Homogeneous and Heterogeneous Traffic of Data Packets on Complex Networks: The Traffic Congestion Phenomenon *

Transcription

1 Communctons nd Network, 2012, 4, Publshed Onlne My 2012 ( Homogeneous nd Heterogeneous Trffc of Dt Pckets on Complex Networks: The Trffc Congeston Phenomenon * Alfonso Frn 1, Antono Grzno 1, Frncesc Mrn 2#, Mr Crstn Recchon 3, Frncesco Zrll 2 1 SEEX Sstem Integrt S.p.A., Rom, Itly 2 CERI, Unverstà d Rom Spenz, Rom, Itly 3 Dprtmento d Scenze Socl, Unverstà Poltecnc delle Mrche, Ancon, Itly Eml: {frn, grzno}@selex-s.com, fr_mrn@lbero.t, f.zrll@cspur.t, m.c.recchon@unvpm.t Receved Februry 24, 2012; revsed Mrch 25, 2012; ccepted Aprl 15, 2012 ABSTRACT We study the congeston phenomenon n mthemtcl model of the dt pckets trffc n trnsmsson networks s functon of the topology nd of the lod of the network. Two types of trffc re consdered: homogeneous nd heterogeneous trffc. The congeston phenomenon s studed n sttonry condtons through the behvour of two qunttes: the men trvel tme of pcket nd the men number of pckets tht hve not reched ther destnton nd re trvelng n the network. We defne trnsformton tht mps network hvng the smll world property (Inet 3037 n our numercl experments) nto (modfed) lttce network tht hs the sme number of nodes. Ths mp chnges the cpcty of the brnches of the grphs representng the networks nd cn be regrded s n nterpolton between the two clsses of networks. Usng ths trnsformton we compre the behvour of Inet 3037 to the behvour of modfed rectngulr lttce nd we study the behvour of the nterpoltng networks. Ths study suggests how to chnge the network topology nd the brnch cpctes n order to llevte the congeston phenomenon. In the webste: some uxlry mterl ncludng nmtons nd stereogrphc scenes tht helps the understndng of ths pper s shown. Keywords: Complex Networks; Network Topology; Congeston Phenomenon; Smll World Property; Phse Trnston 1. Introducton In ths pper we study the dt pcket trffc congeston phenomenon n trnsmsson networks n the cse of homogeneous nd heterogeneous trffc. Ths s n nterestng phenomenon wth severl prctcl pplctons mong whch we menton the study of the behvour of telecommuncton networks nd n prtculr of the behvour of Internet. More n generl the congeston problem n network flows s of gret nterest n severl engneerng felds, such s, for exmple, ppelne flow, electrc power trnsmsson [1], telecommunctons nd hgh-wys or rlwys trffc mngement. To fx the des let us consder the Internet trffc. The Internet network grows n sze every dy nd ts possble congeston s relevnt ssue from the socl nd the economc pont * The numercl experence reported n ths pper hs been obtned usng the computng grd of Ene (Rom, Itly). The support nd the sponsorshp of Ene re grtefully cknowledged. # The work of ths uthor hs been prtlly supported by SEEX Sstem Integrt S.p.A. (Rom, Itly) through reserch contrct grnted to CERI-Unverstà d Rom Spenz. of vew. In fct t s esy to mge the dmge tht the Internet congeston wll cuse to the socl reltons nd to the economc trnsctons tht tke plce on t. In the lst yers severl end-hosts hve been dded to the edge of the Internet network nd these new entres hve ncresed substntlly the lod of the network. The Internet Servce Provders hve dded new routers nd lnks to vod the congeston rsng from these new entres. Severl uthors hve developed trnsmsson network models devotng specl ttenton to the network congeston problem (see for exmple [2] nd the reference theren). Remrkble network models relevnt n the study of Internet re those due to Aello, Chung nd u [3] tht re bsed on power lw rndom grph models, to Fbrknt, Koutsoups nd Ppdmtrou [4] tht propose bcrter optmzton model nd to Brbs nd Albert [5] tht hve ntroduced the preferentl connectvty model. These models hve common feture: power lw dstrbuton of the degree of the network nodes. Furthermore more prctcl models of the Internet network re those bsed on nferences drwn from mesurements of

2 158 A. FARINA ET A. rel Internet dt. In prtculr huge effort hs been produced to construct genertors of grphs tht reproduce the fetures ctully mesured n the Internet network (see for exmple [6,7]). Prevous work tht ddresses the congeston problem n flow problems on grphs nd more specfclly the problem of desgnng routng schemes ble to vod congeston s, for exmple, the work contned n the ppers of Aumnn nd Rbn [8], Okmur nd Seymour [9], eghton [10] nd eonrd [11]. In [8] nd [9] the network congeston problem s formulted s problem concernng the mx-cut rto of grph whle n [10] nd [11] the problem of fndng frctonl routng strteges wth good congeston propertes s ddressed. The dt pcket trffc s descrbed by the routng strtegy nd by the trffc mngement rules. The routng strtegy consdered here to move the pckets from ther orgn to ther destnton s the shortest weghted pth. The trffc mngement rules re the rules used to mnge the dt pcket trffc t the nodes where more thn one dt pcket s wtng to be drected. The trffc mngement rules consdered n our study re smple vrton of the frst n frst out rule. Gong nto detls let n be postve nteger, we model the trnsmsson network usng n undrected (weghted) grph G hvng n nodes. The grph G conssts n the couple (V, E), where V s the set of nodes/vertces nd E s the set of brnches/edges/lnes of the grph. The vertces re numbered, tht s for grph wth n vertces we cn choose V = {1, 2,, n}. A brnch s ndexed by couple of ntegers m, k V wth m k, the brnch e m,k E connects the nodes m (source node) nd k (destnton node). For m, k V such tht m k, f e m,k E then e k,m E nd e k,m denotes the sme brnch thn e m,k. Moreover, for e m,k E we ssocte to the brnch e m,k cpcty c m,k > 0 nd we denote wth p m,k the nverse of c m,k, p m,k s the weght ssocted to the brnch e m,k. For e m,k E the cpcty c m,k s postve nteger nd s the mxmum number of dt pckets tht cn go through the brnch e m,k n tme unt movng from the source node m to the destnton node k. Moreover for e m,k E we ssume tht c m,k = c k,m nd s consequence tht p m,k = p k,m. We extrpolte ths relton nd we sy tht when the weght of brnch e m,k s nfnty ts cpcty s zero nd ths corresponds to the fct tht no pcket cn flow through t. Tht s, gven couple of (dstnct) nodes we cn lwys mgne tht there s brnch wth zero cpcty tht connects them. We consder connected grphs, tht s grphs G such tht for every couple of (dstnct) nodes (m, k) there exsts pth mde of brnches (wth cpcty greter thn zero) connectng the source node m to the destnton node k. The degree of node s the number of brnches (wth cpcty greter thn zero) tht hve the node s source node (or s destnton node). In the generton of the dt pcket trffc the source nd the destnton nodes of the dt pckets re smpled from rndom vrble unformly dstrbuted on the set of the couples of (dstnct) nodes of the grph. A dt pcket cn move from node to nother node n one tme unt f there s brnch (wth cpcty greter thn zero) tht connects the two nodes, tht s, f the nodes re djcent nodes n the grph. The movement of dt pcket from node to one of ts djcent nodes s lwys mde n one tme unt. To ech couple of dstnct nodes (m, k) (source, destnton), m, k = 1, 2,, n, we ssocte route tht s the pth tht dt pcket movng from m to k must follow, ths route s the shortest weghted pth connectng m to k wth some extr specfctons explned lter. The dt pcket trvel tme (or delvery tme) s the tme tht elpses between the creton of the pcket t ts source node m nd the rrvl of the pcket t ts destnton node k. The trvel tme s mesured n tme unts. Ech node s equpped wth queues, whch s n ech node there s queue for ech brnch (wth cpcty greter thn zero) levng the node. When brnch deprtng from node s busy (.e. ts cpcty s sturted) pcket rrvng or lredy present t tht node drected through tht brnch towrd ts destnton s put n the node queue reltve to tht brnch nd wts the successve tme unts to be drected. The node queues re ssumed to be potentlly unbounded nd, of course, eventully, they cn be empty. The mngement of the queues s done through the trffc mngement rules. In the cse of homogeneous trffc, tht s when only one knd of dt pckets s movng on the network, the queue mngement rule s smply bsed on the rrvl order of the pckets t the node, whch s the trffc mngement rules consst n only one rule: the frst n frst out rule wth the followng specfctons dded. et us restrct our ttenton to the pckets present t node tht re levng the node (followng ther routes) through gven brnch. Note tht when two or more pckets must go through the sme brnch nd ths brnch hs suffcent cpcty vlble the pckets go through the brnch smultneously, tht s, they go through the brnch n the sme tme unt. When two or more pckets must go through the sme brnch nd there s no suffcent cpcty vlble on the brnch to let them go smultneously we use the frst n frst out rule, tht s we delver frst the pcket tht rrved frst t the node. When two or more pckets tht must go through the sme brnch re the frst rrved one t the node we delver frst the pcket tht hs been generted erler. Note tht the pcket tht s delvered frst s the one tht goes frst n the queue. When two or more of these frst rrved pckets tht hve been generted erler hve been generted t the sme tme we choose rndomly between these lst pckets the pcket to move frst. In the study of the heterogeneous trffc cse we lmt our ttenton to the cse

