Application of Optimal Control Theory to Dynamic Routing in Data Communication Networks

Abstract

The flow of messages in a message-switched data communication network is modeled in a continuous dynamical state space.The state variables represent message storage at the nodes and the control variables represent message flow rates along the links. A deterministic linear cost functional is defined which is the weighted total message delay in the network when we stipulate that all the message backlogs are emptied at the final time and the inputs are known. The desired minimization of the cost functional results in a linear optimal control problem with linear state and control variable inequality constraints. The remainder of the thesis is devoted to finding the feedback solution to the optimal control problem when all the inputs are constant in time. First, the necessary conditions of optimality are derived and shown to be sufficient. The pointwise minimization in time is a linear program and the optimal control is seen to be of the bangbang variety. Utilizing the necessary conditions it is shown that the feedback regions of interest are convex polyhedral cones in the state space.A method is then described for constructing these regions from a comprehensive set of optimal trajectories constructed backward in time from the final time.Techniques in linear programming are employed freely throughout the development, particularly the geometrical interpretation of linear programs and parametric linear programming.There are several properties of the method which complicate its formulation as a compact algorithm for general network problems. However, in the case of problems involving networks with single destinations and all unity weightings in the cost functional it is shown that these properties do not apply.A computer implementable algorithm is then detailed for the construction of the feedback solution.-2--2a-

Optimal control of the time-varying flows in networks

“…Possibly the most advanced results to date are to be found in [2]. Due to various assumptions still seriously curbing the generality of application of these analyticaî results, less limited numerical solutions have also been sought as an alternative.…”

“…In this model, the control action at a node is analogous to that of a valve [2,4] in the sensé that if / and V are two links incident to node / then increasing the service rate capacity for one direction means decreasing the service rate capacity for the other. This complementary is expressed by the condition:…”

“…A common measure of performance is the sum over a prescribed control period [0, T\ of ail delays incurred by units of commodity for reason of insufficient service capacities at network nodes [2,4]. This total delay is most simply expressed [19] by setting a == 0 in the model just described and Computing, P/-A £ £ x,(n);…”

“…These problems are found in areas as diverse as the routing of data in communication networks [1,2], water management in networks of pipes, canals and reservoirs [3], urban traffic control [4,5], to mention only a few examples.…”

The linear saturated decentralized strategy for constrained flow control is asymptotically optimal

“…Constrained minimum-norm flows play a central role in flow network optimization and find important applications in communication [11,22], water distribution control [19], traffic [12,21], manufacturing [9,27]. In particular, decentralized strategies are appealing in the control of large networks [29,2,[12][13][14][15] in which the existence of a centralized supervisor implies a cost or it is even unrealistic.…”