parametrc Lnear Programmng Mode Descrbng Bandwdth Sharng Poces for BR Traffc I. Moschoos, M. Logothets and G. Kokknaks Wre ommuncatons Laboratory, Dept. of Eectrca & omputer Engneerng, Unversty of Patras, 265 00 Patras, Greece. TEL: +30 61 996 166, +30 61 991722, FX: +30 61 991855, E-ma: {moschoos, m-ogo, gkokkn}@wc.ee.upatras.gr bstract We consder the far bandwdth aocaton probem of BR cas by a onnecton dmsson ontroer () at ca setup. Its genera souton s gven by appyng the max-mn farness aocaton pocy. Extensons of ths pocy take nto account the Mnmum e Rate (MR) and the Peak e Rate (PR) traffc descrpton parameters, whch are essenta for BR. These extensons are dstngushed accordng to the exstence or non-exstence of weghts n the pocy. The weghts may be defned ndependent of MR or dependent on MR. In ths paper frsty we present a near programmng mode, whch descrbes n a parametrc way both weghted and unweghted bandwdth aocaton poces for BR servces. Because of the parametrc nature of the LP mode, t can be easy extended n order to take nto account varous bandwdth aocaton poces. Secondy, we propose an unweghted far bandwdth aocaton pocy n whch BR cas are grouped n categores, accordng to ther traffc descrpton parameters and the max-mn farness aocaton pocy s apped among the groups whe the MR and PR can be taken nto account for each group. Thrdy we propose a weght-based far bandwdth aocaton pocy n whch the number of nks of an BR connecton stands for the weght. 1. Introducton The vaabe Bt Rate (BR) servce, defned by the TM Forum [1] s used to support non-rea tme data appcatons. It can guarantee a mnmum amount of bandwdth, named Mnmum e Rate (MR), and may be mted to a specfed amount of bandwdth, named Peak e Rate (PR). The BR servce uses the amount of bandwdth eft over after handng the guaranteed Quaty of Servce (QoS) sources, onstant Bt Rate (BR) and Varabe Bt Rate (VBR). Ths bandwdth, named avaabe bandwdth needs to be aocated n a far way among competng BR connectons. Farness may be examned on a per nk bass (oca farness) or aong a path (goba farness). To acheve a gobay far aocaton each ca shoud use the mnmum ocay far aocaton aong ts path [2]. In ths
I. Moschoos, M. Logothets, G. Kokknaks paper we refer to the oca far share. The most popuar aocaton pocy of oca farness s that of Max-Mn Farness (MMF), ntay adapted by the TM Forum to aocate avaabe network bandwdth among BR connectons. bandwdth aocaton s sad to be max-mn far f the bandwdth aocated to a connecton cannot be ncreased wthout decreasng at the same tme that of a connecton havng a smaer or equa aocaton [3]. The scope of MMF s to maxmze the avaabe bandwdth of the connecton wth the smaest bandwdth. If there s st avaabe bandwdth, MMF contnues by maxmzng the bandwdth of the connecton wth the second smaest bandwdth and so forth [3]. The man drawback of the MMF s that t doesn t take nto account the MR and PR traffc descrpton parameters, whch are essenta for BR. Furthermore MMF treats each connecton wth the same prorty (weght) whch means that t tres, whenever ths s possbe, to dstrbute bandwdth equay among BR connectons. Ths dea reduces the opton of havng groups of connectons wth hgher and ower prorty. Extensons of MMF take nto account MR, PR and they are further dstngushed accordng to the exstence or non-exstence of weghts [4, 5, 6, 7,8]. In [4] an extenson of MMF, whch supports MR, has been presented. In [5] a generazaton of MMF, named Generazed Max-Mn (GMM) has been proposed whch supports MR, PR for each BR connecton whe n [6] GMM has been extended to ncude a weght for each connecton. Ths extenson caed Weght Proportona Max-Mn (WPMM) ensures each connecton wth each mnmum rate and shares the remanng bandwdth among users dependng on each