3 A. FARINA ET A. 159 tht there re three knds of dt pckets movng on the network. The choce of studyng s heterogeneous trffc cse the trffc of three knds of dt pckets s nspred by the fct tht n rel telecommuncton networks we cn dstngush dt, vdeo nd voce pckets. In ths lst cse the trffc mngement rules re specfed s follows. We ssocte to ech pcket wtng t busy node wtng tme mesured n tme unts bsed on the rrvl order t the node. For the pckets of knd we multply ths wtng tme for the prorty fctor p of the knd pckets, = 1, 2, 3. The prorty fctors re chosen s follows: p 1 = 3 for the pckets hvng hghest prorty (pckets of knd = 1), p 2 = 2 for the pckets hvng ntermedte prorty (pckets of knd = 2) nd p 3 = 1 for the pckets hvng lowest prorty (pckets of knd = 3). The queue mngement rule conssts n the choce of movng frst the pcket hvng the lrgest product wtng tme t the node tmes prorty fctor. When two or more pckets hve the lrgest product wtng tme t the node tmes prorty fctor we delver frst the pcket tht hs been generted erler. If t node two or more pckets hvng the lrgest wtng tme tmes prorty fctor tht hve been generted erler hve been generted t the sme tme we choose rndomly between these lst pckets the pcket to move frst. Of course lso n the cse of heterogenous trffc when there s cpcty vlble more thn one pcket cn go through the sme brnch t the sme tme. Note tht the heterogenous trffc cse reduces to the homogeneous trffc cse when we choose p 1 = p 2 = p 3. For smplcty we present the lod model used lter drectly n the heterogeneous trffc cse. We leve to the reder to fnd out the smple chnges necessry to descrbe the lod model n the homogeneous trffc cse. We model the number of knd pckets, M, generted n the network n tme unt s Posson rndom vrble wth men β λ, = 1, 2, 3, where β j, j = 1, 2, 3, re gven non- negtve constnts such tht 3 j1 1nd λ > 0 s rel prmeter. The choce of the constnts β j, j = 1, 2, 3, determnes dfferent heterogeneous lod condtons on the network. ter we consder three choces of the constnts β j, j = 1, 2, 3. The frst one s β 1 = β 2 = β 3 = 1/3, tht genertes blnced lod condton, tht s genertes lod hvng n verge the sme number of pckets for the three knds of pckets. The second one s β 1 = 5/7, β 2 = 1/7, β 3 = 1/7, tht genertes n unblnced lod condton, tht s genertes lod hvng n verge lrge number of pckets wth hghest prorty (knd = 1) nd such tht the remnng two knds of pckets hve (n verge) smller number of pckets, n prtculr snce β 2 = β 3 they hve (n verge) the sme number of pckets. The lst one s β 1 = 1/6, β 2 = 2/6, β 3 = 3/6, tht genertes the most common lod stuton n rel networks, tht s the stuton where j the number of pckets wth the hghest prorty (knd = 1) s (n verge) smller thn the number of pckets wth ntermedte prorty (knd = 2) nd ths lst number s (n verge) smller thn the number of pckets wth lowest prorty (knd = 3). We cll congeston phenomenon for the (knd ) dt pcket trffc the pssge from free flow to congested flow ( = 1, 2, 3). We study the congeston phenomenon n sttonry condtons usng the followng two qunttes: the men trvel tme of knd dt pckets, t M, nd the men number of knd pckets tht hve not reched ther destnton nd re trvellng on the network, N, = 1, 2, 3. Our nlyss shows tht, n the free trffc regme nd n sttonry condtons, for = 1, 2, 3 the qunttes t M nd N re ndependent of tme nd tht they ncrese when the network men lods β j λ, j = 1, 2, 3, ncrese. Gven the weghted grph G, nd the constnts β j, j = 1, 2, 3, such tht β j 0, j = 1, 2, 3, nd 1, for = 1, 2, 3 3 j1 we study the qunttes t M nd N s functon of the prmeter λ. We wll see from the numercl smultons tht some expressons specfed lter nvolvng t M nd N hve jump for vlue of the prmeter λ tht we denote wth λ = *, = 1, 2, 3. For the trffc of knd * pckets the vlue λ = represents the so clled crtcl vlue of the prmeter λ tht dvdes the zone of congested * trffc (β λ > β ) from the zone of free trffc (β λ < * * β ), = 1, 2, 3. Note tht strctly spekng when λ > the qunttes t M nd N do not hve sttonry vlue n tme nymore, n fct n ths cse these qunttes dverge when tme ncreses, = 1, 2, 3. Under specfed trffc condtons we hve evluted through extensve smultons these crtcl vlues λ = *, = 1, 2, 3, s functon of the network topology. Our nlyss shows tht the networks hvng the smll world property (see, for exmple, the network of Fgure 1) nd the (possbly modfed) lttce networks, tht do not hve the smll world property, (see, for exmple, the network of Fgure 3) behve very dfferently wth respect to the congeston phenomenon. In prtculr we hve compred the behvour of Inet 3037 (Fgure 1) to the behvour of modfed rectngulr network (Fgure 3) mde of rectngulr lttce hvng 3036 nodes nd of n extr node of degree four. Note tht Inet 3037 s network mde of 3037 nodes generted usng the softwre pckge [6] (lgorthm Inet-3.0) tht s supposed to reproduce the propertes of the AS-level Internet grph topology. The emprcl nlyss of the AS-level Internet grph topology goes bck to 1999 wth the work of M. Floutsos, P. Floutsos, C. Floutsos [12] tht hs shown tht the Internet node degree dstrbuton decys s power-lw, tht s the functon F'(k) = {percentge of the nodes wth degree greter or equl to k} ssocted to AS-level Internet j

