Optical Networks - _10_5 Maximum Load Dimensioning Models_121

In this case the task is to compare the number of

Info icon This preview shows pages 2–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 2

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 3
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern