Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter ‘residual’ probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.      

Dielectric behaviour of La substituted BPZT ceramics

Here, we report dielectric behaviour of lanthanum substituted Ba_0.80Pb_0.20Ti_0.90Zr_0.10O₃ (BPZT) ceramics. The material series with compositional formula Ba_0.80−xLa_xPb_0.20Ti_0.90Zr_0.10O₃ (BLPZT) with x varying from 0 to 0.01 in the steps of 0.0025 was chosen for investigations. The material was synthesized by solid state reaction method. Reacted powder compacted in form of circular discs were sintered at 1325 °C. All the samples were subjected to X-ray diffraction (XRD) analysis and found to be single phase. Dielectric behaviour was studied as a function of frequency and temperature and Curie temperature (T_c) was determined. T_c was found to decrease with increasing x. The details are discussed and presented in this paper.     

Erratum to: Determination of metal ion content of beverages and estimation of target hazard quotients: a comparative study

This is a correction to the following paper: Hague T, Petroczi A, Andrews PR, Barker J, Naughton DP: Determination of metal ion content of beverages and estimation of target hazard quotients: a comparative study. Chem Central J 2008, 2:13.     

Attempts to prepare aromatic O-acyl-hydroxylamines—I : Dehydrohalogenation of 1-bromo-4-acetoxyimino-1,2,3,4-tetrahydrophenanthrene by tetramethylammonium dimethylphosphate

The dehydrohalogenation of 1-bromo-4-acetoxyimino-1,2,3,4-tetrahydrophenanthrene by tetramethylammonium dimethylphosphate gave a mixture of products. Ten of these were identified and together account for 70% of the total yield. The proposed reaction mechanism implicates the corresponding O-acetyl-hydroxylamine 15 as an intermediate. Attempts to prepare O-esters or O-ethers of aromatic hydroxylamines by dehydrogenation, with quinones, of the corresponding oximino-derivatives of oxo-1,2,3,4-tetrahydro-naphthalene and oxo-1,2,3,4-tetrahydro-phenanthrene were without success.     

The concert queueing game: strategic arrivals with waiting and tardiness costs

We consider the noncooperative choice of arrival times by individual users, who seek service at a first-come first-served queueing system that opens up at a given time. Each user wishes to obtain service as early as possible, while minimizing the expected wait in the queue. This problem was recently studied within a simplified fluid-scale model. Here, we address the unscaled stochastic system, assuming a finite (possibly random) number of homogeneous users, exponential service times, and linear cost functions. In this setting, we establish that there exists a unique Nash equilibrium, which is symmetric across users, and characterize the equilibrium arrival-time distribution of each user in terms of a corresponding set of differential equations. We further establish convergence of the Nash equilibrium solution to that of the associated fluid model as the number of users is increased. We finally consider the price of anarchy in our system and show that it exceeds 2, but converges to this value for a large population size.