On Two-Periodic Random Walks with Boundaries

Abstract

Two-periodic random walks are models for the one-dimensional motion of particles in which the jump probabilities depend on the parity of the currently occupied state. Such processes have interesting applications, for instance those in chemical physics where they arise as embedded random walks of a special queueing problem. In this paper, we discuss in some detail first-passage time problems of two-periodic walks, the distribution of their maximum, and the transition functions when the motion of the particle is restricted by one or two absorbing boundaries. For particular applications, we show how our results can be used to derive the distribution of the busy period of a chemical queue and give an analysis of a somewhat weird coin-tossing game.

Keywords:

• Absorbing boundaries
• Chemical queue
• Random walk

Mathematics Subject Classification:

• Primary 60G50
• Secondary 60K25

Notes

Here we have used the relation

A trite calculation shows that G has exactly the same discriminant as F.

It is not really a tiresome combinatorial task, as Conolly et al. (1997)^{[ 3 ]} remarked.

We continue the tradition introduced by William Feller (1968, p. 346)^{[ 5 ]}, who has given them these names.