Although product-forms have long been known for the equilibrium probability distribution of queue lengths in Markovian G-Nets, sojourn (or response) time distribution pose a much more difficult problem. This is because customers can be affected by later arrivals, even under FCFS queueing discipline. Here, the Laplace transform of the probability density function of the response time in a single server, M/M/1 queue with negative rarivals is obtained and the result generalised to the end-to-end delay in tandem networks of two queues. The generalisation is complex, the method being based on the solution of a Riemann-Hilbert-Carleman boundary value problem on a closed contour in the complex plane. The solution is that of a Fredholm integral equation of the second kind.
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Information from pubs.doc.ic.ac.uk/g-net-response-times.