# On the area swept under the occupation process of an *M/M/1* queue in a busy period

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## Abstract

We compute in this paper the distribution of the area \(\mathcal{A}\) swept under the occupation process of an *M/M/1* queue during a busy period. For this purpose, we use the expression of the Laplace transform \({\mathcal{A}^ \star }\) of the random variable \(\mathcal{A}\) established in earlier studies as a fraction of Bessel functions. To get information on the poles and the residues of \({\mathcal{A}^ \star }\), we take benefit of the fact that this function can be represented by a continued fraction. We then show that this continued fraction is the even part of an *S* fraction and we identify its successive denominators by means of Lommel polynomials. This allows us to numerically evaluate the poles and the residues. Numerical evidence shows that the poles are very close to the numbers \(\sigma _n = - \left( {1 + \rho } \right)/n\) as \(n \to \infty\). This motivated us to formulate some conjectures, which lead to the derivation of the asymptotic behaviour of the poles and the residues. This is finally used to derive the asymptotic behaviour of the probability survivor function \(P\left\{{\mathcal{A}>x}\right\}\). The outstanding property of the random variable \(\mathcal{A}\) is that the poles accumulate at 0 and its tail does not exhibit a nice exponential decay but a decay of the form \(cx^{{{ - 1} \mathord{\left/ {\vphantom {{ - 1} 2}} \right. \kern-\nulldelimiterspace} 2}} {\text{e}}^{ - \gamma \sqrt x }\) for some positive constants *c* and \(\gamma\), which indicates that the random variable \({\mathcal{A}}\) has a Weibull-like tail.

*M/M/1*queue continued fractions Bessel functions asymptotic expansion

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