A transient symmetry analysis for absorbed random walks on multi-dimensional integer lattices

We develop an exact transient analysis for a special class of continuous time random walks on the d-dimensional integer lattice, driven by an independent family of Poisson processes. We derive closed form solutions to their transition probabilities by applying complex analytic techniques tied directly to their Markovian random walk structure. These results are achieved without the use of generating functions, Laplace transforms, or special functions. Our analysis goes directly from the sample path structure to a spectral decomposition of the eigenvalues for the underlying Markov matrix-generator. These random walks have a canonical structure that leads to a natural set of group actions on the lattice state space. The resulting group symmetries then allow us to solve for the transition probabilities of various stopped random walks. These stopping times correspond to the boundary absorption or interior exit times within distinguished lattice subsets. As a result, these random walk solutions give us an exact transient absorption analysis for a large family of queueing network models. The analysis of this paper extends and simplifies the original work for these transient absorbing probabilities found in papers [J. Appl. Probab. 24 (1987), no. 1, 226–234] and [Theoret. Comput. Sci. 125 (1994), no. 1, 149– 165]. Moreover, papers [Queueing Syst. 103 (2023), no. 1-2, 1–43] and [A transient symmetry analysis for absorbed random walks on two-dimensional integer lattices, 2026] show that one and two dimensional examples of these random walks and queueing networks have transportation service applications.