CALCULATING EXIT TIMES FOR SERIES JACKSON NETWORKS - I.

The paper defines a new family of special functions called lattice Bessel functions. They are indexed by the N-dimensional integer lattice such that they reduce to modified Bessel functions when N equals 1, and the exponential function when N equals O. The transition probabilities for an M/M/1 queue going from one state to another before becoming idle (exiting at zero) can be solved in terms of modified Bessel functions. Lattice Bessel functions are used to solve the analogous problem involving the exit time from the interior of the positive orthant of the N-dimensional lattice for a series Jackson network with N nodes. These special functions allow one to derive asymptotic expansions for the taboo transition probabilities, as well as for the tail of the exit time distribution.