On the eigenstructure of a class of max-plus linear systems

Various applications in scheduling, such as train timetables and multi-legged locomotion, can be modeled using systems of max-plus linear equations. In this framework, the eigenvalue of the system matrix represents the total cycle time, whereas the eigenvector dictates the steady-state behavior. For a class of concurrent two-state cyclic systems, with direct application to legged locomotion, we present closed-form expressions for the max-plus eigenvalue and eigenvector of the system matrix. Additionally, we probe into the transient properties of this class of max-plus linear systems by computing the coupling time.