybrd snce the LP solver s stll responsble for testng the feasblty of a confguraton. Ths method s based on the sparse cuts of the graph. Roughly speakng, a cut s consdered sparse f t has a hgh average load per lnk. Snce ths noton s mostly studed n the context of undrected graphs, we wll consder for ths heurstc the underlyng undrected graph of our ntal symmetrc dgraph. Let us defne the sparsty more formally. Let G = (V, E) be the undrected graph. Let D be the set of demands (pars of vertces). Let c be the capacty functon that gven a subset of edges returns the sum of capactes of all edges n ths subset. Let also h be the functon that for a gven subset of demands returns the sum of volumes of demands n ths subset. For any cut separatng S (a subset of V ) from S, let α(s) denote the sparsty of ths cut, wth: 3

5 Algorthm Maxmze the number of shutdown lnks Requre: G = (V, E) s the ntal graph Requre: c s a capacty functon on the edges Requre: cutslst s the lst of precomputed sparse cuts Requre: D s the set of demands Ensure: G = (V, E ) wth E E s a subgraph ensurng the routng of all demands. nonp otentallnks E E whle nonp otentallnks E do SORT(cutsLst) n descendng order of average load per lnk n up state potentallst for from to SIZE(cutsLst) do potentallst potentallst LINKS(cutsLst[]) potentallst potentallst (E (potentallst nonp otentallnks)) for from SIZE(potentalLst) downto do {n LIFO order} f ROUTING LP((V, E potentallst[]), c, D) s feasble then E E potentallst[] nonp otentallnks nonp otentallnks potentallst[] break else nonp otentallnks nonp otentallnks potentallst[] α(s) = c(e(s, S))/h((S S) D). If we consder a unform verson n whch all edges have capacty and to every par of vertces corresponds a demand wth the same volume, then the sparsty s α(s) = E(S, S) / S S. As sad before, our heurstc s based on sparse cuts; those whch have a small sparsty. Fndng the mnmum-sparsty cut, also called the sparsest cut, s a well-known combnatoral optmzaton problem n the lterature. Although t s NP-complete, many work targetng mproved approxmatons have been carred out [,, 3, 8], and recently an approxmaton that reles on a sem defnte program (SDP) relaxaton to acheve a log n approxmaton factor has been proposed []. In order to obtan many dfferent sparse cuts, we run the approxmaton many tmes whle addng each tme a small randomness n the objectve functon of the SDP. The motvaton behnd usng sparse cuts s the fact that the sparser the cut s, the more loaded ts edges tend to be, and the less effcent deletng one of them would be. So every lnk wll have an estmated load equal to the average load per lnk of the sparsest cut coverng ths edge. The heurstc wll try to remove the lnks n ascendng order of estmated load. At each teraton, the feasblty of routngs wll be checked wth a lnear program and, after removng a lnk, the sparsty of the cuts wll be updated. An effcent way to mplement ths heurstc s descrbed n Algorthm whch performs Ω(m) calls to the LP solver (m beng the number of edges). 5 Smulaton results We wll follow the same scenaro for the dfferent approaches prevously explaned: the exact formulaton usng the lnear program, the hybrd algorthm, and the sparse cuts based algorthm. The smulatons have all been executed on SNDlb topologes whch represent backbones (France wth 45 lnks and Norway wth 5 lnks). The man reason for ths choce s that we were not 4

