Exponential Bounds for Queues with Markovian Arrivals

Exponential bounds P[queue ≥ b] ≤ φe^(-γb) are found for queues whose increments are described by Markovian Additive Processes. This is done application of maximal inequalities to exponential martingales for such processes. Through a thermodynamic approach the constant γ is shown to be the decay rate for an asymptotic lower bound for the queue length distribution. The class of arrival processes considered includes a wide variety of Markovian multiplexer models, and a general treatment of these is given, along with that of Markov modulated arrivals. Particular attention is paid to the calculation of the prefactor φ

Many-Sources Large Deviations for Max-Weight Scheduling

In this paper, a many-sources large deviations principle (LDP) for the transient workload of a multi-queue single-server system is established where the service rates are chosen from a compact, convex and coordinate-convex rate region and where the service discipline is the max-weight policy. Under the assumption that the arrival processes satisfy a many-sources LDP, this is accomplished by employing Garcia's extended contraction principle that is applicable to quasi-continuous mappings. For the simplex rate-region, an LDP for the stationary workload is also established under the additional requirements that the scheduling policy be work-conserving and that the arrival processes satisfy certain mixing conditions. The LDP results can be used to calculate asymptotic buffer overflow probabilities accounting for the multiplexing gain, when the arrival process is an average of \emph{i.i.d.} processes. The rate function for the stationary workload is expressed in term of the rate functions of the finite-horizon workloads when the arrival processes have \emph{i.i.d.} increments.Comment: 44 page

On time-to-buffer overflow distribution in a single-machine discrete-time system with finite capacity

A model of a single-machine production system with finite magazine capacity is investigated. The input flow of jobs is organized according to geometric distribution of interarrival times, while processing times are assumed to be generally distributed. The closed-form formula for the generating function of the time to the first buffer overflow distribution conditioned by the initial buffer state is found. The analytical approach based on the idea of embedded Markov chain, the formula of total probability and linear algebra is applied. The corresponding result for next buffer overflows is also given. Numerical examples are attached as well

A Unified Method of Analysis for Queues with Markovian Arrivals

We deal with finite-buffer queueing systems fed by a Markovian point process. This class includes the queues of type M/G/1/N, /G/1/N, PH/G/1/N, MMPP/G/1/N, MAP/G/1/N, and BMAP/G/1/N and is commonly used in the performance evaluation of network traffic buffering processes. Typically, such queueing systems are studied in the stationary regime using matrix-analytic methods connected with M/G/1-type Markov processes. Herein, another method for finding transient and stationary characteristics of these queues is presented. The approach is based on finding a closed-form formula for the Laplace transform of the time-dependent performance measure of interest. The method can be used for finding all basic characteristics like queue size distribution, workload distribution, loss ratio, time to buffer overflow, and so forth. To demonstrate this, several examples for different combinations of arrival processes and characteristics are presented. In addition, the most complex results are illustrated via numerical calculations based on an IP traffic parameterization

Scheduling analysis with martingales

This paper proposes a new characterization of queueing systems by bounding a suitable exponential transform with a martingale. The constructed martingale is quite versatile in the sense that it captures queueing systems with Markovian and autoregressive arrivals in a unified manner; the second class is particularly relevant due to Wold’s decomposition of stationary processes. Moreover, using the framework of stochastic network calculus, the martingales allow for a simple handling of typical queueing operations: (1) flows’ multiplexing translates into multiplying the corresponding martingales, and (2) scheduling translates into time-shifting the martingales. The emerging calculus is applied to estimate the per-flow delay for FIFO, SP, and EDF scheduling. Unlike state-of-the-art results, our bounds capture a fundamental exponential leading constant in the number of multiplexed flows, and additionally are numerically tight

Some aspects of traffic control and performance evaluation of ATM networks

The emerging high-speed Asynchronous Transfer Mode (ATM) networks are expected to integrate through statistical multiplexing large numbers of traffic sources having a broad range of statistical characteristics and different Quality of Service (QOS) requirements. To achieve high utilisation of network resources while maintaining the QOS, efficient traffic management strategies have to be developed. This thesis considers the problem of traffic control for ATM networks. The thesis studies the application of neural networks to various ATM traffic control issues such as feedback congestion control, traffic characterization, bandwidth estimation, and Call Admission Control (CAC). A novel adaptive congestion control approach based on a neural network that uses reinforcement learning is developed. It is shown that the neural controller is very effective in providing general QOS control. A Finite Impulse Response (FIR) neural network is proposed to adaptively predict the traffic arrival process by learning the relationship between the past and future traffic variations. On the basis of this prediction, a feedback flow control scheme at input access nodes of the network is presented. Simulation results demonstrate significant performance improvement over conventional control mechanisms. In addition, an accurate yet computationally efficient approach to effective bandwidth estimation for multiplexed connections is investigated. In this method, a feed forward neural network is employed to model the nonlinear relationship between the effective bandwidth and the traffic situations and a QOS measure. Applications of this approach to admission control, bandwidth allocation and dynamic routing are also discussed. A detailed investigation has indicated that CAC schemes based on effective bandwidth approximation can be very conservative and prevent optimal use of network resources. A modified effective bandwidth CAC approach is therefore proposed to overcome the drawback of conventional methods. Considering statistical multiplexing between traffic sources, we directly calculate the effective bandwidth of the aggregate traffic which is modelled by a two-state Markov modulated Poisson process via matching four important statistics. We use the theory of large deviations to provide a unified description of effective bandwidths for various traffic sources and the associated ATM multiplexer queueing performance approximations, illustrating their strengths and limitations. In addition, a more accurate estimation method for ATM QOS parameters based on the Bahadur-Rao theorem is proposed, which is a refinement of the original effective bandwidth approximation and can lead to higher link utilisation