Design of Heterogeneous Traffic Networks Using Simulated Annealing ...

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Design of Heterogeneous Traffic Networks Using Simulated Annealing Algorithms Miguel Rios, Vladimir Marianov, and Cristian Abaroa Department of Electrical Engineering, Universidad Catolica de Chile, Casilla 306, Correo 22, Santiago-Chile {mrios, marianov, cabaroa}@ing.puc.cl

Abstract. We propose a global design procedure for heterogeneous networks , which includes the definition of their topology; routing procedures; link capacity assignment; transfer mode; and traffic policy. We discuss the network model, which minimizes the cost of interconnecting a number of nodes whose locations are known; the traffic model, where we use the concept of Equivalent Bandwidth, and the resolution algorithms, where we compare a Simulated annealing algorithm (SAA) with a commercial solver, with good results. Results show that SAA gets to the optimum solution over 10 times faster than the commercial solver. Experiments also show that, for networks with more than 50 nodes, the SAA still delivers good feasible solutions while the commercial solver is unable to deliver results.

1

Introduction

Current communications systems rely largely on networks that are able to carry heterogeneous traffic. INTERNET itself is a communications network that can transmit data, voice and video. ATM (Asynchronous Transfer Mode) networks are an example of broadband networks that also deal with heterogeneous traffic. The design of heterogeneous broadband networks includes the definition of their topology, routing procedures, capacity assignment, transfer mode, fault tolerance methods and traffic policy. Therefore, models for the design of these networks are very complicated, and involve generally a very large number of integer and continuous variables. Most of the known approaches to this problem deal with these issues as separate problems. There is a considerable body of literature that focuses on the design of broadband networks. Models and solution procedures for network design include the capacity assignment through multiple origin-destination pairs [5], fault tolerant network dimensioning [1], and uncapacitated network design [3]. Several network design problems are presented in textbooks such as [8] and [9]. In [6] a virtual hierarchical network is designed that uses Fractional Brownian Motion (FBM). We use the same distribution in this paper. However, there are no design methods that consider all of the aspects together, being at the same time practical and efficient enough for their use by the industry. We address the global design of a heterogeneous broadband network that carries data, voice and video, including all the aspects except for C. Kim (Ed.): ICOIN 2005, LNCS 3391, pp. 520–530, 2005. c Springer-Verlag Berlin Heidelberg 2005


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the fault tolerance, which is left as a subject for future research, and propose and solve models for the design of these networks. Although ATM networks are taken as an example, the proposed methods can be used for the design of any broadband network. In ATM networks, the traffic is divided in cells of a certain length. These cells can become lost or delayed, due to congestion in the network. Thus, in addition to the cost and heterogeneous traffic issues, there is also the Quality of Service (QoS) issue. QoS in these networks refers to the cell loss probability and cell delay. We impose constraints on the QoS, bounding the delay and, consequently, the cell loss and cell delay. Node locations and traffic load for each traffic type are assumed to be known. We first address the network model, minimizing the cost of interconnecting a number of nodes whose locations are known. Assumed known is the amount of traffic of each type between each pair of nodes. In addition, for each information type there must be an acceptable delay. Secondly we address the traffic model, where we use the concept of Equivalent Bandwidth [7]. Although several traffic-modelling techniques can be used, we choose FBM, because it adapts better to the real time network behaviour. Finally we solve the model by using two tools: a) AMPL-CPLEX, a commercial package which finds the optimal solution of the linear model in a time that is exponentially related to the size of the problem, and b) a simulated annealing algorithm (SAA), which does not guarantee an optimal solution, but has a good performance, in terms of finding solutions that are close to the optimum. The main contributions of this paper are: a) a network model that considers most of the aspects of the network design problem starting from the node locations and traffic data; b) we propose an adaptation of the FBM traffic model to the conditions of the network design problem; c) we apply the SAA to solve the model, with good results. We first present and discuss the traffic model. Then, we address the network model. Afterwards, the solution methods are discussed. The next section presents the computational experiments. For this problem, the simulated annealing algorithm finds a solution (which corresponds to the optimum in most cases) around 10 times faster than AMPL-CPLEX. The experiments also show that for networks with more than 50 nodes, AMPL-CPLEX is unable to deliver results due to the enormous processing time, while the SAA still delivers good feasible solutions. Finally, conclusions are drawn and future work is proposed.

