Exact Buffer Overflow Calculations for Queues via Martingales

Let _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 _n and the Laplace transform e^-sn 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.     

Regenerative simulation of TES processes

Regenerative simulation has become a familiar and established tool for simulation-based estimation. However, many applications (e.g., traffic in high-speed communications networks) call for autocorrelated stochastic models to which traditional regenerative theory is not directly applicable. Consequently, extensions of regenerative simulation to dependent time series is increasingly gaining in theoretical and practical interest, with Markov chains constituting an important case. Fortunately, a regenerative structure can be identified in Harris-recurrent Markov chains with minor modification, and this structure can be exploited for standard regenerative estimation. In this paper we focus on a versatile class of Harris-recurrent Markov chains, called TES (Transform-Expand-Sample). TES processes can generate a variety of sample paths with arbitrary marginal distributions, and autocorrelation functions with a variety of functional forms (monotone, oscillating and alternating). A practical advantage of TES processes is that they can simultaneously capture the first and second order statistics of empirical sample paths (raw field measurements). Specifically, the TES modeling methodology can simultaneously match the empirical marginal distribution (histogram), as well as approximate the empirical autocorrelation function. We explicitly identify regenerative structures in TES processes and proceed to address efficiency and accuracy issues of prospective simulations. To show the efficacy of our approach, we report on a TES/M/1 case study. In this study, we used the likelihood ratio method to calculate the mean waiting time performance as a function of the regenerative structure and the intrinsic TES parameter controlling burstiness (degree of autocorrelation) in the arrival process. The score function method was used to estimate the corresponding sensitivity (gradient) with respect to the service rate. Finally, we demonstrated the importance of the particular regenerative structure selected in regard to the estimation efficiency and accuracy induced by the regeneration cycle length.     

On the behavior of solutions of certain differential equations of the fourth order

Summary In this paper three new results are obtained for equations of the form (1.1). Conditions are established which guarantee asymptotic stability, ultimate boundedness, and convergence of solutions of (1.1). This work was supported by the National Science Foundation COSIP (GY 4754). Entrata in Redazione il 12 ottobre 1970.     

Conjugate processes and the silumation of ruin problems

A general methods is developed for giving simulation estimates of boundary crossing probabilities for processes related to random walks in discreate or continuous time. Particular attention is given to the probability ψ(u, T of ruin before time T in cpumpound Poisson risk processes. When the provbabi;ity law P governing the given process is imbedded in an exponentaial family (P_gq), one can write ψ (u, T) + _θR_gq for certain random variables R_gq given by Wald's fundamental identity. Using this to simulate from P_gq rather than P, it is possible not only to overcome the difficulties connected with the case T =∞, but also to obtain a considerable variance reduction.It is shown that the solution of the Lundberg equation determines the asymptotically optimal value of θ in heavy traffic when T = ∞, and some results guidelining the choice of θ when T > ∞ are also given. The potential of the method in different is illustrated by two examples.     

Extreme Value Theory for Queues Via Cycle Maxima

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 of the queueing process during a regenerative cycle . For simple queues, methods for obtaining the distribution of both explicitly and asymptotically are reviewed. In greater generality, the study leads into Wiener–Hopf problems. Extensions to queues in a Markov regime, for example governed by Markov-modulated Poisson arrivals, are also considered.     

Photoinitiation rate profiles during polymerization of a dimethacrylate-based resin photoinitiated with camphorquinone/amine. Influence of initiator photobleaching rate

The photodecomposition of camphorquinone (CQ)/amine during the photo polymerization of a dimethacrylate-based resin under continuous irradiation was investigated in thick samples. The global CQ photoconsumption was measured by monitoring the decrease in light absorption as a function of irradiation time and the kinetics were satisfactory fitted to a first order expression where the rate constant of photobleaching was proportional to the irradiation intensity. In a thick sample, the photobleaching of the photoinitiator is accompanied by a deeper penetration of the light through the underlying layers. These gradients of photoinitiator concentration, light intensity and photoinitiation rate along the path of irradiation were calculated. The photodecomposition reaction was spatially inhomogeneous and the degree of nonuniformity increased with increased initial sample absorbance. The influence of the photobleaching process on the polymerization reaction was examined. The photobleaching rate of CQ was much slower than the polymerization rate and only 20% of the initial amount of CQ was consumed before the polymerization reaction had almost ceased. Results obtained in this research highlight the inherent interlinking of light attenuation and photobleaching rate in bulk polymerizing systems.     

Electron density measurements of argon surface-wave discharges

The electron density of microwave-generated surface-wave discharges in argon have been measured using Stark broadening and calculated from the measured wavelengths of the standing surface wave. Results obtained from these two techniques compare well. The electron density varies from 10¹³ to 10¹⁴/cm³ for pressures ranging from 50 to 800 torr.     

Computational methods in risk theory: A matrix-algorithmic approach

This paper is concerned with the numerical computation of the probability ψ(u) of ruin with initial reserve u. The basic assumption states that the claim size distribution is phase-type in the sense of Neuts. The models considered are: the classical compound Poisson risk process, the Sparre Anderse process and varying environments which are either governed by a Markov process or exhibit periodic fluctuations. The computational steps involve the iterative solution of a non-linear matrix equation Q = Ψ (Q) as well as the evaluation of matrix-exponentials e^Qu. A number of worked-out numerical examples are presented.     

Discretization error for a two-sided reflected Lévy process

An obvious way to simulate a Lévy process X is to sample its increments over time 1 / n, thus constructing an approximating random walk \(X^{(n)}\). This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators L, U at the lower and upper barriers at some fixed time. Under the weak assumption that \(X_\varepsilon /a_\varepsilon \) has a non-trivial weak limit for some scaling function \(a_\varepsilon \) as \(\varepsilon \downarrow 0\), it is proved in particular that \((Y_1-Y^{(n)}_n)/a_{1/n}\) converges weakly to \(\pm \, V\), where the sign depends on the last barrier visited. Here the limit V is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically.