  We develop entanglement swapping protocols and memory allocation methods for
quantum repeater chains. Unlike most of the existing studies, the memory size
of each quantum repeater in this work is a parameter that can be optimized.
Based on Markov chain modeling of the entanglement distribution process, we
determine the trade-off between the entanglement distribution rate and the
memory size for temporal multiplexing techniques. We then propose three memory
allocation methods that achieve entanglement distribution rates decaying
polynomially with respect to the distance while using constant average memory
slots per node. We also quantify the average number of memory slots required
due to classical communication delay, as well as the delay of entanglement
distribution. Our results show that a moderate memory size suffices to achieve
a polynomial decay of entanglement distribution rate with respect to the
distance, which is the scaling achieved by the optimal protocol even with
infinite memory size at each node.