4 A. FARINA 160 ET A. Fgure 2. The rch club of Inet Fgure 1. Inet grphs behves lke k β, wth β = 2.22, for lrge vlues of k (see Secton 2 formul (1)). Ths property s usully stted syng tht Internet t the AS-level grph s scle free network. ter n 2002 Subrmnn, Agrwl, Rexford, Ktz [13] hve shown tht relstc model of the ASlevel Internet grph topology s power lw topology wth core structure. Ths core structure s subset of nodes hvng lrge degree tht s clled rch-club (see, for exmple, Fgure 2 where the rch club of Inet 3037 s shown). The nodes belongng to the rch club re well connected between themselves by pths of short length (nd hgh cpcty), moreover they re connected to the remnng nodes by pths of short length. We recll tht network hvng ths feture s sd to hve the smll world property. When routng strteges such s the shortest pth or the shortest weghted pth re used to delver the pckets the rch club s smlr to set of trffc hubs. The network model due to Brbs nd Albert [5] nd the Inet3.0 lgorthm [6] used to generte grphs wth AS-level Internet grph topology consder scle free networks wth β (2, 3), however n these models the reproducton of the Internet core structure s not completely stsfctory. In fct n these models the rther sophstcted structure of the rch club observed n the rel Internet s replced wth just few nodes hvng very lrge degree, nd these nodes ply the role of network hubs n the study of dt pckets trffc. We consder, for moment, networks where ll the brnches hve the sme cpcty nd the shortest pth routng strtegy. In the free trffc regon we show tht network hvng the smll world property s much more effcent n the delverng of pckets n terms of trvel tme thn lttce network of smlr sze. However when the lod ncreses network hvng the smll world property nd the sme cpcty on ll ts brnches reches the congested regme for smller vlues of the lod thn lttce network of smlr sze. When we consder net Fgure 3. The modfed rectngulr lttce. works where the brnches do not hve necessrly the sme cpcty nd the cpctes re chosen n order to explot the smll world property ths phenomenon occurs n more subtle wy. We nvestgte ths phenomenon studyng Inet 3037 (see Fgure 1) n comprson to modfed rectngulr lttce network (see Fgure 3). Note tht these two networks hve the sme number of nodes. Ths fct motvtes the de of defnng trnsformton, dependng on prmeter α [0, 1], tht mps the network hvng the smll world property Inet 3037 (.e.: α = 0) nto the modfed rectngulr lttce network (.e.: α = 1). Ths trnsformton cts chngng the weghts of the brnches of the networks nd depends on the wy the nodes of the networks re numbered. Note tht n ths pper the node numberng s gven nd tht the dependence of the trnsformton between networks mentoned bove nd of the phenomen studed on the networks from the numberng of the nodes s not consdered. Gven the

5 A. FARINA ET A. 161 constnts β j, j = 1, 2, 3, we study the crtcl vlues of the prmeter λ = *, = 1, 2, 3, s functon of the prmeter α nd of the trffc mngement rules. Ths study suggests some chnges n the trffc mngement rules nd/ or n the network topology ble to llevte the consequences of the congeston phenomenon. Some nloges between the congeston phenomenon studed here nd the phse trnston phenomenon studed n sttstcl mechncs re llustrted. In the web ste: some uxlry mterl ncludng nmtons nd stereogrphc scenes tht helps the understndng of ths pper s shown. The pper s orgnzed s follows. In Secton 2 we descrbe the topologcl propertes of Inet 3037 nd of the modfed rectngulr lttce network. Usng the fct tht Inet 3037 nd the modfed rectngulr lttce network consdered hve the sme number of nodes we defne trnsformton dependng on prmeter α [0, 1] tht mps Inet 3037 (.e. α = 0) nto the modfed rectngulr lttce network (.e. α = 1). In Secton 3 n the homogeneous trffc cse we study the trffc congeston phenomenon on Inet 3037, on the modfed rectngulr lttce network consdered n Secton 2 nd on the α- networks, α (0, 1). In Secton 4 n the heterogeneous trffc cse we study the trffc congeston phenomenon on the prevous networks. Fnlly n Secton 5 we drw some concluson. 2. The Networks nd Ther Topologcl Propertes The topologcl fetures of grph tht wll be consdered n our nlyss re: the node degree dstrbuton, the men length of the shortest pth between the couples of (dstnct) nodes, the betweenness centrlty of the nodes nd the grph connectvty. These purely topologcl notons re dpted to the fct tht the networks consdered re modeled s weghted grphs, tht s re dpted to the fct tht we hve cpctes ssocted to the brnches of the grphs. et G = (V, E) be grph hvng n nodes tht models network. For k = 0, 1,, n 1, let f(k) be the frequency of the nodes of the grph hvng degree k, nd let F(d), d = 0, 1,, n 1, be the node degree cumulte dstrbuton, tht s F( d) f( ), d = 0, 1,, n 1, we defne F (d) = d 1 F(d), d = 0, 1,, n 1. We sy tht network s scle free network when the functon F (d) decys s n nverse power of d when d ncreses (see Fgure 4), tht s when we hve: Fd cd, d 0, 1,, n 1 (1) for some postve constnts c nd β (see [6]). The men length of the shortest pth s the men vlue Fgure 4. F (d) s functon of the node degree d. n n 1 l nn 1 1 1, l s, d (2) s d d s of the length of the shortest pth tht jons the couples of (dstnct) nodes, (s, d), s d, s, d V, tht s: where l s,d s the length of the shortest pth connectng the nodes s nd d, s d, s, d V, tht s l s,d s the smllest number of nodes tht must be vsted to go from s to d, s d, movng on brnches wth cpcty greter thn zero, s, d V. Note tht when evlutng the length of pth tht goes from s to d the node d must be counted between the nodes vsted to go from s to d. We consder connected grphs so tht for every couple of nodes (s, d), s d, s, d V there s t lest one pth tht goes from s to d. Note tht the shortest pth between two nodes my be nonunque. In our nlyss we consder weghted grphs nd s consequence weghted pths so tht we consder the men length of the shortest weghted pth defned s the men vlue of the length of the shortest weghted pth tht jons the couples of network nodes. Tht s, let S s,d be the set of the pths connectng the nodes s nd d, s d, s, d V. The shortest weghted pth between the nodes s nd d, s d s the pth w* pth s, d S s,d tht goes from s to d hvng the smllest sum of the nverse of the cpctes of ts brnches, s, d V. Note tht the shortest weghted pth my be nonunque. w* When t s nonunque we choose s pths, d S s,d tht s, s the shortest weghted pth whose length s used n (5), the pth tht s determned by the lgorthm used to w* compute t. et l be defned s follows: s, d w* w* sd sd l, lenght pth,, s d, s, d V, (3) where wth the prevous specfctons we defne: pth rg mn p, s d, s, d V. w* sd, w pth, sd, S mk sd, w emk, pthsd, (4) The men length of the shortest weghted pth of the

