Motivated by recent development in high speed networks, in this paper we study two types of stability problems: (i) conditions for queueing networks that render bounded queue lengths and bounded delay for customers, and (ii) conditions for queueing networks in which the queue length distribution of a queue has an exponential tail with rate `. To answer these two types of stability problems, we introduce two new notions of traffic characterization: minimum envelope rate (MER) and minimum envelope rate with respect to `. Based on these two new notions of traffic characterization, we develop a set of rules for network operations such as superposition, input-output relation of a single queue, and routing. Specifically, we show that (i) the MER of a superposition process is less than or equal to the sum of the MER of each process, (ii) a queue is stable in the sense of bounded queue length if the MER of the input traffic is smaller than the capacity, (iii) the MER of a departure process from a stable queue is less than or equal to that of the input process (iv) the MER of a routed process from a departure process is less than or equal to the MER of the departure process multiplied by the MER of the routing process. Similar results hold for MER with respect to ` under a further assumption of independence. These rules provide a natural way to analyze feedforward networks with multiple classes of customers. For single class networks with nonfeedforward routing, we provide a new method to show that similar stability results hold for such networks under the FCFS policy. Moreover, when restricting to the family of two-state Markov modulated arrival processes, the notion of MER with respect to ` is shown to be

The main contributions of this paper are two-fold. First, we prove fundamental, similarly behaving lower and upper bounds, and give an approximation based on the bounds, which is effective for analyzing ATM multiplexers, even when the traffic has many, possibly heterogeneous, sources and their models are of high dimension. Second, we apply our analytic approximation to statistical models of video teleconference traffic, obtain the multiplexing system's capacity as determined by the number of admissible sources for given cell loss probability, buffer size and trunk bandwidth, and, finally, compare with results from simulations, which are driven by actual data from coders. The results are surprisingly close. Our bounds are based on Large Deviations theory. The main assumption is that the sources are Markovian and time-reversible. Our approximation to the steady state buffer distribution is called "Chernoff-Dominant Eigenvalue" since one parameter is obtained from Chernoff's theorem and t...

Even though ATM seems to be clearly the wave of the future, one performance analysis indicates that the combination of stringent performance requirements (e.g., 10 - 9 cell blocking probabilities), moderate-size buffers and highly bursty traffic will require that the utilization of the network be quite low. That performance analysis is based on asymptotic decay rates of steady-state distributions used to develop a concept of effective bandwidths for connection admission control. However, we have developed an exact numerical algorithm that shows that the effective-bandwidth approximation can overestimate the target small blocking probabilities by several orders of magnitude when there are many sources that are more bursty than Poisson. The bad news is that the appealing simple connection-admissioncontrol algorithm using effective bandwidths based solely on tailprobability asymptotic decay rates may actually not be as effective as many have hoped. The good news is that the statistical multiplexing gain on ATM networks may actually be higher than some have feared. For one example, thought to be realistic, our analysis indicates that the network actually can support twice as many sources as predicted by the effectivebandwidth approximation. That discrepancy occurs because for a large number of bursty sources the asymptotic constant in the tail probability exponential asymptote is extremely small. That in turn can be explained by the observation that the asymptotic constant decays exponentially in the number of sources when the sources are scaled to keep the total arrival rate fixed. We also show that the effective-bandwidth approximation is not always conservative. Specifically, for sources less bursty than Poisson, the asymptotic constant grows exponentially in the numbe...

In this paper, we propose a new methodology based on economic models to provide Quality of Service (QoS) guarantees to competing traffic classes (classes of sessions) in packet networks. We consider an economic model of a packet network where resources are priced. Traffic classes compete for network resources and they purchase them to satisfy their QoS needs. Our contributions are the following: 1) We provide a new definition for QoS provisioning based on economic models (Pareto efficiency). 2) We obtain the set of optimal resource allocations (Pareto optimal) which provide QoS guarantees to competing traffic classes. 3) We show the impact on equilibrium prices and optimal allocations due to traffic load and variability, and QoS requirements. 4) We propose packet scheduling and admission policies to provide QoS guarantees to traffic classes based on available QoS and prices in the network.

Over the last few years, a substantial number of call admission control (CAC) schemes have been proposed for ATM networks. In this article, we review the salient features of some of these algorithms. Also, we quantitatively compare the performance of three of these schemes.

