| 1. |
- Asmussen, Sören, et al.
(författare)
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A local limit theorem for random walk maxima with heavy tails
- 2002
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Ingår i: Statistics and Probability Letters. - 0167-7152. ; 56:4, s. 399-404
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Tidskriftsartikel (refereegranskat)abstract
- For a random walk with negative mean and heavy-tailed increment distribution F, it is well known that under suitable subexponential assumptions, the distribution pi of the maximum has a tail pi(x, infinity) which is asymptotically proportional to integral(x)(infinity)F(y,infinity) dy. We supplement here this by a local result showing that pi(x, x + z] is asymptotically proportional to zF(x,infinity). (C) 2002 Elsevier Science B.V. All rights reserved.
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| 2. |
- Asmussen, Sören, et al.
(författare)
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A Note on Skewness in Regenerative Simulation
- 2011
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Ingår i: Communications in statistics. Simulation and computation. - : Informa UK Limited. - 0361-0918 .- 1532-4141. ; 40:1, s. 45-57
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Tidskriftsartikel (refereegranskat)abstract
- The purpose of this article is to show, empirically and theoretically, that performance evaluation by means of regenerative simulation often involves random variables with distributions that are heavy tailed and heavily skewed. This, in turn, leads to the variance of estimators being poorly estimated, and confidence intervals having actual coverage quite different from (typically lower than) the nominal one. We illustrate these general ideas by estimating the mean occupancy and tail probabilities in M/G/1 queues, comparing confidence intervals computed from batch means to various intervals computed from regenerative cycles. In addition, we provide theoretical results on skewness to support the empirical findings.
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| 3. |
- Asmussen, Sören
(författare)
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A probabilistic look at the Wiener-Hopf equation
- 1998
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Ingår i: SIAM Review. - 0036-1445. ; 40:2, s. 189-201
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Tidskriftsartikel (refereegranskat)abstract
- Existence, uniqueness, and asymptotic properties of solutions Z to the Wiener-Hopf integral equation Z(x) = z(x) + integral(-infinity)(x) Z(x - y)F(dy), x greater than or equal to 0, are discussed by purely probabilistic methods, involving random walks, supermartingales, coupling, the Hewitt-Savage 0-1 law, ladder heights, and exponential change of measure.
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| 4. |
- Asmussen, Sören, et al.
(författare)
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An operational calculus for matrix-exponential disributions, with applicaions to a Brownian (q,Q) inventory model
- 1998
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Ingår i: Mathematics of Operations Research. - 0364-765X. ; 23:1, s. 166-176
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Tidskriftsartikel (refereegranskat)abstract
- distribution G on [math not displayed] is called matrix-exponential if the density has the form αeTz s where α is a row vector, T a square matrix and s a column vector. Equivalently, the Laplace transform is rational. For such distributions, we develop an operator calculus, where the key step is manipulation of analytic functions f(z) extended to matrix arguments. The technique is illustrated via an inventory model moving according to a reflected Brownian motion with negative drift, such that an order of size Q is placed when the stock process down-crosses some level q. Explicit formulas for the stationary density are found under the assumption that the leadtime Z has a matrix-exponential distribution, and involve expressions of the form f(T) where [math not displayed].
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| 5. |
- Asmussen, Sören, et al.
(författare)
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Asymptotics for sums of random variables with local subexponential behaviour
- 2003
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Ingår i: Journal of Theoretical Probability. - 1572-9230. ; 16:2, s. 489-518
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Tidskriftsartikel (refereegranskat)abstract
- We study distributions F on [0, infinity) such that for some T less than or equal to infinity F*(2)(x, x + T] similar to 2F(x, x + T]. The case T = infinity corresponds to F being subexponential, and our analysis shows that the properties for T < &INFIN; are, in fact, very similar to this classical case. A parallel theory is developed in the presence of densities. Applications are given to random walks, the key renewal theorem, compound Poisson process and Bellman-Harris branching processes.
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| 6. |
- Asmussen, Sören, et al.
