Multidimensional Residues, Generating Functions, and Their Application to Queueing Networks

Abstract

This paper has two goals: the first is to introduce applied mathematicians to a new technique involving several complex variable residue theory. This is a multidimensional extension of a well-known result for functions of one complex variable [P Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, New York, 1978] that yields useful asymptotic information when applied to generating functions. The authors begin by reviewing the residue theory of one complex variable, noting that there are useful sets of relationships between the residues of certain rational functions. They use the one-dimensional theory to rederive and extend a well-known result of Koenigsberg [P G. Harrison, Oper. Res., 33 (1985), pp. 464–468]. These relationships, to a degree, can be found for residues of the analogous rational functions of several complex variables. The second goal is to apply these tools to the study of product form queueing networks. It is well known that for generating functions of a single variable, residue theory turns out to be a very powerful tool for studying the asymptotic behavior of the coefficients. In this paper, the authors investigate how these ideas extend to the study of multidimensional generating functions. Then they demonstrate these techniques on generating functions which arise from the theory of product form queueing networks. In particular, they compute explicitly the generating function of the partition function for a variety of networks. Then, using these generating functions, they demonstrate the utility and limitations of these techniques.

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