OPTIMIZATION OF FLOWS AND ANALYSIS OF EQUILIBRIA IN TELECOMMUNICATION NETWORKS

OPTIMIZATION OF FLOWS AND ANALYSIS OF EQUILIBRIA IN TELECOMMUNICATION NETWORKS Ciro D Apice (a, Rosanna Manzo (b, Luigi Rarità (c (a, (b, (c Department of Information Engineering and Applied Mathematics, Via Ponte Don Melillo, 84084, Fisciano, Salerno, Italy (a dapice@diima.unisa.it, (b manzo@diima.unisa.it, (c lrarita@unisa.it ABSTRACT The aim of this paper is the analysis of flows on data networks in order to improe traffic conditions. In particular, the attention is focused on optimization techniques and equilibrium solutions for a fluid dynamic model of telecommunication networks. The optimization algorithm allows to redistribute the packets at nodes to aoid congestions. The performance analysis is made through a cost functional measuring the aerage elocity of packets and depending on priority and distribution coefficients at nodes. The study of the equilibrium flows gies an understanding of the asymptotic solution on the whole network and permits the inestigation of security issues when some nodes of the network fail. Here, on the basis of analytical studies, simulatie results are obtained. Simulations confirm that the optimization algorithm gies better performances with respect to other choices of the model parameters. For the equilibrium analysis, an iteratie algorithm is studied in order to determine outflows equilibria as function of the input flows. Keywords: conseration laws, simulation, optimization, equilibria.. INTRODUCTION The aim of this work is to present some numerical results about the analysis of packets flows on data networks in order to understand typical phenomena, such as packets congestion. Starting from a fluiddynamic model, introduced in D Apice 006 and refined in D Apice 008, we hae tested some optimization algorithms (Marigo 006 for the optimal choice of the model parameters and faced, from a numerical point of iew, the problem of identifying equilibrium solutions (Marigo 007. As for the model of data networks, looking at an intermediate time scale, the aerage amount of packets is considered consered, hence the density eolution for each line can be described by a conseration law. The dynamics at nodes, in which many lines intersect, is gien by a routing algorithm, according to which packets are processed by arrial time (FIFO policy and sent to the outgoing lines in order to imize the flux on both incoming and outgoing lines. To determine uniquely the eolution at nodes, some parameters (priority parameters and traffic distribution coefficients hae to be introduced. Such parameters hae been considered in Marigo 006 as controls and a functional, which measures the network efficiency in terms of the packets elocity, has been analysed. The optimization approach is of decentralized type, and consists in the following steps: first, the optimal parameters for simple networks, formed by a single node and eery constant initial data, are ealuated; then, for a complex network, the (locally optimal parameters at eery node, updating the alue of the parameters at eery time instant, are used for testing the global performance on the network. Similar optimization approaches hae already been considered for car traffic networks in Cascone 007 and Cascone 008. For the first step, we focus the attention on a simple network characterized by two incoming lines, two outgoing lines (node of type, a traffic distribution coefficient, α, and a priority parameter, p. Simulations proe that, setting the parameters α and p with the optimal ones, the traffic conditions at the node improe. For the second step, we consider an Oriented Manhattan network, characterized by nodes of type. Three different choices for the traffic distribution coefficients and priority parameters are analyzed: (locally optimal, static random and fixed. The first choice is gien by the optimal alues. By static random parameters, we mean a random choice done at