Physical vs. Virtual (Logical) Networks The physical network refers to the set of optical nodes and fiber links connecting them. The Virtual (Logical) network is an overlay onto the physical (transport) network. It consists of a set of "lightpaths" established between a "subset" of node pairs in the network (see Chart #47 in class notes titled "Optical Network Architectures". Of course the pattern of connectivity depends on the traffic demand between node pairs. Hence in a Virtual topology, a node corresponds to a "routing node" in the network and an "edge" corresponds to a "lightpath". So if two nodes are connected by a "lightpath", we say they can communicate in one 'light hop". Note that if two routers are connected by a lightpath, they become "neighbors" regardless whether or not they are connected directly by a fiber link in the "Physical" topology. Note that it may not be possible to set up lightpaths between all node pairs (due to technological limitations on number of available wavelengths). If two nodes are not connected directly by a lightpath but are connected by a sequence of lightpaths, we say that the two nodes are communicating over "multi-hops". The traffic between non-neighbors needs to be processed electronically at every intermediate router. The traffic between node pairs is routed over the "Virtual" topology in one or more hops. The virtual topology is designed to carry such traffic in a way to "optimize" some performance metric (minimize delay, maximize throughput, etc…). Example Consider an 8-node "Physical Ring" and a 3-dimensional hypercube "Virtual TopologyThe embedded virtual topology looks like thisProblem #1, Homework #5 In this case, the virtual (logical) topology is the 3-dimensional hypercube, where as the Physical topology is a Passive 8-node Star coupler. The average number of hops is found as follows: Without loss of generality pick node 000. It is connected to nodes {100, 010 and 001} by a single hop, it is connected to nodes {110, 011 and 101} by 2 hops and it is connected to node {111} by 3 hops, so the average number of hops is [(3x1) + (3X2) + (1x3)]/7 = 12/7. There are 12 edges in the hypercube (remember each edge correspond to a light path) so 12 wavelengths are needed but since the question states that each link represents the ability to transmit in either direction, a total of 24 wavelengths are needed. The question then states that each waveform can support OC-48 = 2.5 Gbps. Since the traffic is uniform, the total "flow" inside the network is 24(2.5G) = 60Gbps. The question is asking, "what is the maximum end-to-end traffic" that can be supported? That means γ (in our earlier charts on Network of Queues, we referred to γ as the total "external" traffic offered into the network. The total flow = (γ x average path length) and hence, γ ≤ 60G/(12/7) = 35Gbps. Problem #2, Homework #5 Here the Physical network is a 5-node "unidirectional ring". A single wavelength, supporting 1n OC-192 is used between adjacent nodes. At each node there is an ADM and a router. The first Virtual topology is a Bi-directional 5 node ring as shown below L51 L45 L34 L23 L12 1 23 4 5 Physical Topology 1 23 45Virtual TopologyA circuit is a connection for example 1Æ 2 is a circuit and so is 2Æ 1. So from the Virtual topology it is clear that we have a total of 4 circuits. The circuit between 1Æ 2 traverses one link (or segment), which is L12 where as the circuit 2Æ 1 traverses 4 segments namely L23Æ L34Æ L45Æ L51 and similarly for other circuits. Note that we have 5 circuits of length 1 segment and 5 circuits of length 4 segments. The average number of segments is hence = [5(1) + (5x4)]/10 = 5/2. We need to figure out now what is the average # of circuits per segment. Take the segment L12 for example. That segment is used to support circuits 1Æ 2, 3Æ 2 , 4Æ 3, 5Æ 4 and 1Æ 5. Hence segment L12 can support 5 circuits. Similarly for other segments (remember the traffic is uniform). Since the capacity of a wavelength is OC-192, each wavelength can supports roughly 5 circuits (or channels) each at OC-38 (Note that OC-38 is not part of the SONET standard but the question is assuming all partitioning of OC-192 are possible) Now back to the Virtual topology. Remember a node in a Virtual topology, a node corresponds to a "routing node" in the network. Take node for example. The average number of router hops is = [(1x1) + (1x1) + (1x2) + (1x2)]/4 = 3/2. Since there are 10 circuits, and assuming the total external traffic is γ (and the traffic is uniform), the average flow per circuit = (γ)(3/2)/10 = 3γ/20 The flow matrix is found by considering the 'logical links" (There are 10 of them) and finding the flows that go over them. For example f12 = flow between node 1 and node 2 = γ12 + γ13+ γ52 f23 = flow between node 2 and node 3 = γ13 + γ23+ γ24 and so on for other logical links. The Average network delay (Review the charts on flows in network of queues) is given by γγγγγ32030203203.10.11)(3838log1038−=−=−=∑CCfCfTElinksicalijijSimilar analysis can be carried for the case of fully connected Virtual Topology (for that case, the average number path length in router hops is 1). Try it and see me in my office Tuesday if you are having