user s weght. The weght for each connecton s ether proportona to the connecton s MR or proportona to the dfference between PR and MR. revew paper of the ast two generazatons of MMF can be found n [7]. In [8] nstead of havng one weght, two sets of weghts may be used. The frst one may be the same to those mentoned above whe the second one s dynamcay adjusted so as to take nto account factors such as a change n the number of nks used by a connecton, the round trp deay of Resource Management ces (RM ces) etc. The aforementoned works examne networks supportng a fxed number of connectons. In a case of a network n whch the number of users changes over tme, the nterested reader may resort to [9] and the references theren. In [9] the authors examne the stabty of a network mode, whch ncudes stochastc arrvas and departures, under severa bandwdth aocaton poces. We assume networks supportng a fxed number of connectons. In ths paper, we start wth a bref revew of the MMF pocy. We contnue by proposng a near programmng (LP) mode wth whch we can mpement the MMF pocy. Based on ths LP mode we extend the MMF pocy to the MMBR pocy, whch takes nto account the MR, PR traffc descrpton parameters. We contnue by proposng two new poces, whch aso ncude the MR, PR and can be weghted or not. In the frst pocy, named MMF per BR connecton pocy, BR cas are grouped n categores accordng to ther traffc descrpton parameters and the max-mn farness aocaton pocy s apped among them. Ths pocy may aso ncude weghts. In the second one, named MMF nk based pocy, the man characterstc s that the number of nks that consttute an BR connecton stands for the weght. The dea behnd ths pocy s to favor BR connectons that have ong paths n an TM network. It can be seen [11] that BR connectons wth ong paths are more key to have ther rates reduced than other BR connectons, whch have short paths, because they face hgh feedback deays and therefore they respond sowy to acqure bandwdth. The remander of ths paper s organzed as foows. In secton 2, we revew the MMF pocy and present an eght-step agorthm wth whch MMF may be mpemented. In secton 3, frsty we present the LP mode, secondy we extend the MMF pocy to the MMBR pocy and thrdy we present the MMF per BR connecton and the MMF nk based poces. In secton 4, we present numerca resuts and compare the aforementoned poces together wth the GMM pocy. Fnay n secton 5 we concude.
parametrc Lnear Programmng Mode 2. The Max-Mn Farness Pocy The foowng mathematca notaton s used n order to express the MMF pocy. We assume that P s the set of servces whch use a network N. network N conssts of a number of swtches connected through a set of nks L. Each servce p P has a fxed path n the network and uses one or more nks of L. We denote by P the set of servces whch use nk L. Lettng r the vector of aocated rates, where r = (r 1, r 2,, r p ) and r p the aocated rate of servce p, then the tota fow through a nk of the network s gven by: F =. r p p P If we denote wth the capacty of nk then we have the foowng constrants on the vector r : a) r p 0 for a p P b) F for a L If a vector r satsfes these constrants then t s feasbe. vector r w be max-mn far f t s feasbe and for each p P, r p cannot be ncreased whe mantanng feasbty wthout decreasng r z for some servce z for whch r z r p [3]. Gven a feasbe rate vector r, we say that a nk s a botteneck nk wth respect to r for a servce p usng f: a) F = b) r p r z for a servces z usng nk. It can be proved [3] that a feasbe rate vector r s max-mn far f and ony f each servce has a botteneck wth respect to r. The 4nk - 5node network shown beow w be used to ustrate MMF pocy and the noton of a botteneck nk. Ths network has been adopted by the TM Forum [12] for the comparson of the