the beginning of the simulation and then kept constant. Finally, the fixed choice is represented by the simulation with fixed priority parameters and distribution coefficients. Simulations proe that the optimal algorithm, with respect to other traffic choices, gies the best performances as for the packets traffic eolutions. The optimization analysis is completed by some numerical results on network equilibrium flows, that are reached in the asymptotic state (see Marigo 007. Equilibria for data networks, which represent solutions constant in time on the whole network, can be used to face security issues in case of node failures. The analysis of equilibria is carried out in the following way. First, we determine the possible types of equilibria at each node, depending on the combination of good and bad alues at the incident lines. Then, we deduce the types for subnetworks, which are considered Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 3

the building blocks of the whole network. In this way, we understand which are all types of equilibria configuration on a generic network. From the knowledge of the type of equilibria configuration, we define a matrix equation that inoles incoming and outgoing flows of the network and describes analytically the total asymptotic solution. The paper is organized as follows. In Section, we introduce the model and gie basic definitions and results for equilibrium solutions. Section 3 is about the optimization algorithm for a junction of type. Section 4 considers equilibria configuration on an Oriented Manhattan Network. The paper ends with simulation results in Section 5.. A FLUID DYNAMIC MODEL FOR TELECOMMUNICATION NETWORKS A network is a finite collection of transmission lines and nodes (or routers. It is modeled by a finite set of interals Ii = [ ai, bi] R, i =,..., L, ai < bi. We assume that the transmission lines are connected by some nodes. We identify each node J with (( i,..., im,( j,..., j n, where the first m -tuple indicates the set of incoming lines and the second n - tuple the set of outgoing ones. Each transmission line is considered incoming at most for one node and outgoing at most for one node. Hence, the complete model is gien by the couple (, IJ, where I = { Ii : i =,..., L} and J are the collections of transmission lines and nodes, respectiely. We set N to be the cardinality of J. We assume that each packet traels on the network with a fixed speed and an assigned final destination. Considering an intermediate time scale and assuming the conseration of packets, the load dynamics for a single line of the network is described by the conseration law (see D Apice 006: t ( ρ 0, ρ + f = ( x where ρ = ρ( tx, [ 0, ρ ], ( tx, R, is the density of packets, ρ is the imal density, f ( ρ = ρ is the flux with the aerage elocity. In what follows, we suppose that: (F the flux f is a strictly concae function, with f(0 = f( ρ = 0. Setting, for simplicity, ρ =, a flux function ensuring (F is: which has a unique imum [0, ρ ]. σ = oer the interal In order to sole the dynamics at nodes, we follow the strategy proposed in D Apice 006 and consider the routing algorithm: (RA Packets are processed by arrial time and are sent to outgoing lines in order to imize the flux on incoming and outgoing lines. The main ingredient for finding an approximate solution of the Cauchy problem at a node with the wae-front tracking algorithm (see Garaello 006 is gien by the solution of a Riemann Problem (RP, which is a Cauchy problem with a constant initial datum on each incoming and outgoing line. Using ϕ as index for the incoming lines and ψ for the outgoing ones, we introduce the following: Definition. A Riemann Soler for a node J is a map RS ρ = ρ, ρ at J a that associates to Riemann data 0 ( ϕ,0 ψ,0 ector ˆ ρ ( ˆ ρϕ, ˆ ρψ incoming line I ϕ is gien by the wae ( ϕ,0, ˆϕ on an outgoing one I ψ by the wae ( ˆ ψ, ψ,0 = so that the solution on an following conditions are required: (C RS ( RS ( ρ0 RS ( ρ0 =. ρ ρ and ρ ρ. The (C On each incoming line, the wae ( ϕ,0, ˆϕ negatie speed, while on each outgoing line ( ˆ ρψ, ρ ψ,0 has positie speed. ρ ρ has Definition. Let τ :0, [ ] [ 0,] be the map such that f( ρ = f( τ ( ρ and τ( ρ =/ ρ if ρ =/ σ. From condition (C, we hae the following: Proposition. Let RS be the Riemann Soler for a node. Assuming an initial data ρ0 = ( ρϕ,0, ρψ,0 and the flux function (, then { ρϕ,0 } τ( ρϕ,0,, if 0 ρϕ,0 <, ˆ ρϕ,, if ρϕ,0, ϕ =,..., m, and (3 ( ( f ρ = ρ ρ. ( Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 33

