Reduced-load equivalence for queues with Gaussian input

In this note, we consider a queue fed by a number of independent heterogeneous Gaussian sources. We study under what conditions a reduced load equivalence holds, i.e., when a subset of the sources becomes asymptotically dominant as the buffer size increases. For this, recent results on extremes of Gaussian processes are combined with de Haan theory. We explain how the results of this note relate to square root insensitivity and moderately heavy tail

Applications of factorization embeddings for L

We give three applications of the first factorization identity for

Extremes of Markov-additive processes with one-sided jumps, with queueing applications

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive pro- cess with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is \xe2\x80\x98attained\xe2\x80\x99. For this, we investigate discrete-time Markov-additive pro- cesses and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. Then we turn to networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property

On spectral simulation of fractional Brownian motion

This paper focuses on simulating fractional Brownian motion (fBm). Despite the availability of several exact simulation methods, attention has been paid to approximate simulation (i.e., the output is approximately fBm), particularly because of possible time savings. In this paper, we study the class of approximate methods that are based on the spectral properties of fBm's stationary incremental process, usually called fractional Gaussian noise (fGn). The main contribution is a proof of asymptotical exactness (in a sense that is made precise) of these spectral methods. Moreover, we establish the connection between the spectral simulation approach and a widely used method, originally proposed by Paxson, that lacked a formal mathematical justification. The insights enable us to evaluate the Paxson method in more detail. It is also shown that spectral simulation is related to the fastest known exact method

On asymptotically efficient simulation of large deviation probabilities

Consider a family of probabilities for which the decay is governed by a large deviation principle. To find an estimate for a fixed member of this family, one is often forced to use simulation techniques. Direct Monte Carlo simulation, however, is often impractical, particularly if the probability that should be estimated is extremely small. Importance sampling is a technique in which samples are drawn from an alternative distribution, and an unbiased estimate is found after a likelihood ratio correction. Specific exponentially twisted distributions were shown to be good sampling distributions under fairly general circumstances. In this paper, we present necessary and sufficient conditions for asymptotic efficiency of a single exponentially twisted distribution, sharpening previously established conditions. Using the insights that these conditions provide, we construct an example for which we explicitly compute the `best' change of measure. However, simulation using the new measure faces exactly the same difficulties as direct Monte Carlo simulation. We discuss the relation between this example and other counterexamples in the liturature. We also apply the conditions to find necessary and sufficient conditions for asymptotic efficiency of the exponential twist in a Mogul'skii sample-path problem. An important special case of this problem is the probability of ruin within finite time

Fast simulation of overflow probabilities in a queue with Gaussian input

In this paper, we study a queue fed by a large number $n$ of independent discrete-time Gaussian processes with stationary increments. We consider the {it many sources} asymptotic regime, i.e., the buffer exceedance threshold $B$ and the service capacity $C$ are scaled by the number of sources ($Bequiv nb$ and $Cequiv nc$). We discuss three methods for simulating the steady-state probability that the buffer threshold is exceeded: the single twist method (suggested by large deviation theory), the cut-and-twist method (simulating timeslot by timeslot), and the sequential twist method (simulating source by source). The asymptotic efficiency of these three methods is investigated as $n oinfty$: for instance, a necessary and sufficient condition is derived for the efficiency of the method based on a single exponential twist. It turns out that this method is asymptotically inefficient in practice, but the other two methods are asymptotically efficient. We evaluate the three methods by performing a simulation study

Quasi-product forms for L

We study stochastic tree fluid networks driven by a multidimensional