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Effective bandwidth of general Markovian traffic sources and admission control of high speed networks
 IEEE/ACM Transactions on Networking
, 1993
"... Absfruct The emerging highspeed networks, notably the ATMbased 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 ..."
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Cited by 269 (5 self)
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Absfruct The emerging highspeed networks, notably the ATMbased 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 identified, 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 bandwidthreducing 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.
Effective Bandwidths for Multiclass Markov Fluids and Other ATM Sources
, 1993
"... 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 l ..."
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Cited by 187 (14 self)
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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 circuitswitched 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.
Large Deviations, the Shape of the Loss Curve, and Economies of Scale in Large Multiplexers
, 1995
"... 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 ..."
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Cited by 115 (11 self)
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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.
Queue Response to Input Correlation Functions: Continuous Spectral Analysis
 IEEE/ACM Trans. Networking
, 1993
"... 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 higherorder input statistics on queues, and develops new concepts of traffic measurement, netw ..."
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Cited by 114 (28 self)
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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 higherorder 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 highfrequency power to a large extent can be neglected. The research reported here was supported by NSF under grant NCR9015757 and by Texas Advanced Research Program under gr...
Exponential approximations for tail probabilities in queues, I: waiting times
 Oper. Res
, 1995
"... In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate waiting time, sojourn time and workload. We evaluate the exponential approximations based on th ..."
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Cited by 39 (20 self)
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In this paper, we focus on simple exponential approximations for steadystate tail probabilities in G/GI/1 queues based on largetime asymptotics. We relate the largetime asymptotics for the steadystate 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.
Statistical Properties of a NearOptimal MeasurementBased CAC Algorithm
, 1997
"... Our algorithm, called Mosquito, allows sources to be ignorant of their statistics but offers nearoptimal utilisation of the network. Our approach is based on Large Deviation Theory: the large deviation ratefunction (entropy) of bursty ATM traffic can be estimated from measurements of traffic act ..."
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Cited by 17 (1 self)
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Our algorithm, called Mosquito, allows sources to be ignorant of their statistics but offers nearoptimal utilisation of the network. Our approach is based on Large Deviation Theory: the large deviation ratefunction (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.
On Input State Space Reduction and Buffer Noneffective Region
 In Proc. IEEE INFOCOM
, 1994
"... Consider a singleserver finitebuffer system. Its stationary random input process is characterized by power spectrum P (!) and input rate steady state distribution f(x). The two functions represent secondorder and steadystate input statistics. Here we use the superposition of heterogeneous 2stat ..."
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Cited by 14 (7 self)
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Consider a singleserver finitebuffer system. Its stationary random input process is characterized by power spectrum P (!) and input rate steady state distribution f(x). The two functions represent secondorder and steadystate input statistics. Here we use the superposition of heterogeneous 2state 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 2state MCs with each individual source, our 2state 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 finitebuffer system to support P (!) and f(x) subject to a desired average loss rate L. Our numer...
Second Order Effect of Binary Sources on Characteristics of Queue and Loss Rate
 IEEE
, 1995
"... A wideband source in high speed networks is typically represented by a binary random process. In this paper we characterize the secondorder properties of each binary source by a multistate MMPP. A comprehensive numerical study is carried out to identify the individual effect of the source secondo ..."
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Cited by 13 (8 self)
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A wideband source in high speed networks is typically represented by a binary random process. In this paper we characterize the secondorder properties of each binary source by a multistate MMPP. A comprehensive numerical study is carried out to identify the individual effect of the source secondorder dynamics on the queue length and loss rate. The results can be used to verify the validity of the twostate 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 NCR9009926 and NCR9015757, and by Texas Advanced Research Program under grant TARP129. This paper was submitted to IEEE Transaction on Communications in Jan. 1992. 1 Introduction A wideband source in high speed networks is expected ...
Analysis of a Correlated Queue in a Communication System
 IN PROC. INFOCOM'93
, 1993
"... 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 communicatio ..."
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Cited by 7 (2 self)
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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 ONO...
Bursty Traffic and Finite Capacity Queues
 Annals of Operations Research
, 1996
"... Traffic on highspeed networks is expected to be `bursty'. This has spurred interest in the queueing properties of bursty traffic models, such as Markov additive processes. These processes have by now been studied extensively by many authors, but almost exclusively in the infinite buffer setting. Th ..."
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Cited by 7 (0 self)
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Traffic on highspeed networks is expected to be `bursty'. This has spurred interest in the queueing properties of bursty traffic models, such as Markov additive processes. These processes have by now been studied extensively by many authors, but almost exclusively in the infinite buffer setting. This paper deals with the case of a finitebuffered queue loaded with bursty traffic. It shows how to calculate the exact solution for the queue length distribution and overflow rate in equilibrium, and also how to find efficient upper and lower bounds for these quantities. In addition, the large buffer asymptotic behaviour of the queue length distribution and overflow rate are studied for a queue loaded with general traffic, using large deviation techniques. 1 Work supported in part by the European Commission under ESPRIT LTR project 20113. 1 Introduction This paper extends several results known to hold for the infinitebuffered singleserver queue to the case of a queue with finite waitin...