Routing Optimization in Optical Burst Switching Networks: a Multi-path Routing
6.4 Resolution Methods and Numerical Examples .1 Formulation of the Optimization Problem.1 Formulation of the Optimization Problem
burst x1
x2
path 1
path 2
Fig. 6.2 An example of OBS network with source-based routing; x1and x2are the traffic splitting factors and x1+x2=1
0≤xp≤1 p∈P, (6.8a)
∑
p∈Pstxp=1 s,t∈V,s=t. (6.8b)Thus trafficρpoffered to path p∈Pst can be calculated as
ρp=xpτp, (6.9)
whereτp=γst is the total traffic offered between s and t.
Here vector x= (x1, . . . ,x|P|)determines the distribution of traffic over the net-work; this vector should be optimized to reduce congestion and to improve overall performance.
6.4 Resolution Methods and Numerical Examples 6.4.1 Formulation of the Optimization Problem
Taking into account different methods of the link load and the network loss cal-culation presented in Section 6.3, several network loss models with corresponding objective functions can be defined.
1. NL-RL. The link load is calculated according to the RL model given by (6.2), and the network loss is calculated according to the NL model given by (6.6), with the objective function given by
BNL−RL(x) =
∑
p∈P
xpτpLp. (6.10)
2. NL-NRL. The link load is calculated according to the NRL model given by (6.4), and the network loss is calculated according to the NL model given by (6.6), with the objective function given by
BNL−NRL(x) =
∑
p∈P
xpτpLp. (6.11)
3. LL-NRL. The link load is calculated according to the NRL model given by (6.4), and the network loss is calculated according to the LL model given by (6.7), with the objective function given by
The last possible combination of the link load and the network loss calculation is LL-RL. Because such a model does not bring much gain with respect to the NL-RL model, as it does not avoid the complexity of fixed-point calculation, we do not study it.
In each case the normalization factor&
∑p∈Pρp
'−1
has been omitted because we assume it to be a constant value.
The optimization problem is the same for each method, and is formulated as follows:
minx B(x) (6.13)
subject to the multi-path routing constraints given by (6.8a) and (6.8b).
Since in each case B(x)is a nonlinear function of vector x, the optimization prob-lem is nonlinear. Taking into account the form of both constraints (6.2) and (6.4), a particularly convenient optimization method is the Frank-Wolfe reduced gradient method (algorithm 5.10 in [19]); this algorithm was used for a similar problem in circuit-switched (CS) networks [7].
6.4.2 Calculation of Partial Derivatives
In general, gradient methods are iterative methods used in the optimization of con-vex functions. Gradient methods need to employ the calculation of partial deriva-tives of the cost function to find the direction for its improvement. Below, we pro-vide adequate formulas for the partial derivatives for each of the models.
NL-RL model. The partial derivative of BNL−RLwith respect to xq, q∈P, can be derived directly by a standard method involving the solution of a system of linear equations. It follows from (6.2) and (6.1) that
∂ρe(x)
whereαqe=1 if e∈q, andαqe=0 otherwise, and
In order to solve the system of equations (6.14)–(6.15), a fixed-point calculation procedure, i.e., repeated substitution of (6.14) in (6.15), has to be applied.
From (6.5) we have
The calculation of partial derivatives (6.14)–(6.17) in NL-NRL model is extremely time consuming since it involves an iterative fixed-point approximation procedure.
NL-NRL model. The partial derivative of BNL−NRLwith respect to xq, q∈P, could be derived directly from formulas (6.1) and (6.4)–(6.6) by a standard method involv-ing resolution of a system of linear equations, similarly to (6.14)–(6.17). Although there is no need for a fixed-point calculation in NL-NRL model, still such a compu-tation would be time consuming.