6 162 A. FARINA ET A. grph G cn be defned s follows: l w* n n 1 l nn 1 s1 d1, ds w* s, d. (5) w* Note tht lenght pth s, d s not well defned when the shortest weghted pth s not unque, tht s the defnton gven bove s loose. In fct dfferent shortest weghted pths my hve dfferent lengths nd dfferent lgorthms to determne them my determne dfferent pths. More stsfctory defntons cn be gven t the prce of mkng the exposton more complcted. These defntons wll be omtted. Remnd tht when we consder Inet 3037 for every brnch e m,k E the weght p m,k of the brnch s postve number equl to the nverse of the cpcty c m,k of the brnch tht s ssgned by the Inet-3.0 lgorthm [6]. The betweenness centrlty of node v V s the followng quntty: n n 2 CB() v rs, d(), v vv, n3, ( n1)( n2) s1, sv d1, ds, dv (6) where r s,d (v), s d, s v, d v, s the number of the shortest pths jonng s to d pssng through the node v dvded by the totl number of the shortest pths jonng the node s to the node d, s, d, v V. Note tht we hve: n lsd, rsd, (), v s d, s, dv. (7) v1, vs We generlze the betweenness centrlty defned n (7) ntroducng the weghted betweenness centrlty of node v V defned s follows: n n w 2 w CB v rs, d v, vv, n 3, ( n1)( n2) s1, sv d1, ds, dv (8) w where r sd, v, s d, s v, d v, s the number of the shortest weghted pths jonng s to d pssng through the node v dvded by the totl number of the shortest weghted pths jonng the node s to the node d, s, d, v V. The weghted betweenness centrlty dpts the noton of betweenness centrlty to the fct tht we consder weghted grphs. Note tht the betweenness centrlty nd the weghted betweenness centrlty of node re mesures of the relevnce of the node for the trffc on the network when we use s routng strtegy respectvely the shortest pth nd the shortest weghted pth. Roughly spekng they re mesures of how much node cts s hub n the network when we use the routng strteges mentoned bove. et us consder grph G = (V, E) nd the nonempty sets S V such tht the grph G\S = (V\S, E ((V\S) (V\S))) s dsconnected grph nd the set Σ of the crdnltes of the sets S tht hve the prevous property. The connectvty of the grph G s the mnmum of the set Σ. Roughly spekng the connectvty of grph s the smllest number of nodes tht must be removed from the grph to dsconnect t. et us descrbe the Inet 3037 network. The Inet 3037 network hs been generted usng the pckge Inet (Internet Topology Genertor) (see [6] for further detls). The lgorthm used by the Inet pckge s the Inet-3.0 lgorthm nd genertes rndom grphs modelng the ASlevel Internet grph topology. In fct Internet cn be vewed s n AS-level topology grph where ech AS (Autonomous Systems) s node, nd the BGP (Border Gtewy Protocol) peerng between two ASes s lnk. Inet-3.0 tkes nto ccount the Internet topologcl propertes s observed by the Unversty of Mchgn reserch tem tht hs mplemented nd dstrbutes the Inet pckge. For exmple, t consders the node degree dstrbuton nd the network connectvty of the grphs generted. In detl, the Inet-3.0 lgorthm tkes s nput the number n of nodes tht the network generted must hve nd the number of nodes of the network tht must hve degree one. Tken these two nput dt Inet-3.0 genertes grph whose lnks re chosen n order to reproduce the result of study of the node degree dstrbuton, of the connectvty nd of some other propertes of Internet s regstered n the BGP (Border Gtewy Protocol) routng tbles of the Oregon server: route-vews.oregon-x.net. The Oregon server collects nformton on the trffc over utonomous systems belongng to dstnct Internet servce provders nd on ther topology communctng wth these provders through the BGP protocol. The output of the Inet-3.0 lgorthm re the lnks of the grph generted nd set of postve ntegers ssocted to the lnks tht we nterpret s cpctes of the lnks. We hve used Inet-3.0 to generte weghted grph wth n = 3037 nodes nd we hve requred tht pproxmtely 30% of the nodes of the grph generted (to be precse 911 nodes) must hve degree one. The connectvty of the grph generted by Inet-3.0 s 3, tht s the nteger prt of n. The resultng grph s Inet 3037 (see Fgure 1). We note tht n Inet 3037 the men vlue of the node degree s pproxmtely 3 nd tht the mxmum vlue of the node degree s 684. The nodes hvng degree greter thn 60 re 11 nd re the red nodes shown n Fgures 1 nd 2. These nodes re the nodes of the rch club of Inet Fgure 2 shows the nodes of the rch club of Inet 3037 nd the brnches of Inet 3037 tht connect them. The red nodes pperng n Fgures 1 nd 2 cn be consdered s representtves of sutbly defned subgrphs. A node y of Inet 3037 ( yellow node n Fgure 1) belongs to subgrph whose representtve node s node k of the rch club (red node n Fgure 1) f there exsts t lest one pth (mde of brnches wth cpcty greter thn zero) jonng y to k nd there re no pths jonng y to node of the rch club dfferent from k tht do not go through k. The nodes such tht there re pths