[0, ], if 0 ρψ,0, ˆ ρ ψ { ρψ,0 } 0, τ( ρψ,0, if < ρψ,0, ψ = m+,..., m+ n. (4 To describe the solution on each transmission line, ˆ γ = f ˆ ρ and it is enough to specify the flux alues ϕ ( ϕ ˆ γψ = f ( ˆ ρψ... Riemann Soler according to rule (RA In order to hae a unique solution of a RP at a node, based on rule (RA, some additional parameters, called, respectiely, priority and traffic distribution parameters hae to be defined. Precisely, we hae m priority parameters ( p, p,..., p m, i= m pi =, p i ] 0,[, for i= the incoming lines and n traffic distribution parameters n ( α, α,..., α n, αi =, α i ] 0,[, for the outgoing ones. Assume ( ρϕ,0, ρ ψ,0 the initial data and use the notation γi f ( ρi,0 =, i = a, b, c, d. Let the imum fluxes, γ ϕ and γ ψ, and the imal through flux Γ at J be defined as follows: γ ϕ ψ γ γϕ, if ρϕ,0 0,, = ϕ =,, m, (5 f ( σ, if ρϕ,0,, f ( σ, if ρψ,0 0,, = ψ = m+,, m+ n. γψ, if ρψ,0,, (6 { in out } Γ= min Γ, Γ, (7 where: in m γϕ ϕ = Γ =, out m+ n γψ ψ = m+ Γ =. (8 We gie the solution of a RP at a node with two incoming lines, a and b, and two outgoing lines, c and d. Introduce the conditions: (A αγ< γ c ; (A ( α γ d Γ<. In the case Γ in =Γ, we get that: if (A and (A are both satisfied, then ˆ γ, ˆ γ, ˆ γ, ˆ γ = γ, γ, αγ, α Γ ; ( ( ( a b c d a b if (A does not hold and (A is satisfied, then ˆ γ, ˆ γ, ˆ γ, ˆ γ = γ, γ, γ, Γ γ ; ( a b c d ( a b c c if (A holds and (A does not hold, then ˆ γ, ˆ γ, ˆ γ, ˆ γ = γ, γ, Γ γ, Γ γ. ( a b c d ( a b c d In the case Γ out =Γ, the solution of the RP depends on the priority parameter p (for details, see D Apice 006... Equilibrium solutions Now, we gie a classification of the solutions to the RP at a node. Definition. A component of the solution at a node, ˆ ρϕ, ϕ =,,m, is: bad if ˆ ρ ϕ 0, ; good if ˆ ρ ϕ, ; a component of the solution, ˆ ρψ, ψ = m+,, m+ n, is: bad if ˆ ρ ψ, ; good if ˆ ρ ψ 0,. We are interested in equilibrium solutions. Definition. An equilibrium is a solution ρ(, tx = ( ρ,, ρl, which is constant in time. We also assume that ρ(, t is BV, thus we can define, for eery i =,, L, the following: ( tx ( tx i i x ai x bi ρ = lim ρ,, ρ = lim ρ,. (9 m Since ρ is a solution, then m+ n f( ρ = f( ρj, (0 ψ jϕ ϕ= ψ= m+ is satisfied at each node J j J, j =,, N. We distinguish two cases: (i there exists i =,, L, such that ρ i ρi + i i =/. In this case, ρ = τ( ρ and Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 34

the fluxes γ i = f ( ρ ± i, are anyhow constant in time and along the whole line I ; i + (ii for all i =,, L, ρi = ρi and we call this alue ρ i. Definition. Let ρ = ( ρ,, ρ L be an equilibrium for the network (, IJ, satisfying (ii. We say that ρ,, ρl are the alues of the equilibrium. Moreoer, if ρ i is of type τ i, with τi { bad, good}, then we say that T = ( τ,, τ L is the equilibrium type..3. Examples For simplicity, we consider a telecommunication network consisting of only one node, o, two sources, {, }, two destinations, {3, 4} and four lines, {a, b, c, d}, as in Figure. Since ˆ ρϕ = ρϕ,0 <, ϕ = ab,, ˆ ρϕ, ϕ = ab,, is a bad data; ˆ ρψ <, ψ = cd,, thus ˆ ρψ, ψ = cd,, is a good data. Assume now α = 0.6, p = 0.4 and an initial data ρ 0 = ( 0.55, 0.75, 0.65, 0.85. We hae that Γ =Γ out <Γ in, thus the fluxes, solution to the RP, at J are gien by: = ( pγ ( p Γ c d = ( 0.4, 0.3, 0.75, 0.75. ˆ γ,, γ, γ = (4 Obsere that the solution does not depend on the distribution coefficient α. The