The linearity of low frequency traffic flow: an intrinsic I/O property in queueing systems

Consider a single node queueing system which can be modeled by a finite quasi-birth-death (QBD) process. We present a computational technique for spectral analyses (i.e., second-order statistics) of output, queue, and loss. The emphasis is placed on the performance evaluation of output power spectrum and input-output coherence function with respect to various input power spectral properties and system parameters. The coherence function is defined to measure the linear relationship between input and output processes. Through the evaluation of the coherence function, we identify a so-called nonlinear break frequency, ω_b, under which the low-frequency traffic stay intact via a queueing system. Such a low-frequency I/O linearity plays an important role in characterizing the output process, which may form a partial input to other “downstream” queues of the network. In particular, the unchanged “upstream” low-frequency traffic characteristics are expected to have a significant impact on the “downstream” queues as well. Our numerical analysis examines the sensitivity of ω_b to traffic characteristics and system parameters. The study further indicates that the link capacity requirement of traffic at a given buffer system is essentially characterized by its maximum input rate filtered at ω _b