A Queue with Semi-periodic Traffic

This paper analyzes the diffusion limit of a discrete time queueing system with constant service rate and connections that randomly enter and depart the system. Each connection generates periodic traffic while it is active, and a connection’s lifetime has finite mean. This can model a TDMA system with constant bit rate connections. The diffusion scaling retains the semiperiodic behavior, allowing for short time (within one frame) and long time (multiple frames) analysis in the limit. Weak convergence of the cumulative arrival process and stationary buffer length distribution is proved. It is shown that the limit of the cumulative arrival process can be viewed as a discretetime stationary increment Gaussian process interpolated by Brownian bridges. Bounds on the overflow probability of the limit queueing process as a function of the arrival rate and connection lifetime distribution are presented. Numerical and simulation results are presented for geometrically distributed connection lifetimes.