This paper considers a tollbooth queueing system where customers arrive according to a Poisson process and there are two heterogeneous servers of exponential service times. We show that eigenvalues can be found explicitly for the characteristic matrix polynomial associated with the Markov chain characterizing the system. We derive a closed-form solution for the steady state probabilities to make the straightforward computation of performance measures. 
A closed-form solution for a tollbooth tandem queue with two heterogeneous servers and exponential service times