6 E- NLF(5%) E- NLF(%) H- SA(5%).5 E- NLF(5%) E- NLF(%) H- SA(5%) H- SA(5%) H- SA(5%) H- SA(75%) H- SA(75%).5 H- SA(%).5 H- SA(%) execuon me (s) execuon me (s) (a) France H-SA vs E-NLF (b) Norway H-SA vs E-NLF E- NLF(5%) E- NLF(%) H- SC(5%).5 E- NLF(5%) E- NLF(%) H- SC(5%) H- SC(5%) H- SC(5%) H- SC(75%) H- SC(75%).5 H- SC(%).5 H- SC(%) execuon me (s) execuon me (s) (c) France H-SC vs E-NLF (d) Norway H-SC vs E-NLF Fgure : The saved power as a functon of the executon tme able to fnd nstances of topologes for fxed wreless networks n the ltterature. Lnk capactes and energy consumpton were set for a typcal fxed wreless network scenaro. For the traffc matrx, we consder a unform traffc matrx. The demand volume for every topology wll take four values. Supposng that P s the maxmum weght that we can acheve for that topology n an all-to-all scheme, these values would be { %P, 75%P, 5%P, 5%P }. All the smulatons have been executed on the same knd of machne equpped wth dual core processors operatng at 3GHz and GB of RAM. As MIP solver, we used CPLEX. to whch MIP emphass was set to feasblty over optmalty. In order to solve SDPs, we use the CSDP software. For notaton, we use E-NLF for the exact (node-lnk) formulaton. The two heurstcs are labeled H-SA for the smulated-annealng-based heurstc and H-SC for the sparse cuts based heurstc. Snce our objectve s to show that some heurstcs can yeld good solutons when the decson makng s constraned by a lmted executon tme, we are not nterested n the best solutons found by those heurstcs. Hence, n Fgure we have reported the behavor of the algorthms before the Exact algorthm fnds the optmal soluton or near optmal (for the topology of Norway, n some cases, there was a slght mprovement of the objectve functon after a thousand of seconds). 5

7 6 Concluson We frst gave a mathematcal formulaton to the problem of mnmzng energy consumpton n fxed broadband wreless networks. Then we compared the behavor of two heurstc algorthms wth that of an LP solver. We found out that heurstcs would gve a better soluton f the executon tme s very lmted, especally when the sze of the network ncreases. Thanks to the sparse cuts based algorthm we were able to verfy that these cuts are an mportant argument to consder whle desgnng heurstcs for the studed problem. Therefore, one of our perspectves s to fnd an effcent way of choosng k cuts out of the sparsest n a gven graph. References [] Harry Anderson. Fxed Broadband Wreless System Desgn. John Wley & Sons, 3. [] Sanjeev Arora, Satsh Rao, and Umesh Vazran. Geometry, flows, and graph-parttonng algorthms. Commun. ACM, 5:96 5, October 8. [3] Shuch Chawla, Anupam Gupta, and Harald Racke. Embeddngs of negatve-type metrcs and an mproved approxmaton to generalzed sparsest cut. In 6th Annual ACM-SIAM Symposum on Dscrete Algorthms (SODA), pages, 5. [4] Luca Charavglo, Marco Mella, and Fabo Ner. Reducng power consumpton n backbone networks. In IEEE Internatonal Conference on Communcatons, pages 6, 9. [5] D. Coudert, N. Nepomuceno, and H. Rvano. Power-effcent rado confguraton n fxed broadband wreless networks. Computer Communcatons, 33(8):898 96, 5. [6] F. Grore, D. Mazaurc, J. Moulerac, and B. Onfroy. Mnmzng routng energy consumpton: from theoretcal to practcal results. In Internatonal Conference on Green Computng and Communcatons (GreenCom ), page 8p, Hangzhou, Chna,. [7] F. Idzkowsk, S. Orlowsk, C. Raack, H. Woesner, and A. Wolsz. Savng energy n p-overwdm networks by swtchng off lne cards n low-demand scenaros. In Proceedngs of the 4th conference on Optcal network desgn and modelng, ONDM, pages 4 47,. [8] Roht Khandekar, Satsh Rao, and Umesh Vazran. Graph parttonng usng sngle commodty flows. In STOC 6: Proceedngs of the thrty-eghth annual ACM symposum on Theory of computng, pages , New York, NY, USA, 6. ACM. [9] S. Krkpatrck, C. D. Gelatt, and M. P. Vecch. Optmzaton by smulated annealng. Scence, (4598):67 68, 983. [] Tom Leghton and Satsh Rao. Multcommodty max-flow mn-cut theorems and ther use n desgnng approxmaton algorthms. J. ACM, 46(6):787 83, 999. [] Davd B. Shmoys. Cut problems and ther applcaton to dvde-and-conquer,