In this thesis, we have improved a QoS routing problem using a Lagrangean Relaxation based approach augmented with an efficient primal Heuristic algorithm, called LRH. With the aid of generated Lagrangean multipliers and lower bound indexes, the primal heuristic algorithm of LRH achieves a near-optimal upper-bound solution. In our thesis, we have three major characteristics which are compared with other proposed methods. First, we start to consider user’s perspective and system’s perspective jointly. Second, in our routing procedure, the candidate path set does not need to be prepared in advance and the best paths are generated while solving the subproblems in our approach. Third, our method can both provide the upper bound and lower bound to the problem, this distinguishing feature can help us to verify the performance of our solutions. A performance study delineated that the performance trade-off between accuracy and convergence speed can be manipulated via adjusting the Unimproved Count (UC) parameter in the algorithm. We have drawn comparisons of accuracy and computation time between LRH and the Linear Programming Relaxation (LPR)-based method, under three networks NSFNET, PACBELL, and GTE and three random networks. Experimental results demonstrated that the LRH is superior to the other approach, namely the LPR method in both accuracy and computational time complexity, particularly for larger size networks.
7.1. Future Works
A future work we can do is that we are able to reconfigure the virtual topology to adapt to changing traffic patterns. Some reconfiguration studies on virtual networks have been reported before [7, 12, 9]; however, these studies assumed that the new virtual topology was
known a priori, and were concerned with the cost and sequence of branch-exchange
operations to transform from the original virtual topology to the new virtual topology. We propose a methodology to obtain the new virtual topology, based on optimizing a given objective function, as well as minimizing the changes required to obtain the new virtual topology from the current virtual topology. This approach would result in the minimum number of switch re-tunings, thus minimizing the number of disrupted virtual paths.
Consequently, this approach also minimizes the time it takes to complete the reconfiguration process. Some discussions on the control mechanisms required to perform re-tunings of virtual paths can be found in [21].
In the ideal situation, given a small change in the traffic matrix, we would prefer for the new virtual topology to be largely similar to the previous virtual topology, in terms of the constituent virtual paths and the routes for these virtual paths, i.e., we would prefer to minimize the changes needed to adapt from the existing virtual topology to the new topology.
More formally, it would be preferable if a large number of the δpl variables retain the same
values in the two solutions, without compromising the quality of the solution (in terms of minimizing the congested link utilization).
Let us consider the snapshot of two traffic matrices, λsd1 and λsd2 , taken at two not-too-distant time instants. We assume that there is a certain amount of correlation between
these two traffic matrices. Given a certain traffic, there may be many different virtual topologies, each of which has the same optimal value with regard to the objective function.
But we will terminate after the first such optimal optimal solution is found. Our reconfiguration algorithm finds the virtual topology corresponding to λsd2 which matches
“closest” with the virtual topology corresponding to λsd1 (based on our above definition of
“closeness”).
7.2. Reconfiguration Algorithm
We perform the following sequence of actions:
1). Generate formulations F(1) and (2)F corresponding to traffic matrices λsd1 and λsd2 , respectively, based on the formulation in Section 3.
2). Derive solutions S(1) and S(2), corresponding to F(1) and F(2), respectively.
Denote the variables’ values in (1)S as xp(1) and ywl(1), and those in (2)S as
p(1)
x and ywl(1), respectively. Let the value of the objective function for (1)S and (2)
S be OPT1 and OPT2, respectively.
3). Modify (F(2) to F'(2)) by adding the new constraint
α =OPT2 (10)
This ensures that all the virtual topologies generated by F'(2) would be optimal with
regard to the objective function.
The new objective function for F'(2) is
Minimize: (1)wl wl(2)
w l
y −y
∑∑
(12)Note that the mod operation, x , is a nonlinear function. If we assume that y can only wl take on binary values, then (12) become linear, i.e., if ywl(1)=1 , then
(1) (2) (1 (2))
wl wl wl
y −y ≡ −y ; else if ywl(1)= , then 0 ywl(1)−ywl(2) ≡ ywl(2). Hence, F'(2) may be solved directly.
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