For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.
Tijms, H. C., M. H. Van Hoorn, and Awi Federgruen. "Approximations for the steady-state probabilities in the M/G/c queue." Advances in Applied Probability 13, no. 1 (March 1981): 186-206.
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