We consider the efficient estimation, via simulation, of very low buffer overflow probabilities in certain acyclic ATM queueing networks. We apply the theory of effective bandwidths and Markov additive processes to derive an asymptotically optimal simulation scheme for estimating such probabilities for a single queue with multiple independent sources, each of which may be either a Markov modulated process or an autoregressive processes. This result extends earlier work on queues with either independent arrivals or with a single Markov modulated arrival source. The results are then extended to estimating loss probabilities for intree networks of such queues. Experimental results show that the method can provide many orders of magnitude reduction in variance in complex queueing systems that are not amenable to analysis.

Observations of both Ethernet traffic and variable bit rate (VBR) video traffic have demonstrated that these traffics exhibit “self-similarity ” and/or infinit l e asymptotic index of dispersion for counts (IDC). We report here on measurements of traffic an a commercialpublic broadband network where similar characteristics have been observed. For the purpose of analysis and dimensioning of the central links of an ATM network we analyse in this paper the performance of a single server queue fed by Gaussian trafic with infjlnite IDC. The analysis leads to an approximation fo,r the performance of a queue in which the arriving trafic is “fractal ” Gaussian and consequently where there does not exist a dominant negative-exponential tail. The term ufractal ” is used here in the sense thtzt the autocovariance of the traffic exhibits self-similarity, that is to say, where the autocovariance of an aggregate of the trafic is the same, or asymptotically the same for large time lags, as the original traffic. We are not concerned with proving or exploiting this self-similarity property as such, but only with performancle analysis techniques which are effective for such processes. In order to be able to test the performance analysis formulae, we show that trafic with the same arutocovariawe as measured an a real network over a wide range of lags (sufficiently wide a range for the traffic to be equivalent from the point of view of queueing performance) can be generated as a mixture of two Gaussian AR(1) processes. In this way we demonstrate that the analytic performance formulae are accurate. 1

The theory of large deviations provides a simple unified basis for statistical mechanics, information theory and queueing theory. The objective of this paper is to use large deviation theory and the Laplace method of integration to provide an simple intuitive overview of the recently developed theory of effective bandwidth for high speed digital networks, especially ATM networks. This includes (i) identification of the appropriate energy function, entropy function and effective bandwidth function of a source, (ii) the calculus of the effective bandwidth functions, (iii) bandwidth allocation and buffer management, (iv) traffic descriptors, and (v) envelope processes and conjugate processes for fast simulations and bounds.

versatile Markovian point process, tail probabilities in queues, asymptotic decay rate, Perron-Frobenius eigenvalue, asymptotic expansion, caudal characteristic curve, heavy traffic In great generality, the basic steady-state distributions in the BMAP / G /1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the derivatives of the Perron-Frobenius eigenvalue δ(z) of the BMAP matrix generating function D(z) at z = 1. We give recursive formulas for the derivatives δ (k) ( 1). The asymptotic expansions provide the basis for efficiently computing the asymptotic decay rates as functions of the traffic intensity, i.e., the caudal characteristic curves. The asymptotic expansions also reveal what features of the model the asymptotic decay rates primarily depend upon. In particular, δ(z) coincides with the limiting time-average of the factorial cumulant generating function (the logarithm of the generating function) of the arrival counting process, and the derivatives δ (k) ( 1) coincide with the asymptotic factorial cumulants of the arrival counting process. This insight is important for admission control schemes in multi-service networks based in part on asymptotic decay rates. The interpretation helps identify appropriate statistics to compute from network traffic data in order to predict performance. 1.

Consider a single-server finite-buffer system. Its stationary random input process is characterized by power spectrum P (!) and input rate steady state distribution f(x). The two functions represent second-order and steady-state input statistics. Here we use the superposition of heterogeneous 2-state Markov chains for construction of P (!) and f(x). The resulting P (!) is a monotone function of j!j, and f(x) is the convolution of heterogeneous binomial functions. The first part of this paper shows how to eliminate the state space explosion in input modeling. Unlike the existing modeling technique which matches 2-state MCs with each individual source, our 2-state MCs are built to statistically match with functions P (!) and f(x) of the aggregate input. The input state space is then reduced by many orders of magnitude. In the second part of this paper, we examine the maximum throughput of a finite-buffer system to support P (!) and f(x) subject to a desired average loss rate L. Our numer...