Computing system-time and system-length distributions for MAP/D/1 queue using distributional Little’s law

In this paper, we first present (in terms of roots) a simple closed-form analysis for evaluating virtual queueing-time and actual system-time distributions for the MAP/D/1 queue. Then we obtain the system-length distribution, using the distributional Little’s law. The analysis proposed here is based on the roots of the characteristic equation of the Laplace–Stieltjes transform of the virtual queueing-time distribution. Numerical aspects have been tested for a variety of arrival and service-time parameters and a sample of numerical outputs along with detailed discussion on accuracy and computation-time is presented.