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

Consider a class of queueing systems which can be modeled by a finite quasi-birth-death (QBD) process. In this paper we develop a powerful computational technique for spectral analyses (i.e. second-order statistics) of output, queue and loss. Emphasis is placed on output power spectrum and input-output coherence function in response to various input power spectral properties and system parameters. The coherence function is defined to measure linear relationship between input and output processes. A key technical contribution of this paper is the exploration of linearity of low frequency traffic flow. Through the evaluation of the coherence function, one can 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. After all, it is the “upstream” output low frequency characteristics that will have most impact on the “downstream” queueing performance. Our study further indicates that the link capacity required by an input process is essentially characterized by its maximum input rate filtered at ω_b