Therefore, we propose instead a straightforward exact calculation based on the approach for CS networks by Kelly [10]; a detailed derivation of formulas is pre-sented in [11]. In particular, for each path q∈Pwe have
∂ Due to assumption (6.4) we have managed to simplify the model (6.2) and make the calculation of partial derivatives defined by (6.18) and (6.19) straightforward, not involving any iterations. Indeed, once|E|of unknowns(ce)are pre-calculated they can be used in (6.18) to obtain the partial derivatives. Calculating the gradient in this method, therefore, is not longer an issue.
1,E-06
Fig. 6.3 Validation of optimization models
LL-NRL model. The partial derivative of BLL−NRLwith respect to xq, q∈P, can be derived directly from formulas (6.1), (6.4), and (6.12),
∂
whereηeis given by equation (6.20).
6.4.3 Numerical Results
We evaluated the performance of our multi-path source routing scheme in an event-driven simulator. In order to find a splitting vector x specifying near-optimal rout-ing we used a solverfminconfor constrained nonlinear multivariable functions available in the MATLABoptimization toolbox. Then we applied this vector in the simulator.
The evaluation was performed for NSFNET, an American backbone network topology of 15 nodes and 23 links [17]; each link had C=32 wavelengths and the transmission bitrate in each wavelength channel was 10 Gbit/s. Besides the results of optimized multi-path routing (MR) we provide, for comparison, the results of two other routing strategies: simple shortest path routing (SPR) and pure alternative routing (AR). We considered two shortest paths per source-destination pair of nodes in MR; they were not necessarily disjoint. In SPR only one path was available, while in the case of AR we considered two different scenarios: with two and six paths available. Uniform traffic matrix and exponential burst inter-arrivals and durations were considered. All the simulation results had 99% level of confidence.
In Figure 6.3 we show the overall burst loss probability results of the MR strategy, which was optimized with the assistance of NL-NRL, NL-RL, and LL-NRL models,
1,E-06
Fig. 6.4 Performance comparison of routing strategies (simulation results)
successively. The characteristics are obtained in the function of offered traffic load, which is normalized to the wavelength bitrate and expressed in Erlangs (e.g., 12.8 Erlangs means that each node generates 128 Gbit/s). As a reference, we provide the results of SPR.
In the studied scenario, we can see that the burst loss probability results of opti-mized MR evaluated in the MATLABenvironment are (almost) the same regardless of the network loss model used. Moreover, the analytical results obtained for NL-NRL model agree very well with simulation results (’(sim)’ in Figure 6.3).
In Figure 6.4 we compare simulation results obtained for different routing sce-narios. We see that the optimized multi-path routing outperforms the shortest path routing in the whole range of traffic loads. Also, it offers at least as good results as the alternative routing if the same number of routing paths is available.
6.5 Discussion
In this section we investigate the accuracy of network loss models and the charac-teristics of the objective function. We also discuss the computational effort of the optimization procedure.
6.5.1 Accuracy of Loss Models
We study the accurary of both NR-NRL and LL-NRL network loss approximations relative to the NL-RL network loss model. To do that we define the approximation error as
Approximation errors relative to NL-RL model
Fig. 6.5 Approximation errors relative to NL-RL model vs. blocking probability; NSFNET, short-est path routing, eight and 32 wavelengths
ErXBX−BNL−RL BNL−RL
, (6.22)
where X refers to either NL−NRL or LL−NRL; so BX means the result of the objective function for model X .
In Figure 6.5 we present the results of ErX obtained in NSFNET network, with a different number of wavelengths per link considered and the shortest path routing used. We can see that the accuracy of both network loss approximate models is very strict for the blocking probability in the network BNL−RLbelow 10−2.
6.5.2 Properties of the Objective Function
NL-RL model. In [10], Kelly demonstrated that the reduced load loss model of a CS network is in general not convex. Taking into account an analogy of the reduced-load calculation in both CS and OBS networks, we can expect that function (6.10) is not convex as well. Therefore, a solution of optimization problem (6.13) may not be unique.
NL-NRL model. As in the case of the RL-NL model, it can be shown numerically that the objective function (6.11) is not necessarily convex; in particular, under high traffic load conditions, two feasible vectors x1, x2can be found such that
BNL−NRL(λx2+ (1−λ)x2)>λBNL−NRL(x2) + (1−λ)BNL−NRL(x1), (6.23) where 0≤λ≤1.
