Markov processes are an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered
to them. This paper is concerned with Markovian traffic, i.e., traffic processes
whose inter-arrival times (separating the time points of discrete arrivals) form a
real-valued Markov chain. As such this paper aims to extend the classical results
of renewal traffic, where interarrival times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The
paper derives integral equations for these functions in the transform domain.
These are then specialized to a subclass, TES+, of a versatile class of random sequences, called TES (Transform-Expand-Sample), consisting of marginally uniform autoregressive schemes with modulo-1 reduction, followed by various transformations. TES models are designed to simultaneously capture both first-order
and second-order statistics of empirical records, and consequently can produce
high-fidelity models. Two theoretical solutions for TES+ traffic functions are derived: an operator-based solution and a matric solution, both in the transform
domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to
obtain instructive closed-form representations for two measures of traffic
burstiness: peakedness and index of dispersion, elucidating the relationship
between them.
Correlations in Output and Overflow Traffic Processes in Simple Queues
