A matrix-algebraic solution to two K m servers in a loop

Abstract

An explicit steady-state solution is given for any queuing loop made up of two general servers, whose distribution functions have rational Laplace transforms. The solution is in matrix geometric form over a vector space that is itself a direct or Kronecker product of the internal state spaces of the two servers. The algebraic properties of relevant entities in this space are given in an appendix. The closed-form solution yields simple recursive relations that in turn lead to an efficient algorithm for calculating various performance measures such as queue length and throughput. A computational-complexity analysis shows that the algorithm requires at least an order of magnitude less computational effort than any previously reported algorithm.