## Abstract

In this paper, an inhomogeneous queuing system with an unlimited number of servers operating in a random environment is considered. The arrival process is a Poisson Process, the process of changing the state of the environment is a Markov chain, which is given by the matrix of infinitesimal characteristics. The service discipline is defined as follows: if the customer comes with some intensity, then it is served by a random time distributed according to an exponential distribution with the corresponding parameter, which does not change when, the state of the environment changes. We proposed the characteristic functions method and the asymptotic analysis method to study the system. Using partial characteristic functions, we obtained the matrix equation which allows us to determine the main characteristics of the system». Applying the asymptotic analysis method, we obtained the solution of this equation under the condition of an infinitely growing servicing time. It determines the average number of occupied servers of each type in the system. For a more detailed study, we used an asymptotic analysis of the second order, as a result we showed that the asymptotic characteristic function of the number of occupied servers of each type in the system has the form of a Gaussian characteristic function and the probability distribution of the number of occupied servers of each type in the system under the condition of an infinitely growing service time is a multidimensional Gaussian distribution.

Original language | English |
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Pages (from-to) | 75-83 |

Number of pages | 9 |

Journal | Vestnik Tomskogo Gosudarstvennogo Universiteta - Upravlenie, Vychislitel'naya Tekhnika i Informatika |

Issue number | 47 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Asymptotic analysis method
- Infinite-server queue
- Random environment

## ASJC Scopus subject areas

- Computer Science Applications
- Information Systems
- Computer Networks and Communications