Consider a three layer protocol in which Layer 3 encapsulates Layer 2 and Layer 2 encapsulates Layer 1. Assume minimalist headers with fixed length packets. Assume the following characteristics of the layers: Layer 1, 6 octet address length, 512 octet payload; Layer 2, 4 octet address length , 256 octet payload; Layer 3, 8 octet address length, 1024 octet payload. Note that in the minimalist header arrangement, no error detection or correction will be used; however, there must be a scheme (that you must devise) to allow a multipacket datagram at each separate layer. You may assume that there is some sort of routing or other address translation protocol that will identify which addresses are to be used. Assume that the data communications channel in use do form a data communications network. (Hint: do recall what is needed for a data communications network as contrasted with an arbitrary graph.) 1.1. For each of the three layers, separately calculated, how many items (nodes) can be addressed? Do not simply show an answer, but ex-plain your reasoning (hint: combinatorics). 1.2. We have discussed functors as a formal, theoretical description be-tween layers of a network. In terms of these three layers, illustrate the functors between the layers, and explain how functors address the di?erences in topology and relevant information content at the di?erent layers 1.3. Assume that the datagram has 5 layer 1 packets. How does this datagram encapsulate in layer 2? 1.4. Taking the above result, or a made-up one of your choosing if you cannot calculate such a result, how does this data from layer 2 en-capsulating layer 1 data then encapsulate in layer 3? 1.5. Assuming that the only information of interest is the payload of layer 1, what is the overall e?ciency of the final encapsulated data stream in layer 3 for the layer 1 datagram described above? 1.6. Assuming that only the information in the starting layer 1 datagram payload is the signal, and the rest of all packets at all layers is noise with respect to this very restricted view of the information content of a channel, what is the e?ective Shannon-Hartley theorem relation between “channel capacity” and “bandwidth” for this specific data-gram? 2. . [50 points] Consider a set of intersecting rings as in the following figure. Here, a small square represents a node, including nodes that can transfer packets between rings, and each ring has an arrow that indicates the di-rection of packet flow on that ring. Each ring is labeled by a lower case Greek letter with the first ring labeled. 2.1. Display the adjacency matrix for the network in the figure. 2.2. Which, if any, nodes are equivalent on the network and why? (Hint: recall that equivalence in this context has to do with the number of links to a node as well as the information flow directions to and from a node.) 2.3. Which nodes, if any, represent single points of failure of the network? 2.4. Display the weight matrix for the network. 2.5. Using Dykstra’s algorithm, calculate a route from node S to node D on the network as displayed in Fig. 1. Show each step of the algorithm as you develop the route.

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