Tails of waiting times and their bounds

Tails of distributions having the form of the geometric convolution are considered. In the case of light-tailed summands, a simple proof of the famous Cramér asymptotic formula is given via the change of probability measure. Some related results are obtained, namely, bounds of the tails of geometric convolutions, expressions for the distribution of the 1st failure time and failure rate in regenerative systems, and others. In the case of heavy-tailed summands, two-sided bounds of the tail of the geometric convolution are given in the cases where the summands have either Pareto or Weibull distributions. The results obtained have the property that the corresponding lower and upper bounds are tailed-equivalent. This revised version was published online in June 2006 with corrections to the Cover Date.     

Regenerative queueing processes and their qualitative and quantitative analysis

Recently developed methods of qualitative analysis for regenerative processes arising in queueing are presented. These methods are essentially qualitative and use notions such as coupling, probability metrics, etc. They are developed for studying various properties of regenerative models, including convergence rate to a stationary regime, continuity of their characteristics with respect to some parameters and first-occurrence time of an event such as queue overflowing. In spite of their qualitative nature they lead to good quantitative estimates of underlying properties with computer methods available to calculate them.     

Thermodynamically stable states in eutectic systems

It is shown that in a binary system in which the liquid-solid phase transition is described by a eutectic phase diagram, there are temperature and concentration regions in the liquid state with different thermodynamic stability. These regions are separated by bell-shaped decay and spinodal curves. Zh. Tekh. Fiz. 67, 7–12 (April 1997)     

Ordered three-dimensional structures resulting from instability of two-dimensional flow in crossed channels

Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–8, March–April, 1991.     

Spatial No-Waiting Stations with Moving Customers

We analyze spatial MAP/G/∞-, spatial MAP/G/c/0¹ and spatial Cox/G/∞-stations with group arrivals over some Polish space X (with Borel σ-algebra X), including the aspect of customer motion in space. For models with MAP-input, characteristic differential equations are set up that describe the dynamics of phase dependent random functions Q _r;ij (u,t;S′), where Q _r;ij (u,t;S′) is the probability to observe, at time u≤t, the number r of those customers in some source set S′∈X, who will be in a destination set S∈X at time t. For Cox/G/∞-stations, i.e., infinite server stations with doubly stochastic input, the arrival intensities as well as service times may depend on some general stochastic process (J′ _t ) _t≥0 with countable state space. For that case we obtain explicit expressions for space–time distributions as well as stationary and non-stationary characteristics.     

Application of the Padé approximations to calculation of rotational energies of a triatomic linear molecule

A possibility of applying the Padé approximations to calculation of rotational-level energies of degenerate vibrational states is studied for linear molecules with various quantum numbers of the vibrational angular momentum 1. The calculations of rotational energy levels for the CO ₂ molecule show that the Padé form has a better predictability as compared to a power series. Spectroscopic constants of the Padé approximations for the vibrational states (04⁰0, 04²0, 04⁴0), (14⁰0, 14²0, 14⁴0), (15¹0, 15³0, 15⁵0), and (06⁰0, 06²0, 06⁴0, 06⁶0) for the CO ₂ molecule are determined. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 54–57, September, 2006.     

The 5v 3 bands of 18O enriched ozone: line positions of 16O16O18O, 16O18O16O, 16O18O18O and 18O16O18O

The four 5v ₃ bands of ¹⁸O enriched ozone have been observed and analysed for the first time. Two species (¹⁶O¹⁸O¹⁶O and ¹⁸O¹⁶O¹⁸O) belong to the C_2v symmetry group and two other (¹⁸O¹⁸O¹⁶O and ¹⁶O¹⁶O¹⁸O) to the C_s symmetry group. They have been recorded at a resolution of 0.008 cm^?1 with a pathlength of 32.16 m. Despite the very weak absorptions observed, almost 250 energy levels have been derived for each of the 4 species, with J ? 35 and K _a ? 13, and suitable sets of Hamiltonian parameters have been determined. For 3 species it has been necessary to account for the resonance between the (005) and (311) states to correctly reproduce the spectra observed. These resonances, anharmonic for C_2v, and hybrid (both anhar-monic and Coriolis) for C_s symmetry confirm the accidentally extremely strong coupling between the (005) and (311) states for ¹⁶O₃, due in that case to the very close distance between unperturbed energy levels. This work also confirms the excellent prediction of band centres of these four species derived from the recently determined isotopically invariant molecular potential function.