Simple approximations for the gi/g/c queue—i: The steady-state probabilities

Viewing the GI/G/c queue as a service system alternating between two basic states—that of a loaded (non-empty) GI/G/1 queue and that of a GI/G/oo queue (dependent, respectively, on whether all servers in the GI/G/c queue are busy or otherwise)—approximations for the components of the mixture distribution of the steady-state probabilities are derived. The M/G/c queue is separately treated. Two imposed prerequisites, that only minimal prior information about the queue will be required and that no numeric method be needed other than a root-finding algorithm, are strictly adhered to. The accuracy attained is generally satisfactory, while remarkable algebraic simplicity is preserved.