An efficient method to compute the rate matrix for retrial queues with large number of servers

The approximate solution technique for the main M/M/c retrial queue based on the homogenization of the model employs a quasi-birth-death (QBD) process in which the maximum retrial rate is restricted above a certain level. This approximated continuous-time Markov chain (CTMC) can be solved by the matrixgeometric method, which involves the computation of the rate matrix R. This paper is motivated by two observations. Firstly, retrial queues for the performability analysis of telecommunication systems often involve the number of servers in the order of several hundreds of thousands. Secondly, there are no workable solutions till now for systems with such large number of servers, due ill-conditioning or prohibitively large computation times. Our paper is the first to tackle the problem of large number of servers, very efficiently, in the homogenized M/M/c retrial queue which has paramount applications in networks. We present an efficient algorithm with the time complexity of only O(c) to compute the rate matrix R.