Retrial BMAP/PH/N Queueing System with a Threshold-Dependent Inter-Retrial Time Distribution

In this paper, we study a multi-server queueing system with retrials and an infinite orbit. The arrival of primary customers is described by a batch Markovian arrival process (BMAP), and the service times have a phase-type (PH) distribution. Previously, in the literature, such a system was mainly considered under the strict assumption that the intervals between the repeated attempts from the orbit have an exponential distribution. Only a few publications dealt with retrial queueing systems with non-exponential inter-retrial times. These publications assumed either the rate of retrials is constant regardless of the number of customers in the orbit or this rate is constant when the number of orbital customers exceeds a certain threshold. Such assumptions essentially simplify the mathematical analysis of the system, but do not reflect the nature of the majority of real-life retrial processes. The main feature of the model under study is that we considered the classical retrial strategy under which the retrial rate is proportional to the number of orbital customers. However, in this case, the assumption of the non-exponential distribution of inter-retrial times leads to insurmountable computational difficulties. To overcome these difficulties, we supposed that inter-retrial times have a phase-type distribution if the number of customers in the orbit is less than or equal to some non-negative integer (threshold) and have an exponential distribution in the contrary case. By appropriately choosing the threshold, one can obtain a sufficiently accurate approximation of the system with a PH distribution of the inter-retrial times. Thus, the model under study takes into account the realistic nature of the retrial process and, at the same time, does not resort to restrictions such as a constant retrial rate or to rough truncation methods often applied to the analysis of retrial queueing systems with an infinite orbit. We describe the behavior of the system by a multi-dimensional Markov chain, derive the stability condition, and calculate the steady-state distribution and the main performance indicators of the system. We made sure numerically that there was a reasonable value of the threshold under which our model can be served as a good approximation of the BMAP/PH/N queueing system with the PH distribution of inter-retrial times. We also numerically compared the system under consideration with the corresponding queueing system having exponentially distributed inter-retrial times and saw that the latter is a poor approximation of the system with the PH distribution of inter-retrial times. We present a number of illustrative numerical examples to analyze the behavior of the system performance indicators depending on the system parameters, the variance of inter-retrial times, and the correlation in the input flow.

Reliability modeling for dependent competing failure processes with phase-type distribution considering changing degradation rate

In this paper, a system reliability model subject to Dependent Competing Failure Processes (DCFP) with phase-type (PH) distribution considering changing degradation rate is proposed. When the sum of continuous degradation and sudden degradation exceeds the soft failure threshold, soft failure occurs. The interarrival time between two successive shocks and total number of shocks before hard failure occurring follow the continuous PH distribution and discrete PH distribution, respectively. The hard failure reliability is calculated using the PH distribution survival function. Due to the shock on soft failure process, the degradation rate of soft failure will increase. When the number of shocks reaches a specific value, degradation rate changes. The hard failure is calculated by the extreme shock model, cumulative shock model, and run shock model, respectively. The closed-form reliability function is derived combining with the hard and soft failure reliability model. Finally, a Micro-Electro-Mechanical System (MEMS) demonstrates the effectiveness of the proposed model.

Study on Acidity and Neutralizing Ability of Ions in the Chemical Composition of Rainwater

The acidity in rainwater is mainly controlled by the presence of H2SO4, HNO3 in combination with the ability to neutralize cations in rainwater. pH is an important value in the evaluation of acidity in rainwater. The research used a series of rainwater quality monitoring data from 2005 to 2018 in Vietnam. The research showed that the average pH distribution at 23 stations ranged from 5.83 ± 0.62. The rains with pH <5.6 appear in all years at the research stations. Considering the ability of acid neutralizing to various ions shows that Ca2+ is the main contributor to acid neutralization processes in rainwater, followed by Mg2 +, NH4+, and K+. While Ca2+ always play the highest acid neutralizing role at all stations; Depending on each station, Mg2+ and NH4+ ions play a role in neutralizing acidity in rainwater. The research also shows a match between the trend of H+ concentration and the tendency of cations to contribute to acid neutralization in rainwater.

Relationship of Trace Metal Covariates and pH Distribution in Groundwater within Gold mining and Non-Gold mining Areas in Ghana

One of the most important defining characteristics of groundwater quality is pH as it fundamentally controls the amount and chemical form of many organic and inorganic solutes in groundwater. Groundwater data are frequently characterized by a wide degree of variability of the factors which possibly influence pH distribution. For this reason, it is challenging to link the spatio-temporal dynamics of pH to a single environmental factor by the ordinary least squares regression technique of the conditional mean. In this study, quantile regression was used to estimate the response of pH to nine environmental factors (As, Cd, Fe, Mn, Pb, turbidity, electrical conductivity, total dissolved solids and nitrates). Results of 25%, 50%, 75% quantile regression and ordinary least squares (OLS) regression were compared. The standard regression of the conditional means (OLS) underestimated the rates of change of pH due to the selected factors in comparison with the regression quantiles. The effect of arsenic increased for sampling locations with higher pH values (higher quantiles) likewise the influence of Pb and Mn. However, the effects of Cd and Fe decreased for sampling locations in higher quantiles. It can be concluded that these detected heterogeneities would be missed if this study had focused exclusively on the conditional means of the pH values. Consequently, quantile regression provides a more comprehensive account of possible spatio-temporal relationships between environmental covariates in groundwater. This study is one of the first to apply this technique on groundwater systems in sub-Saharan Africa. The approach is useful and interesting and has broad application for other mining environments especially tropical low-income countries where climatic conditions can drive rapid cycling or transformations of pollutants. It is also pertinent to geopolitical contexts where regulatory; monitoring and management capacities are weak and where mining pollution of groundwater largely occur.

A Parameter Estimation Method of Shock Model Constructed with Phase-Type Distribution on the Condition of Interval Data

The phase-type distribution (also known as PH distribution) has mathematical properties of denseness and closure in calculation and is, therefore, widely used in shock model constructions describing occurrence time of a shock or its damage. However, in the case of samples with only interval data, modeling with PH distribution will cause decoupling issues in parameter estimation. Aiming at this problem, an approximate parameter estimation method based on building PH distribution with dynamic order is proposed. Firstly, the shock model established by PH distribution and the likelihood function under samples with only interval data are briefly introduced. Then, the principle and steps of the method are introduced in detail, and the derivation processes of some related formulas are also given. Finally, the performance of the algorithm is illustrated by a case with three different types of distributions.