Absfruct- The emerging high-speed networks, notably the ATM-based Broadband ISDN, are expected to integrate through statistical multiplexing large numbers of traffic sources having a broad range of burstiness characteristics. A prime instrument for controlling congestion in the network is admission control, which limits calls and guarantees a grade of service determined by delay and loss probability in the multiplexer. We show, for general Markovian traffic sources, that it is possible to assign a notional effective bandwidth to each source which is an explicitly identi-fied, simply computed quantity with provably correct properties in the natural asymptotic regime of small loss probabilities. It is the maximal real eigenvalue of a matrix which is directly obtained from the source characteristics and the admission criterion, and for several sources it is simply additive. We consider both fluid and point process models and obtain parallel results. Numerical results show that the acceptance set for heterogeneous classes of sources is closely approximated and conservatively bounded by the set obtained from the effective bandwidth approximation. Also, the bandwidth-reducing properties of the Leaky Bucket regulator are exhibited numerically. For a source model of video teleconferencing due to Heyman et al. with a large number of states, the effective bandwidth is easily computed. The equivalent bandwidth is bounded by the peak and mean source rates, and is monotonic and concave with respect to a parameter of the admission criterion. Coupling of state transitions of two related asynchronous sources always increases their effective bandwidth. 1.

We show the existence of effective bandwidths for multiclass Markov fluids and other types of sources that are used to model ATM traffic. More precisely,we show that when such sources share a buffer with deterministic service rate, a constraint on the tail of the buffer occupancy distribution is a linear constraint on the number of sources. That is, for a small loss probability one can assume that each source transmits at a fixed rate called its effective bandwidth. When traffic parameters are known, effective bandwidths can be calculated and may be used to obtain a circuit-switched style call acceptance and routing algorithm for ATM networks. The important feature of the effective bandwidth of a source is that it is a characteristic of that source and the acceptable loss probability only.Thus, the effective bandwidth of a source does not depend on the number of sources sharing the buffer nor on the model parameters of other types of sources sharing the buffer.

We analyse the queue Q L at a multiplexer with L inputs. We obtain a large deviation result, namely that under very general conditions lim L!1 L \Gamma1 log P[Q L ? Lb] = \GammaI (b) provided the offered load is held constant, where the shape function I is expressed in terms of the cumulant generating functions of the input traffic. This provides an improvement on the usual effective bandwidth approximation P[Q L ? b] e \Gammaffib , replacing it with P[Q L ? b] e \GammaLI(b=L) . The difference I(b) \Gamma ffi b determines the economies of scale which are to be obtained in large multiplexers. If the limit = \Gamma lim t!1 t t (ffi) exists (here t is the finite time cumulant of the workload process) then lim b!1 (I(b) \Gamma ffi b) = . We apply this idea to a number of examples of arrivals processes: heterogeneous superpositions, Gaussian processes, Markovian additive processes and Poisson processes. We obtain expressions for in these cases. is zero for independent arrivals, but positive for arrivals with positive correlations. Thus economies of scale are obtainable for highly bursty traffic expected in ATM multiplexing.

This paper, together with [1] and [2], opens a new window for the study of queueing performance in a richer, heterogeneous input environment. It offers a unique way to understand the effect of second- and higher-order input statistics on queues, and develops new concepts of traffic measurement, network control and resource allocation for high speed networks in the frequency domain. The technique developed in this paper applies to the analysis of queue response to the individual effects of input power spectrum, bispectrum, trispectrum, and input rate steady state distribution. Our study provides clear evidence that of the four input statistics, the input power spectrum is most essential to queueing analysis. Furthermore, input power in the lowfrequency band has a dominant impact on queueing performance, whereas high-frequency power to a large extent can be neglected. The research reported here was supported by NSF under grant NCR-9015757 and by Texas Advanced Research Program under gr...

In this paper, we focus on simple exponential approximations for steady-state tail probabilities in G/GI/1 queues based on large-time asymptotics. We relate the large-time asymptotics for the steady-state waiting time, sojourn time and workload. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/GI/1 queues. Numerical examples show that the exponential approximations are remarkably accurate at the 90 th percentile and beyond. Key words: queues; approximations; asymptotics; tail probabilities; sojourn time and workload.

Our algorithm, called Mosquito, allows sources to be ignorant of their statistics but offers near-optimal utilisation of the network. Our approach is based on Large Deviation Theory: the large deviation rate-function (entropy) of bursty ATM traffic can be estimated from measurements of traffic activity. The entropy can be used to determine the bandwidth requirement of the traffic. In this paper, we explain the basic ideas behind the algorithm and describe its implementation. We present some results of a statistical investigation of the performance of the Mosquito CAC algorithm comparing it with that of various modifications of the algorithm.

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...

A wideband source in high speed networks is typically represented by a binary random process. In this paper we characterize the second-order properties of each binary source by a multi-state MMPP. A comprehensive numerical study is carried out to identify the individual effect of the source second-order dynamics on the queue length and loss rate. The results can be used to verify the validity of the two-state Markov chain binary source assumption which is commonly made within the framework of input rate control and bandwidth allocation in high speed networks. The concept of input power spectrum is then developed as a unified source characterization for multimedia traffic queueing analyses. The research reported here was supported by NSF under Grants NCR-9009926 and NCR-9015757, and by Texas Advanced Research Program under grant TARP-129. This paper was submitted to IEEE Transaction on Communications in Jan. 1992. 1 Introduction A wideband source in high speed networks is expected ...

In this paper we study a family of queues where the service time B n of customer n depends on the interarrival time I n between customers n \Gamma 1 and n. In particular, we focus on dependencies that arise naturally in the context of communication systems, where the finite speed of the communication links constrains the amount of data that can be received in a given time interval. Specifically, we study queues where the random variables I n and B n exhibit some form of proportionality relation. Such dependencies can have significant impact on system performance and it is, therefore, critical to develop tractable models that account for them. The paper starts with the simple case of a deterministic proportionality relation between the service time of a customer and its preceding interarrival time. This is then extended to allow for the addition of an independent, generally distributed overhead to the service time of each customer. Next, we consider several models that capture the ON-O...

This paper uses generating function approach with spectral decomposition to analyze discrete queues with arrival and service processes characterized by Markov chain (MC). Both generating function and distribution function of the queue are constructed from vanishing and non-vanishing roots. The vanishing roots are used to obtain linear solutions for the boundary probabilities; each non-vanishing root constructs a geometric term in the queue distribution function. The queue asymptotic behavior is expressed in a simple geometric form, which is determined by the minimum non-vanishing root. A key condition for the success of this approach is that all the eigenvalues of both arrival and service MC generating function matrices are distinct and given in explicit analytic form. In order to express eigenvalues in explicit analytic form, both arrival and service MCs must be a special class of MCs which are decomposable into a set of independent MC elements, and each element has no more than four ...