Table 6.1 Comparison of computation times
NL-RL NL-NRL LL-NRL
Network Paths Tol. BLP OF SOLV OF SOLV OF SOLV SIMPLE 2 10−6 2.4·10−3 64 sec 1.5 sec 0.1 sec 1.4 sec 0.1 sec 1.5 sec SIMPLE 4 10−6 2.4·10−3 243 sec 3 sec 0.1 sec 3.4 sec 0.1 sec 3.1 sec NSFNET 2 10−6 4.6·10−2 >5h 0.38 sec 22.3 sec 0.37 sec 24.3 sec NSFNET 4 10−6 3.1·10−2 >5h 1.6 sec 937 sec 1.5 sec 952 sec
EON 2 10−61.76·10−2 >5h 5.5 sec 803 sec 5.3 sec 837 sec EON 2 10−31.77·10−2 >5h 1.1 sec 260 sec 1.0 sec 263 sec
LL-NRL model. An advantageous property of the LL-NRL model is the convexity of its objective function (6.12); a detailed proof can be found in [13]. For this reason, a corresponding optimization problem has a unique solution.
6.5.3 Computational Effort
In Table 6.1 we compare the computation times of both the objective function (with the partial derivatives calculation included) and thefminconsolver function of the MATLABenvironment; in the table they are denoted as OF and SOLV, respec-tively. The evaluation is performed on a Pentium D, 3 GHz computer. The results are obtained for SIMPLE (six nodes, eight links, and 60 paths), NSFNET (15 nodes, 23 links, and 420 paths), and EON (28 nodes, 39 links, and 1,512 paths) network topologies; the number of wavelengths per link is 32, each source-destination pair of nodes has two or four shortest paths available, the traffic load is equal to 25.6 Er-langs and 19.2 ErEr-langs, respectively, for SIMPLE/NSFNET and EON scenarios. In case the iterative procedure of the Erlang fixed-point approximation is used, it ends if the maximal discrepancy between two consecutive link loss calculations is smaller then 10−6. The starting traffic splitting vector is x=0.5·(1, ...,1), meaning that the traffic is equally distributed on the paths for each demand.
We can see that the calculation of the objective function (and of partial deriva-tives) is highly time consuming in the NL-RL model even in a small network sce-nario. In contrast, such a calculation is not an issue if either the NL-NRL or the LL-NRL model is used. It is worth noting that by decreasing the value of a termi-nation tolerance parameter (’Tol.’ in the table), which decides on the termitermi-nation of the solver function, we significantly accelerate the optimization procedure (more than three times) without substantial decrease of routing performance (compare BLP value in both EON scenarios). Moreover, we can see that by increasing the number of paths the computation time of the solver function increases considerably in a larger (NSFNET) network scenario.
6.6 Conclusions
In this chapter we have studied a nonlinear optimization method for the multi-path source routing problem in OBS networks. In this method we calculate a traffic split-ting vector that determines a near-optimal distribution of traffic over rousplit-ting paths.
Since a conventional network loss model of an OBS network is complex, we have in-troduced some simplifications. The proposed models are computationally effective and are still highly accurate compared to the basic model. The obtained formulas for partial derivatives are straightforward and very fast to compute. It makes the proposed nonlinear optimization method a viable alternative to linear programming formulations based on piecewise linear approximations of the cost function.
The simulation results demonstrate that our method effectively distributes the traffic over the network and the overall burst loss probability can be significantly reduced compared with the shortest path routing.
Acknowledgements Part of the results have been achieved during a Short Term Scientific Mission of EU COST action 293 – Graphs and Algorithms in Communication Networks (GRAAL) – and EU COST action 291 – Towards Digital Optical Networks. The work was supported by the Spanish Ministry of Education and Science under the CATARO project (Ref. TEC2005-08051-C03-01).
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