performance of dfferent farness crtera. F Β Ε Α 50 150 150 Mbps Mbps Mbps 100 Mbps Β D Ε node 1 node 2 node 3 node 4 node 5 D F Α Fgure 1: generc farness confguraton s t can be seen there are sx dfferent servces (,B,,D,E and F) that use the network. There are three cas of that use nks 1-2, 2-3 and 3-4, three cas of B that use nks 2-3, 3-4 and 4-5, three cas of that use nk 3-4, sx cas of D that use nk 1-2, sx cas of E that use nk 4-5 and two cas of F that use nk 2-3. n MMF souton to the aocaton of network bandwdth w gve a rate of 5.55 Mbps to each ca of servces, D, a rate of 11.1 Mbps to each ca of servces B, E, a rate of 33.3 Mbps to each ca of servce and a rate of 50 Mbps to each ca of servce F. Based on the noton of a botteneck nk, the botteneck nk of servce s 1-2, snce: a) F 12 = 12 b) r = r D and not the nks 2-3 or 3-4 snce: a) F 23 = 23 but b) r B > r, r F > r, for nk 2-3 and
I. Moschoos, M. Logothets, G. Kokknaks a) F 34 = 34 but b) r > r, for nk 3-4. Smary, the botteneck nk of servce D s agan 1-2 snce: a) F 12 = 12 b) r D = r, the botteneck nk of servce F s 2-3 snce: a) F 23 = 23 b) r F > r, r F >r B, the botteneck nk of servce B s 4-5 snce: a) F 45 = 45 b) r B = r E, the botteneck nk of servce s 3-4 snce: a) F 34 = 34 b) r > r, r >r B and fnay, the botteneck nk of servce E s 4-5 snce: a) F 45 = 45 b) r E = r B It can be proved [3] that a feasbe rate vector r s max-mn far f and ony f each servce has a botteneck nk wth respect to r. The next step s to present an agorthm found n [3] wth whch we can compute a max-mn far rate vector. The dea of the agorthm s to start wth an a zero rate vector and to ncrease the rates on a servces together unt F = for one or more nks. t ths pont, each servce usng a saturated nk (F = ) has the same rate wth every other servce usng that nk. Thus, these saturated nks serve as botteneck nks for a servces usng them. t the next step of the agorthm, a servces not usng the saturated nks are ncremented equay n rate unt one or more new nks become saturated. Note that the servces usng the prevousy saturated nks mght aso be usng these newy saturated nks at a ower rate. The newy saturated nks serve as botteneck nks for those servces that pass through them but do not use the prevousy saturated nks. The agorthm contnues from step to step, aways ncrementng a servces not passng through any saturated nks; when a servces pass through at east one saturated nk the agorthm stops. To contnue, we gve the foowng mathematca notaton, whch s necessary n order to present the steps of the MMF agorthm. We assume that L denotes the set of nks not saturated at the begnnng of teraton, P s the set of servces not passng through any saturated nk at the begnnng of teraton, n denotes the number of servces, whch are n P, that use nk (.e. n s the number of servces that w share nk s yet unused capacty) and r~ denotes the ncrement of rate added to a of the servces n P at teraton. ccordng to the above mathematca notaton the MMF agorthm can be presented n eght steps: 0 0 Inta condtons: =1, F = 0, rp = 0, P 1 = P, L 1 = L Step 1: n : the number of servces p P usng nk 1 F Step 2: r~ = mn L n 1 rp + r~ for p P Step 3: r p = 1 rp otherwse Step 4: = F rp (for those p, whch use nk ) 1 Step 5: L { + = F > 0} L +1 denotes the set of nks not saturated at the begnnng of teraton +1 +1 P = p p crosses any nk L +1 and does not cross any saturated nk Step 6: { }
