Extracting asymptotics from multivariate generating functions is an open problem even for apparently nice functions, such as algebraic or even rational functions. A queuing theory application, partially analyzed by Bertozzi and McKenna (1993), concerns rational generating functions whose denominators are products of linear terms. This case, which may be solved in a completely algorithmic manner, is the subject of my talk. Asymptotics are extracted by computing iterated residues. It is necessary to determine the topology of the complement of the set of (complex) poles. Stratified Morse Theory was practically invented with this application in mind. I will briefly summarize the necessary tools, which may be found in Goresky and MacPherson (1998). I will then show how to apply these to problems of combinatorial interest, such as asymptotic analysis for queuing problems and lattice enumeration (the latter problem is taken from de Loera and Sturmfels 2001).