2

Traffic Model

An ATM broadband network, allocates its resources according to the amount of traffic of each type, present in the network, which means the resources to be allocated need to be determined in real time. Since the decision has to be taken very fast, the idea of using an equivalent bandwidth (EB) has become attractive [16] [7]. In fact, methods are required that generate timesavings at switches, while maintaining an adequate QoS. The EB is the result of applying statistical theories, and it is defined as the bandwidth ensuring a certain QoS (usually taken as the cell loss probability, α).

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Equivalent Bandwidth and the Fractional Brownian Model

There are four basic models of EB, each one with several variants: Poisson Model [2], Gaussian Model [7], ON/OFF Model [7] and FBM [12]. We use the FBM model because it better represents the network behaviour, and because it allows the characterisation of different traffic types using a small number of parameters. Several studies have pointed out the existence of a fractal or self-similar behaviour in data traffic [4]. Regarding traffic modelling, the variance of burst length is so large that, in many cases, it tends to infinity. This makes the decay of α, when the buffer size is increased, to be much lower than the one inferred from a Poisson model. Several researchers have attempted to model this behaviour, and the one used in this paper is due to Norros [12]. The model describes the traffic for a connectionless network. The Norros model can be applied to allocate resources in variable bit rate (VBR) and available bit rate (ABR) broadband networks. To model an accumulative system that complies with the self-similarity characteristics, the following equation is used: √ (1) At = mt + amZt t ∈ (−∞, ∞) where At represents the amount of traffic (number of packets) arrived until time t, m is the mean arrival rate, a is the variance coefficient of the mean per time period (bps per sec). Since t initially has no units, it has to be normalized as t = T /tu where T is the real time and tu is the unit. Zt is a non-dimensional number representing a normalized self-similar Gaussian process (or FBM) with parameter H. The Hurst parameter H, defines the level of self-similarity of the traffic. Its range goes from 0.5 to 1. The fractional Brownian storage with input parameters m, a and H and output capacity C > m is the stochastic process Xt defined as: (2) Xt = sup(At − As − C(t − s)), t ∈ (−∞, ∞) s≤t

Then, the approximate queue length distribution is given by a lower bound [12]:
 (C − m)H x1−H √ P (Xt > x) ≥ Φ (3) κ(H) am where κ(H) = H H · (1 − H)1−H

and Φ = P (Z1 > y)

(4)

Function Φ is approximated by a Weinbull distribution, and a lower bound for the probability is obtained. Considering B as the maximum queue length (in bits), the cell loss probability (α) is:   2H α = P (x > B) ≥ e

−

(C−m) 2κ(H)2 am

·B 2−2H

(5)

From this equation, the equivalent bandwidth can be written as:  H1   (1−H) 1 1 · a 2H · B − H · m 2H EB = C = m + κ(H) · −2 · ln(α)

(6)

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It can be seen that the equivalent bandwidth is a function of a small number of parameters (m, a, H, B and α), and then it can be used to solve the network model with less complexity. 2.2

Traffic and Network Parameters

Each one of the types of traffic (data, audio and video) is characterized by its flow mean and variance (parameter a of the FBM model), network distribution type, Hurst parameter, α and maximum buffer length B. Most of parameters are related to the EB chosen model. Since in a broadband network resource reservation is used, the parameters are defined according to the EB, allowing a controlled α. The nodes of the network represent switches that generate, receive or redirect the traffic. The arcs of the network represent the links that connect the switches, and they have a certain capacity or bandwidth. Three discrete capacities (622, 155 and 45 Mbps) were considered. One of the outcomes of the problem is deciding what links have to be built, what is the capacity of each link, and which switches are to be located. 2.3

Delay

The delay considers two components: queuing delay at switches and transmission delay at links. We use a typical optical fibre network, with a transmission delay of 4[microsec/Km]. The queuing delay depends on the traffic model. For the FBM model, each switch has a service queue and independent virtual connections (VC). Then, every connection has its own queue with a Weinbull distribution (equation 9), and a service rate equal to the EB. Defining β and γ as follows:
β=

(C − m)2H 2(H H (1 − H)1−H )am

−1/γ

γ = 2 − 2H  0 x B) = e−( β )

x≥0

(10)

Using these equations, it is possible to obtain the expected number of individuals in the queue as:
 1 E(X) = β · Γ 1 + (11) γ The average queue length x is obtained by replacing the EB (equation 6) into C of equation 7. This value does not depend of the mean or variance of the traffic

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type, but only on the maximum buffer size B and the cell loss probability, α. Finally using Little’s law, we obtain the queuing delay at the switches, DS , as:   − 1 1 (− ln(α)) 2−2·H · B · Γ 1 + 2−2·H N x = = (12) Ds = λ m m

