Offered-load model for Pareto inter-arrival network traffic

Video-on-Demand (VoD) and video sharing services account for a large percentage of the total downstream Internet traffic. In order to provide a better understanding of the load on these services, we analyze and model a workload trace from a VoD service provided by a major Swedish TV broadcaster. The trace contains over half a million requests generated by more than 20000 unique users. Among other things, we study the request arrival rate, the inter-arrival time, the spikes in the workload, the video popularity distribution, the streaming bit-rate distribution and the video duration distribution. Our results show that the and the session arrival rates for the TV4 workload does not follow a Poisson process. The arrival rate distribution is modeled using a lognormal distribution while the inter-arrival time distribution is modeled using a stretched exponential distribution. We observe the user behavior where users abandon streaming sessions after minutes or even seconds of starting them. Both very popular videos and non-popular videos are particularly affected by impatient users. We investigate if this behavior is an invariant for VoD workloads.

We consider the arrival rate of a Markovian Arrival Process with time-varying periodic transition rates. The arrival rate can vary widely for a MAP with fixed average transition rates by selecting appropriate transition rate functions over the period.

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.

The insurance has often been involved to minimize financial losses. As the product providers, the insurance companies must effectively manage risks to prevent errors in risk measurement. The amount of risk or loss experienced by the policyholder refers to the claim amount. The Value at Risk (VaR) is commonly used to measure risk. The VaR is calculated from the probability function, which can be obtained by evaluating the distribution of claims data. Most claim frequencies are small, but occasionally, huge claims appear. Therefore, the appropriate distribution would be characterized by a heavy-tailed. Thus, this research aims to model and evaluate insurance claims data using exponential, Weibull, Pareto, and lognormal distributions to assess financial risk through VaR. The insurance claims data were collected from a single insurance company and include 1,326 claims. This research specifically examines variables such as gender, diabetic status, smoking status, the number of claims, and the level of confidence. The data were analysed using descriptive statistics, Maximum Likelihood Estimation for parameter estimation, and Goodness of Fit tests to determine the best-fitting distribution, along with VaR calculations based on the results. The suitability of the distribution model is assessed through the VaR and is analysed based on the appropriate distribution of insurance claims data. It is obtained that the Weibull and lognormal distributions appropriately model insurance claims data. The highest VaR is observed in the claim data for female non-diabetic smokers, with a level of confidence of 99.5%. The lowest VaR is obtained from the claim data for male diabetic non-smokers, with a level of confidence of 90%. This approach enhances the prediction of large potential losses for specific demographic groups, aiding more informed decision-making in premium pricing and risk management. The integration of heavy-tailed distributions in risk assessment, with a particular focus on demographic specificity, constitutes a substantial and novel contribution to this research.

Performance analysis of an integrated scheduling scheme in the presence of bursty MMPP traffic

We apply supervised learning to a general problem in queueing theory: using a neural net, we develop a fast and accurate predictor of the stationary system-length distribution of a GI/GI/1 queue—a fundamental queueing model for which no analytical solutions are available. To this end, we must overcome three main challenges: (i) generating a large library of training instances that cover a wide range of arbitrary interarrival and service time distributions, (ii) labeling the training instances, and (iii) providing continuous arrival and service distributions as inputs to the neural net. To overcome (i), we develop an algorithm to sample phase-type interarrival and service time distributions with complex transition structures. We demonstrate that our distribution-generating algorithm indeed covers a wide range of possible positive-valued distributions. For (ii), we label our training instances via quasi-birth-and-death(QBD) that was used to approximate PH/PH/1 (with phase-type arrival and service process) as labels for the training data. For (iii), we find that using only the first five moments of both the interarrival and service times distribution as inputs is sufficient to train the neural net. Our empirical results show that our neural model can estimate the stationary behavior of the GI/GI/1—far exceeding other available methods in terms of both accuracy and runtimes. History: Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: O. Baron received financial support from the Natural Sciences and Engineering Research Council of Canada (NERC) [Grant 458051]. D. Krass received financial support from the NERC [Grant 458395]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0263 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Probabilistic seismic hazard analysis (PSHA) is a methodology with a long history and has been widely implemented. However, in the conventional PSHA and sequence-based probabilistic seismic hazard analysis (SPSHA) approaches, the occurrence of mainshocks is modeled as the homogeneous Poisson process, which is unsuitable for large earthquakes. To account for the stationary occurrence of small-to-moderate (STM) mainshocks and the nonstationary behavior of large mainshocks, we propose a time-dependent sequence-based probabilistic seismic hazard analysis (TD-SPSHA) approach by combining the time-dependent mainshock probabilistic seismic hazard analysis (TD-PSHA) and aftershock probabilistic seismic hazard analysis, consisting of four components: (1) STM mainshocks, (2) aftershocks associated with STM mainshocks, (3) large mainshocks, and (4) aftershocks associated with large mainshocks. The approach incorporates an exponential-magnitude, exponential-time model for STM mainshocks, and a renewal-time, characteristic-magnitude model for large mainshocks to assess the time-dependent hazard for mainshocks. Then nonhomogeneous Poisson process is used to model the occurrence of associated aftershocks, in which the aftershock sequences can be modeled using the Reasenberg and Jones (RJ) model or the epidemic-type aftershock sequence (ETAS) model. To demonstrate the proposed TD-SPSHA approach, a representative site of the San Andreas fault is selected as a benchmark case, for which five time-dependent recurrence models, including normal, lognormal, gamma, Weibull, and Brownian passage time (BPT) distributions, are chosen to determine the occurrence of large mainshocks. Then sensitivity tests are presented to show the effects on TD-SPSHA, including (1) time-dependent recurrence models, (2) mainshock magnitude, (3) rupture distance, (4) aftershock duration, (5) escaped time since the last event, and (6) future time interval. Furthermore, the bimodal hybrid renewal model is utilized by TD-SPSHA for another case site. The comparison results illustrate that the sequence hazard analysis approach ignoring time-varying properties of large earthquakes for long periods and the effects of associated aftershocks will result in a significantly underestimated hazard. The TD-SPSHA-based hazard curves using the ETAS model are larger than those of the RJ model. The proposed TD-SPSHA approach may be of significant interest to the field of earthquake engineering, particularly in the context of structural design or seismic risk analysis for the long term.

