Performance Evaluation of the Priority Multi-Server System MMAP/PH/M/N Using Machine Learning Methods

In this paper, we present the results of a study of a priority multi-server queuing system with heterogeneous customers arriving according to a marked Markovian arrival process (MMAP), phase-type service times (PH), and a queue with finite capacity. Priority traffic classes differ in PH distributions of the service time and the probability of joining the queue, which depends on the current length of the queue. If the queue is full, the customer does not enter the system. An analytical model has been developed and studied for a particular case of a queueing system with two priority classes. We present an algorithm for calculating stationary probabilities of the system state, loss probabilities, the average number of customers in the queue, and other performance characteristics for this particular case. For the general case with K priority classes, a new method for assessing the performance characteristics of complex priority systems has been developed, based on a combination of machine learning and simulation methods. We demonstrate the high efficiency of the new method by providing numerical examples.

Description of the Markov chain using the producing function

In article the way of creation of an assumed function of transformation with the help of a generating function and also a method of use of characteristic numbers and vectors for creation of a matrix which elements well describe conditions of process at any moment is described. This approach differs from preceding that, having used a concept of characteristic numbers and characteristic vectors of a matrix of the transitional probabilities, it is possible to simplify considerably calculation of the elements characterizing process. In article methods of stochastic model operation, ways of the description of a generating function, the solution of matrixes of the equations by means of characteristic numbers and vectors are used. Using properties of a generating function, made “dictionary” of z-transformations which helped to define an assumed function of transformation. The generating function of a vector was applied to a research of behavior of a vector of absolute probabilities which elements represent stationary probabilities. For definition of degree of a matrix of transition of probabilities used a concept of characteristic numbers and characteristic vectors of the transitional probabilities. Determined by such way an unlimited set of latent vectors of which made matrixes which describe a condition of a system at any moment. Reception of definition of latent vectors in more difficult examples which is that along with required coefficients of secular equations the system of auxiliary matrixes and an inverse matrix is under construction is also described.

A model of indirect contagion based on a news similarity network

Our objective is to model indirect contagion among companies. We use pieces of news about companies to measure the similarities between them. We assume that two companies are similar if we may associate a story about one company with the story about another company and vice-versa. First, after statistically eliminating arbitrary similarities between companies, we introduce a network based on the news similarities. Second, we evaluate a vector of stationary probabilities associated with the process of contagion that takes place in the network and we use these pieces of information to define the notion of centrality. Third, we use this vector of stationary probabilities to build an associated graph that reveals the most important paths of information contagion. Finally, we build a model of indirect contagion and evaluate the size of the information avalanches that take place in the similarity network.

Application of Decomposable Semi-Regenerative Processes to the Study of k-out-of-n Systems

This paper aimed to demonstrate the capabilities of decomposable semi-regenerative processes for the investigation of the k-out-of-n system. Proposed in 1955 by W. Smith, the regeneration idea has come a long way in terms of development and has found widespread applications. First, we briefly recall the history of the development of the regeneration idea and the main results of the theory of regenerative, semi-regenerative, and decomposable semi-regenerative processes. Then, the methods of the theory of decomposable semi-regenerative processes are used for the study of a k-out-of-n renewable system with exponentially distributed life and generally distributed repair times of its components. This system is very important for practice and its special cases have previously been considered (including by the authors); however, only special cases and using other methods are considered herein. In the current paper, two scenarios of system repair after its failure are considered for the first time: the partial and the full system repair scenarios. For both scenarios, the time-dependent system state probabilities are calculated in terms of their Laplace transforms. The closed form representation of the stationary probabilities for both scenarios are also presented. These latest results represent a new contribution to the study of this system.

Modeling behaviour of wet and dry days in Iran from the perspective of Markov chains

The current study aims to model the behaviour of wet and dry days in Iran using Markov Chain Models. To this end, data related to daily precipitation of 44 synoptic stations for a 25-years interval (1991-2015) was obtained from Iran Meteorological Organization. Then, the Markov features of dry and wet days of Iran including stationary probabilities of dry and wet days occurrence, the expected length of dry periods, the expected length of wet periods, dry-wet spells cycle, return periods for dry or wet episodes and finally, the possibility of occurrence of the continuity of dry days for 5, 10, 15, 20, 25 and 30 days were calculated for all the synoptic stations in a seasonal scale. The results showed that there is the occurrence of dry short continuities (5 and 10 days) in three seasons of autumn, winter and spring with different possibilities all over Iran. However, the possibility of occurrence of long-term dry continuities (more than 20 days) is variable in terms of season and place so that in winter, no possibility of occurrence of this type of continuities is obvious in the northern half of Iran. As in autumn and spring those are the end and beginning of long-term stability conditions of the atmosphere in the upper atmosphere levels of Iran, the possibility of periodical occurrence of 30-days dry days, particularly in the southern half of Iran increases. In addition, the expected return periods for dry days is almost steady for every part of Iran and is in the range between 1 and 2 days. However, the number of return days to a precipitation period does not follow this rule and varies for every part of Iran so that from 2.15 days in autumn to 79 days in spring is variable, pointing to the climate diversity of Iran.

