An Alternative Design Topology for Metropolitan Area Networks

Transcription

1 ITB J. ICT Vol. 6 No A Alteative Desig Topology fo Metopolita Aea Netwos Atoius Suhatomo Study Pogam of Eleletial Egieeig Faulty of Egieeig Pesidet Uiveisty Kota Jababea Beasi 7550 Idoesia asuhato77@yahoo.om Abstat. Oe of the ey issues i desigig a etwo topology is vuleability. The vuleability paamete measues the esistae of a etwo to disuptio of opeatio afte the failue of etai statios o ommuiatio lis. Oe oute-measue to addess the vuleability of a etwo is edge oetivity. I this pape a moe seletive oept of edge oetivity is itodued alled ompoet ode edge oetivity fo MAN topology desig. This paamete equals the smallest umbe of edges that must be emoved i ode to esue that the ode of eah ompoet of the esultig sub-etwo o sub-gaph is less tha. Keywods: edge oetivity; -ompoet edge oetivity; -ompoet edgefailue set; -ompoet edge-failue state; miimum degee. Itodutio A metopolita aea etwo MAN is a ompute etwo that usually oves a ity o a lage ampus whih has a geogaphi sope that falls betwee a wide aea etwo AN ad a loal aea etwo LAN. A MAN usually iteoets a umbe of odes epesetig LANs that ae iteoeted usig high-apaity baboe tehology suh as fibe optis ad povides upli sevies to ANs ad the Iteet. Some implemetatio tehologies used fo this pupose ae the asyhoous tasfe mode ATM the fibe distibuted data itefae FDDI ad the swithed multimegabit data sevie SMDS. I most aeas these tehologies ae i the poess of beig displaed by etheet-based oetios e.g. Meto Etheet. MAN lis betwee LANs have bee built without ables usig miowave adio o ifaed lase lis. Most ompaies et o lease iuits fom ommo aies due to the fat that layig log stethes of able a be expesive. Distibuted-queue dual-bus DQDB is the metopolita aea etwo stadad fo data ommuiatio that is speified i the IEEE stadad. Usig Reeived July 26 th 200 Revised May 3 th 20 2 d Revisio August 29 th 20 3 d Revisio May 28 th 202 Aepted fo publiatio Jue 22 d 202. Copyight 202 Published by LPPM ITB ISSN: DOI: 0.564/itbj.it

2 04 Atoius Suhatomo DQDB etwos a be up to 30 m log ad opeate at speeds up to 55 Mbit/s. Poit-to-poit etwoig that suppots MAN oetios a be built based o etwo ofiguatios suh as a bus etwo a sta etwo a ig etwo o a mesh etwo. ith the widespead depedee upo suh etwos it beomes impotat to loo fo topologies that yield a high level of eliability ad a low level of vuleability to disuptio. I this pape the etwo ofiguatios wheel ad fa will be disussed. It is desiable to oside quatitative oute-measues to addess a etwo s vuleability whe desigig a MAN. I ode to obtai suh measues we model the etwo by a gaph i whih the statio temials ae epeseted by the odes of the gaph ad the lis by the edges. I what follows we assume G = V E a simple gaph whee V is the o-empty set of odes ad E is the set of edges. e use the otatio G V fo the ode of the gaph G ad e G E fo the size of the gaph G. Uless speifially stated othewise we follow the stadad gaph theoy otatio foud i []. Defiitio. The edge-oetivity of G deoted by G o simply is defied to be G mi F : F E F is a edge failue set [2]. Oe dawba of the taditioal edge-failue model is that the gaph G F is a edge-failue state if it is disoeted ad o osideatio is give to whethe o ot thee exists a lage ompoet that i itself may be viable. It is easoable to oside a model i whih it is ot eessay that the suvivig edges fom a oeted sub-etwo as log as they fom a subetwo with a ompoet of some pedetemied ode. Theefoe we itodue a ew edge-failue model the -ompoet ode edge-failue model. I this model whe a set of edges F fails we efe to F as a -ompoet edgefailue set ad the suvivig sub-etwo G F as a -ompoet edge-failue state if G F otais o ompoet of ode at least whee is a pedetemied theshold value. Defiitio 2. Let 2 be a pedetemied theshold value. The - ompoet ode edge-oetivity o ompoet ode edge oetivity of G deoted by G o simply is defied to be G mi F : F E F is - ompoet edge failue set i.e. all ompoets of G F have ode [3][4].

