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Martin Day - Although the curves in this figure might resemble the veins of a leaf, it actually has little to do with botany. In reality this is the solution to a partial differential equation arising in a simple queueing network problem. Although the figure does not make it apparent, in some places the solution illustrated fails to have derivatives. This requires the differential equation to be understood in an unusual way, using the method of "viscosity solutions" developed in the 1980s. Modern enhancements to the classical method of characteristic curves allows even such nondifferentiable solutions to be built up from a family of curves as in the illustration. Problems in the design and control of networks can be studied these and other methods from nonlinear differential equations, as well as probability and stochastic processes.

Click here for more information about network control problems and partial differential equations methods.