The variance function of the Erlang process

Summary A relatively simple exact expression of closed form is obtained for the varianceσ ²(t) of the asynchronous counting distribution for a counting period of lengtht,t>0, in an Erlang process. Useful bounds are placed upon the error of the linear approximation toσ ²(t). Implications of these results are examined. In particular, a new exact expression and related bounds are obtained for the mean function of the synchronous counts (also known as the renewal function of the process). All bounds given are sharp in asymptotic order of magnitude as the length of the counting period is allowed to increase. Prepared under Contract No. FH-11-6890, Bureau of Public Roads, Federal Highway Administration, U.S. Department of Transportation. The opinions, findings and conclusions expressed herein are those of the author and not necessarily those of the Bureau of Public Roads.