7 A. FARINA ET A. 163 jonng them drectly to more thn one node of the rch club re represented s green nodes n Fgure 1. The nodes of Inet 3037 re numbered n ncresng order for decresng vlues of the node degree. The nodes tht hve the sme degree re numbered rndomly wth consecutve numbers. Inet 3037 hs 4788 brnches. The smllest nd lrgest cpctes of the brnches re 67 nd respectvely. The men vlue of the cpctes of the brnches of Inet 3037 s pproxmtely An esy comw w putton shows tht Inet 3037 hs l linet et us denote wth R I (v) the sum of the cpctes of the brnches of Inet 3037 rrvng t (or deprtng from) the node v, v V. Fgure 5 shows the quntty R I (v) s functon of v V. We note tht the frst eleven nodes (.e. the nodes numbered from one to eleven tht re the nodes of the rch club) re those wth hghest degrees nd wth the hghest vlues of R I. et us descrbe the modfed rectngulr lttce network tht we use s comprson term n the study of the behvour of Inet The modfed rectngulr lttce network (see Fgure 3) hs 3037 nodes nd s chrcterzed by the degree of the nner nodes tht s equl to four nd by the cpcty of ts brnches tht s the sme for ll brnches. The modfed rectngulr lttce network shown n Fgure 3 hs 5964 brnches. The cpcty of ts brnches, c R, s chosen s the nteger prt of the sum of the cpctes of the brnches of Inet 3037 dvded by the number of brnches of the modfed rectngulr lttce network. An esy computton gves c R = Note tht snce ll the brnches of the modfed rectngulr lttce hve the sme cpcty the choce of c R mde gurntees tht the totl cpcty nstlled on Inet 3037 nd the totl cpcty nstlled on the modfed rectngulr lttce network re roughly the sme. They re roughly the sme nd not exctly the sme snce we hve used the nteger prt to defne the brnch cpcty of the modfed rec- Fgure 5. R I (v) s functon of the node number v. tngulr lttce. The use of the nteger prt n the defnton of the brnch cpcty s due to the fct tht the cpctes re supposed to be nteger numbers. In m l rectngulr lttce network the nner nodes hve the sme betweenness centrlty nd the men length of the shortest pth s gven by: m 1 l 1 l Rect. (9) 3 m l To buld the modfed rectngulr lttce network we consder m l rectngulr lttce network wth m = 46 nd l = 66, tht s network wth 3036 nodes, nd we dd to t n extr node hvng degree four (see Fgure 3). The nodes of the m l rectngulr lttce re numbered from rght to left, nd from the bottom row to the frst row (see Fgure 3) n ncresng order begnnng wth v = 1 correspondng to the rght bottom corner of the network. In prtculr let A R = ((A R,j)) R be the weghted djcency mtrx of the modfed rectngulr lttce network defned strtng from the weghted djcency mtrx of the rectngulr lttce network (wth ll the brnches of cpcty c R ) ddng to t n extr node, v = 3037, connected to the rectngulr lttce network s shown n Fgure 3, tht s ddng n extr row nd column to the djcency mtrx of the rectngulr lttce s follows: R A3037, cr, 3033, 3034, 3035, 3036, R R A,3037 A3037,, 3033, 3034, 3035, 3036, (10) R R A3037, A,3037 0, 1, 2,,3032, R A ,3037 It s esy to see tht the network modeled by the grph correspondng to the djcency mtrx A R s the one shown n Fgure 3. The length of the men shortest pth n the modfed rectngulr lttce network shown n Fgure 3 s l R l w INET. et us defne the weghts of the brnches of the modfed rectngulr lttce network s the w nverse of ther cpctes nd let l R be the men length of the shortest weghted pth of the modfed rectngulr lttce, we note tht snce ll the brnches of the modfed rectngulr lttce network hve the sme cpctes, they hve lso the sme weghts so tht we hve l In the numercl smultons we scle the cpctes of the brnches of Inet 3037 nd of the modfed rectngulr lttce by dvdng them by the smllest cpcty of the brnches of Inet 3037 tht s dvdng them by 67. Ths choce reduces the computtonl work needed to crry out the numercl smultons. Note tht Tble 1 shows tht the nodes of the modfed rectngulr lttce hve pproxmtely the sme relevnce n terms of betweenness centrlty, n fct for the modfed rectngulr lttce the men vlue of the betweenness centrlty s of the sme order of mgntude of ts mx- w R lr.

8 164 A. FARINA ET A. Tble 1. Some fetures of Inet 3037 nd of the modfed rectngulr lttce. Feture Inet 3037 Modfed Rectngulr lttce men length shortest pth men length shortest weghted pth mxmum vlue betweenness centrlty men betweenness centrlty mxmum vlue weghted betweenness centrlty men vlue weghted betweenness centrlty mum vlue. Ths s consequence of the fct tht the modfed rectngulr lttce does not hve the smll world property. The cse of Inet 3037 s substntlly dfferent, n fct for Inet 3037 the men vlue of the betweenness centrlty s three orders of mgntude smller thn ts mxmum vlue. Roughly spekng, n the cse of Inet 3037, the node where the betweenness centrlty reches ts mxmum vlue (.e. the node v = 1) bsorbs pproxmtely 58% of the trffc when we use the shortest pth routng, tht s bout 58% of the shortest pths goes through ths node. Ths percentge reduces to 46% f we consder the weghted shortest pth routng nd s consequence the weghted betweenness centrlty nsted thn the betweenness centrlty. Fnlly, for Inet 3037 we note tht the sum of the betweenness centrlty of the nodes of the rch club s 52% of the sum of the betweenness centrlty of ll the nodes nd tht the sum of the cpctes of the brnches connected to the nodes of the rch club s 41% of the sum of the cpctes of ll the brnches of the network. In the cse of the modfed rectngulr lttce (when we consder s rch club of the modfed rectngulr lttce the grph ssocted to ts frst eleven nodes, tht s the nodes numbered from 1 to 11) the sme rtos re pproxmtely equl to 0.003% nd to 0.03% respectvely. Note tht the correspondence between the rch club of Inet 3037 nd the grph ssocted to the frst eleven nodes of the modfed rectngulr lttce s rther rbtrry one. In fct when we consder the modfed rectngulr lttce there s no nturl cnddte to ply the role of the rch club. Tble 1 summrzes the topologcl propertes of Inet 3037 nd of the modfed rectngulr lttce dscussed bove. et E I denote the set of brnches of Inet 3037 nd A I R denote the weghted djcency mtrx of Inet I I 3037, tht s the mtrx whose entres re: A, j c, j, j I I I nd A,, where,, f e,j E I 0 c j c j or c, j 0 f e,j E I,, j = 1, 2,, et α [0,1] be rel prmeter, we denote wth A α = (( A, j)) R the followng combnton of the weghted djcency mtrces A R nd A I reltve to the modfed rectngulr lttce nd to Inet 3037 respectvely: I R A 1 A A, 0, 1, (11) where s the celng prt of nd the celng prt of mtrx s defned s the mtrx whose entres re the celng prts of the entres of the orgnl mtrx. Remember tht the celng prt of rel number x s the smllest nteger greter or equl to x. The mtrx A α s the weghted djcency mtrx of grph tht we cll α-network, α [0, 1]. Note tht when α = 0 we hve A α equl to the weghted djcency mtrx of Inet 3037 nd when α = 1 we hve A α equl to the weghted djcency mtrx of the modfed rectngulr lttce. We note tht n the trnsformton (11) the most mportnt hub of Inet 3037, tht s the node v = 1, s mpped nto the node v = 1 of the modfed rectngulr lttce (tht s, t s mpped n the bottom rght corner of the modfed rectngulr lttce, see Fgure 3) nd so on. Ths correspondence between the nodes s somehow rbtrry nd depends on the wy the nodes re numbered. Note tht snce the men shortest weghted pth of the modfed lttce network s bout seven tmes greter thn the men length of the shortest weghted pth of Inet 3037 n the numercl smulton of the trffc flow (n the free flow regme) on these networks we expect tht smlr reltonshp holds between the smulton tmes needed to rech the sttonry condton. We cn conclude tht the tme ntervl smulted must be chosen pproprtely n order to gurntee tht on both networks the sttonry regme hs been reched. et us mke some comments bout the topologcl fetures of the α-networks. et us denote wth G I = (V I, E I ), G R = (V R, E R ) nd G α = (V α, E α ), α [0, 1], the grphs ssocted respectvely to Inet 3037, to the modfed rectngulr lttce nd to the α-network, α [0, 1]. We hve V I = V R = V α, α [0, 1], nd E I = E 0, E R = E 1, moreover for ny α (0, 1) we hve E α = E I E R, α (0, 1). Ths mples tht for α (0, 1) the men length of the shortest pth of the α-network l s smller or equl thn l INET nd l R. Tble 2 shows tht, t lest for the vlues of α consdered n Tble 2, ths property contnues to hold when we consder the shortest weghted pth. Ths s not obvous nd depends on the weghts ssocted to the brnches. Note tht between the α-networks consdered n Tble 2 the one hvng the smllest men length of the shortest weghted pth s the α = 0.5-network. The cpctes of the brnches of the α-network re I R A, j 1 c, jc, e,j E α nd the correspondng weghts used n (4) re p, j 1 A, j, e,j E α, α [0, 1]. Note tht when α goes from zero to one n (11) n the α- network we hve reducton of the cpctes of the