parametrc Lnear Programmng Mode Step 7: =+1 Step 8: If P s empty then stop; ese return to step 1 Note: In step 3 of teraton the servce rate of each servce p P receves a mnmum ncrement of rate (cacuated n step 2). If servce p s not ncuded n P (.e. servce p aready uses a saturated nk from teraton -1) then ts rate remans the same. Havng cacuated the tota fow F (step 4 of teraton ) we then determne the new sets L +1, P +1 at steps 5,6 respectvey by removng a the saturated nks, the servces that use them and the capacty assocated wth servces. If we appy the aforementoned agorthm to the network of fg.1 then after four teratons we w have the foowng resuts: The cas of servces, D w be aocated a rate of 5.55 Mbps, the cas of servces B, E a rate of 11.1 Mbps, the cas of servce a rate of 33.3 Mbps and the cas of servce F a rate of 50 Mbps. The noton of farness can be expaned by observng that servces, D, that have the same botteneck nk (1-2) and servces B, E that aso have the same botteneck nk (4-5) take an equa amount of bandwdth (5.55 and 11.1 Mbps respectvey). s t has aready been shown the computaton of a max-mn far rate vector depends on an eght step agorthm whose teratons are equa to the number of the nks of the network. dfferent approach for the cacuaton of a max-mn far rate vector s proposed n secton 3. In that secton we propose the LP mode, whch can descrbe the MMF pocy n a parametrc way. Wth the ad of the LP mode we present n secton 3 extensons of the MMF pocy. More precsey, we propose the MMBR pocy, the MMF per BR connecton pocy and the MMF nk based pocy whose man characterstc s that they ncude the MR and PR traffc descrpton parameters. 3. The Lnear Programmng mode and the extensons of the MMF Pocy The am of the near programmng mode s to maxmze the functon: w 1 r 1 +w 2 r 2 + +w p r p or w pr p p P where w p s the weght (or prorty) assocated wth each connecton p P, under the genera constrants: 1. n p P p r p, for a L 1 2 L 2. r mn,,...,, for a P cp cp cp where the parameter cp denotes the chosen pocy. Note: The frst set of constrants s named set of network constrants, whe the second one s named set of apped pocy constrants. In our case the chosen pocy s the MMF, therefore the second set of constrants takes the form: 1 2 L r mn,,..., n n n n n n P P P Note: For the MMF pocy, w p = 1 for every p P. The MMBR pocy ncudes the MR, PR traffc descrpton parameters and can be descrbed by ntroducng the foowng notaton:
I. Moschoos, M. Logothets, G. Kokknaks We denote wth MR p and PR p the mnmum and the maxmum rate for each servce p P. We assume that the sum of the mnmum rates of the servces, whch use nk, does not exceed the nk s capacty,.e. MR, for every L. p P p In order to compute a MMBR far rate vector we shoud ncude a thrd set of constrants, named set of traffc parameters constrants, whose form s the foowng: MRp rp PRp, for a p P. The MMF per BR connecton pocy may be descrbed wth the ad of the foowng fgure: Tw o BR onnectons consstng of 4 and 2 V s, respectvey VP Lnk VP Lnk a) M M F a n d M M B R p o c e s V P onnecton / B R onnecton V rtua hannes Vrtua Paths b) M M F per BR connecton pocy Fgure 2: The noton of MMF, MMBR and MMF per BR connecton poces s we can see, we have a 2nk - 3node network used by two BR connectons. The frst one conssts of four Vrtua onnectons (Vs) whe the second one conssts of two Vs. The dea behnd the MMF and the MMBR poces can be ustrated by fg. 2a. s one can see the capacty of the nks s dvded nto sx parts (equa to the number of Vs). Four out of these sx parts are gven to the frst BR connecton whe two out of the sx parts are gven to the second BR connecton. In contrast to the above, n the MMF per BR connecton the capacty of the nks s dvded nto two parts (equa agan to the number of BR connectons) and then the MMF pocy s apped. MMF per BR connecton far rate vector can be cacuated by usng the aforementoned three set of constrants. The ony change s n the set of apped pocy constrants, whch takes the form: 1 2 L r mn,,..., n n n P P P Note: The MMF per BR connecton can be weghted or not.
