Explicit Solutions of M/G/C/ /N-type Queueing Loops with Generalizations

Using recently developed Matrix-Algebraic techniques, we find explicit steady-state solutions of M/G/C/ /N-type loops that depend on a set of recursively defined matrices. We show that such systems are special cases of arbitrary service centers that contain (load dependent) exponential servers in which no more than C customers can be active simultaneously. We also outline a recursive algorithm that can be used to evaluate the properties of small- to moderate-sized systems. The solution to the M/G/C open system is then found by letting the overall customer population N go to infinity. The solutions for closed systems in general are not of the matrix geometric type, and only in the limit does the solution become geometric in form.