Departure Processes of a Tandem Network

Consider a 2-stage single-server tandem queue with a MAP to the first stage and the ex- ponential service times. Using the DREB scheme, we formulate the joint queue length process into a single-dimensional level-dependent quasi-birth-death (LDQBD) process with expanding blocks. This allows us to show that the departure process from stage 1 is a MAP with infinite phases or IMAP and that the departure process of any IMAP/M/1 is still an IMAP. We consider a two-stage tandem queueing network. Each stage has a single expo- nential server with an unlimited input buffer. Customers arrive to stage 1 following a Markovian arrival process (MAP). Using the DREB formulation scheme, Lian and Liu (1) construct a single-dimensional level-dependent quasi-birth-death (LDQBD) process with expanding blocks to study the queue length processes and the system sojourn time process. In this paper, we study the busy period and idle period in stage 2. We also study the departure process from each stage. To do this, we introduce a useful arrival process, the IMAP, which stands for infinite dimensional Markov arrival process. MAP with an infinite dimension has not been formally studied in the literature, although it has appeared in a number of works including Sadre and Havercourt (2), Miyazawa (3), Miyazawa and Zhao (4), and Zhang, Heindl, and Smirni (5). One interesting work is by Green (6) on the output process from a MAP/M/1 queue. He finds a set of conditions on the input MAP under which the output process is still a MAP (with a finite dimension). Though in general, the departure process of an MAP/M/1 queue is not a finite MAP, Green restricts his study to the finite cases only. We pick up from where Green left untouched. We found that an IMAP arises naturally as the departure process of a MAP/M/1 queue (Theorem 1), and certain properties of the IMAP are important and are very helpful to queueing analysis. For example, the fact that the departure process from an IMAP/M/1 queue is still an IMAP (Theorem 2) shows that IMAP has good closure properties. As such, it is natural and important to define and discuss IMAP formally. This paper is organized as follows. In Section 2, we define the 2-station tandem net- work. In Section 3, we define IMAP and show that the departure process from IMAP/M/1 is still an IMAP. We conclude the paper in Section 4.