Transient distribution of the length of a queueing model with input and service dependent on the system state

A queueing system is studied where the arrival times α and service times β are independent and not identically distributed random variables. Both the distribution of every α and the distribution of every β depend on the transient state of the system. Clearly, the queueing system is a generalization of GI/G/1 queue. Using the supplementary variable technique for stochastic models, a multi-dimensional Markov process which includes the transient queue length of the queue is constructed. Then, the transient distribution of the queue length is obtained.