I have some difficulties with drawing up a birth death process for the following special case:
Suppose that customers arrive at a system with a total of $n$ servers according to a Poisson process with rate $\lambda$ and the service time is exponentially distributed with mean 1/$\mu$. When all servers are occupied, customers do not enter the system. In this situation, it is easy to draw up a birth death process and calculate the steady state distribution. But suppose now that a customer, after entering the system, has to search for a free server first and that the search time is inversely proportional to the number of free servers. I do not see how to draw up a birth death process in this case. Any ideas? Thanks in advance!