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Keywords = state-dependent marked Markovian arrival process

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27 pages, 1266 KB  
Article
Performance Evaluation of the Priority Multi-Server System MMAP/PH/M/N Using Machine Learning Methods
by Vladimir Vishnevsky, Valentina Klimenok, Alexander Sokolov and Andrey Larionov
Mathematics 2021, 9(24), 3236; https://doi.org/10.3390/math9243236 - 14 Dec 2021
Cited by 14 | Viewed by 2827
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section E: Applied Mathematics)
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23 pages, 446 KB  
Article
A Multi-Server Heterogeneous Queuing-Inventory System with Class-Dependent Inventory Access
by Karumbathil Rasmi, Machuveettil Joseph Jacob, Alexander S. Rumyantsev and Achyutha Krishnamoorthy
Mathematics 2021, 9(9), 1037; https://doi.org/10.3390/math9091037 - 3 May 2021
Cited by 16 | Viewed by 2790
Abstract
In this paper, we consider a queuing inventory system with heterogeneous customers of K types arriving according to a marked Markovian arrival process. Each class of customers differs by nature of the service they seek and different priorities are assigned for each class [...] Read more.
In this paper, we consider a queuing inventory system with heterogeneous customers of K types arriving according to a marked Markovian arrival process. Each class of customers differs by nature of the service they seek and different priorities are assigned for each class resulting in different levels of inventory admitted to exhaust for customers of each class. A single service node is provided for each class with exponential services having class-dependent service rates. All classes of customers are served from a single source of inventory replenished according to (s,S) policy with exponentially distributed lead time. Stability condition and steady state probabilities are obtained by matrix-analytic method. Some important performance measures are also derived. Inventory recycle time was analyzed in detail. Useful cost function and numerical illustrations are also given. The optimization problem is interesting and can be solved in similar real scenario. Full article
(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)
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17 pages, 316 KB  
Article
Queueing Network with Moving Servers as a Model of Car Sharing Systems
by Chesoong Kim, Sergei Dudin and Olga Dudina
Mathematics 2019, 7(9), 825; https://doi.org/10.3390/math7090825 - 6 Sep 2019
Cited by 2 | Viewed by 3026
Abstract
We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives [...] Read more.
We consider a queueing network with a finite number of nodes and servers moving between the nodes as a model of car sharing. The arrival process of customers to various nodes is defined by a marked Markovian arrival process. The customer that arrives at a certain node when there is no idle server (car) is lost. Otherwise, he/she is able to start the service. With known probability, which depends on the node and the number of available cars, this customer can balk the service and leave the system. The service time of a customer has an exponential distribution. Location of the server in the network after service completion is random with the known probability distribution. The behaviour of the network is described by a multi-dimensional continuous-time Markov chain. The generator of this chain is derived which allows us to compute the stationary distribution of the network states. The formulas for computing the key performance indicators of the system are given. Numerical results are presented. They characterize the dependence of some performance measures of the network and the nodes on the total number of cars (fleet size of the car sharing system) and correlation in the arrival process. Full article
(This article belongs to the Special Issue Queueing Methods in Reliability Theory and Information Technology Systems)
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19 pages, 930 KB  
Article
Analysis of a Semi-Open Queuing Network with a State Dependent Marked Markovian Arrival Process, Customers Retrials and Impatience
by Chesoong Kim, Sergey Dudin, Alexander Dudin and Konstantin Samouylov
Mathematics 2019, 7(8), 715; https://doi.org/10.3390/math7080715 - 7 Aug 2019
Cited by 11 | Viewed by 3296
Abstract
We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the [...] Read more.
We consider a queuing network with single-server nodes and heterogeneous customers. The number of customers, which can obtain service simultaneously, is restricted. Customers that cannot be admitted to the network upon arrival make repeated attempts to obtain service. The service time at the nodes is exponentially distributed. After service completion at a node, the serviced customer can transit to another node or leave the network forever. The main features of the model are the mutual dependence of processes of customer arrivals and retrials and the impatience and non-persistence of customers. Dynamics of the network are described by a multidimensional Markov chain with infinite state space, state inhomogeneous behavior and special structure of the infinitesimal generator. The explicit form of the generator is derived. An effective algorithm for computing the stationary distribution of this chain is recommended. The expressions for computation of the key performance measures of the network are given. Numerical results illustrating the importance of the account of the mentioned features of the model are presented. The model can be useful for capacity planning, performance evaluation and optimization of various wireless telecommunication networks, transportation and manufacturing systems. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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