3

Network Model

The model minimizes the costs of establishing the network, given the node locations, traffic loads of each origin - destination pair for the three types of traffic, subject to QoS constraints (node and arc delay). Its formulation follows. First we introduce the notation used. The following terms are given as inputs to the problem: n : number of nodes in the network N ode : Set of all nodes Arc : Set of all possible undirected arcs among network nodes. Link : Set of all possible directed arcs among nodes. Cap : Set of arc capacities: 622, 155 or 45 Mbps. OD : Set of all possible origin-destination (od) pairs. Ca : Cost of arc, per length unit. Cs : Fixed costs of node Cc : Cost of link capacity hardware xij : Binary variable that takes value 1 if the arc between nodes i and j exists, and 0 otherwise. r yij : Integer variable. It is the quantity of links of type r in arc ij. ni : Binary variable that counts the number of terminal equipments located at node i. od,k Ef lowij : Binary variable equal to 1 if there is flow of traffic type k through arc ij of od pair of type k, 0 otherwise. od,k : Real variable. Corresponds to the flow through link i, j of pair od and f lowij traffic type k. k : Amount of traffic of type k going from origin to destination node. trafod dij : Distance between node i and j. Dsk : Queuing delay at switches, for traffic type k DT : Transmission delay in the optical fiber, per distance unit V capr : Capacity of link of type r (45, 155 or 622 Mbps) 3.1

Mathematical Formulation

The objective function takes into account the total topology building cost, the total switching equipment cost of nodes, and the total transmission equipment cost. Objective: Minimize the network building cost:     r Ca dij xij + Cs ni + Ccr yij (13) min (i,j)∈Arc

i∈N ode

(i,j)∈Link r∈Cap

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Subject to: i) Connection of nodes. This restriction tells that if an arc exists, switches must exist at both ends of the arc. ni + nj ≥ 2xij

∀ (i, j) ∈ Arc

(14)

ii) Traffic flow. This restriction sets the flow passing for a given arc of a given od pair. od,k od,k k = trafod Ef lowij f lowij

∀ (i, j) ∈ Arc, (o, d) ∈ OD, k ∈ typeT

(15)

iii) Arc presence. If flow exists, in any direction and of any type, then an arc must exist.     od,k od,k k f lowij ≤ + f lowji trafod xij ∀ (i, j) ∈ Arc, (o, d) ∈ OD k ∈ typeT

k ∈ typeT

(16) iv) Flow balance. There are three cases for the nodes: they can be origin, destination or transfer nodes. The restriction indicates that the traffic is generated, absorbed or transferred at the node, according to its type.  k if j = d  trafod   od,k od,k k f lowij − f lowjl = −trafod if j = o ;  0 otherwise (i,j) ∈ Arc (j,l) ∈ Arc ∀ j ∈ N ode, (o, d) ∈ OD, k ∈ typeT

(17)

v) Link capacity assignment. The sum of all flows passing through an arc has to be less or equal than the total capacity of the arc.      od,k od,k r f lowij ≤ + f lowji V capr yij ∀ (i, j) ∈ Arc r ∈ cap

(o,d) ∈ P ath k ∈ typeT

(18) vi) Delay constraint. The delay is composed of two components: one is related with the number of nodes of the path and the other with the distance.     od,k od,k Dsk  Ef lowij − 1 + DT dij Ef lowij ≤ Delay k (i,j) ∈ Arc

(i,j) ∈ Arc

∀ (o, d) ∈ OD, k ∈ typeT

(19)

vii) Unidirectional flow constraint. This constraint forces the flow of an od pair to go through a given arc in only one direction. od,k od,k + Ef lowji ≤1 Ef lowij

∀ (o, d) ∈ OD, (i, j) ∈ Arc, k ∈ typeT

(20)

The distance was calculated using the Euclidian formula, since node coordinates are known. Traffic was defined by using equation 6.

