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Abstract In this paper, we discuss the applications of queuing theory in working vacation state and normal busy state, then establish a M/M/1 queuing model with impatient customers and single working vacation. The model provides theoretical guidance for practical problems such as library reservation system. In a queuing system, the customers' arrival follows Poisson distribution, while the service time is a negative exponential distribution. We consider the situation that there is a stationary distribution when the queuing system reaches steady state at any time. Using the constant variation method of differential equation (Bernoulli equation) to solve the stationary equation, the average queue length in two states of probability generation function is obtained. Finally, we improve the model to acquire Q matrix of the single working vacation model, which can be used to calculate the average queue staying time of the system in two states. By construction and improvement of M/M/1/WV queuing models, ingenious proofs of three important theorems are given accordingly, which will definitely make a difference in many practical problems.
Huang et al. (Wed,) studied this question.
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