Self-inflating tire and method of constructing a tire

A tire comprising a tire carcass; an elongate integral air passageway contained within a flexible tire component, the air passageway extending between an air inlet and an air outlet cavity in the flexible tire component, the air passageway extending at least a partial circumferential path around the tire carcass; and a connector assembly inserted within at least a first of the cavities is disclosed. The connector assembly includes a hollow body having a central chamber, a protruding funnel housing portion extending from the hollow body, and a through-channel extending through the funnel housing portion to the central chamber, the through-channel having a sectional profile accommodating receipt of a free end of the strip core therein. Furthermore, a method of constructing a tire is disclosed.