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Simulated Annealing Methods

Simulated annealing, a technique introduced by Kirkpatrik [10], is a Monte Carlo approach for minimizing multivariate functions. The term simulated annealing derives from the roughly analogous physical process of heating and then slowly cooling a substance to obtain a strong crystalline structure. In simulation, a minimum of the cost function corresponds to this ground state of the substance. The simulated annealing process lowers the temperature by slow stages until the system “freezes” and no further changes occur. At each temperature the simulation must proceed long enough for the system to reach a steady state or equilibrium. This is known as thermalisation. The time required for thermalisation is the decorrelation time; correlated microstates are eliminated. The sequence of temperatures and the number of iterations applied to thermalise the system at each temperature comprise an annealing schedule. To apply simulated annealing, the system is initialised with a particular configuration. A new configuration is constructed by imposing a random displacement. If the energy of this new state is lower than that of the previous one, the change is accepted unconditionally and the system is updated. If the energy is greater, the new configuration is accepted probabilistically. This is the Metropolis step, the fundamental procedure of simulated annealing [17]. This procedure allows the system to move consistently towards lower energy states, yet still ”jump” out of local minima due to the probabilistic acceptance of some upward moves, using the Boltzmann probability distribution. If the temperature is decreased logarithmically, simulated annealing guarantees an optimal solution.

5

Computer Experiments

The proposed model was tested by first trying it with small networks, comparing the results of a commercial package AMPL-CPLEX and a SAA heuristic on MATLAB. Then the heuristic was used on a large 48 node network. The first step validated the heuristic showing its efficiency and speed. All the tests were done in a Digital DEC Alpha 433 cluster. To test the model with small networks, the TSP library [15] was used. From that library 8 problems were chosen that appear to have random node locations (Att48, Bier127, Ch130, Eil51, KroB100, Rd100 and St70). From these problems the coordinates and Euclidean distances were calculated. In all the problems, a subset of the first 8 node locations was used to define each small network. The solving time for the proposed model using AMPL-CPLEX was quite large. Up to 8 nodes, the run time was reasonable (hundreds of thousands of seconds). However, for 9 nodes, the run time grew to over 1 million seconds. In the implementation of the SAA, some adaptations were performed to specialize it to the proposed model. Two variables that are important are the number of heating cycles and the MetropolisR function adaptation constant [14]. In the first case, the method consists in re-heating the system, with the best solution found in the previous processing cycle as the initial network. Using an

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8-node subset of the ATT48 problem, the optimum solution of AMPL-CPLEX was compared with the results of running 100 times the SAA with two cycles per run. It was observed that increasing the number of cycles improves the percentage of cases reaching the optimum. This result is very important in large networks. One of the characteristics of the SAA is its ability of jumping out from local optima by using the probability function defined by Metropolis [14]. This constant was defined in such a way that, in some cases, a new higher cost state can be accepted. We tried values of 0.3 and 0.4 observing that the latter improves the results. 5.1

Running Tests

The first observed result of the tests is that SAA solution times are clearly shorter than AMPL-CPLEX, though not always reaching the optimum. In 5 of the 8 networks more than 77% of the SAA runs reached the optimum.The AMPL-CPLEX results were found by using an uppercut restriction between 0.5 and 10% of the optimum. Cases with poor results (Bier127, Krob100 and Rd100) have all a common characteristic, which is that the optimum solution is clearly a star, far from the initial MST solution used as a starting solution. That explains the SAA not providing very good results. Other cases such as Ch130 and Ch150, show very high percentages of reaching the optimum, since the final solution topology is very similar to the initial MST solution. The main conclusion of all the tests is that the SAA delivers good results when the networks have more uniform traffic distributions and without very hard delay restrictions.Table 1 shows for the Att48 test, and links connected to node n1, the link capacity assignment and flows for each traffic type. All tests used the same traffic matrices for each data type. Table 1. Tests results for node n1 connections on 8-node subset of problem ATT48. Link Number of work cards 622 155 n1 - n3 2 1 n1 - n7 2 3 n1 - n8 0 3

5.2

net- Total capacity [Mbps] 45 0 1399 0 1709 0 465

Traffic flow [Mbps] Data 6.71 5.59 1.68

Sound 526.15 394.61 230.19

Video 804.93 1207.39 201.23

Full-Scale Problem

To evaluate the model for a larger size network the full ATT48 problem was solved. Here the optimal solution is not known, and the number of runs is limited as each cycle takes around 35 hours.

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Fig. 1. Initial network and best feasible solution for the ATT48 problem.

