ABHELSINKI UNIVERSITY OF TECHNOLOGY Networking Laboratory

Transcription

1 ABHELSINKI UNIVERSITY OF TECHNOLOGY Networkng Laboratory Load Balanng n Cellular Networks Usng Frst Poly Iteraton Johan an Leeuwaarden Samul Aalto & Jorma Vrtamo Networkng Laboratory Helsnk Unersty of Tehnology samul.aalto@hut.f loadbal.ppt COST79/FIT semnar Otanem..

2 Problem formulaton: Routng of new alls n the oerlappng area of two BS s BS BS new all arrng

3 Sgnal based routng BS BS hoose base staton 3

4 Load based routng BS BS hoose the optmal base staton 4

5 Model Assumptons: new alls arre aordng to Posson proesses wth rates and onneton holdng tmes exponentally dstrbuted wth mean /µ no moblty modellng no handoers Notaton: k apaty of BS k BS BS µ µ apply routng poly α k state of BS k 5

6 Stat state-ndependent routng poles Randomzed Routng RRp arrng all n the oerlappng area s routed to BS wth probablty p BS wth probablty p as a result there are two ndependent Erlang loss systems wth parameters p and p p BS µ p BS µ 6

7 7 Optmal Randomzed Routng ORR For RRp the blokng probablty Bp s learly gen by The blokng probablty s mnmzed by some p* RRp* salledtheoptmal Randomzed Routng ORR For the ase there s an explt soluton: Idea: Balane the loads as far as possble Erl Erl p p p p p B < f f * p

8 Dynam state-dependent routng poles Dynam routng poly α: whennstate arrng all n the oerlappng area s routed to staton α Polyα s greedy f t hooses the other staton wheneer one s full.e. α α Polyα s a swth-oer strategy f there s a non-dereasng swth ure s suh that α f s α f < s 8

9 Optmal dynam poly In prnple the optmal dynam routng poly an be determned e.g. by the poly teraton algorthm deeloped n the theory of Marko Deson Proesses: fx the mmedate ost rate for eah state hoose a bas poly α determne the relate osts of states for the bas poly from the Howard equatons when teratng the deson of the terated poly α made n state mnmzes the relate osts of the post-deson state j j for the bas poly α by ths way we get a new better poly α wth smaller aerage ost used as the bas poly for the next teraton an optmal poly s found as soon as the poly does not hange anymore n ths teraton 9

10 Optmal dynam poly Target: mnmze blokng probablty mmedate ost rate r s as follows: Howard equatons for relate osts α : Iterated poly α : < < for for for r j j j j j j q r r α α α > f f α α α α α

11 Relate osts for stat poles If the bas poly s RRp we hae two ndependent subsystems. Thus Iterated polyα s therefore: Moreoer these subsystems are Erlang loss systems for whh the relate osts are easly found: α Erl Erl > f f α

12 Frst poly teraton: poly FPI All terated poles are dynam The alulaton of the relate osts for dynam poles s muh more demandng albet possble lnear equaton system of arables On the other hand t s known that typally the frst teraton step s the most sgnfant Straghtforward dea: Use ORR as the bas poly The two ndependent Erlang loss systems are p* and p* Iterate only one Call ths FPI It s easly seen to be a greedy swth-oer strategy

13 FPI s. Optmal dynam poly 5 a b 5 5 a. 5 a. 5 a.3. a B # teratons 5 b. 5 b. 5 b.3 4 x 3 b B # teratons 3

14 Frst poly teraton: poly FPI* But s ORR an optmal bas poly among stat poles mnmzng the blokng probablty of the terated poly after one step? Johan s dea: ORR tres to balane the loads thus gnorng the mpat of possbly dfferent dedated streams What f we smply gnore the flexble arral stream wth rate? Ths bas poly rejets all new alls n the oerlappng area! The two ndependent Erlang systems are and Iterate only one Call ths FPI* It s also easly seen to be a greedy swth-oer strategy 4

15 FPI s. FPI* s. Optmal dynam poly a b 4 x d FPI 3 FPI* B # teratons 5

16 Frst poly teraton: bas poly optmzaton Consder a ombned admsson & routng poly RARfp that aepts the new all arrng n the oerlappng area wth prob. f routes an aepted all to staton wth prob. p Ths s a stat poly the two ndependent Erlang loss systems are pf and p f Note that FPI s the terated poly orrespondng to bas poly RARp* FPI* s the terated poly orrespondng to bas poly RAR Parameters f and p anbeoptmzedsothattheblokng probablty of the terated poly after one step s mnmzed the optmal alue seems to be f leadngtofpi* 6

17 Optmal bas poly 5 5 x B f.5 7 p

18 Greedy heurst poles Least rato routng LRR Oerflow routng OFR Greedy ORR GORR / p* 8

19 ORR s. greedy heurst poles s. FPI* 5 5 a b ORR OFR GORR LRR FPI*

20 Open questons Rgorous proofs Optmal dynam poly s a greedy swth-oer poly sn t t? FPI* s the optmal one-step-terated poly based on stat bas poles sn t t? Senstty analyss What f and are only approxmately known? IsFPI*stllagoodpoly? Moblty modellng & handoers geometr approah stohast approah

21 THE END