parametrc Lnear Programmng Mode The MMF nk based pocy can be ustrated wth the ad of the foowng fgure: B MMF nk-based D E MMF nk-based D E B Fgure 3: MMF nk based pocy We have agan a 2nk 3node network used by fve BR connectons. The BR connectons B, traverse the frst nk whe the BR connectons D, E traverse the second nk. Ony the BR connecton traverses both nks. s one can see the capacty of each nk s dvded nto two parts (equa to the number of nks). One part s gven to the BR connecton and the other part s gven to the connectons B, (frst nk) and the connectons D, E (second nk). The reason for dong so s n order to favor BR connecton that has a onger path and may face hgher feedback deays from the other connectons, whch may resut n a sow response to acqure bandwdth. fter ths dvson the MMF pocy s apped. MMFD nk based far rate vector can be cacuated by changng the set of apped pocy constrants, whch w have the form: r mn n 1 2 P, n P,..., n L P The MMF nk based pocy can aso be weghted or not. The weght can be anaogous to the number of nks used by each BR connecton. The foowng fgure ustrates the LP mode, that s the three sets of constrants and the objectve functon that s to be maxmzed. Network onstrants T raffc Param eter onstrants (MR-PR) pped P ocy U n w egh ted o r W egh ted S U M o f V rates M X Fgure 4: The LP mode
I. Moschoos, M. Logothets, G. Kokknaks The advantage of the LP mode s that t can ncude varous bandwdth aocaton poces by changng the set of apped pocy constrants. 4. Numerca resuts In ths secton we use the LP mode to compare the aforementoned bandwdth aocaton poces together wth the GMM pocy. In order to do so we use two dfferent generc farness network confguratons. The frst one, shown n fgure 5, s a 2nk- 3node network used by four BR connectons. D B Node 1 1 Mbps Node 2 1 Mbps Node 3 D B Fgure 5: Test-bed network 1 The second one s that of fgure 1. The foowng tabe presents the traffc parameters for test-bed network 1 [7]. Tabe 1: MR, PR traffc parameters for test-bed network 1. BR connecton Number of Vs MR (Mbps) PR (Mbps) 1 0.40 1.00 B 1 0.10 0.25 1 0.05 0.50 D 1 0.10 1.00 Tabe 2 presents the bandwdth aocaton (n Mbps) for the four BR connectons. Tabe 2: Bandwdth aocaton of the BR connectons of test-bed network 1. MMBR MMF nk based MMF nk weght GMM No Farness 0.40 0.40 0.65 0.40 0.40 B 0.25 0.25 0.35 0.25 0.10 0.35 0.35 0.25 0.35 0.50 D 0.60 0.60 0.40 0.60 0.60
parametrc Lnear Programmng Mode The coumn named MMF nk weght presents the resuts obtaned by the MMF nk-based pocy when we appy an extra weght dependng on the number of nks that each BR connecton traverses. For ths exampe the functon that s to be maxmzed takes the form: 2r +r B +r +r D max snce BR connecton traverses two nks. The coumn named GMM presents the resuts found n [7] whe the coumn named No Farness presents the resuts when no pocy s apped,.e. when we don t take nto consderaton the set of apped pocy constrants. Note: The MMF per BR connecton pocy w gve the same resuts as the MMBR pocy snce the BR connectons, B,, D consst of one V. Tabe 3 presents a comparson between the varous poces concernng the tota V bandwdth, whe tabe 4 compares the varous poces as far as the farness s concerned. Tabe 3: arred traffc- Test-bed network 1 Pocy arred Traffc (Index: Tota V bandwdth) 1 No Farness 1.60 2 MMF nk-based 1.60 3 MMBR 1.60 4 MMF nk-weght 1.60 5 GMM 1.60 Tabe 4: Farness - Test-bed network 1 Pocy Farness (Index: Standard devaton n the congested nk 1-2) 1 No Farness 0.06 2 MMF nk-based 0.06 3 MMBR 0.06 4 MMF nk-weght 0.06 5 GMM 0.17 S.D = where, r M = The farness ndex s the standard devaton (S.D) n the congested nk 1-2 cacuated by the formua: 2 2 ( r M ) + ( r M ) + ( r M ) + rb + r 3 B 3 2 ccordng to the resuts presented n the ast two tabes, t seems that there s no dfference between the apped bandwdth aocaton poces. The ony dfference s when no pocy s apped (.e. there s no set of apped pocy constrants) whch resuts n the worst farness ndex. The man reason for not obtanng dfferent resuts has to do ony wth the seected vaues of the traffc parameters, whch are the same as those, presented n [7].
