An asymptotic analysis of a queueing system with Markov-modulated arrivals
Operations Research
Ordinary CLT and WLLN versions of L=λW
Mathematics of Operations Research
Heavy-traffic analysis of a data-handling system with many sources
SIAM Journal on Applied Mathematics
IEEE/ACM Transactions on Networking (TON)
Proof of a fundamental result in self-similar traffic modeling
ACM SIGCOMM Computer Communication Review
Generating representative Web workloads for network and server performance evaluation
SIGMETRICS '98/PERFORMANCE '98 Proceedings of the 1998 ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems
A central-limit-theorem-based approach for analyzing queue behavior in high-speed networks
IEEE/ACM Transactions on Networking (TON)
Macroscopic models for long-range dependent network traffic
Queueing Systems: Theory and Applications
An invariance principle for semimartingale reflecting Brownian motions in an orthant
Queueing Systems: Theory and Applications
Queueing Systems: Theory and Applications
State space collapse with application to heavy traffic limits for multiclass queueing networks
Queueing Systems: Theory and Applications
Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions
Queueing Systems: Theory and Applications
Heavy traffic limits associated with M/G/∞ input processes
Queueing Systems: Theory and Applications
LIMITS FOR CUMULATIVE INPUT PROCESSES TO QUEUES
Probability in the Engineering and Informational Sciences
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Explicit M/G/1 waiting-time distributions for a class of long-tail service-time distributions
Operations Research Letters
The M/G/1 queue with heavy-tailed service time distribution
IEEE Journal on Selected Areas in Communications
On a Class of Lévy Stochastic Networks
Queueing Systems: Theory and Applications
Heavy-Traffic Limits for Loss Proportions in Single-Server Queues
Queueing Systems: Theory and Applications
Queueing networks with discrete time scale: explicit expressions for the steady state behavior of discrete time stochastic networks
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We review functional central limit theorems (FCLTs) for the queue-content process in a single-server queue with finite waiting room and the first-come first-served service discipline. We emphasize alternatives to the familiar heavy-traffic FCLTs with reflected Brownian motion (RBM) limit process that arise with heavy-tailed probability distributions and strong dependence. Just as for the familiar convergence to RBM, the alternative FCLTs are obtained by applying the continuous mapping theorem with the reflection map to previously established FCLTs for partial sums. We consider a discrete-time model and first assume that the cumulative net-input process has stationary and independent increments, with jumps up allowed to have infinite variance or even infinite mean. For essentially a single model, the queue must be in heavy traffic and the limit is a reflected stable process, whose steady-state distribution can be calculated by numerically inverting its Laplace transform. For a sequence of models, the queue need not be in heavy traffic, and the limit can be a general reflected Lévy process. When the Lévy process representing the net input has no negative jumps, the steady-state distribution of the reflected Lévy process again can be calculated by numerically inverting its Laplace transform. We also establish FCLTs for the queue-content process when the input process is a superposition of many independent component arrival processes, each of which may exhibit complex dependence. Then the limiting input process is a Gaussian process. When the limiting net-input process is also a Gaussian process and there is unlimited waiting room, the steady-state distribution of the limiting reflected Gaussian process can be conveniently approximated.