Optimization of network mesh topologies and link capacities for congestion relief

Transcription

1 Optmzaton of networ mesh topologes and ln capactes for congeston relef D. de Vllers * J.M. Hattngh School of Computer-, Statstcal- and Mathematcal Scences Potchefstroom Unversty for CHE * E-mal: [email protected] Abstract - Networ Desgn problems usually nclude the selecton of nodes and arcs from lsts of potental sets to accomplsh certan desrable propertes. Foremost s often the capablty to accommodate the flow demands at a reasonable cost. In many applcatons t s also mperatve to have bult-n relablty or survvablty of the networ. Delays of traffc are undesrable snce t affects QoS to clents of the networ. It s seldom possble to start a desgn for a new networ and have the luxury of desgnng the topology as well as the optmal flow (routng). In ths paper we consder the constructon of an optmzaton system that may help n the plannng of mesh topologes and ln capactes to avod costly desgns and congeston (or to gve advce on congeston relef n exstng networs.) I. BACKGROUND In desgnng mesh networs, topology optmzaton s a dffcult problem and even component sub-problems le routng capacty assgnment have been shown to be dffcult. Integer programmng technques have been used to produce optmal solutons or tght bounds on optmal solutons only for problems of modest sze (a few dozen nodes) (see Gavsh []). For larger problems many researchers have opted for heurstc approaches. In ths paper we use exact nteger programmng methods to solve smaller (sub) problems and have managed to create a decson support system that can be used as an ad n analyzng strateges for topologcal decsons and routng plans under varous condtons le arc avalablty and traffc demands. II. NETWORK DESIGNS A. Some Classc models Consder a model that ncorporates two types of costs. The desgn cost and flow cost f an arc s ntroduced. The requrement s to route (multple) commodtes on the networ. Ths wor forms part of the research done at PU for CHE and UNW wthn the TELKOM CoE research programme, funded by TELKOM, GRINTEK TELECOM and THRIP. It also forms part of a Masters dssertaton n preparaton by the frst author. If the desgn cost of an arc s denoted by f, t s thus the obectve to fnd the topology and routng that mnmzes the total cost of desgn and flow on the networ. The potental node and lns are presumed to be gven by a drected networ G = (N,A) Ths type of model arses n many applcaton contexts e.g. the desgn of telecommuncaton or computer networs, load plannng n the trucng ndustry and the desgn of producton schedules. B. Uncapactated Networ Desgn In the basc uncapactated form, t s assumed that when an arc ( s ntroduced to form part of the desgn, t has suffcent capacty to route all of the flow by all commodtes. We further assume that we need to route multple commodtes (K n number) on the networ and that each commodty has (as a frst smplfcaton) a sngle source node s and a sngle destnaton d. f x denotes the vector of flows of commodty on the networ we wll assume that the elements of x vz. x denote the fracton of the requred flow commodty to be routed from s to d that flows on arc (. We let c denote the cost vector for, whch we scale to reflect the defnton of x. Also we let y be a zero-one vector ndcatng whether arc ( s selected as part of the networ desgn. In mathematcal terms the problem becomes Mnmze c x + fy K subectto x x { : ( { :(, ) x y x = y = or f = s f = d forall N, =,,..., K, otherwse =,,... K, A, and all =,,... K, A (B) (B) (B) (B) (B)

