Spectral decomposition approach for transient analysis of multi-server discrete-time queues

Previous work by the authors (See Proceedings Conf. on Information Sciences and Systems, Baltimore, USA, 1990) where the spectral decomposition method was used for the transient analysis of a single-server queue is generalized. For the arrival process a general discrete-time Markovian batch arrival process is assumed, where the batch size distribution of the arrivals in successive slots is governed by a N-state discrete-time Markov chain. It is shown that once the N eigenvalues of the probability generating matrix of the arrival process are obtained, the complete solution in the transform domain may be given. Using the complex analysis technique and Cauchy´s integral formula, an efficient numerical method is presented for the calculation of a few performance measures. The numerical method is generalized to the situations where the superpositions of a number of independent arrival sources are fed to the queue. It is shown that the numerical complexity are fed to the queue. It is shown that the numerical complexity of obtaining the transient solution in this case can be substantially reduced by using an approach based on the Kronecker product