An alternative approach in finding the stationary queue length distribution of a queueing system with negative customers

A single-server queueing system with negative customers is considered in this paper. One positive customer will be removed from the head of the queue if any negative customer is present. The distribution of the interarrival time for the positive customer is assumed to have a rate that tends to a constant as time t tends to infinity. An alternative approach will be proposed to derive a set of equations to find the stationary probabilities. The stationary probabilities will then be used to find the stationary queue length distribution. Numerical examples will be presented and compared to the results found using the analytical method and simulation procedure. The advantage of using the proposed alternative approach will be discussed in this paper.

Investigations of the Potential Application of k-out-of-n Systems in Oil and Gas Industry Objects

The purpose of this paper was to demonstrate the possibilities of assessing the reliability of oil and gas industry structures with the help of mathematical models of k-out-of-n systems. We show how the reliability of various structures in the oil and gas complex can be described and investigated using k-out-of-n models. Because the initial information about the life and repair time of components of systems is only usually known on the scale of one and/or two moments, we focus on the problem of the sensitivity analysis of the system reliability indices to the shape of its components repair time distributions. To address this problem, we used the so-called markovization method, based on the introduction of supplementary variables, to model the system behavior with the help of the two-dimensional Markov process with discrete-continuous states. On the basis of the forward Kolmogorov equations for the time-dependent process’ state probabilities, relevant balance equations for the process’ stationary probabilities are presented. Using these equations, stationary probabilities and some reliability indices for two examples from the oil and gas industry were calculated and their sensitivity to the system component’s repair time distributions was analyzed. Calculations show that under “rare” component failures, most system reliability indices become practically insensitive to the shape of the components repair time distributions.

On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

In this paper, we revisit level-dependent quasi-birth-death processes with finitely many possible values of the level and phase variables by complementing the work of Gaver, Jacobs, and Latouche (Adv. Appl. Probab. 1984), where the emphasis is upon obtaining numerical methods for evaluating stationary probabilities and moments of first-passage times to higher and lower levels. We provide a matrix-analytic scheme for numerically computing hitting probabilities, the number of upcrossings, sojourn time analysis, and the random area under the level trajectory. Our algorithmic solution is inspired from Gaussian elimination, which is applicable in all our descriptors since the underlying rate matrices have a block-structured form. Using the results obtained, numerical examples are given in the context of varicella-zoster virus infections.

Cached Files Updating Revisited: The Distribution of Popularity-Weighted Average File Age

In this paper, using the discrete time model, we consider the average age of all files for a cached-files-updating system where a server generates N files and transmits them to a local cache. In order that the cached files are fresh, in each time slot the server updates files with certain probabilities. The age of one file or its age of information (AoI) is defined as the time the file stays in cache since it was last time sent to cache. Assume that each file in cache has corresponding request popularity. In this paper, we obtain the distribution function of the popularity-weighted average age over all files, which gives a complete description of this average age. For the random age of single file, both the mean and its distribution have been derived before by establishing a simple Markov chain. Using the same idea, we show that an N dimensional stochastic process can be constituted to characterize the changes of N file ages simultaneously. By solving the steady-state of the resulting process, we obtain the explicit expression of stationary probability for an arbitrary state-vector. Then, the distribution function of the popularity-weighted average age can be derived by mergering a proper set of stationary probabilities. For the possible applications, the distribution function can be utilized to calculate the probability that the average age violates certain statistical guarantee.

JOINT OPTIMIZATION OF TRANSITION RULES AND THE PREMIUM SCALE IN A BONUS-MALUS SYSTEM

Bonus-malus systems (BMSs) are widely used in actuarial sciences. These systems are applied by insurance companies to distinguish the policyholders by their risks. The most known application of BMS is in automobile third-party liability insurance. In BMS, there are several classes, and the premium of a policyholder depends on the class he/she is assigned to. The classification of policyholders over the periods of the insurance depends on the transition rules. In general, optimization of these systems involves the calculation of an appropriate premium scale considering the number of classes and transition rules as external parameters. Usually, the stationary distribution is used in the optimization process. In this article, we present a mixed integer linear programming (MILP) formulation for determining the premium scale and the transition rules. We present two versions of the model, one with the calculation of stationary probabilities and another with the consideration of multiple periods of the insurance. Furthermore, numerical examples will also be given to demonstrate that the MILP technique is suitable for handling existing BMSs.