An M/G/1 queueing model with k sequential heterogeneous service steps and vacations in the transient state

This article studies a transient state M/G/1 queueing model with k sequential heterogeneous service phase and vacations. According to a Poisson process, entrants, each arrival is serviced sequentially in k phases in the order of arrival. The service times follow a general distribution function. Upon completion of the service, the server could take a vacation with the probability θ and remain in the system to provide service to other customers with the probability 1−θ. For this model, first, we obtain the Laplace transform of the probability generating functions. The probability generating functions of system sizes are obtained for a special case in the transient state. The mean system size is obtained by using numerical analysis. Then, the sensitivity of the model is considered.