Figure 1 shows the best-found solution and the initial MST. While this solution may not be the optimum solution, the SAA heuristic ensures the solution found would be a network that will comply with all the QoS requirements and with building costs within reasonable limits. A larger number of cycles may improve this solution but the processing time necessary to find feasible networks will grow considerably. This means that in the time assigned to a cycle less and less low cost solutions are found. Table 2 shows the capacity assignment and traffic flow obtained in the best solution for node n1. Table 2. Capacity assignment and resulting flow for node n1 of the ATT48 problem. Link

Number of work cards 622 155 n1 - n8 5 0 n1 - n40 23 3

6

net- Total capacity Traffic flow [Mbps] 45 Data Sound 0 3110 9.5 1545.55 0 14771 19 3025.34

[Mbps] Video 1408.62 11671.46

Conclusions

The model presented considers all the basic elements used in a broadband network design: topology, routing and capacity assignment. It only requires knowledge about node locations and traffic distributions for each information type. The solution algorithm, which uses Simulated Annealing techniques, provides good solutions in reasonable computer times, even for a 48-node example carrying three information types. Three topics were discussed: a) the network model, which minimizes the cost of interconnecting a number of nodes whose locations

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are known, b) the traffic model, where we used the concept of Equivalent Bandwidth and c) the resolution algorithms, where we used SAA and a commercial solver. The main contributions of this paper are: a) a network model that includes most of the aspects of the network design problem starting from the node locations and traffic data, b) we propose an adaptation of the Fractional Brownian Motion traffic model to the conditions of the network design problem, and c) we apply the simulated annealing algorithm to solve the model. Computational experiments show that SAA gets to the optimum solution over 10 times faster than a commercial AMPL-CPLEX solver. The experiments also show that for networks with more than 50 nodes, AMPL-CPLEX is unable to deliver results due to the enormous processing time, while the SAA still delivers good feasible solutions. In future developments of the model we will try to incorporate fault tolerance capability to the design1 .

References 1. Alevras, D., Grotschel, M., Wessaly, R.: Capacity and Survivability Models for Telecommunication Networks. Preprint SC97-24. Konrad-Zuse-Zentrum fur Informationstechnik. Berlin (1997). 2. Babic, G., Vandelore, B., and Jain, R.: Analysis and Modeling of Traffic in Modern Data Communication Networks. Ohio State University, Department of Computer and Information Science, Technical Report OSU-CISRC-1/98-TR02, Feb 1998. 3. Balakrishnan, A., Magnanti, T., and Wong, T.: A Dual-Ascent Procedure for LargeScale uncapacitated Network Design. INFORMS Operation Research, 37,(1989) 716740. 4. Beran, J. et al.: Long Range Dependence in Variable Bit Rate Video Traffic. IEEE Transactions on Communications , 43,(1995) 1566-1579. 5. Bienstock, D. et al.: Minimum Cost Capacity Installations for multicommodity network flows. Mathematical Programming 81 , no. 2-1,(1998) 177-199. 6. Bienstock, D., and Saniee, I.: ATM Network Design: Traffic Models and OptimationBased Heuristics. DIMACS Report 98-20, Telecom Systems, June 2001. 7. Courcoubetis, C., Fouskas, G., and Weber, R.: On Performance of an Effective Bandwidths Formula. Proc. 14th Int. Teletraffic Cong., 6-10 June 1994 North-Holland Elsevier Science, (1994) 201-212. 8. Ball , M.O. et al. (Eds.): Network Models, Handbook in Operations Research and Management Science, Vol. 7, Amsterdam: Elsevier, (1995). 9. Ball , M.O. et al. (Eds.): Network Routing, Handbook in Operations Research and Management Science, Vol. 8, Amsterdam: Elsevier, (1995). 10. Kirkpatrik, S., Gelatt, C.D., and Vecchi, M.P.: Optimization by Simulated Annealing. Science, 220, 4598,(1983) 671-680. 11. Laarhoven, P. J. M. Van., and Aarts, E. H. L.: Simulated annealing : theory and applications. Kluwer Academic (1987). 12. Norros, I. : On the Use of Fractional Brownian Motion in The Theory of Conneccionless Networks”. JSAC, 13-6, August 1995 13. Nurmela, K.J., and Ostergard R.J. :Constructing Covering Designs by Simulated Annealing. Technical report B10, Digital Systems Laboratory, Helsinki University of Technology, (1993). 1

This work was funded, in part, by grant 1040577 from Fondecyt

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14. Press, W. H., et al.: Numerical Recipes in C (2 Ed.). Cambridge University Press (1992). 15. Reinelt, G.:TSPLIB95. Universitat Heidelberg, Institut fur Angewandte Mathematik,Im Neuenheimer Feld 294(1995). 16. De Veciana, G., Kesidis, G., and Walrand, J.: Resourse Management in Wide-Area ATM Networks Using Effective Bandwidths. JSAC, 13-6, August 1995. 17. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., and Teller, E.: Simulated Annealing. Journal of Chemical Physics, Vol. 21, (1953) 1087-1092.