Moderate Deviations for Queues in Critical Loading

We establish logarithmic asymptotics of moderate deviations for the processes of queue length and waiting times in single server queues and open queueing networks in critical loading. Our results complement earlier heavy-traffic approximation results

About the cumulative idle time in multiphase queues

The paper is designated to the analysis of queueing systems, arising in the network theory and communications theory (called multiphase queueing systems, tandem queues or series of queueing systems). Also we note that multiphase queueing systems can be useful for modelling practical multi-stage service systems in a variety of disciplines, especially on manufacturing (assembly lines), computer networking (packet switch structures), and in telecommunications (e.g. cellular mobile networks), etc. This research presents heavy traffic limit theorems for the cumulative idle time in multiphase queues. In this work, functional limit theorems are proved for the values of important probability characteristics of the queueing system (a cumulative idle time of a customer)

Sample path large deviations for multiclass feedforward queueing networks in critical loading

We consider multiclass feedforward queueing networks with first in first out and priority service disciplines at the nodes, and class dependent deterministic routing between nodes. The random behavior of the network is constructed from cumulative arrival and service time processes which are assumed to satisfy an appropriate sample path large deviation principle. We establish logarithmic asymptotics of large deviations for waiting time, idle time, queue length, departure and sojourn-time processes in critical loading. This transfers similar results from Puhalskii about single class queueing networks with feedback to multiclass feedforward queueing networks, and complements diffusion approximation results from Peterson. An example with renewal inter arrival and service time processes yields the rate function of a reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Control of the multiclass G/G/1G/G/1 queue in the moderate deviation regime

A multi-class single-server system with general service time distributions is studied in a moderate deviation heavy traffic regime. In the scaling limit, an optimal control problem associated with the model is shown to be governed by a differential game that can be explicitly solved. While the characterization of the limit by a differential game is akin to results at the large deviation scale, the analysis of the problem is closely related to the much studied area of control in heavy traffic at the diffusion scale.Comment: Published in at http://dx.doi.org/10.1214/13-AAP971 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Moderate deviations of many-server queues in the Halfin-Whitt regime and weak convergence methods

This paper obtains logarithmic asymptotics of moderate deviations of the stochastic process of the number of customers in a many--server queue with generally distributed interarrival and service times in the Halfin--Whitt heavy traffic regime. The deviation function is expressed in terms of the solution to a Fredholm equation of the second kind. The proof uses characterisation of large deviation relatively compact sequences of probability measures as exponentially tight ones

Sample path large deviations for single and multi class queues in the many sources asymptotic

In this thesis we consider prove large deviations results for two kinds of queuing systems. In the first case, we consider a queuing system fed by traffic from N independent and identically distributed marked point processes. We establish novel one-dimensional large deviations results for such a system in the previously unexplored lightly loaded case (the load vanishes as N → ∞). This case requires the introduction of novel speed scalings for such queueing systems. We also prove some important properties about the sample paths of such systems in the scaled uniform topology. However, we are unable to prove sample path large deviations principles in this case because the log moment-generating function in this case is not steep, and we are unable to find tools in the literature that enable us to deal with such scenarios. This part of the work is done using the framework introduced by Cruise [1] and Cruise et al. [2] to explore this scaling. In the second case, we consider a two-class queuing network, with each class fed by traffic from N independent and identically distributed marked point processes. We introduce a new, probabilistic interpretation of state-space collapse, and show that under a given scaling of the system, the probability of the vector of stationary queue lengths being a given distance from the identity line in R2 decreases exponentially as the distance increases, and therefore the most likely sample paths are those which stay close to the identity line in R2.James Watt scholarshi