Ward Whitt

We establish functional central limit theorems (FCLTs) for a cumulative input process to a fluid queue from the superposition of independent on–off sources, where the on periods and off periods may have heavy-tailed probability distributions. Variants of these FCLTs hold for cumulative busy-time and idle-time processes associated with standard queueing models. The heavy-tailed on-period and off-period distributions can cause the limit process to have discontinuous sample paths (e.g., to be a non-Brownian stable process or more general Lévy process) even though the converging processes have continuous sample paths. Consequently, we exploit the Skorohod M₁ topology on the function space D of right-continuous functions with left limits. The limits here combined with the previously established continuity of the reflection map in the M₁ topology imply both heavy-traffic and non-heavy-traffic FCLTs for buffer-content processes in stochastic fluid networks.

We establish functional large deviation principles (FLDPs) for waiting and departure processes in single-server queues with unlimited waiting space and the first-in first-out service discipline. We apply the extended contraction principle to show that these processes obey FLDPs in the function space D with one of the nonuniform Skorohod topologies whenever the arrival and service processes obey FLDPs and the rate function is finite for appropriate discontinuous functions. We apply our previous FLDPs for inverse processes to obtain an FLDP for the waiting times in a queue with a superposition arrival process. We obtain FLDPs for queues within acyclic networks by showing that FLDPs are inherited by processes arising from the network operations of departure, superposition, and random splitting. For this purpose, we also obtain FLDPs for split point processes. For the special cases of deterministic arrival processes and deterministic service processes, we obtain convenient explicit expressions for the rate function of the departure process, but not more generally. In general, the rate function for the departure process evidently must be calculated numerically. We also obtain an FLDP for the departure process of completed work, which has important application to the concept of effective bandwidths for admission control and capacity planning in packet communication networks.

Traffic measurements from communication networks have shown that network traffic is quite complex, exhibiting phenomena such as long-tail probability distributions, long-range dependence and self similarity. Thus, in order to design and control new communication networks, we are motivated to consider new source traffic models and new ways to analyze network performance when there are many independent sources each with traffic that can be described by such models. We present a candidate source traffic model: The required bandwidth (arrival rate) as a function of time for each source is represented as the sum of two stochastic processes: (1) a macroscopic (longer-time-scale) level process and (2) a microscopic (shorter-time-scale) within-level variation process. We let the level process be a finite-state semi-Markov process (SMP), allowing general (possibly long-tail) level holding-time distributions, and we let the within-level variation process be a zero-mean piecewise-stationary process. To cope with the added model complexity, we suggest making design and control decisions based on the likelihood that aggregate demand (the input rate from a set of sources) will exceed capacity (the maximum possible output rate). This approach to model analysis avoids the customary queue-ing detail (focus on buffer content and overflow). We propose using transient analysis for control, conditioning on the level and elapsed level holding time of each source. In doing so, we exploit asymptotics associated with multiplexing a large number of sources

We derive the two-dimensional transforms of the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes. Numerical results are obtained via two-dimensional transform inversion algorithms based on the Fourier-series method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different