A tandem GI/PH/1→•/PH/1/0 queue with blocking

Tandem queues are widely used in mathematical modeling of random processes describing the operation of manufacturing systems, supply chains, computer and telecommunication networks. Although there exists a lot of publications on tandem queueing systems, analytical research on tandem queues with non-Markovian input is very limited. In this paper, the results of analytical investigation of two-node tandem queue with arbitrary distribution of inter-arrival times are presented. The first station of the tandem is represented by a single-server queue with infinite waiting room. After service at the first station, a customer proceeds to the second station that is described by a single-server queue without a buffer. Service times of a customer at the first and the second server have PH (Phase-type) distributions. A customer, who completes service at the first server and meets a busy second server, is forced to wait at the first server until the second server becomes available. During the waiting period, the first server becomes blocked, i.e., not available for service of customers. We calculate the joint stationary distribution of the system states at the embedded epochs and at arbitrary time. The Laplace-Stieltjes transform of the sojourn time distribution is derived. Key performance measures are calculated and numerical results presented.