Computing the Stationary Distribution of Queueing Systems with Random Resource Requirements via Fast Fourier Transform

• Published in: Mathematics (Basel) 2020-05, Vol.8 (5), p.772
• Main Authors: Naumov, Valeriy A., Gaidamaka, Yuliya V., Samouylov, Konstantin E.
• Format: Article
• Language: English
• Subjects: discretization; fast Fourier transform; queueing system; random resource requirements
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math8050772

Queueing systems with random resource requirements, in which an arriving customer, in addition to a server, demands a random amount of resources from a shared resource pool, have proved useful to analyze wireless communication networks. The stationary distributions of such queuing systems are expressed in terms of truncated convolution powers of the cumulative distribution function of the resource requirements. Discretization of the cumulative distribution function and the application of the fast Fourier transform are a traditional way of calculating convolutions. We suggest finding truncated convolution powers of the cumulative distribution functions by calculating the convolution powers of the truncated cumulative distribution functions via fast Fourier transform. This radically decreases computational complexity. We introduce the concept of resource load and investigate the accuracy of the proposed method at low and high resource loads. It is shown that the proposed method makes it possible to quickly and accurately calculate truncated convolution powers required for the analysis of queuing systems with random resource requirements.