3 Alteative Desig Topology fo Metopolita Aea Netwo 05 Defiitio 3. A set of edges F of gaph G is -edge set if ad oly if it is a - ompoet ode edge-failue set ad F [3]-[5]. Fom this shot explaatio we offe the optio of etwos o gaphs i desigig a MAN etwo topology with vuleability issues by omputig G fo a speifi type of gaphs. 2 Pelimiay Results The followig etwos suggest whih MAN topology is the best optio egadig its degee of vuleability. The fist type of gaph we oside is the yle C. The fomula fo C a be deived i a simila mae to that of P [3]. he oe edge of the yle is emoved it beomes a path with odes P. Thus C P ad sie get the followig esult [6]. Theoem. Give 2 C [4][6]. we The seod type of etwo that has bee osideed is the omplete gaph o odes K. Let F EK be a edge set. e a ompute F by alulatig the maximum umbe of edges that a emai i the -ompoet edge-failue state K F. It is easy to see that ay edge i F must have its edpoits i two diffeet ompoets of K F; theefoe eah ompoet of K F must itself be omplete. Lemma 2. Give [3]-[5]. 2 let whee 0 Fom this it immediately follows that a maximum-size -ompoet edgefailue state of K osists of omplete ompoets eah of ode -

4 06 Atoius Suhatomo ad possibly oe additioal ompoet of ode less tha. Thus we have the followig: Theoem 3: Give 2 K whee [][4][5]. 3 Mai Results Model of Netwo Topology The pelimiay esults suggest the etwo topologies wheel ad fa. Sie we do ot have a fomula o a algoithm that omputes G of a abitay gaph G epesetig the etwo theefoe we wat to fid bouds both lowe ad uppe that may be applied to establish the age of possible values of G. Befoe we itodue a set of bouds we eed to establish some otatio ad temiology. Fist we oside the value of H whee H is a sub-etwo of G but H is ot spaig o oeted. If H is a gaph o m odes ad m the evey ompoet of H has ode so it follows that H 0. If H is disoeted ad H H. p i H H 2 H i p ae the ompoets of H the Defiitio 4. If F E is a set of edges G[F] deotes the sub-etwo of G with ode set V ad edge set F i.e. G[F] = G E F [4]. Defiitio 5. If U V with U 0[ U } deotes the set of all edges with oe edpoit i U ad the othe edpoit i [4]. The ext theoem is the basis of ou lowe boud. Theoem 4. If E E2 is a patitio of E the G[ E ] G[ E ] G [4]. 2 Poof. Let D E be a -edge set ad let Di Ei D i 2. Sie Ei Di E D eah ompoet of G[ E i ] Di is otaied i a ompoet of GD ad theefoe is of ode. It follows that

5 Alteative Desig Topology fo Metopolita Aea Netwo 07 G[ E ] D. Theefoe G E ] G[ E ] D D D G. i i Thus we obtai the followig lowe boud: G[ E ] G[ E ] G 2 whee E2 [ 2 2 E is a patitio of E. I patiula if u is a ode of full degee the G u G. Now let U V be a set of odes with U =. e ostut a -ompoet edge-failue set as follows: delete all edges that oet U to the est of the gaph ad the delete edges fom V U util all ompoets have ode <. Thus we obtai the followig uppe boud: G [ U V U] V U whee U V with U =. e will apply these bouds to ompute G whe G is eithe the wheel of ode o F the fa of ode. Defiitio 6. The wheel of ode is the etwo o gaph fomed by oetig a sigle vetex to all the veties of a C [4]. Figue A wheel gaph.

