This paper studies an M/G/1 retrial queue with two types of breakdowns. When the server is idle, it is subject to breakdowns according to a Poisson process with rate δ and it cannot be repaired immediately. While when the server is busy, it may break down according to a Poisson process with rate θ and can be immediately repaired. Firstly, based on embedded Markov chain technique and probability generating function (PGF) method, we present the necessary and sufficient condition for the system to be stable and the PGF of the orbit size at the departure epochs. Secondly, we give the steady-state joint queue length distribution by supplementary variable method, and present some important performance measures and reliability indices. Thirdly, we provide the analysis of sojourn time of an arbitrary customer in the system when the system is in stable state. Finally, some numerical examples are presented to illustrate the effect of the some system parameters on important performance measures and reliability indices.

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As an important tool in the field of mathematics, higher-order derivation problems are widely used in differentials, quantum mechanics, and engineering applications. However, in the electronic computer (EC), due to the existence of the carry in the calculation, the computational efficiency is low when solving the higher-order derivation problem. In response to this problem, the ternary optical computer (TOC) has the advantages of no carry-in and the characteristics of numerous data bits, reconfigurable processors and parallel computing. Solve the higher-order derivation problems with complex operations by constructing multipliers and adders on the TOC platform, and copying multiple composite operator units (COUs). This article introduces the design of the higher-order derivative algorithm based on TOC in detail, the reconfiguration process of the multiplier and adder, and the number of bits of the multiplier and adder required in the implementation is given. Besides, the hardware resources and clock cycles in the operation are analyzed. The feasibility of the implementation scheme is verified by experiments. Compared with the traditional higherorder derivative, the higher-order derivative based on the TOC is superior in time performance, computational efficiency, and processing of complex operations. Due to the limitation of the research stage, the algorithm is only applicable to the function of polynomials, which lays a foundation for the further research of higherorder derivative algorithms, and has certain application significance.

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