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This article studies a transient statequeueing model withsequential 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 probabilityand remain in the system to provide service to other customers with the probability. 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.

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