Recursive solution technique in a multi-server bi-level queueing system with server breakdowns

The authors present a multiserver, first-come first-served queuing system that alternates between two modes of system operation. In one mode, all s servers are available, and in the other mode, only s-1 servers are available for serving the customers. This is due to breakdown of one of the servers. The random variables representing the system with s servers and s-1 servers have exponential distributions. In such a system, the steady-state birth/death equations are coupled because of the two modes of operation. A recursive solution is presented for computing the steady-state probabilities of such a system. Once these probabilities are known, the performance measures of interest can be easily obtained. Two practical examples validate the results and show the utility of this method. A distinct advantage of the recursive technique is that it is much faster and requires much less memory than the existing nonrecursive techniques. In a bilevel situation, the system performance measures are always bounded by two independent queuing systems with s and s-1 servers. A procedure has been outlined for extension to multiple modes of system operation