A Matrix Solution for the Two Server Queue with Overflow - Disney, Ralph L. (Management Science, 1972)
A complete solution to the two-server Markov queue in which arrivals to the second server are the stream of overflows from the first server is given. It is shown that the solution can be determined from a particular second order difference equation in vectors subject to boundary conditions defined by the problem. An example provides more details. A means of finding various marginal and conditional probabilities is given.