(författare)
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Exact buffer overflow calculations for queues via martingales
- 2002
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Ingår i: Queueing Systems. - 0257-0130. ; 42:1, s. 63-90
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Tidskriftsartikel (refereegranskat)abstract
- Let tau(n) be the first time a queueing process like the queue length or workload exceeds a level n. For the M/M/1 queue length process, the mean Etaun and the Laplace transform Ee(-staun) is derived in closed form using a martingale introduced in Kella and Whitt (1992). For workload processes and more general systems like MAP/PH/1, we use a Markov additive extension given in Asmussen and Kella (2000) to derive sets of linear equations determining the same quantities. Numerical illustrations are presented in the framework of M/M/1 and MMPP/M/1 with an application to performance evaluation of telecommunication systems with long-range dependent properties in the packet arrival process. Different approximations that are obtained from asymptotic theory are compared with exact numerical results.
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| 7. |
- Asmussen, Sören
(författare)
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Extreme value theory for queues via cycle maxima
- 1998
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Ingår i: Extremes. - 1572-915X. ; 1:2, s. 137-168
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Tidskriftsartikel (refereegranskat)abstract
- The present state of extreme value theory for queues is surveyed. The exposition focuses on the regenerative properties of queueing systems, which reduces the problem to the study of the tail of the maximum $overline X( au)$ of the queueing process ${X(t)}$ during a regenerative cycle $ au$. For simple queues, methods for obtaining the distribution of $overline X( au)$ both explicitly and asymptotically are reviewed. In greater generality, the study leads to Wiener-Hopf problems. Extensions to queues in a Markov regime, for example, to those governed by Markov-modulated Poisson arrivals, are also considered.
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| 8. |
- Asmussen, Sören, et al.
(författare)
-
Large deviations and fast simulation in the presence of boundaries
- 2002
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Ingår i: Stochastic Processes and their Applications. - 1879-209X. ; 102:1, s. 1-23
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Tidskriftsartikel (refereegranskat)abstract
- Let c(x) = inf {t > 0: Q(t) greater than or equal to x} be the time of first overflow of a queueing process 1001 over level x (the buffer size) and Z = P(T(X) less than or equal to T). Assuming that {Q(t)) is the reflected version of a Levy process {X(t)} or a Markov additive process, we study a variety of algorithms for estimating z by simulation when the event {tau(X) less than or equal to T} is rare, and analyse their performance. In particular, we exhibit an estimator using a filtered Monte Carlo argument which is logarithmically efficient whenever an efficient estimator for the probability of overflow within a busy cycle (i.e., for first passage probabilities for the unrestricted netput process) is available, thereby providing a way out of counterexamples in the literature on the scope of the large deviations approach to rare events simulation. We also add a counterexample of this type and give various theoretical results on asymptotic properties of Z=P(tau(x) less than or equal to T), both in the reflected Levy process setting and more generally for regenerative processes in a regime where T is so small that the exponential approximation for T(x) is not a priori valid.
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| 9. |
- Asmussen, Sören
(författare)
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Large deviations in rare events simulation: examples, counterexamples and alternatives
- 2002
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Ingår i: Monte-Carlo and Quasi-Monte Carlo Methods 2000. Proceedings of a Conference. - 354042718X ; , s. 1-9
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Konferensbidrag (refereegranskat)abstract
- When simulating small probabilities, say of order 10-6 or less, by importance sampling, an established principle is to choose the importance sampling distribution as close to the conditional distribution given the rare event as possible. Implementing this often leads into large deviations calculations and exponential change of measure. We survey some of the standard examples where this approach works and supplement existing counterexamples with new ones. Difficulties often arise as consequence of reflecting barriers and we present an algorithm which at least in simple cases is able to deal with this problem. Also the case of heavy-tailed distributions is considered
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| 10. |
- Asmussen, Sören, et al.
(författare)
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Loss rates for Lévy processes with two reflecting barriers
- 2007
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Ingår i: Mathematics of Operations Research. - : Institute for Operations Research and the Management Sciences (INFORMS). - 0364-765X .- 1526-5471. ; 32:2, s. 308-321
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- Let {Xt} be a Lévy process which is reflected at 0 and K>0. The reflected process {VtK} is constructed as VtK = V0K + Xt + Lt0 - LtK where{Lt0} and {LtK} are the local times at 0 and K, respectively. We consider the loss rate lK, defined by lK=E KL1K where E K is the expectation under the stationary measure K. The main result of the paper is the identification of lK in terms of K and the characteristic triplet of {Xt}. We also derive asymptotics of lK as K when EXt <0 and the Lévy measure of {Xt} is light-tailed.
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