6 08 Atoius Suhatomo Defiitio 7. The fa of ode F is the gaph fomed by oetig a sigle vetex to all the veties of a P [4]. Figue 2 A fa gaph F. Sie the appliatio of the bouds to ad F ae doe similaly we will oly demostate them fo ad state the oespodig esult fo F. Let G = ad let u be the full degee ode. The C u G ad the lowe boud implies C G. Now if } { 2 u u u U } { u u U V the ] [ U V U = = + 3 ad P U V. Thus the uppe boud implies 3 3 P. 2 Iequalities ad 2 togethe yield the followig bouds o : 3. 3 e will fid the fomula fo by ompaig the lowe ad uppe bouds. 3. Subtatig + fom eah side gives

7 Alteative Desig Topology fo Metopolita Aea Netwo Applyig the divisio algoithm we wite whee 0 2 thus. Hee = +. If i.e. does ot divide the 2 2. If = 0 i.e. divides the. Theoem 5. Let G be a wheel o -odes the [4] if does ot divide 5a 2 if divides. 5b Poof. By usig boud 3 3 If does ot divide the 2 3. Thus if does ot divide ad holds. If divides the 2 thus

8 0 Atoius Suhatomo 2. Claim. If divides the if we emove at most edges fom we do ot get a -ompoet edge-failue state. Let if we emove edges fom the C we get ompoets ad i ode to have a -ompoet edge-failue state eah has at most odes. Sie divides eah must have exatly odes. Thus we eed to isolate the full degee ode u fom the C whih meas we must delete additioal edges. Hee we must emove at least edges fom the C. Sie a emove at most edges fom the full degee ode whih leaves at least edges emaiig fom u to the C ad thus thee is a ompoet of ode at least. Thus if divides the holds. 2 ad 2 As peviously stated the fomula fo F a be deived i a simila mae as that fo so the followig esult is stated without poof. Theoem 6. Give 2 the [4] F if does ot divide if divides 4 Colusio The pimay esults disussed i [3] suggest ivestigatio of ew etwo models that mae it possible to desig aman etwo topology with the best solutio fo vuleability issues.

9 Alteative Desig Topology fo Metopolita Aea Netwo he ompaig etwo ofiguatios with the same umbe of statios o odes that eed to be istalled a omplete etwo is the best topology but this maes it impossible to desig a eal MAN etwo. The yle etwo is the simplest oe i tems of istallatio but it yields the wost esults i tems of etwo vuleability. e a see the ompaiso esults i Table below whih is based o the assumptio that a MAN etwo is established with 0 statios o odes while = 4 5 ad 6 espetively. Table Results of vuleability ompaiso betwee diffeet etwo ofiguatios yle omplete wheel ad fa with the same umbe of odes. Type Fomula = 4 = 5 = 6 C C K K if does ot divide if divides 9 7 F 0 F if does ot divide if divides Refeees [] Haay F. Gaph Theoy Addiso esley Readig MA 969. [2] Boesh F. Goss D. Suffel C. Saoma Joh T. Kazmieza L.. & Suhatomo A. A Geealizatio of A Theoem of Chatad Netwos 2009 DOI 0.002/et. [3] Boesh F. Goss D. Kazmieza L. Suhatomo A. & Suffel C. Compoet Ode Edge Coetivity-A Itodutio Cogessus Numeatium 78 pp [4] Suhatomo A. Compoet Ode Edge Coetivity: A Vuleability Paamete fo Commuiatio Netwos Dotoal Thesis Steves Istitute of Tehology Hoboe NJ May [5] Suhatomo A. A Measue of Vuleability fo Commuiatio Netwos: Compoet Ode Edge Coetivity Poeedig of CITEE UGM ISBN: pp [6] Boesh F. Goss D. Kazmieza L.. Suhatomo A. & Suffel C. Bouds Compoet Ode Edge Coetivity Cogessus Numeatium 85 pp