Sizing exit buffers in ATM networks: an intriguing coexistence of instability and tiny cell loss rates

This paper deals with the sizing of end buffers in ATM networks for sessions subject to constant bit rate (CBR) traffic. Our objective is to predict the cell-loss rate at the end buffer as a function of the system parameters. We introduce the D+G/D/1 queue as a generic model to represent exit buffers in telecommunications networks under constant rate traffic, and use it to model the end buffer. This is a queue whose arrival rate is equal to its service rate and whose arrivals are generated at regular intervals and materialize after a generally distributed random amount of time. We reveal that under the infinite buffer assumption, the system possesses rather intriguing properties: on the one hand, the system is unstable in the sense that the buffer content is monotonically nondecreasing as a function of time. On the other hand, the likelihood that the buffer contents will exceed certain level B by time t diminishes with B. Improper simulation of such systems may therefore lead to false results. We turn to analyze this system under finite buffer assumption and derive bounds on the cell-loss rates. The bounds are expressed in terms of simple formulae of the system parameters. We carry out the analysis for two major types of networks: (1) datagram networks, where the packets (cells) traverse the network via independent paths and (2) virtual circuit networks, where all cells of a connection traverse the same path. Numerical, examination of ATM-like examples show that the bounds are very good for practical prediction of cell loss and the selection of buffer size.