Stationary and transient solution of Markovian queues - an alternate approach

Abstract: This paper aims at presenting an alternate approach to derive the exact transient solution of Markovian queues. The continued fraction (Jacobi fraction) is expressed as power series and the power series coefficients are connected by a two-dimensional recurrence relation associated with infinite Stieltjes matrix equation. The recurrence relation is solved by using generating functions. A novel state-dependent birth-death queueing model is taken and expressed as continued fractions by employing integral transforms. The stationary probabilities of general state-dependent Markovian queues are obtained from the continued fractions and its associated tridiagonal determinants. As a special case the time-dependent system size probabilities and busy period distribution of a classical single server queue are deduced using continued fraction and its power series. Numerical illustrations are also presented.