Abstract
<p> We develop an exact transient analysis for a fundamental class of continuous time random walks on the two-dimensional integer lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . They arise from a finite sum of lattice vectors, where each one is scaled by an independent Poisson process of some given rate. This sum, added to an initial vector state, results in a continuous time Markov random walk. </p> <p> In [Queueing Syst. 103 (2023), no. 1-2, 1–43], we do this analysis for the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M slash upper M slash 1 slash k"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">M/M/1/k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> queue by using a special one-dimensional random walk. In this paper, we generalize this approach and derive similar closed form solutions for the transition probabilities of our two-dimensional random walk. This is achieved by applying complex analytic techniques directly to its two-dimensional Markovian sample path structure. One outcome is a complete spectral analysis of the Markov matrix-generator for this random walk process. We obtain all these results without the use of generating functions, Laplace transforms, or special functions. </p> <p>Moreover, the study of the underlying group symmetries for these processes gives us an exact transient analysis for many fundamental transportation queueing times that can be modelled as random walk absorption times. As shown in [Queueing Syst. 103 (2023), no. 1-2, 1–43], the one-dimensional case describes the times to rebalance bicycle sharing stations. Our two-dimensional case corresponds to similar times of interest for car rental stations.</p>