Optical Networks - _10_5 Maximum Load Dimensioning Models_121

# In this case the task is to compare the number of

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wavelengths on each link would be able to support all the requests. In this case, the task is to compare the number of additional wavelengths that would be required to support the same sets of lightpaths with nonwavelength-converting crossconnects. One shortcoming of this maximum load model is that the number of wave- lengths required may be excessively large in order to support all sets of lightpaths with maximum load L . If we are permitted not to support a small fraction of these sets of lightpaths, it may be possible to considerably reduce the number of wave- lengths required. In this sense, the maximum load model is a worst-case dimensioning method. 10.5.1 Ofﬂine Lightpath Requests In this section, we will survey the results for ofﬂine lightpath requests. Theorem 10.1 [ABC + 94] Given a routing of a set of lightpaths with load L in a network G with M edges, with the maximum number of hops in a lightpath being D , the number of wavelengths sufficient to satisfy this request is W min[ (L 1 )D + 1 , ( 2 L 1 ) M L + 2] . Proof. Observe that each lightpath can intersect with at most (L 1 )D other lightpaths. Thus the maximum degree of the path graph P(G) is (L 1 )D . Any graph with maximum degree can be colored using + 1 colors by a simple greedy coloring algorithm, and hence the path graph can be colored using (L 1 )D + 1 colors. So W (L 1 )D + 1 . To prove the remainder of the theorem, suppose there are K lightpaths of length M hops. The average load due to these lightpaths on an edge is K M M L

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10.5 Maximum Load Dimensioning Models 611 l 1 l 2 l 3 L (a) (b) Figure 10.18 (a) A line network with a set of lightpaths, also called an interval graph. (b) Wave- length assignment done by Algorithm 10.3. so that K L M . Assign L M separate wavelengths to these lightpaths. Next consider the lightpaths of length M 1 hops. Each of these inter- sects with at most (L 1 )( M 1 ) other such lightpaths, and so will need at most (L 1 )( M 1 ) + 1 additional wavelengths. So we have W L M + (L 1 )( M 1 ) + 1 = ( 2 L 1 ) M L + 2 , which proves the theorem. A line network, shown in Figure 10.18, is simply a network of nodes intercon- nected in a line. A sample set of lightpath requests is also shown in the figure. In this case, there is no routing aspect; only the wavelength assignment problem remains. We study this topology because the results will be useful in analyzing ring networks, which are practically important. Our WA-NC problem (see Section 10.2.2) is equivalent to the problem of coloring intervals on a line. The following greedy algorithm accomplishes the coloring using L wavelengths. The algorithm is greedy in the sense that it never backtracks and changes a color that it has already assigned when assigning a color to a new interval. Algorithm 10.3 [Ber76, Section 16.5] 1. Number the wavelengths from 1 to L . Start with the first lightpath from the left and assign to it wavelength 1. 2. Go to the next lightpath starting from the left and assign to it the least numbered wavelength possible, until all lightpaths are colored.
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