scholarly journals FAST SIMULATION OF A QUEUE FED BY A SUPERPOSITION OF MANY (HEAVY-TAILED) SOURCES

2002 ◽  
Vol 16 (2) ◽  
pp. 205-232 ◽  
Author(s):  
Nam Kyoo Boots ◽  
Michel Mandjes
Keyword(s):  
Short Range ◽  
Fast Simulation ◽  
Change Of Measure ◽  
Simulation Techniques ◽  
Performance Metric ◽  
Overflow Probability ◽  
Link Rate ◽  
Heavy Tailed ◽  
Exponential Change ◽  
Optimal Change

We consider a queue fed by a large number, say n, on–off sources with generally distributed on- and off-times. The queueing resources are scaled by n: The buffer is B ≡ nb and the link rate is C ≡ nc. The model is versatile. It allows one to model both long-range-dependent traffic (by using heavy-tailed on-periods) and short-range-dependent traffic (by using light-tailed on-periods). A crucial performance metric in this model is the steady state buffer overflow probability.This probability decays exponentially in n. Therefore, if n grows large, naive simulation is too time-consuming and fast simulation techniques have to be used. Due to the exponential decay (in n), importance sampling with an exponential change of measure goes through, irrespective of the on-times being heavy or light tailed. An asymptotically optimal change of measure is found by using large deviations arguments. Notably, the change of measure is not constant during the simulation run, which is different from many other studies (usually relying on large buffer asymptotics).Numerical examples show that our procedure improves considerably over naive simulation. We present accelerations, we discuss the influence of the shape of the distributions on the overflow probability, and we describe the limitations of our technique.

scholarly journals Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting

1999 ◽  
Vol 29 (2) ◽  
pp. 197-214 ◽  
Author(s):  
Rudolf Grübel ◽  
Renate Hermesmeier
Keyword(s):  
Numerical Evaluation ◽  
Change Of Measure ◽  
Exponential Tilting ◽  
Transform Methods ◽  
Compound Distributions ◽  
Heavy Tailed Distributions ◽  
Panjer Recursion ◽  
Heavy Tailed ◽  
Suitable Change

AbstractNumerical evaluation of compound distributions is one of the central numerical tasks in insurance mathematics. Two widely used techniques are Panjer recursion and transform methods. Many authors have pointed out that aliasing errors imply the need to consider the whole distribution if transform methods are used, a potential drawback especially for heavy-tailed distributions. We investigate the magnitude of aliasing errors and show that this problem can be solved by a suitable change of measure.

An exponential change of measure for the Feller diffusion and some implications for superprocesses

2000 ◽  
Vol 37 (1) ◽  
pp. 236-245
Author(s):  
Robert J. Adler ◽  
Srikanth K. Iyer
Keyword(s):  
Finite Time ◽  
Total Mass ◽  
Probability Measures ◽  
Change Of Measure ◽  
Exponential Functions ◽  
Branching Diffusion ◽  
Exponential Change

Let Xt be a Feller (branching) diffusion with drift αx. We consider new processes, the probability measures of which are obtained from that of X via changes of measure involving suitably normalized exponential functions of with λ > 0. The new processes can be thought of as ‘self-reinforcing’ versions of the old.Depending on the values of α, T and λ, the process under the new measure is shown to exhibit explosion in finite time. We also obtain a number of other results related to the new processes.Since the Feller diffusion is also the total mass process of a superprocess, we relate the finite-time explosion property to the behaviour of superprocesses with local self-interaction, and raise some interesting questions for these.

scholarly journals Finite-time ruin probabilities under large-claim reinsurance treaties for heavy-tailed claim sizes

2020 ◽  
Vol 57 (2) ◽  
pp. 513-530
Author(s):  
Hansjörg Albrecher ◽  
Bohan Chen ◽  
Eleni Vatamidou ◽  
Bert Zwart
Keyword(s):  
Finite Time ◽  
Heavy Tails ◽  
Asymptotic Expression ◽  
Sample Path ◽  
Rare Event ◽  
Large Claim ◽  
Simulation Techniques ◽  
Insurance Portfolio ◽  
Regularly Varying ◽  
Heavy Tailed

AbstractWe investigate the probability that an insurance portfolio gets ruined within a finite time period under the assumption that the r largest claims are (partly) reinsured. We show that for regularly varying claim sizes the probability of ruin after reinsurance is also regularly varying in terms of the initial capital, and derive an explicit asymptotic expression for the latter. We establish this result by leveraging recent developments on sample-path large deviations for heavy tails. Our results allow, on the asymptotic level, for an explicit comparison between two well-known large-claim reinsurance contracts, namely LCR and ECOMOR. Finally, we assess the accuracy of the resulting approximations using state-of-the-art rare event simulation techniques.

scholarly journals Overflow behavior in queues with many long-tailed inputs

2000 ◽  
Vol 32 (4) ◽  
pp. 1150-1167 ◽  
Author(s):  
Michel Mandjes ◽  
Sem Borst
Keyword(s):  
Buffer Size ◽  
Qualitative Behavior ◽  
Input Rate ◽  
Substantial Fraction ◽  
Fluid Queue ◽  
Overflow Probability ◽  
Typical Time ◽  
Link Rate ◽  
Regularly Varying ◽  
Insight Into

We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B and link rate C are scaled by n, so that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where the on periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull.The results show a sharp dichotomy in the qualitative behavior, depending on the shape of the function v(t) := - logP(A* > t) for large t, A* representing the residual on period. If v(.) is regularly varying of index 0 (e.g., Pareto, Lognormal), then, during the path to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will be ‘more than linear’ in the buffer size. In contrast, if v(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is roughly proportional to the buffer size.In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribute at their mean rates. These observations lead to approximations for the overflow probability. The approximations may be extended to the case of heterogeneous sources. The results provide further insight into the so-called reduced-load approximation.

An exponential change of measure for the Feller diffusion and some implications for superprocesses

2000 ◽  
Vol 37 (01) ◽  
pp. 236-245
Author(s):  
Robert J. Adler ◽  
Srikanth K. Iyer
Keyword(s):  
Finite Time ◽  
Total Mass ◽  
Probability Measures ◽  
Change Of Measure ◽  
Exponential Functions ◽  
Branching Diffusion ◽  
Exponential Change

Let X t be a Feller (branching) diffusion with drift αx. We consider new processes, the probability measures of which are obtained from that of X via changes of measure involving suitably normalized exponential functions of with λ > 0. The new processes can be thought of as ‘self-reinforcing’ versions of the old. Depending on the values of α, T and λ, the process under the new measure is shown to exhibit explosion in finite time. We also obtain a number of other results related to the new processes. Since the Feller diffusion is also the total mass process of a superprocess, we relate the finite-time explosion property to the behaviour of superprocesses with local self-interaction, and raise some interesting questions for these.

Sign in / Sign up

Export Citation Format

Share Document