Abstract
In this paper, a multi-server retrial queue with two orbits is considered. There are two arrival processes of positive customers (with two types of customers) and one process of negative customers. Every positive customer requires some amount of resource whose total capacity is limited in the system. The service time does not depend on the customer’s resource requirement and is exponentially distributed with parameters depending on the customer’s type. If there is not enough amount of resource for the arriving customer, the customer goes to one of the two orbits, according to his type. The duration of the customer delay in the orbit is exponentially distributed. A negative customer removes all the customers that are served during his arrival and leaves the system. The objects of the study are the number of customers in each orbit and the number of customers of each type being served in the stationary regime. The method of asymptotic analysis under the long delay of the customers in the orbits is applied for the study. Numerical analysis of the obtained results is performed to show the influence of the system parameters on its performance measures.
Highlights
The theory of queuing systems with repeated calls (Retrial Queue) is an important section of the modern teletraffic theory, the relevance of which is due to wide practical applications, such as the performance evaluation and design of broadcast, radio, and cellular networks, as well as local networks with the random multiple access protocols
We research the queuing system with all mentioned above features: repeated calls, negative customers, and resource
Using the result of the first-order asymptotic analysis 1.(d), we rewrite the characteristic function as (11); rewriting of Equations (3) and (4) for this notations; introducing of an infinitesimal parameter ε2 and new asymptotic function notation (13); rewriting of equations obtained in 2.(b) for the asymptotic notations; approximating asymptotic functions by its 2th-degree Maclaurin series with respect to ε as (14); deriving of a limit solution of the asymptotic equations for ε → 0; using the inverse substitutions, we obtain the form of the second-order asymptotic characteristic function, which gives the value of the asymptotic variance of the considered process
Summary
The theory of queuing systems with repeated calls (Retrial Queue) is an important section of the modern teletraffic theory, the relevance of which is due to wide practical applications, such as the performance evaluation and design of broadcast, radio, and cellular networks, as well as local networks with the random multiple access protocols. In classical queuing theory, the evaluation of almost all performance characteristics leads us to the analysis of a stochastic process of the number of customers in the system It is insufficient if we would like to determine a buffer space capacity of a communication network’s node which guarantees small losses of transmitted data [20,21,23]. We research the queuing system with all mentioned above features: repeated calls, negative customers, and resource Such a model can be applied, for example, for 5G New Radio systems. The key feature and the main problem of the 5G New Radio network is that people themselves, cars, buildings, etc., are signal blockers, which is the cause of service interruptions In this reasoning, the scientific community has the task of analyzing the performance of these systems and improving it in the future. The problems and discussions about the applicability of the obtained approximations are presented in the conclusion
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