Abstract
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.
Highlights
When queuing theory is applied to modeling modern information technology systems, one should take into account various system’s features, such as the reliability of a radio channel in wireless communication networks
The steady state distributions of such systems are expressed in terms of truncated convolution powers of the cumulative distribution function (CDF) of the resource requirement
In order to obtain the stationary distribution for a resource queueing system of capacity L it is necessary to compute L truncated convolution powers of the resource requirement CDF
Summary
When queuing theory is applied to modeling modern information technology systems, one should take into account various system’s features, such as the reliability of a radio channel in wireless communication networks. The method of numerical transform inversion can be successfully applied for the analysis of some resource queueing systems with discrete CDF of resource requirements, such as product-form loss networks [9,10,11]. A more accurate h i approximation, F3 (x) = G τx τ + 0.5τ , was used in reference [18] to compute the CDF of stationary waiting time in a single server queue with general inter-arrival and service time distributions by means of the fast Fourier transform (FFT). In order to obtain the stationary distribution for a resource queueing system of capacity L it is necessary to compute L truncated convolution powers of the resource requirement CDF. In this paper we develop an algorithm for the computation of sequences of truncated convolution powers of continuous CDF and truncated compound distributions via FFT.
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