9 A. FARINA ET A. 165 α-network Tble 2. Some fetures of the α network. Men length shortest pth Men length shortest p w* weghted pth emk, pthsd, α = α = α = α = α = α = α = α = α = α = brnches connected to the hubs of Inet 3037 nd tht ths mples the reducton of the weghted betweenness centrlty of the hubs. Ths fct mkes possble to reduce the men length of the shortest pth snce the dvntge of gong through the hubs n the shortest weghted pths (see [14]) s reduced. Fnlly, let α [0, 1] for the grph G α consder the w weghted betweenness centrlty C,, v V α B () v defned s n (8). For severl choces of α, such tht α [0, 1], Tble 4 shows the correspondng vlues of the mxmum, w w, wth respect to v of () v, v V α. b,mx CB, 3. The Trffc Congeston Phenomenon n the Homogeneous Trffc Cse et us study the congeston phenomenon n the homogeneous trffc cse for the networks descrbed by the weghted djcency mtrces defned n (11). Note tht the homogeneous trffc cse cn be obtned s specl cse of the heterogeneous trffc cse, for exmple, wth the choce of the prorty fctors: p 1 = p 2 = p 3 = 1 or wth the choce β 1 = 1, β 2 = β 3 = 0. Then n the homogeneous trffc cse the men lod of pckets generton s equl to β 1 λ = λ. At tme zero the networks studed re empty, tht s they do not contn dt pckets. Ths fct genertes trnsent behvour tht connects the ntl stte (.e. empty network) to the lrge tme behvour of the trffc on the network. For the homogeneous nd the heterogeneous trffc cse the smulton of the dt pcket trffc s crred out n the tme ntervl [0, T]. The tme T s chosen to be 400 tme unts n order to gurntee tht n the free flow regme the trffc on the networks consdered hs (pproxmtely) reched the sttonry condton. The smulton procedure n the homogeneous trffc cse conssts of the followng steps: 1) Choose the number N α + 1 of the network topologes tht wll be consdered n the smultons, we consder N α = 10; mk, 2) Choose the network topologes, tht s, ssgn the vlues of α used n the smultons, we consder α = α j = (j 1)/N α, j = 1, 2,, N α +1; 3) Defne the topology of the networks studed, for j = 1, 2,, N α +1, we consder the network ssocted to the j A weghted djcency mtrx defned by (11) when α = α j, ths s the α = α j -network topology; 4) Pckets generton: n ech tme unt of the smulton we frst smple from Posson rndom vrble wth men λ the number of pckets generted n tht tme unt. The vlues of λ consdered n the smultons re 201 vlues equspced n the ntervl [10, 3000], tht s we choose λ = λ k = 10 + ( )k/200, k = 0, 1,, 200. A couple of nodes re ssgned to ech pcket generted. Ths couple of nodes s smpled from rndom vrble unformly dstrbuted on the set of couples (s, d), such tht s d, s, d = 1, 2,, For s, d = 1, 2,, 3037, s d, the pcket ssocted to the couple (s, d), orgntes n the node s nd ts destnton s the node d; 5) Routng: the routng conssts n ssoctng to ech pcket shortest weghted pth connectng the node where the pcket orgntes wth ts destnton node. The pcket s forwrded long ths pth durng the tme steps tht follow the tme of ts generton. When the shortest weghted pth between the orgn nd destnton nodes of pcket s not unque the pcket follows the shortest weghted pth determned by the lgorthm used to compute the routng of the pckets (see [14]). When the pcket reches ts destnton s removed from the trffc smulton; 6) Queue mngement: every node hs queue for ech brnch levng the node, we ssume the queues to be potentlly unbounded. When pcket s generted n node or rrves t node, t s plced n the pproprte queue of the node n the process of beng delvered towrd ts destnton long ts route. The queue mngement rule s: frst n frst out. When two or more pckets tht must contnue ther route on the sme brnch rrve smultneously t node the one tht goes frst n the queue s the pcket tht hs been generted erler. When two or more pckets tht must contnue ther route on the sme brnch nd tht hve been generted smultneously rrve smultneously t node we choose rndomly between them the one tht goes frst n the queue; 7) Pckets movement: n tme unt, when there s cpcty vlble on the relevnt brnch, pcket moves from the node where t s to the djcent node long ts route (.e. the shortest weghted pth). When there s no cpcty vlble on the relevnt brnch the pcket wts the next tme unt t the node where t s. Remember tht n ech tme unt, when n node we hve severl pckets tht (ccordng wth ther routes) must be delvered to the sme djcent node j we delver them n the sme tme unt s long s the number of pckets delvered n the tme