corresponding densities are: ˆ ρ = ( 0.88634,0.6934,0.65,0.85. (5 Since ˆ ρ ϕ >, ϕ = ab,, we get that ˆ ρϕ, ϕ = ab,, is a good data; moreoer, ˆ ρψ = ρψ,0 >, ψ = cd,, thus ˆ ρ, ψ = cd,, is a bad data. ψ Figure : A node of type Remark. The RS for the node o depends on one priority coefficient, α, and one distribution parameter, p. Fix the flux function (, the distribution coefficient α = 0.6, the priority parameter p = 0., and the initial data ρ ( ρ, ρ, ρ, ρ ( 0.3,0.,0.5,0.55 0 = a,0 b,0 c,0 d,0 =. From (8 we hae that: in γa γb out γc γd Γ = + = 0.37; Γ = + = 0.4975. ( As Γ=Γ in <Γ out, the fluxes, solution to the RP, at J are gien by: ( a b ( ( 0., 0.6, 0., 0.48. ˆ γ = γ, γ, αγ, α Γ = = ( Notice that, in this case, the solution does not depend on the priority parameter p. The corresponding densities are: ˆ ρ = ( 0.3,0.,0.33668,0.8066. (3 3. OPTIMIZATION OF DATA NETWORKS Consider a junction J with m incoming lines and n outgoing lines (junction of type m n. For an initial ρ = ρ, ρ, we define the cost functional data 0 ( ϕ,0 ψ,0 W ( t as: ( ϕ ( ρψ ( ( ρ ( W t = t, x dx+ t, x dx, Iϕ Iψ ϕ =,..., m, ψ = m+,..., m+ n. (6 Such functional indicates the aerage elocity of packets that, from an incoming line Iϕ, ϕ =,..., m, are directed to the outgoing line Iψ, ψ = n+,..., n+ m. For a fixed time horizon [ 0,T ], we want to imize T W t dt ( by a suitable choice of distribution 0 coefficients αψ ]0,[, ψ = m+,..., m+ n if Γ in =Γ or priority parameters p ϕ ]0,[, ϕ =,...,m if Γ out =Γ. 3.. The case m = and n = Fix the flux function ( and focus on a network with a junction of type. For T sufficiently big, the cost W T can be written as: functional ( Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 35

W ( T = ( ˆ ρi = si 4 ˆ γ i, i { abcd,,, } i { abcd,,, } (7 where: for ϕ = ab, : s ϕ = if ρϕ,0 and Γ =Γ in, or ρϕ,0, p ϕ Γ= γ ϕ and Γ=Γ out ; s ϕ =+ if ρ ϕ,0 >, or ρϕ,0, p ϕ Γ< γ ϕ and Γ=Γ out ; for ψ = cd, : s ψ =+ if ρψ,0 and Γ =Γ out, or ρϕ,0, αψγ= γψ and Γ=Γ in ; s ψ = if ρ ψ,0 <, or ρψ,0, αψγ< γψ and Γ=Γ in, with: p ϕ p, if ϕ = a, α, if ψ = c, = αψ = p, if ϕ = b, α, if ψ = d. We report results for the optimal choice of α. Regarding the choice of the optimal priority parameter, refer to Cascone 007 and Marigo 006. If Γ=Γ in, γˆϕ = γ ϕ, ϕ = ab,, hence imizing (7 is equialent to imize ( Wˆ T = s 4ˆ γ s 4ˆ γ. (8 c c d d The optimal choice of α, α opt, for different choice of s c and s d is as follows: if β β and sc = sd = or s = = s or s = + = s, α = 0.5 ; c d c if β β and sc = sd = or sc =+ = sd, α opt 0, β + ; + if β β and sc = = sd, α opt does not exist and one can choose αopt = + ε ; + β + if β β and sc = sd = or sc = = sd, α opt, β ; + d opt if β β and sc =+ = sd, α opt does not exist and one can choose where β = ε, + β Γ γc γ d = and β + =. γ Γ γ αopt c 4. EQUILIBRIA IN MANHATTAN TYPE NETWORKS Let us focus on oriented square networks with s t nodes of type as in the Figure, which we call Oriented Manhattan (OM briefly. Figure : topology of an OM network In Marigo 007, we got the following. Proposition. Consider a network (, IJ, where N = s t is the cardinality of J, while L = 4 s t is the cardinality of I. The set of equilibrium alues is a L N = 3 s t dimensional subspace of d 4 s t R. According to such proposition, at the equilibrium, on each line I I, there is a flow among two adjacent nodes which is constant in time and along the line I ( L ariables with the constraint (0 at each node J (for a total of N constraints. This result is based only on the topological property of the network. We consider an equilibrium at one node of a square network, and describe componentwise its type good or bad. The possible equilibrium types at one node are the followings (we use the short notations b for bad and g for good: I :((,,(, bb bb ; II0:(( b, b,( g, g ; II.: (( b, b,( b, g ; II. : (( b, b,( g, b ; III 0:(( g, g,( b, b ; III.: (( b, g,( b, b ; III.:(( g, b,( b, b. Definition. The set of all possible equilibria types at a node is M = { I, II 0, II., II., III0, III., III.}. The set of all possible equilibria types for the whole network is s t N = { δ ij M, i =,, s, j =,, t} = M. Clearly, the cardinality of N s t is M. Howeer, not all the elements of N may arise. Indeed, two kinds of equilibria oer the whole network Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 36

