Routing is essential for quantum communication since it controls the effectiveness, dependability, and scalability of quantum networks. Although research on routing algorithms using quantum memories has been published in the literature, there aren't any quantum memories that can simultaneously achieve high fidelity, long storage duration, wide bandwidth, multimode capacity, and high efficiency. Therefore, in this study, we developed the fidelity-guaranteed MEDIRA routing algorithm, utilizing a distributed routing approach in a memoryless quantum network. As a result of the algorithm, we achieve a throughput of 17.2 qubits per second, for a demand fixed between 1 and 20. To the best of our knowledge, this is the first work that utilizes a queuing approach in a memoryless distributed routing algorithm with fidelity assurance.