The authors consider an M/G/1 queue with a time-varying arrival rate. The objective is to obtain infinitesimal perturbation analysis (IPA) gradient estimates for various performance measures of interest with respect to certain system parameters. In particular, the authors consider the mean system time over n arrivals and an arrival rate alternating between two values. By choosing a convenient sample path representation of this system, they derive an unbiased IPA gradient estimator which, however, is not consistent, and investigate the nature of this problem. >

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

In this paper, we study infinite server queues with time-varying rates. Under the assumption of non homogenous Poisson process for the arrivals and service processes, in addition to deriving the standard Kolmogorov forward equation, we introduce and develop a generalised backward equation for the M (t)/M (t)/∞ queue using martingales. Explicit solution for these two equations are provided. We also introduce two more Kolmogorov type equations namely, the quasi Kolmogorov Backward Equation and the quasi Generalised Forward Equation for the M (t)/M (t)/∞ queue. Based on the solutions developed for the M (t)/M (t)/∞ queue, we provide an alternative and easy to interpret justification for the transition probabilities of the M (t)/G(t)/∞ queue. Finally we develop expressions to calculate the time dependent version of the mean and variance of the queue length processes.

The ON/OFF model is considered, known as the packet series model, in which the source states alternate between ON and OFF states, where the time spent in at least one of these states is described by distributions with heavy tails - the Pareto distribution. It is shown that the correct implementation of the Pareto distribution is associated with specific computational difficulties caused by the slow decay of the heavy power-law tail of the distribution.

Heavy traffic limit theorems are established for a class of single server queueing models including those with heavy-tailed or long-range dependent arrivals and time-varying service rates. The models are motivated by wireless queueing systems for which there is an increasing evidence of the presence of heavy-tailed or long-range dependent arrivals, and where the service rates vary with the changes in the wireless medium. The main focus of the paper is to obtain the different possible limit processes that can arise depending on the relationship between scalings for both the arrival and departure processes. The limit processes obtained here are driven by either Brownian motion (when the contribution from the departure process dominates the limit) or the limits of properly scaled arrivals (when the contribution from the arrival process dominates the limit), typical examples being stable Lévy motion or fractional Brownian motion. In particular, for the case where arrival process is given by the infinite source Poisson process, this relationship, which determines the type of the limiting queue-length process, is a simple condition involving the heavy tail exponent, arrival rate and channel variation parameter in the wireless medium model. To establish these limit results, two approaches are studied. In one approach, when the limit is driven by Brownian motion, the perturbed test function method is extended to incorporate reflection. In contrast, the second approach allows for non-Markovian and/or non-Gaussian driving processes in the limit. Both approaches involve averaging in the drift term arising from random service rates at the departures. In the second approach, this averaging is carried out directly and pathwise, thus sidestepping the assumption of driving Brownian motion used in the perturbed test function method.

We focus on correlated queues where service and inter-arrival times are strongly correlated, and the input and output link speeds are the same. We present an analytical method to derive the LST (Laplace Stieltjes transform) of the (actual) waiting time distribution. To investigate the effect of correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where inter-arrival time and service time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlations have significant impact on the system performance.

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.

A previous method for deciding if a set of time-to-fail data follows a lognormal distribution or a Weibull distribution is expanded upon. Pearson's s-correlation coefficient is calculated for lognormal and Weibull probability plots of the time-to-fail data. The test statistic is the ratio of the two s-correlation coefficients. When standardized, the lognormal and Weibull variables map into 1 of 2 gamma distributions with no dependence on the shape or scaling factors, confirming earlier observations. Using a set of Monte Carlo simulations, the test statistic was found to be s-normally distributed to good approximation. Formulas for estimating the mean and standard deviation of the test statistic were derived, allowing for an estimate of the probability of hypothesis test errors. As anticipated, the test capability increases with increasing sample size, but only if a substantial fraction of the parts actually fail. If less than 10% of the parts are stressed to failure, then it is almost impossible to distinguish between lognormal and Weibull distributions. If all parts are stressed to failure, the probability of making a correct choice is fair for sample sizes as small as 10, and becomes quite good if the sample size is at least 50. The statistical technique for distinguishing lognormal from Weibull distributions is presented. Its theoretical foundation is given at a qualitative level, and the range of useful application is explored. An approximate form for the distribution of the test statistic is inferred from Monte Carlo simulation.