Maximum queue lengths during a fixed time interval in the M/M/cM/M/c retrial queue

We are concerned with the problem of characterizing the distribution of the maximum number Z(t0)Z(t0) of customers during a fixed time interval [0,t0][0,t0] in the M/M/cM/M/c retrial queue, which is shown to have a matrix exponential form. We present a simple condition on the service and retrial rates for the matrix exponential solution to be explicit or algorithmically tractable. Our methodology is based on splitting methods and the use of eigenvalues and eigenvectors. A particularly appealing feature of our solution is that it allows us to obtain global error control. Specifically, we derive an approximating solution p(x;t0)≡p(x;t0;ε)p(x;t0)≡p(x;t0;ε) verifying |P(Z(t0)⩽x|X(0)=(i,j))-p(x;t0)|<ε|P(Z(t0)⩽x|X(0)=(i,j))-p(x;t0)|<ε uniformly in x⩾i+jx⩾i+j, for any ε>0ε>0 and initial numbers i of busy servers and j of customers in orbit.