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Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 269
Author(s):  
Valentina I. Klimenok ◽  
Alexander N. Dudin ◽  
Vladimir M. Vishnevsky ◽  
Olga V. Semenova

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.


Author(s):  
Meng Gao ◽  
Wenhai Qi ◽  
Jinde Cao ◽  
Jun Cheng ◽  
Kaibo Shi ◽  
...  

2022 ◽  
pp. 1-32
Author(s):  
Martin Bladt

Abstract This paper addresses the task of modeling severity losses using segmentation when the data distribution does not fall into the usual regression frameworks. This situation is not uncommon in lines of business such as third-party liability insurance, where heavy-tails and multimodality often hamper a direct statistical analysis. We propose to use regression models based on phase-type distributions, regressing on their underlying inhomogeneous Markov intensity and using an extension of the expectation–maximization algorithm. These models are interpretable and tractable in terms of multistate processes and generalize the proportional hazards specification when the dimension of the state space is larger than 1. We show that the combination of matrix parameters, inhomogeneity transforms, and covariate information provides flexible regression models that effectively capture the entire distribution of loss severities.


2021 ◽  
pp. 265-272
Author(s):  
Evgeniia S. Sevasteeva ◽  
Sergei A. Plotnikov ◽  
Volodymyr Lynnyk

The brain is processing information 24 hours a day. There are millions of processes proceeding in it accompanied by various spectra of rhythms. This paper tests the hypothesis that the slow delta rhythm excites the gamma rhythm oscillations. Unlike other papers, we determine the slow rhythm spectrum not at the hypothesis stage but during the experiment. We design algorithms of filtering, envelope extraction, and correlation coefficient calculation for signal processing. Moreover, we examine the data on all electroencephalogram channels, which allows us to make a more reasonable conclusion. We confirm that a slow delta rhythm excites a fast gamma rhythm with an amplitude-phase type of interaction and calculate a delay between these two signals equal to about half a second.


Crystals ◽  
2021 ◽  
Vol 12 (1) ◽  
pp. 47
Author(s):  
Zia Ur Rehman ◽  
Mohsan Nawaz ◽  
Hameed Ullah ◽  
Pervaiz Ahmad ◽  
Mayeen Uddin Khandaker ◽  
...  

In the quasi-binary system CaNi2-MgNi2 solid-solutions CaxMg1−xNi2 (0 ≤ x ≤ 1) were prepared from the elements. They crystallize in the hexagonal Laves phase type (MgNi2, C36) for x ≤ 0.33 (P63/mmc, a = 482.51(7) pm, c = 1582.1(3) pm for x = 0, a = 482.59 (3), c = 1583.1(1) for x = 0.33) and in the cubic Laves phase type (MgCu2, C15) for 0.33 < x (Fd−3m, a = 697.12(3) pm for x = 0.5, a = 705.11(2) pm for x = 0.67, a = 724.80(2) pm for x = 1). After hydrogenation in an autoclave the X-ray diffraction patterns changed completely. Reflections assigned to CaNiH3, and Ni and Rietveld refinement confirmed this. The hydrogenation properties of CaxMg1−xNi2 (0 ≤ x ≤ 1) compounds were also studied in situ by X-ray powder diffraction. In situ X-ray powder diffraction of CaxMg1−xNi2 (0 ≤ x ≤ 1) compounds under 0.3 MPa hydrogen gas flow (15 sccm), data collected on a Rigaku SmartLab diffractometer in an Anton Paar XRK 900 Reactor Chamber using Cu-Kα1 radiation. Scanning electron microscopy and EDX spectroscopy confirmed the entitled materials and elemental composition, respectively. From the Transmission electron microscopy and Selected area electron diffraction concluded that the CaxMg1−xNi2 (0 ≤ x ≤ 1) compounds were crystalline.


Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 104
Author(s):  
Jaison Jacob ◽  
Dhanya Shajin ◽  
Achyutha Krishnamoorthy ◽  
Vladimir Vishnevsky ◽  
Dmitry Kozyrev

We consider a queueing inventory with one essential and m optional items for sale. The system evolves in environments that change randomly. There are n environments that appear in a random fashion governed by a Marked Markovian Environment change process. Customers demand the main item plus none, one, or more of the optional items, but were restricted to at most one unit of each optional item. Service time of the main item is phase type distributed and that of optional items have exponential distributions with parameters that depend on the type of the item, as well as the environment under consideration. If the essential item is not available, service will not be provided. The lead times of optional and main items have exponential distributions having parameters that depend on the type of the item. The condition for stability of the system is analyzed by considering a multi-dimensional continuous time Markov chain that represent the evolution of the system. Under this condition, various performance characteristics of the system are derived. In terms of these, a cost function is constructed and optimal control policies of the different types of commodities are investigated. Numerical results are provided to give a glimpse of the system performance.


2021 ◽  
Author(s):  
Samaa Adel Ibrahim Hussein ◽  
Fayez Wanis Zaki ◽  
Mohammed Ashour

Abstract In recent years, SDN technology has been applied to several networks such as wide area network (WAN). IT provides many benefits, such as: enhancing data transfer, promoting Application performance and reducing deployment costs. Software Defined-WAN networks lack studies and references. This paper introduced a system for SD-WAN network using PH/PH/C queues. It concentrates on the study of algebraic estimates the probability distribution of the system states. The Matrix-Geometric solution procedure of a phase type distribution queue with first-come first-served discipline is used.


2021 ◽  
Vol 58 (4) ◽  
pp. 880-889
Author(s):  
Qi-Ming He

AbstractWe consider a class of phase-type distributions (PH-distributions), to be called the MMPP class of PH-distributions, and find bounds of their mean and squared coefficient of variation (SCV). As an application, we have shown that the SCV of the event-stationary inter-event time for Markov modulated Poisson processes (MMPPs) is greater than or equal to unity, which answers an open problem for MMPPs. The results are useful for selecting proper PH-distributions and counting processes in stochastic modeling.


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