In this paper, we present a model for TCP/IP congestion control mechanism. The rate at which data is transmitted increases linearly in time until a packet loss is detected. At this point, the transmission rate is divided by a constant factor. Losses are generated by some exogenous random process which is assumed to be stationary ergodic. This allows us to account for any correlation and any distribution of inter-loss times. We obtain an explicit expression for the throughput of a TCP connection and bounds on the throughput when there is a limit on the window size. In addition, we study the effect of the Timeout mechanism on the throughput. A set of experiments is conducted over the real Internet and a comparison is provided with other models that make simple assumptions on the inter-loss time process. The comparison shows that our model approximates well the throughput of TCP for many distributions of inter-loss times.

We propose to characterize the burstiness of packet arrival processes with indices of dispersion for intervals and for counts. These indices, which are functions of the variance of intervals and counts, are relatively straightforward to estimate and convey much more information than simpler indices, such as the coefficient of variation, that are often used to describe burstiness quantitatively.

In this paper we establish asymptotics for the basic steady-state distributions in a large class of single-server queues. We consider the waiting time, the workload (virtual waiting time) and the steady-state queue lengths at an arbitrary time, just before an arrival and just after a departure. We start by establishing asymptotics for steady-state distributions of Markov chains of M/GI/1 type. Then we treat steady-state distributions in the BMAP/GI/1 queue, which has a batch Markovian arrival process (BMAP). The BMAP is equivalent to the versatile Markovian point process or Neuts (N) process; it generalizes the Markovian arrival process (MAP) by allowing batch arrivals. The MAP includes the Markov-modulated Poisson process (MMPP), the phase-type renewal process (PH) and independent superpositions of these as special cases. We also establish asymptotics for steady-state distributions in the MAP/MSP/1 queue, which has a Markovian service process (MSP). The MSP is a MAP independent of the arrival process generating service completions during the time the server is busy. In great generality (but not always), the basic steady-state distributions have asymptotically exponential tails in all these models. When they do, the asymptotic parameters of the different distributions are closely related. 1.

In this paper we examine three different approximation techniques for modeling packet loss in finite-buffer voice multiplexers. The performance models studied differ primarily in the manner in which the superposition of the voice sources (i.e., the arrival process) is modeled. The first approach models the superimposed voice sources as a renewal process and performance calculations are based only on the first two moments of the renewal process. The second approach is based on modeling the superimposed voice sources as a Markov Modulated Poisson Process (MMPP). Our choice of parameters for the MMPP attempts to capture aspects of the arrival process in an alternate, more intuitive, manner than previously proposed approaches for determining the MMPP parameters and is shown to compute loss more accurately. Finally, we also evaluate a fluid flow approximation for computing packet loss. For all three approaches, we consider as a unifying example, the case of multiplexing voice sou...

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.

Abstract-- We propose a novel approach, called parallel rollout, to solving (partially observable) Markov decision pro-cesses. Our approach generalizes the rollout algorithm of Bertsekas and Castanon (1999) by rolling out a set of multiple heuristic policies rather than a single policy. In particular, the parallel rollout approach aims at the class of problems where we have multiple heuristic policies available such that each policy performs near-optimal for a different set of system paths. Parallel rollout automatically combines the given multiple policies to create a new policy that adapts to the different system paths and improves the performance of each policy in the set. We formally prove this claim for two criteria: total expected reward and infinite horizon discounted reward. The parallel rollout approach also resolves the key issue of select-ing which policy to roll out among multiple heuristic policies whose performances cannot be predicted in advance. We present two example problems to illustrate the effectiveness of the parallel rollout approach: a buffer management prob-lem and a multiclass scheduling problem. I.

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one dimensional martingale results for L'evy processes. This martingale is then applied to various storage processes, queues and Brownian motion models. AMS 1991 Subject Classification 60J30; 60K25 Keywords L' evy process, EOQ model, Markov modulation, multivariate martingale, reflected Brownian motion, storage process Department of Mathematical Statistics; University of Lund; Box 118; S-221 00 Lund; Sweden (asmus@maths.lth.se) y Department of Statistics; The Hebrew University of Jerusalem; Mount Scopus, Jerusalem 91905; Israel (mskella@olive.mscc.huji.ac.il) 1 Introduction With X t being a L'evy process with exponent ' (see Section 2), e ffX t \Gammat'(ff) is the well known Wald martingale. From a recent paper [14], it follows that under certain conditions '(ff) Z t 0...

Extensive studies indicate that traffic in high-speed communication networks exhibits long-range dependence (LRD) and impulsiveness, thus posing new challenges in network engineering. While many models have recently appeared for capturing the traffic LRD, far less models exist that account for impulsiveness as well as LRD. One of the few existing constructive models for network traffic is the celebrated On/Off model, or Alternating Fractal Renewal Process (AFRP). However, while the AFRP results in aggregated traffic with LRD, it fails to capture impulsiveness, yielding traffic with Gaussian marginal distribution. A new constructive model, namely the Extended AFRP (EAFRP), is proposed here, which overcomes the limitations of the AFRP model. We show that, for both single-user and aggregated traffic, it results in impulsiveness and long-range dependence, the LRD being defined here in a generalized sense. We provide queueing analysis of the proposed model, which clearly demonstrates the implications of the impulsiveness in traffic engineering, and validate all theoretical findings based on real traffic data.

We derive the two-dimensional transforms of the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, and allows correlated interarrival times and batch sizes. Numerical results are obtained via two-dimensional transform inversion algorithms based on the Fourier-series method. From the numerical examples we see that predictions of system performance based on transient and stationary performance measures can be quite different. n

This paper is concerned with the explicit evaluation of the steady--state distribution of the waiting time W in a many--server queue with (possibly heterogeneous) servers each having a phase--type service time distribution. Consider first the most classical case GI=PH het =c of renewal arrivals (we later generalize to MAP=PH het =c, i.e. a Markovian arrival process as in Neuts [15]). Denote the interarrival distribution by H and the service time distribution of server i by F i . It is assumed that any F i is phase--type, say with representation (fi i ; S i ) where fi i is a m i --dimensional row vector and S i is a m i \Theta m i --matrix. 2 S. Asmussen, J.R. Mller / Calculation of the waiting time That is, the density is \Gammafi i e S i x S i e (throughout, e denotes the column vector of ones with the appropriate dimension depending on the context). In the homogeneous case F 1 = : : : = F c = F , m i = m, we write GI=PH=c. The service time is FCFS in the sense that the customers form a single line in the order of arrival and joins the next server to become available. In the homogeneous case, this suffices for a complete model description, whereas in the heterogenous case, we also need to describe the rule for the allocation of a server to a customer arriving in a not--all--busy period, that is, a period in which some or all servers are idle. We will consider two situations, one where any of s ! c idle servers is selected with equal probability 1=s, and one where the servers have priorities in the sense that the customer selects server i before server j when i ! j (other Markovian rules can easily be treated with small modifications of the analysis given for these two cases). The waiting time of a customer is the time from his arrival until service starts; it can b...