2 The constrant B prevents flow on an arc that s not selected and s often called forcng constrants. If the forcng constrants are removed from, the resultng problem decomposes nto a number of ndependent shortest path problems. We have consdered ths basc model and varous extensons of t n ths paper. C. Capactated networ models A verson of capactated models can be obtaned by alterng the forcng constrants (B) to ncorporate bounds on the flow along arc ( for every commodty by replacng B wth l y x u y for all( =,,... K (C) D. Models wth dscrete flow cost alternatves In many real lfe stuatons the cost of flow along an arc ( s gven by a dscrete functon and not a contnuous one snce addtonal capacty has to be nstalled f flow exceeds a certan level. Ths model s ndcated n cases where certan faclty lnes (e.g. T and T) can be ntroduced at a cost (desgn cost) to enable the networ to handle the ncreased flow more economcally. In papers by Leung et al [] and Magnant et al [] such a desgn problem s dscussed. These models arse n problems such as the followng: In desgnng telecommuncaton networs, we would le to nstall suffcent capacty to carry requred traffc (telephone calls, data transmssons) smultaneously between varous source-sn locatons. Suppose that (s,d ) for K denote K pars of source-sn locatons and r denotes the number of messages sent from the source s to sn t. We can nstall ether of two dfferent types of facltes on each ln of the transmsson networ, so-called T and T lnes. Each T lne can carry unt of message and each T lne can carry unts of messages; nstallng a T lne on arc ( ncurs a cost of a and nstallng a T lne on arc ( ncurs a cost of b. Once we have nstalled the lnes we ncur no addtonal costs n sendng flow on them. Ths problem arses n practce because companes wth large telecommuncaton requrements mght be able to lease lnes more cost-effectvely than payng publc tarffs. E. Model consdered for emprcal wor and system (Netdesgner) development To create a platform for a system that has the capablty that enables one to study mesh topologes and ln capactes for congeston relef we consdered the followng gudelnes: Dscrete alternatves for ln selecton (T,T,T) each wth fxed desgn costs and dfferent capactes and flow costs. Capactes on ln flows for each ln type T,T and T A desgn cost g assocated wth the development of a node. The model used n the system can be represented by Mnmze s.t ( c x + c x + c x ) + f y + fy + f y + K g z N x x { :( A} p= { :( p= ) f = s = f = d N, =,,... K Otherwse l p y p x p u p y p p =,,; y p E. z p= { :( x p for y p p {,} {,} ( =,,..., K N p =,,; ( A; =,,... K for p =,,;( A z for N where E s the maxmumoutdegree for all nodes. In ths model the notaton n II B s extended to ncorporate capactes, the dfferent lns T, T and T and node desgn decsons. p III. SYSTEM DEVELOPMENT ASPECTS A. Software development aspects The software was developed n Qt a multplatform C++ applcaton development framewor. One source runs natvely on four dfferent platforms (Wndows, Unx/Lnux, Mac OS X, embedded Lnux. The software can be ported to multple platforms wth a smple recomple. The system outputs s a plan text MIP problem that can be transferred to a machne capable of solvng mxed nteger problems.

3 B. Netdesgner system specfcatons and bult-n capabltes The networ desgner s propertes and functons are descrbed below Add Node / Ln Edt Node/Ln Cost Enable/Dsable Node or Ln Increase or Decrease Ln Constrants Alter requred flow of commodty No Other Alteratons Requred? Yes Choose Alteraton Yes Flow Chart of networ Desgner System Start Open or create networ Construct MILP Solve MILP (CPLEX) Further Expermentaton? No Save & Ext Vew/Prnt/Save Networ Problem Vew /Prnt MILP Vew MILP Soluton Vew/Prnt/ Save Networ Soluton Graph Vew/ Prnt Networ Soluton Table Fgure Flow chart of networ desgner system ) Open or Create a Networ An exstng networ can be opened and altered or a new networ can be constructed. In the constructon phase the number of commodtes that wll be present n the networ as well as the constructon (desgn) cost of the commodtes must be specfed. ) Addng a Potental node A new potental (transhpment) node can be added to or removed from the networ. A node may be assgned a desgn cost or the desgn cost can be set equal to zero. ) Addng a Potental Ln Potental lns onng node pars n the networ may also be added to or removed from the networ desgn. Three types of potental lns may be ncluded. These potental lns can be assgned dfferent desgn and flow costs as well as dfferent upper and lower bounds. The desgn cost s the cost assocated for establshng the type of ln between the node pars and the flow for a specfc commodty can also be specfed for the ln. If no desgn or flow cost s needed n the desgn t can be set equal to zero. ) Edtng Nodes The desgn cost of a node may be edted or a node may be dsabled (excluded from the potental confguraton). ) Edt Potental lns The desgn and flow cost of a specfc ln may be altered as well as the ln capacty (upper and lower bounds on flow). Potental ln may also be dsabled (excluded from potental confguraton f one wshes to study the effect of a ln falure) ) Edt Capactes The flow requrement of a specfc commodty may be altered; ether ncreased or decreased to study the effect on the resultng networ and assocated (mnmum) cost. ) Construct MILP The current networ desgn problem s modelled wth mathematcal modellng technques as an optmzaton problem (MILP), and the MILP may be vewed, prnted and solved as ndcated below. ) Solve MILP The MILP s solved by means of a software product of ILOG vz CPLEX. The soluton to the problem may be vewed n text or as a graphcal representaton. In the graphcal representaton t s possble to vew only a certan ln type or all ln types ncluded n the soluton. It s also possble to save and prnt graphcal representatons. A table showng the lns to be ncluded n the desgn as well as the ln utlzaton s also avalable. ) Save The networ layout can be saved to a fle for later use. ) Networ Alteraton Alteratons to the networ desgn may be done va the edt optons for example dsablng a ln or node ncluded n the optmal answer or ncreasng the requred flow of a certan commodty through the networ. IV. ILLUSTRATION OF NETWORK DESIGNER In the followng llustraton only two ln types where used namely T (wth upper bound of. unts) and T (wth an upper bound of. unts) on flow respectvely. The desgn cost and flow cost of each commodty n the networ s represented n the followng fgure. We assume that unt of flow for commodty s to be routed from node to node and smlarly for commodty from node to. T(f,c,c T(f,c,c T (,,.) T (,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,,.) T(,.,.) T(,.,,) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.) T(,,.) T(,.,.) T(,.,.) T(,.,.) T(,.,.)

