Asymptotic analysis of multiserver retrial queueing system with \(\pi\)-defeat of negative arrivals under heavy load

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The paper studies a multiserver retrial queuing system with \(\pi\)-defeat as a mathematical model of cloud services. The arrival processes of “positive” calls are Poisson. The system has a finite number of servers and the service time for calls at the servers is exponentially distributed. When all servers are busy, calls entering the system transfer to an orbit, where they experience a random delay. After the delay, calls from the orbit attempt to access the service unit according to a multiple access policy. The system also receives a stream of negative calls. Negative calls do not require the service. An negative call “deletes” a random number of calls is the service unit. For the considered model, the Kolmogorov equations are written in the steady state. The method of asymptotic analysis under a heavy load condition is applied for deriving the stationary probability distribution of the number of calls in the orbit. The results of the numerical analysis are presented.

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1. Introduction Cloud technologies represent a model for providing computing resources on demand over the Internet, where infrastructure, software, and data are located on remote servers (in the “cloud”) rather than on the user’s local devices. Users access these resources via the Internet using various devices, such as computers, smartphones, and tablets, and pay only for the resources actually consumed, making cloud technologies a flexible, scalable, and cost-effective solution. Cloud technologies encompass a wide range of services delivered over the internet, including Infrastructure as a Service (IaaS) from AWS, Azure, and Google Cloud; Platform as a Service (PaaS) such as Heroku and App Engine; Software as a Service (SaaS) like Microsoft 365 and Salesforce; Functions as a Service (FaaS) such as AWS Lambda; as well as Database as a Service (DBaaS) and cloud storage solutions such as Amazon S3, Azure Blob Storage, and Google Cloud Storage, providing users with flexible, scalable, and cost-effective computing resources [1]. Cloud technologies, as and development of new telecommunication networks, remains a priority for science and technology, reflected in diverse approaches ranging from analytical methods [2],the development of architectures for managing network slicing [3] and the performance modeling of mmWave networks [4], requiring the advancement of analytical tools. Mathematical modeling is critically important for the optimization of costs, the enhancement of performance, and the assurance of reliability by predicting resource consumption, identifying bottlenecks, and planning for scaling. It is particularly important that modeling allows for the consideration of potential negative factors, such as software failures, cyberattacks, and accidents, in order to develop effective protection and redundancy strategies, guaranteeing the uninterrupted operation of the system. In this paper, we present a mathematical model of a cloud in the form of a multi-server retrial queueing system (RQ system) with negative calls. Retrial queueing systems are mathematical models of queueing theory widely used to analyze and optimize various telecommunications systems, mobile communication networks, and call centers [5- 7]. The main feature of such models is the presence of repeated calls to the server after an unsuccessful attempt to receive the service. There is not a queue in the system, unserved calls go to an orbit (some virtual place), where they perform a random delay. There is a random access protocol for all calls in an orbit. J. Artalejo and A. Gomez-Corral [5], G. Falin and J. Templeton [6] offered comprehensive treatments in retrial queueing systems, establishing the groundwork for analyzing queues with repeated calls. T. Phung-Duc [7] provides a survey of retrial queueing models, highlighting their theoretical developments and diverse applications. The concept of negative calls was pioneered by E. Gelenbe [8]. He introduced negative signals that can remove calls from the system, providing a framework for modeling complex interactions in various systems. This concept was further explored in [9, 10], solidifying the mathematical foundation for this area, such models began to be called as G-networks and G-systems. Do [11] offers a valuable bibliography on G-networks, negative calls, and their applications. M. Caglayan [12] highlighted the applications of G-networks to machine learning and energy packet networks. Later research has expanded upon these foundations, considering various aspects of G-networks and retrial queues with negative calls. P. Bocharov and V. Vishnevsky [13] discussed the development of the theory of multiplicative networks in the context of G-networks. Y. Shin [14] investigated multiserver retrial queues with both negative calls and disasters, while R. Razumchik [15] studied queueing systems with negative arrivals, a “bunker” for displaced calls, and varying service intensities. M. Matalytski and V. Naumenko [16] analyzed queueing networks with bounded waiting times for both positive and negative calls under non-stationary conditions. Further contributions to this field include research on related queuing models. Liu et al. [17] explore a multiserver two-way communication retrial queue subject to disasters and synchronous working vacations, offering insights into the impact of disruptive events and service strategies on system performance. A. Melikov [18] considers inventory queuing systems with negative arrivals. E. Lisovskaya et al. [19] investigate a resource retrial queue with two orbits and negative customers. These works demonstrate a great interest and expansion of the theoretical understanding and practical applications of G-systems with negative calls. In this paper, we consider a multiserver retrial queuing system with
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About the authors

Natalya P. Meloshnikova

National Research Tomsk State University

Email: meloshnikovana@gmail.com
ORCID iD: 0000-0002-8708-124X
Scopus Author ID: 58304893200
ResearcherId: MTF-1866-2025

PhD-student, Junior researcher of Laboratory of queueing theory and teletraffic theory

36 Lenin Ave, Tomsk, 634050, Russian Federation

Ekaterina A. Fedorova

National Research Tomsk State University

Author for correspondence.
Email: ekat_fedorova@mail.ru
ORCID iD: 0000-0001-8933-5322
Scopus Author ID: 56439120600
ResearcherId: E-3161-2017

PhD in Physical and Mathematical Sciences, Associate Professor of Department of Probability Theory and Mathematical Statistic

36 Lenin Ave, Tomsk, 634050, Russian Federation

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