10 166 A. FARINA ET A. unt on the brnch e,j s smller or equl thn the cpcty of the brnch e,j. For exmple for the α-network the mxmum number of pckets tht cn be forwrded n the sme tme unt long the brnch e,j E α s A, j, α [0, 1]. Ths procedure pples to every node nd to every brnch. We note tht the prmeter λ s the men vlue of the number of pckets generted n the entre network n tme unt, so tht gven the fct tht the orgn nd the destnton nodes ttrbuted to the pckets re smpled from n unformly dstrbuted rndom vrble nd the fct tht there re n nodes n the network we cn conclude tht every node on verge genertes λ/n pckets per tme unt. Note tht we hve chosen s routng strtegy the shortest weghted pth (see formul (3)) nsted thn the more usul shortest pth. Ths choce s mde n order to tke dvntge of the network complexty, tht s of the smll world property enjoyed by Inet 3037 (nd by the α-networks, α (0, 1)), nd of the fct tht the brnches of Inet 3037 hve cpctes rngng n wde ntervl n wy tht mkes possble to tke dvntge of the smll world property. In fct n the cse of Inet 3037 the use of the shortest pth route wll drect lmost ll the pckets through the nodes wth hgher betweenness centrlty resultng n n ndequte explotton of the brnch cpctes nd ultmtely n trffc congeston. Note tht n the modfed rectngulr lttce cse (.e.: α = 1), due to the choce of gvng the sme cpcty to ll the brnches, the shortest pth nd the shortest weghted pth routng strteges concde. In the homogeneous trffc cse let N, t M be respectvely the qunttes nlogous to the qunttes N, t, M = 1, 2, 3, ntroduced prevously n the heterogenous trffc cse. Note tht n N, t M we drop the subscrpt, snce n the homogeneous trffc cse there s only one knd of dt pckets trvellng on the network, tht s we hve only = 1. The trffc smulton shows tht, when there s no congeston, fter trnsent tme, whose durton depends on the network topology, the trffc flow reches sttonry stte where, n verge, the totl number of pckets creted nd delvered re equl, tht s N s fnte quntty nd the men trvel tme t M mesured n tme unts s of the order of mgntude of the men length of the shortest weghted pth. Ths lst fct s nturl consequence of ttle s lw of queueng theory [15]. Ths sttonry stte s the free flow stte. When λ ncreses we observe tht the trffc on the network goes from the free flow stte to congested stte nd tht ths trnston cn be chrcterzed by vlue λ * of the prmeter λ, tht seprtes these two dfferent sttes. The vlue λ * s clled crtcl vlue of λ. Note tht when λ s greter λ * there s no sttonry stte (n tme) for the trffc on the network, tht s, the qunttes N nd t M ncrese monotonclly towrds nfnty when the smulton tme goes to nfnty. et us defne the crter tht estblsh n the numercl smulton when vlue of the lod prmeter λ s crtcl. et us denote wth t α the men length of the shortest weghted pth expressed n tme unts. Moreover, snce we wnt to put n evdence the dependence of the qunttes N nd t M on the prmeters α nd λ let us use the notton N N ( ) nd tm t M( ), λ > 0, α [0, 1]. Note tht when the lod prmeter λ s below the crtcl vlue (tht s the network trffc s fter trnsent perod n the free flow stte) we hve t M ( ) t α nd N N ( ) λt α. In the numercl smultons we use two crter to recognze the crtcl vlues of λ: Crteron 1: Gven n α-network we sy tht λ * s the crtcl vlue of the lod prmeter λ for the α-network when λ * s the smllest vlue of λ such tht the dervtve wth respect to λ of N ( ) mesured usng λ s unt, tht s the quntty (1 / )d N ( ) / d, s greter thn gven postve constnt γ. Crteron 2: Gven n α-network we sy tht λ * s the crtcl vlue of the lod prmeter λ for the α-network when λ * s the smllest vlue of λ such tht the dervtve wth respect to λ of the men trvel tme mesured usng t α s unt, tht s the quntty (1/ t )d t M ( ) / d, s greter thn gven postve constnt γ. The choce of the threshold γ depends on the network nd on the crtclty crteron consdered. et us study the crtcl vlue of λ s functon of α, α [0, 1]. The numercl smultons show tht gong from Inet 3037 (α = 0) to the modfed rectngulr lttce (α = 1) the crtcl vlue of the prmeter λ, λ = λ * (α), s functon of α hs mxmzer for vlue of α (0, 1). Ths mxmzer depends on the vlue of γ. In prtculr, Fgure 6 shows the crtcl vlues of λ s functon of α for four choces of the threshold γ, tht s γ = 0.04, 0.08, 0.12 nd 0.16, when we use Crteron 1 to recognze crtclty. Note tht when we use Crteron 1 the vlue ssgned to γ cn be seen s rough mesure of the percentge of the pckets tht hve not reched ther destnton nd re trvellng on the network. Fgure 7 shows the crtcl vlues of λ dvded by t α s functon of α for four vlues of γ (γ = 0.2, 0.4, 0.6, 0.8) when we use Crteron 2 to recognze crtclty. The crtclty recognzed usng Crteron 2 (see Fgure 7) s somehow dfferent from the crtclty recognzed usng Crteron 1 (see Fgure 6). In fct when we use Crteron 2 when λ * s the crtcl vlue determned by gven choce of γ, for exmple, let us sy by the choce γ = 0.4, t mens tht when λ = λ * there s n ncrement of t lest 40% n the delvery tme wth respect to the men delvery tme n the noncongested regme, tht s for λ < λ *. Furthermore, usng Crteron 2 the choce of the best performng network, tht s the choce of the network tht hs the bggest vlue of λ *, s not cler cut

11 A. FARINA ET A. 167 Fgure 6. The quntty λ * (α) s functon of the prmeter α for severl choces of the threshold γ (Crteron 1). Fgure 7. The quntty λ * (α)/t α s functon of the prmeter α for severl choces of the threshold γ (Crteron 2). choce s the correspondng choce when we use Crteron 1. However, for exmple, Fgures 6 nd 7 show tht the choce α = 0.5 defnes hghly performng network for both Crter. Fgure 8 shows the qunttes N t nd t ( ) M s functon of λ for severl vlues of the prmeter α, tht s for α = 0, 0.3, 0.5, 0.7, 0.9, 1. We note tht the quntty N t s pproxmtely constnt up to gven vlue of λ nd tht fter tht vlue of λ strts to ncrese (see Fgure 8). The results shown n Fgure 8 suggests tht the networks wth α (0.3, 0.7) re the best performng ones between those studed when we use the qunttes N nd t M to judge the trffc qulty. In fct n Fgure 8 we cn see tht for λ [100, 1100] when α = 0.3, 0.5, 0.7 the vlues tken by the qunttes N t nd t M re smller thn the correspondng vlues of the sme qunttes when we hve α = 0, 0.9, 1. et us study some nloges between the network congeston phenomenon consdered here nd the phse trnston phenomenon studed n sttstcl mechncs usng the rndom vrble D ( ), tht s the number of

12 168 A. FARINA ET A. α Fgure 8. The qunttes N t nd t λ s functon of λ for severl choces of α. M pckets tht re trvellng on the network mnus λt α. Note tht the expected vlue of D ( ) s N ( ) t. We perform the sttstcl nlyss of D ( )/, tht s we pproxmte the probblty densty functon of D ( ) / wth the emprcl dstrbuton of the smulted dt s functon of the prmeters α nd λ n order to estblsh the exstence of szble probblty of hvng lrge congeston nd we compre the results obtned n ths wy wth those obtned usng the Crter ntroduced bove to study crtclty. For severl choces of the prmeters α nd λ we perform number of smultons suffcent to obtn n emprcl pproxmton of the probblty densty functon of D ( )/. When α = 0, Fgures 9 nd 10 show the emprcl probblty densty functons obtned, tht s they show the reltve frequences of D ( )/ obtned n the smultons for fxed vlue of the men lod λ, tht s for λ = 30, 300, 3000 nd * * et mn, mx be lower nd n upper bound respectvely for the crtcl vlue λ * of the prmeter λ. When α = 0 sgn of congeston ppers when the (emprcl) probblty densty functon of D ( )/ chnges ts form from beng unmodl functon ndependent of tme peked round zero (for λ * mn, see Fgure 9) to beng b-modl functon tht trnsltes to the rght when tme goes on (for λ * mx, see Fgure 10 where the probblty densty functons obtned for t = T re shown). ookng * t Fgures 9 nd 10, we cn see tht mn should hve vlue smller thn λ = 3000 nd greter thn λ = 300 (t * seems resonble to be conservtve nd choose mn = * 300) nd tht mx should hve vlue greter or equl thn 3000 nd smller or equl thn 6000 (t seems resonble to be conservtve nd choose = 3000). In * mx fct, Fgure 10 shows tht when λ 3000 the emprcl probblty densty functon of D ( )/ s not peked n zero (.e. the number N s substntlly greter thn the vlue λt α whch s the pproxmte vlue of N n the free flow stte n sttonry condtons). We cn conclude tht the phse trnston from the free flow regme to the congested regme tkes plce for * vlue of λ belongng to the ntervl [, * mn mx ]. Indeed, more refned numercl smultons hve shown tht the * * ntervl [ mn, mx ] cn be reduced to the ntervl: [500, 600]. Note tht ths s n substntl greement wth the crtcl vlue of λ for the network Inet 3037 determned usng Crteron 1 (see Fgure 6). Fgure 11 shows the emprcl probblty densty functon of D ( ) / for α = 0.5 when λ = 300 (Fgure 11()) nd λ = 3000 (Fgure 11(b)). et us compre Fgure 9(b), 11() (.e. λ = 300) nd Fgure 10(), 11(b) (.e. λ = 3000). Note tht when α = 0.5 (.e. Fgures 11() nd (b)) the emprcl probblty densty functon of D ( )/ s peked n zero nd the frequency correspondng to the wndow contnng zero s pproxmtely 70% (λ = 300) or 60% (λ = 3000) nd tht when α = 0 (see Fgure 9(b)) the emprcl probblty densty functon of D ( )/ s peked n zero nd tht the frequency correspondng to the wndow contnng zero s pproxmtely 35% (λ = 300), moreover when α = 0 nd λ = 3000, the probblty densty functon of D ( )/ s not peked n zero (see Fgure 10 ()). Ths fct suggests tht the α = 0-network strts chngng regme erler thn the α = 0.5-network, tht s we should expect tht the α = 0-network reches the congested regme for smller vlue of the lod prmeter λ thn the α = 0.5-network. Ths s confrmed by the nlyss done usng Crteron 1

