Transient queue length distribution of the shortest queue model

In this paper, we consider a service systems consisting of two parallel servers. Each server has a queue with infinite capacity. The arrival process of customers is a renewal process and the service times of customers are independent and exponentially distributed with different parameter in different queue. A new arrival join the shortest of two queues, where in case of both queues have equal length, the arrival join any of the two queues according to some arbitrary probability distribution. Jockeying between the queues is not allowed. By Markov skeleton processes theory, we obtain the transient queue length distribution, and show that it is the minimal nonnegative solution of a backward equation.