Abstract: We study the well-posedness of the Gaver's parallel system sustained by a cold standby unit and attended by a repairman with a single vacation by utilizing the linear operator semigroup theory. It is assumed that the operating times of the units satisfy exponential distributions, the repair times and the vacation time of the repairman satisfy general continuous distributions. By normalizing the system described by differential equations, we convert the system equations into an abstract Cauchy problem in the Banach space through introducing a state space, main operators and their domains. With the help of the Hille-Yosida theorem, Phillips theorem and Fattorini theorem in functional analysis, we prove that the parallel system has a unique and positive time-dependent solution which satisfies probability condition.

Key words: repairman with a single vacation, Gaver parallel system, Dispersive operator, $C_0$-semigroup, quasi-compact operator