4 Fgure Potental drected networ wth T and T lns The optmal soluton of the networ wth an obectve value of. s represented n the followng fgure T(x,x T(x,x T(,) T(,) Fgure Optmal networ soluton T(,) T(.,.) T(,) T(,) If node s dsabled the optmal cost ncreases to. and the resultng networ s gven by the followng T(x,x T(x,x T(.,.) T(.,.) Fgure Networ soluton wth node dsabled The followng fgure shows the effect f the T ln between node and n the optmal soluton (gven n fgure ) s dsabled. The optmal cost ncreases to. T(x,x T(x,x Fgure Networ soluton wth node T ln between node and dsabled Fgure shows the effect on cost of ncreasng commodty s flow requrement whle eepng commodty flow requrement constant. Each pont on ths graph s a result of a model optmsaton. Fgure dsplays the number of lns (contanng T and T lns) obtaned n the relevant optmsaton. Total mnmum cost Effect on cost of ncreasng commodty flow requrement Commodty Increase Fgure Effect on cost of ncreasng commodty flow requrement Total Number of Lns (T and T) Effect on total number of dfferent lns at ncreased commodty flow requrement Commodty Increase Fgure Effect on total number of dfferent lns at ncreased commodty flow requrement If we ncrease the upper lmt on the flow of the T lnes from. to. and study the effect on the ncrease n the flow requrement of commodty we get the followng result dsplayed n fgure. Effect of ncreasng commodty flow requrement T(.,.) T(.,.) Total mnmum cost Commodty Increase

5 Fgure Effect on cost of ncreasng commodty flow requrement when capacty of T lne s ncreased to.. From these graphs t can be seen that t may happen that a desgn of equal cost can accommodate more flow of a commodty and could thus be preferred. See e.g. a % ncrease of commodty flow from to n fgure. V. CONCLUSIONS The results we obtaned wth the expermental system ndcates that such a decson support system, can be useful n mesh desgn and dmensonng of certan types of networs. Some flow congeston and relef measures can also be nvestgated by usng the system. VI. FUTURE WORK The system developed has lmtatons and relaxng some of the constrants n the current verson may contrbute to mae the system more applcable to real lfe problems. Ths, however always leads to a trade off between realsm and the feasblty of fndng solutons. One such constrant s that the lns are not consdered bdrectonal. Another s that all nodes are not consdered to be generators or recpents of commodty flow. These aspects certanly would mert further nvestgaton. VII. REFERENCES [] Gavsh, B.: Bacbone networ desgn tools wth economc tradeoffs, ORSA Journal on Computng, :-,. [] Ahua, R.K, Magnant T.L., Orln, J.B.: Networ flows, [] Leung, J.,T.L. Magnant and V. Snghal.. Routng n pont to pont delvery systems. Transportaton Scence, -. [] Magnant T.L., Mrchandan P., Vachan R.,. Modelng and solvng the two-faclty capactated networ loadng problem. Operatons Research Vol :-