may be considered. In case (ii, the following compatibility rule must be satisfied: (H if a line I i is incoming for some node J and outgoing for some other node J, then wheneer ˆi ρ is of type bad for J, then it must be of type good for J and iceersa. Rule (H gies rise to a compatibility relation among equilibria at adjacent nodes, which determines the subset N N of admissible equilibrium states for the whole network. This fact does not hold for case (i. Consider now the equilibria (ii and gie a characterization of N. Indicate by a and b the incoming lines while by c and d the outgoing lines. Then, the following holds: Proposition. Consider a node J of type and assume that ( ρa,0, ρb,0, ρc,0, ρ d,0 is an equilibrium. If J is of type II 0, γ ϕ = γϕ, ϕ = ab,, while γ = αγ and γ = ( α Γ. c in d For other types of equilibria, similar proposition hold (see Marigo 007. Remark. Note that if ρ 0 = ( ρa,0, ρb,0, ρc,0, ρd,0 is an ˆ ρ = ˆ ρ, ˆ ρ, ˆ ρ, ˆ ρ = ρ = ρ equilibrium, then ( ( ˆ = = f, i= abcd,,,. and γi γi ( ρi,0 in a b c d RS 0 0 A s t network is described by 4 s t h h parameters inij, inij, outij, out ij, i =,, s, j =,, t, together with 4s t s t ω constraints, where ω min( s, t is the number of nodes of type I, while h h inij, inij, outij, out ij are, respectiely: incoming flows for horizontal and ertical lines; outgoing flows for horizontal and ertical lines. In what follows, we consider only networks of type II 0. For such a network, it is shown that a matrix equation rules the behaiour of the outgoing flows as function of incoming flows (for the proof see Marigo 007: AOUT + OUTB + in = 0, (9 h ij ij = T with ( where we hae: OUT ( out, = A = α( Is ( + Ls (, B ( I( t L( t, I n the identity matrix of order n, Ln ( the sub-diagonal matrix of order n, in the inflow ector: h h h h h h x + x x x3 x4 x5 x t 0 x 0 0 x3 0 L =, in =. x4 0 0 xs 0 0 0 0 and x h i, i=,,t and x i, i=,,s are the incoming flows for horizontal and ertical line. The equation (9 is a Sylester matrix equation, and its solution is found recursiely as: OUT = C, (0 where: A = A, B = B, C = in, 0 0 0 k+ = k + k k k C ( C A C B. ( Notice that (0 gies the analytical solution on the whole network in terms of outgoing flows, under the assumption of equal distribution coefficients for all nodes of the network. 5. SIMULATIONS In this section, we present some simulation results in order to test the optimization algorithms either for single nodes or complex networks. The aim is to erify the correctness of the analytical results and to analyse the effects of different choices of distribution and priority parameters, applied locally at each junction, on the global performances of the network. 5.. Single junctions Consider a telecommunication network with only one node of type. We compare the different behaiours of the cost functional (6 in such situations: use of the optimization algorithm (optimal case; adoption of fixed priority parameters and distribution coefficients (fixed case, namely parameters chosen by the user. The eolution of packets traffic is simulated in a time interal [ 0,T ], where T = 0 min for the flux function (. We assume that, at the starting instant of simulation (t = 0, the initial datum is ρ 0 = ( 0.4,0.35,0.3,0.. Moreoer, for lines a, b, c and d, we choose the following Dirichlet boundary data: ρ =, ρ, = 0.35, ρ, = ρ, = 0. ab, 0.4 bb cb Notice that the initial conditions and boundary data are such that the dynamics of the network is ruled only by the distribution coefficient α, whose optimal alue is 0.5. Figures 3 and 4 show that, setting the network with the optimal alue α, traffic conditions are improed db Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 37

