Part III Essay Routing and wavelength assignment in optical networks

In optical networks the vast bandwidth available in an optical bre may be utilized by splitting it into several channels each of a di erent wavelength This allows signals to be routed entirely in the optical domain This es say studies such routings and examines the number of di erent wavelengths needed for irregular networks We nd the number of wavelengths needed in large random networks in terms of the proportion of edges present A simple greedy routing algorithm which uses no wavelength conversion is su cient We give an alternative lower bound on the number of wavelengths needed which performs poorly for large random networks but which is more appro priate for certain existing networks It can sometimes be used to show that heuristically found routings are optimal Introduction Contemporary bre optic networks transmit data at speeds orders of magnitude faster than standard electronic networks But they have as yet only realised a small fraction of their potential If they are to do more than replace copper wires they must go beyond the limitations imposed by electronic technology Currently optical signals are converted into electronic signals whenever they need to be switched and then back into optical signals for transmission Electronic processing becomes a bottleneck in fast networks Routing a signal using exclusively optical techniques may eventually be both cheaper and faster The simplest technique for optical routing and also one of the most promising is wavelength division multiplexing This exploits the ability of an optical bre to carry a number of independent signals of di erent wavelengths These di erent wavelengths can be separated out and switched independently The huge capacities available with optical bres and the simpler routing tech nology lead to design considerations di erent to those for conventional electronic networks In an electronic circuit switched network calls are typically routed as they come on line in order to fully exploit the limited capacities But in an optically routed network there would more commonly be xed routes for calls to take and the routing pattern would only change over longer time scales to cope with changes in average long term tra c This system is called quasi static routing It is well suited for national backbone networks such as that proposed in optical link lightpath routing node end node Figure A wavelength routed optical network A wavelength division multiplexed optical network consists of end nodes and routing nodes connected by bre optic links as in Figure Each link can carry a certain number of wavelengths A routing node can route each wavelength from an input port to an output port independently of other wavelengths it may also change the wavelength of a signal with a wavelength converter An end node consists of one or more tunable optical transmitters and receivers and is connected to a routing node usually by a single link but sometimes by more We are interested in setting up connections between pairs of end nodes assigning to each pair of interest a route and a wavelength in such a way that on any link each wavelength is used by at most one connection We call the collection of routes and wavelengths assigned a routing The practicability of a routing will depend on the number of di erent wavelengths it uses current limits are of the order of tens of wavelengths The e ciency of a network is that proportion of the total throughput able to be carried simultaneously In this context it is the largest proportion of node pairs that may be connected at any one time For example for e ciency each end node must have as many transmitters and receivers as there are other end nodes in the network If in addition there is a single optical link from an end node to its routing node there must be at least this many wavelengths available With less than full e ciency there arises the problem of coordination the transmitting end node and the receiving end node must both be tuned to the correct wavelength Previous Work A permutation routing is a routing in which each end node has one transmitter and one receiver and each transmitter is connected to a single receiver and each reciever to a single transmitter Aggarwal et al consider the problem of constructing networks which can route all permutations by di erent tunings of the receivers and transmitters in the case where each end node is connected to a routing node by a single link They nd bounds on the number of wavelengths necessary and also relate the number of wavelengths to the number of routing nodes Ramaswami and Sivarajan show how to construct a linear programming problem for an arbitrary network in order to maximize the number of simultaneous connections in their approach the number of variables grows exponentially with network size and the constraint of the number of wavelengths is ignored Ramaswami and Svarajan and Zhang and Acampora consider packet switched networks and separate the routing problem into two parts First they connect certain pairs of nodes to form a logical topology by solving a linear pro gramming problem or by a variety of heuristic methods Then they overlay a packet switched network on top of this logical topology an optical route in the physical network corresponding to a single logical link When some node pairs are not connected by an optical route a packet may have to pass through several intermediate optical routes to reach its destination this is called multi hop rout ing Zhang and Acampora simulate various dynamic call admission schemes and nd that although the logical topology is xed in advance and may not change to accomodate new requests there is little loss of throughput Baroni gives some exact results for standard network architectures such as the de Bruijn graph and Shu eNet But such networks are not easily scalable their numerology permits only certain numbers of nodes and links in speci c con gura tions If these new technologies are to be applied to practical large scale networks it is important to understand in general how the number of wavelengths depends on the network topology Baroni Bayvel and Midwinter examine empirically the number of wavelengths needed in random networks This essay will analytically justify their results The Model For simplicity all links will be taken to consist of a single bre which can transmit signals in both directions Each routing node will be identi ed with its end node This might correspond to a full e ciency network in which each end node has many transmitters and receivers and is connected by a large number of bres to its routing node or to a network of e ciency n where n is the number of nodes in which each end node has a single transmitter and receiver and a single bre connecting it to its routing node The mathematical formulation is as follows Let G be a graph assumed to be connected with vertex set V G and edge set E G We de ne a lightpath between vertices v and w in V to be a path in G between v and w together with an associated colour A lightedge is an edge with an associated colour We wish to assign lightpaths to all pairs of distinct vertices so that all lightpaths through a given edge are of di erent colours Let G be the least number of colours for which such an assignment is possible We will examine the relationship between the proportion of edges present in the graph and G Two di erent sorts of bounds on G will be presented Each bound will be de ned motivated by comparison to computer simulated results and justi ed analytically Random graph preliminaries In what follows let n jV j be the order of the graph M jEj the size of the graph N n the maximumnumber of edges in a graph of order n and M N be the proportion of edges present also called the connectivity of G We will be concerned with two types of random graphs G n is the space of random graphs of order n in which each edge is independently present with probability or absent with probability G n M is the space of random graphs of order n and size M in which each graph is equally likely We will write Pn and Pn M for these two cases and supress n and or M when it is clear from the context which is intended In either of the two cases we say that almost all graphs have property Q if limn Pn Q The following result provides a useful link between the two spaces We say that a property Q of graphs is increasing if G H and Q G imply Q H Proposition Let Q be an increasing property If almost all graphs in G have Q then almost all graphs in G b Nc have Q Proof First note that if M M then PM Q PM Q This can be seen by coupling the two spaces Consider adding M edges one by one If Q holds after M edges have been added then it holds when M have been added Suppose it is not the case that almost all graphs in G b Nc have Q Then for some there is a sequence of orders nk such that Pnk bN c Q for all k So