We study a tandem queueing system with K servers and no waiting space in between. A customer needs service from one server but can leave the system only if all down-stream servers are unoccupied. Such a system is often observed in toll collection during rush hours in transportation networks, and we call it a tollbooth tandem queue. We apply matrix-analytic methods to study this queueing system, and obtain explicit results for various performance measures. Using these results, we can efficiently compute the mean and variance of the queue lengths, waiting time, sojourn time, and departure delays. Numerical examples are presented to gain insights into the performance and design of the tollbooth tandem queue. In particular, it reveals that the intuitive result of arranging servers in decreasing order of service speed (i.e., arrange faster servers at downstream stations) is not always optimal for minimizing the mean queue length or mean waiting time. 
A tollbooth tandem queue with heterogeneous servers