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Renewal theory and queueing algorithms for matrix-exponential distributions. (English) Zbl 0872.60064
Chakravarthy, Srinivas R. (ed.) et al., Matrix-analytic methods in stochastic models. Proceedings of the first international conference held in Detroit, MI, USA, August 24–25, 1995. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 183, 313-341 (1997).
Summary: A probability density \(b(x)\) is called matrix-exponential if \(b(x)= \alpha e^{Tx} s\) where \(\alpha\) is a row vector, \(s\) is a column vector and \(T\) is a matrix (complex entries are allowed). An equivalent characterization is that the Laplace transform is rational. For matrix-exponential distributions, we show how to compute explicitly a number of quantities in renewal theory, like the renewal density, the density of the overshoot and the variance of the number \(N_t\) of counts up to time \(t\). Further we give an algorithm for computing the waiting time distribution of a queue with matrix-exponential service times and general interarrival times, which is a literal generalization of recent results obtained for the special case of phase-type services. Also some discussion of basic structural properties of matrix-exponential distributions are given, in particular of the question of minimal representations.
For the entire collection see [Zbl 0852.00031].

60K05 Renewal theory
60K25 Queueing theory (aspects of probability theory)
15A03 Vector spaces, linear dependence, rank, lineability
15A15 Determinants, permanents, traces, other special matrix functions