A common accepted feature of the purple bacteria antenna systems seems to be a ring structure of their subunits LH2 and LH1. We concentrate our investigation on a delivery time of the energy through the subunits LH2 and LH1 of the antenna system to the reaction center. We are dealing with a model system consisting of one ring LH2 and one ring LH1. We have used structure data from Rhodopseudomonas acidophila. We investigate the exciton transfer inside and between LH2 and LH1 rings in the presence of the interaction with a bath. One can expect the incoherent (hopping) regime of the exciton transfer among the LH2 and LH1 rings. The exciton transfer inside the rings LH2 and LH1 is treated in a quasicoherent regime. One is therefore forced to deal with the time development of the full exciton density matrix of the exciton to take into account phase relations given by off-diagonal elements, completing in such a way information given by the site occupation probabilities P-m(t), diagonal elements of the exciton density matrix. Time dependence of the exciton density matrix is governed by dynamic equations which form an extension of the stochastic Liouville equation method with a Haken-Strobl-Reineker parametrization. Some known shortcomings of the original HSR-SLE treatment are removed in our model: (a) we replace a classical stochastic field by a quantum field and (b) we introduce a new parameter A to provide a correct imbalance among the extended states at finite temperatures for long times due to energy relaxation. We discuss the influence of a local energy and a transfer integral heterogeneity, distance, and orientation dependence of the transfer rates between rings, relaxation, etc., on the energy delivery time.