We study a quantum switch that distributes maximally entangled multipartite states to sets of users. e entanglement switching process requires two steps: rst, each user a empts to generate bipartite entanglement between itself and the switch; and second, the switch performs local operations and a measurement to create multipartite entanglement for a set of users. In this work, we study a simple variant of this system, wherein the switch has in nite memory and the links that connect the users to the switch are identical. Further, we assume that all quantum states, if generated successfully, have perfect delity and that decoherence is negligible. is problem formulation is of interest to several distributed quantum applications, while the technical aspects of this work result in new contributions within queueing theory. Via extensive use of Lyapunov functions, we derive necessary and su cient conditions for the stability of the system and closed-form expressions for the switch capacity and the expected number of qubits in memory.