We present a practical approach to routing and anycasting with near-optimum delays taking into account the processing loads at routers and processing elements of a computer network. To accomplish this, the minimum-delay routing problem formulated by Gallager is generalized into the problem of minimum-delay routing with load balancing to account for processing delays in network nodes (servers and routers). Gallager's theorem for necessary and sufficient conditions for minimum-delay routing is modified to include processing delays and changes of traffic levels at network nodes. The first distributed algorithm for load balanced anycasting and routing in computer networks is presented. This algorithm, named MIDAS, provides approximate solutions to the modified necessary and sufficient conditions for minimum-delay routing. Simulations are use to compare the performance of the new algorithm with the performance of a traditional approach to sever load balancing.