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This paper proposes a novel medium access control scheme for low cost, single-hop wireless networks where the source nodes have a transmitter module but no receiver module and hence they can only transmit data to a sink but cannot receive any control signals, like an ACK or NAK, from any other node. The goal of the proposed scheme is to provide QoS (in terms of packet delivery probability) to the nodes in such a network, where the existing schemes like polling or scheduled transmissions, CSMA and ARQ will be ineffective because of the unavailability of a receiver module at the nodes. The proposed scheme uses distributed control and allows the nodes to transmit each packet an optimal number of times at random instants in time within the packet generation interval. We define two optimization problems based on minimizing total network traffic and maximizing the delivery probability of the class of nodes requiring the highest QoS, respectively, and develop mathematical formulae and efficient algorithms to solve them. Numerical analysis and simulation results show that our scheme can provide high QoS to networks of different sizes.