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We consider fertile three-state Hard-Core (HC) models with the activityparameter $\lambda>0$ on a Cayley tree. It is known that there exist four typesof such models: wrench, wand, hinge, and pipe. These models arise as simpleexamples of loss networks with nearest-neighbor exclusion. In the case wand ona Cayley tree of order $k\geq2$, exact critical values $\lambda>0$ are foundfor which two-periodic Gibbs measures are not unique. Moreover, we study theextremality of the existing two-periodic Gibbs measures on a Cayley tree oforder two.

Periodic Gibbs measures for three-state hard-core models in the case Wand