An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.

The DNA-directed assembly of nano-objects has been the subject of many recent studies as a means to construct advanced nanomaterial architectures. Although much experimental in silico work has been presented and discussed, there has been no in-depth consideration of the proper design of single-strand sticky termination of DNA sequences, noted as ssST, which is important in avoiding self-folding within one DNA strand, unwanted strand-to-strand interaction, and mismatching. In this work, a new comprehensive and computationally efficient optimization algorithm is presented for the construction of all possible DNA sequences that specifically prevents these issues. This optimization procedure is also effective when a spacer section is used, typically repeated sequences of thymine or adenine placed between the ssST and the nano-object, to address the most conventional experimental protocols. We systematically discuss the fundamental statistics of DNA sequences considering complementarities limited to two (or three) adjacent pairs to avoid self-folding and hybridization of identical strands due to unwanted complements and mismatching. The optimized DNA sequences can reach maximum lengths of 9 to 34 bases depending on the level of applied constraints. The thermodynamic properties of the allowed sequences are used to develop a ranking for each design. For instance, we show that the maximum melting temperature saturates with 14 bases under typical solvation and concentration conditions. Thus, DNA ssST with optimized sequences are developed for segments ranging from 4 to 40 bases, providing a very useful guide for all technological protocols. An experimental test is presented and discussed using the aggregation of Al and CuO nanoparticles and is shown to validate and illustrate the importance of the proposed DNA coding sequence optimization.

An analytical expression of the time-dependent probability distribution of M/D/1/N queues initialised in an arbitrary deterministic state is derived. We also obtain a simple analytical expression of the differential equation governing the transient average traffic which only involves probabilities of boundary states. As a by-product, a closed form solution of the departure rate from the system is also determined.

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