with respect to fixed cases: the optimal cost functional is higher with respect to others. In particular, Figure 3 depicts the temporal behaiour of W in [ 0,T ], while Figure 4 represents a zoom of the cost functional behaiour. The latter Figure shows clearly that, in some contexts, although the asymptotic state is not reached (the final T, the optimization algorithm gies better performances with respect to other simulation cases. Figure 5: a complex network of OM type Figure 3: behaiour of W ( t for optimal choice of α (solid line, α = 0.5 (dashed line and α = 0.65 (dot dashed line Figure 4: zoom of W ( t for optimal choice of α (solid line, α = 0.5 (dashed line and α = 0.65 (dot dashed line 5.. Complex networks Here, we describe the simulation results for a complex OM network. The network, represented in Figure 5, consists in 4 lines, diided into two subsets, L ={ a, a 3, d, d, d 3, c, c 3, f, f, f 3, e, e 3 }, and L ={ a, a 4, b, b, b 3, c, c 4, e, e 4, g, g, g 3 } that are respectiely, the set of inner and external lines. All nodes are of type and are labelled by the couple ( i, j, where i and j are, respectiely, the row and the column indices. The eolution of traffic flows is simulated in a time interal [ 0,T ], where T = 30 min. Initial conditions for all lines and boundary data are in the following table: Table : Initial conditions and boundary data for all lines Line ρ i,0 ρ ib, Line ρ i,0 ρ ib, a 0. 0.3 d 0. / a 0.3 / d 3 0.5 / a 3 0. / e 0. 0.35 a 4 0.4 0.3 e 0. / b 0. 0.3 e 3 0. / b 0.5 0.5 e 4 0.3 0.5 b 3 0.5 0.35 f 0.3 / c 0.3 0.4 f 0. / c 0.5 / f 3 0. / c 3 0.3 / g 0.4 0.5 c 4 0.5 0. g 0. 0. d 0.3 / g 3 0.5 0.5 We analyze different type of simulation cases: priority parameters and distribution coefficients, that optimize the cost functional (optimal case in each node; fixed parameters (fixed case, which means that networks parameters are fixed for each node by the user at the beginning of the simulation process (in what follows, priority parameters are chosen equal to 0.3, while distribution coefficients are fixed at 0.; random parameters (random case, where the parameters are randomly chosen when the simulation starts. The aim is to understand the effects of the optimal algorithm of local type on the whole network. Figure 6 shows the temporal behaiour of the cost functional W ( t for arious choices of parameters at nodes. The optimal cost functional is higher than other simulation cases, hence the principal aim is achieed. Moreoer, Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 38

when simulation begins, W with optimal alue parameters is lower than the fixed case. In the steady state, instead, the optimal configuration is the highest (Figure 7. Figure 6: W ( t for optimal choice of network parameters (solid line, random parameters (dashed line and fixed parameters (dot dashed line Figure 7: zoom of W ( t in the time interal [ ],5 ; optimal choice of network parameters (solid line, random parameters (dashed line and fixed parameters (dot dashed line The optimization procedure can be also applied to telecommunication networks with hundreds of nodes and arcs and the computational times are not ery expensie. This is due to the few parameters, which characterize fluid dynamic models. 5.3. Equilibria In this section, we consider the analysis of equilibria of type II0,(( b, b,( g, g. We first analyze a node of type. An equilibrium of type II 0 is gien by ρ = with α = 0.733. In this case, 0 ( 0.5,0.3,0.45,0. Γ=Γ in <Γ out, ˆρ ρ0 = =. Obsere that ˆ ρi < σ, i = a, b, c, d, hence the equilibrium is bbg g. (,,, = and ˆi γ γ i f ( ρ i,0 Consider now an OM network (Figure 5 and apply the iteratie algorithm in order to find the equilibrium outflows. Assume that, for each node, the distribution coefficient is α = 0.3, and the inflow matrix, in, is the following: h h h x + x x x 3 0.6 0.5 0.35 in = x 0 0 = 0.4 0 0, x 0.35 