13 A. FARINA ET A. 169 Fgure 9. Frequency dstrbuton of D when α = 0 nd λ = 30 () or λ = 300 (b). Fgure 10. Frequency dstrbuton of D when when α = 0 nd λ = 3000 () or λ = 6000 (b). Fgure 11. Frequency dstrbuton of D when α = 0.5 nd λ = 300 () or λ = 3000 (b).

14 170 A. FARINA ET A. nd 2 (see Fgures 6 nd 7). et us study the congeston phenomenon from dfferent pont of vew. et us consder the functon z ( ) t t M ( ), λ > 0. From the behvour of t M ( ), λ > 0, shown n Fgure 8 t s resonble to ssume tht z α (λ) s functon of λ behves s follows: q,mn,,mx z, (12) 1,mn where ξ α s postve constnt dependng on the network topology, q α s negtve exponent, λ α,mn, λ α,mx re, respectvely, lower nd n upper bound defnng n ntervl where the trnston from the free flow to the congested regme tkes plce. Ths trnston from the free flow to the congested regme cn be ssmlted to the phse trnston phenomenon studed n sttstcl mechncs. Equton (12) s motvted by the fct tht when the network hs not reched the congested regme t M s pproxmtely equl to t α nd the probblty densty functon of D ( )/ s peked n zero. On the contrry when the network hs reched the congested regme we hve tht tm ncreses wth λ nd tht the probblty densty functon of D ( )/ s not peked n zero. Note tht when λ s greter thn λ * (remember tht λ * < λ α,mx ) the network does not hve sttonry regme (n tme) nymore nd the qunttes t M nd D re not defned, n fct they dverge wth tme. Tht s n ths lst cse the qunttes t M, D shown n the fgures re those observed t tme T, the tme where the numercl smulton ends. We choose λ α,mx = 3000 nd we clbrte the prmeters q α, ξ α, λ α,mn pperng n formul (12) from the smulton dt reltve to the functon z α (λ) usng lest squres procedure. et us denote wth e MS the reltve men squre error t the soluton of the lest squres procedure. Fgures 12(), (b), (c), (d) show the error e MS s functon of λ α,mn when α = 0, α = 0.5, α = 0.7 nd α = 1 respectvely. Note tht ll the functons e MS (λ α,mn ), λ α,mn > 0, shown n Fgure 12 hve locl mnmzer except the one shown n Fgure 12(d) tht s reltve to the α = 1-network. Remnd tht the α = 1-network corresponds to the modfed rectngulr lttce. The exstence of locl mnmzer ndctes the trnston from the free flow regme to the congested regme. In fct, when λ s suffcently lrge the functon z ( ) t t M ( ) decreses monotonclly nd behves lke so tht q ncresng λ α,mn we cn mprove the mtchng of the smulted dt wth the curve gven n (12). et [λ α,mn, λ α,mx ] be the ntervl where the functon z α (λ), s functon of λ behves s negtve power. For α = 0, α = 0.1, = 2, 3,, 7, α = 0.9, we cn observe tht the trnston from free flow regme to congested regme (ccordng wth Crteron 2) s locted n the ntervl [ λ α,mn, λ α,mx ] when we choose,mn to be the frst locl mnmzer of e MS (see Fgure 12). Tble 3 shows the prmeters obtned usng the clbrton procedure descrbed bove nd choosng λ α,mn =,mn. We mprove the choce of λ α,mn, λ α,mx mde prevously to obtn n ntervl of crt- * * clty [ mn, mx ] smller thn the ntervl [λ α,mn, λ α,mx ]. * * A possble choce of the ntervl [ mn, mx ] s * * mn λα,mn nd mx, where s the lrgest v- lue of λ such tht we hve MS e e MS,mn, α = 0, α = 0.1, = 2, 3,, 7, α = 0.9. Fnlly Fgure 12(d) shows tht when α = 1 the phse trnston does not occur for the vlues of λ consdered n the smultons. In fct, n the modfed rectngulr lttce network (.e. α = 1), the congeston phenomenon tkes plce for greter vlues of λ thn those consdered n the numercl smulton presented here. Tht s the modfed rectngulr lttce hs free flow regon greter thn tht of the remnng networks consdered. Ths desrble property s obtned t the prce of delvery tme n the free flow regme pproxmtely seven tmes greter thn the delvery tme of the other networks consdered. Note tht Fgure 8 shows tht the delvery tme of the modfed rectngulr lttce network s constnt n the ntervl of λ consdered nd s bgger thn the bggest delvery tme of the remnng α-networks. Tble 3 shows tht the exponent q α obtned from the lest squres procedure tkes substntlly three dfferent vlues: q α for α = 0, q α 0.6 for α = 0.2, 0.3, 0.4 nd q α 0.64 for α = 0.5, 0.6, 0.7, 0.9. These numercl fndngs nd Tble 4 suggest tht the exponent q α depends from the mxmum vlue of the weghted betweenness centrlty, b,mx w, wth respect to the node ndex v. It wll be nterestng to nvestgte more deeply the exstence nd eventully the menng of ths relton. We do not pursue ths nvestgton n ths pper. Our nlyss shows tht when the lod λ ncreses the trffc flow on the network frst tres to contnue to delver ll the pckets ncresng the delvery tme of ech pcket nd then strts to leve some pckets undelvered. In fct Fgure 8 shows tht even when the ncrement of the men trvel tme t ( M ), α [0, 1) s relevnt (see, for exmple, the ntervl λ [150, 300]) the correspondng vlue of N t, α [0, 1) s pproxmtely zero. Tht s, for α [0, 1) the α-network strts to undelver pckets only when the men trvel tme s becomng t lest twce the (pproxmte) delvery tme t α of the free flow regme. The behvour of the modfed rectngulr lttce (α = 1) s dfferent from the behvour of the remnng networks. Ths s probbly due to the fct tht when we go from α [0, 1) to α = 1 the rch club vnshes. As consequence for α = 1 the delvery tme remns substntlly unchnged even for lrge vlues of λ, but there s sgnfctve quntty of undelvered pckets. In fct, when α [0, 1), due to the choce of the routng strtegy, the mjorty of