Optical Networks - _Problems9_123

2 ? 1 ? 2 ? 1 ? 2 ? 1 ? 2 ? 3 ? 4 ? 3 ? 4 ? 3 ? 4 a b

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2 λ 1 λ 2 λ 1 λ 2 λ 1 λ 2 λ 3 λ 4 λ 3 λ 4 λ 3 λ 4 A B C A B C Figure 10.21 Two different scenarios of wavelength assignment in networks with bidi- rectional links. 10.11 Show that a network having P fiber pairs between nodes and W wavelengths on each fiber with no wavelength conversion is equivalent to a network with one fiber pair between nodes with PW wavelengths, and degree P wavelength conversion capability at the nodes. 10.12 Generalize the example of Figure 10.13 to the case when the number of nodes is arbitrary, say, N . Compare the number of wavelengths required in this general case to the upper bound given by Theorem 10.1. 10.13 In order to prove that W ( 2 L 1 ) M L + 2 in Theorem 10.1, we supposed that there were K lightpaths of length M hops. Instead, suppose there are K(x) lightpaths of length x hops, and derive an upper bound for W that holds for every x . Now, optimize x to get the least upper bound for W . Compare this bound with the bound obtained in Theorem 10.1. 10.14 Show that Algorithm 10.3 always does the wavelength assignment using L wave- lengths. Hint: Use induction on the number of nodes. 10.15 Consider the following modified version of Algorithm 10.3. In step 2, the algorithm is permitted to assign any free wavelength from a fixed set of L wavelengths, instead of the least numbered wavelength. Show that this algorithm always succeeds in performing the wavelength assignment. 10.16 Prove that Theorem 10.3 can be tight in some cases. In other words, give an example of a ring network and a set of lightpath requests and routing with load L that requires 2 L 1 wavelengths. Hint: First, give an example that requires 2 L 2 wavelengths and then modify it by adding an additional lightpath without increasing the load. Note that the example in Figure 10.19 shows such an example for the case L = 2 . Obtain an example for the case L > 2 .
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622 WDM Network Design 10.17 Consider a ring network with a lightpath request set of one lightpath between each source-destination pair. Compute the number of wavelengths sufficient to support this set with full wavelength conversion and without wavelength conversion. What do you conclude from this?
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