On the largest negative eigenvalue of the infinitesimal generator associated with M/M/n/n queues

In this note, properties of the largest negative eigenvalue r1(λ,n) of the infinitesimal generator associated with an M/M/n/n queue are under examination having arrival rate λ and service rate μ. It will be shown that r1(λ,n) is strictly decreasing in λ with r1(0,n)=¸-μ and r1(nμ,n)=¸-2μ. An efficient algorithm to find r1(λ,n) is presented. Various upper and lower bounds are evaluated based on the exact value obtained by the algorithm.