I. Moschoos, M. Logothets, G. Kokknaks The next generc farness network confguraton s that of fgure 1 (test-bed network 2). Tabe 5 presents the traffc parameters for the test-bed network 2. Tabe 5: MR, PR traffc parameters for test-bed network 2 BR connecton Number of Vs MR (Mbps) PR (Mbps) 3 2 12 B 3 4 18 3 12 20 D 6 1 5 E 6 3 14 F 2 10 100 Tabe 6 presents the bandwdth aocaton for the sx BR connectons. Tabe 6: Bandwdth aocaton of the BR connectons of test-bed network 2. MMBR MMF per BR MMF nk based MMF nk weght GMM No Farness connecton 5.55 8.33 4.16 12.00 6.66 2.00 B 11.11 16.66 7.14 18.00 11.11 4.00 20.00 20.00 20.00 20.00 20.00 20.00 D 5.00 4.16 5.00 2.33 5.00 5.00 E 11.11 8.33 13.09 7.67 11.00 14.00 F 50.00 37.50 58.03 30.00 48.37 66.00 Tabe 7 presents a comparson between the varous poces concernng the tota V bandwdth, whe tabe 8 shows a comparson between the varous poces as far as the farness s concerned. Tabe 7: arred traffc- Test-bed network 2. Pocy arred Traffc (Index: Tota V bandwdth) 1 No farness 111.00 2 MMF 107.44 nk-based 3 MMBR 102.77 4 GMM 102.25 5 MMF per BR connecton 95.00 6 MMF nk-weght 90.00
parametrc Lnear Programmng Mode Tabe 8: Farness- Test-bed network 2. Pocy Farness (Index: Standard devaton n the congested nk 2-3) 1 MMF nk-weght 7.48 2 MMF per BR connecton 12.26 3 GMM 18.70 4 MMBR 19.77 5 MMF nk- based 24.72 6 No farness 29.70 When no pocy s apped the tota V bandwdth s hgher (111.00 Mbps out of 150 Mbps) but based on the standard devaton n the congested nk 2-3 we see that the resuts we get as far as farness s concerned are poor. On the other hand n the case of MMF nk weght we have the best resuts as far as farness s concerned. By observng the ast two tabes t s apparent that the rankng n tabe 7 s reversed n tabe 8. 5. oncuson We address the far bandwdth aocaton probem for BR cas and presented ts souton, based on the MMF pocy. Snce the MMF pocy does not take nto account the traffc descrpton parameters of BR cas t has been extended to cover them and therefore become meanngfu for BR servces. We show that the MMF pocy (extended or not) can be ceary expressed as a near programmng mode. The varous (weghted or unweghted) bandwdth aocaton poces for BR servces can be descrbed n a parametrc way n the LP mode. Besdes we propose two bandwdth aocaton poces, namey the MMF per BR connecton pocy and the MMF nk based pocy. The MMF per BR connecton pocy ams at a far bandwdth aocaton among the BR connectons, takng nto account the number of BR connectons. The MMF nk based pocy ams at a far bandwdth aocaton, takng nto account the number of nks used by the BR connectons. We examne the proposed new poces and compare them wth other poces n two test-bed networks. The resuts show that the MMF nk based pocy performs best as far as farness s concerned, when not ony the number of nks used by each BR connecton s taken nto account but aso an extra nk-weght s apped to the objectve functon of the LP mode. cknowedgement The authors woud ke to thank the Research ommttee of the Unversty of Patras, Greece, because ths work was supported by the research programme aratheodory. References [1] TM Forum Technca ommttee, Traffc Management Specfcaton, Verson 4.0, TM Forum ontrbuton, F-TM 96-0056.00, pr 1996.
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