0 0 3 0 0 h where xi, i =,,3, are the inflows for lines, respectiely, b, b, b 3, while xi, i =,,3, are the inflows for a, c, e. Using equation (9, we obtain the following outflows: Table : equilibrium outflows alues Line outflow a 0.46538 a 3 0.547337 a 4 0.69059 c 0.440 c 3 0.44495 c 4 0.5054 e 0.36486 e 3 0.38330 e 0.409588 4 6. CONCLUSIONS In this paper, the analysis of packets flows on telecommunication networks has been considered. Some optimization procedures for parameters at junctions and equilibrium solutions hae been inestigated. Simulations proed that the optimization algorithm for packets elocities allows to redistribute data flows so as to aoid congestions. The algorithm was tested either on a single junction or on a network with more nodes, concluding that real benefits on traffic performances are possible. The study of the equilibrium flows, through a matrix equation, gae the possibility of knowing asymptotic solutions on a test complex network. Such results can be used to face the security issues when some nodes of the network fail. REFERENCES Cascone A., D Apice, C., Piccoli, B., and Rarità, L. Optimization of traffic on road networks. Mathematical Models and Methods in Applied Sciences, 7 (0, 587 67. Cascone A., D Apice, C., Piccoli, B., and Rarità, L. Circulation of traffic in congested urban areas. Communication in Mathematical Sciences, 6 (3, 765 784. D Apice, C., Manzo, R., and Piccoli, B., 006. Packets flow on telecommunication networks. SIAM Journal on Mathematical Analysis, 38, 77 740. D Apice, C., Manzo, R., and Piccoli, B., 008. A Fluid Dynamic Model for Telecommunication Networks Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 39

with Sources and Destination. SIAM Journal on Applied Mathematics, 68 (4, 98 003. Marigo, A., 006. Optimal Traffic Distribution and Priority Coefficients for Telecommunication Networks. Networks and Heterogeneous Media, (, 35 336. Marigo, A., 007. Equilibria for Data Networks. Networks and Heterogeneous Media, (3, 497 58. AUTHORS BIOGRAPHY CIRO D'APICE was born in Castellammare di Stabia, Italy and obtained PhD degrees in Mathematics in 996. He is associate professor at the Department of Information Engineering and Applied Mathematics of the Uniersity of Salerno. He is author of approximately 30 publications, with more than 70 journal papers about homogenization and optimal control; conseration laws models for ehicular traffic, telecommunications and supply chains; spatial behaiour for dynamic problems; queueing systems and networks. His e-mail address is dapice@diima.unisa.it. ROSANNA MANZO was born in Polla, Salerno, Italy. She graduated in Mathematics in 996. She is a researcher at the Department of Information Engineering and Applied Mathematics of the Uniersity of Salerno. Her research areas include fluid dynamic models for traffic flows on road, telecommunication and supply networks, optimal control, queueing theory, self similar processes, computer aided learning. She is author of about 30 papers appeared on international journals and many publications on proceedings. Her e-mail address is manzo@diima.unisa.it. LUIGI RARITÁ was born in Salerno, Italy, in 98. He graduated cum laude in Electronic Engineering in 004, with a thesis on mathematical models for telecommunication networks, in particular tandem queueing networks with negatie customers and blocking. He obtained PhD in Information Engineering in 008 at the Uniersity of Salerno with a thesis about control problems for flows on networks. He is actually a research assistant at the Uniersity of Salerno. His scientific interests are about numerical schemes and optimization techniques for fluid dynamic models, and queueing theory. His e-mail address is lrarita@unisa.it. This work is partially supported by MIUR-FIRB Integrated System for Emergency (InSyEme project under the grant RBIP063BPH. Proceedings of the European Modeling and Simulation Symposium, EMSS 009 Vol I - ISBN 978-84-69-544- 40