Analysis of congestion periods of an m/m/-queue

A c-congestion period of an m/m/ ∞ -queue is a period during which the number of customers in the system is continuously above level c. Interesting quantities related to a c-congestion period are, besides its duration D c , the total area A c above c, and the number of arrived customers N c . In the literature Laplace transforms for these quantities have been derived, as well as explicit formulae for their means. Explicit expressions for higher moments and covariances (between D c , N c and A c ), however, have not been found so far. aper presents recursive relations through which all moments and covariances can be obtained. Up to a starting condition, we explicitly solve these equations; for instance, we write E D c 2 explicitly in terms of E D 0 2 . We then find formulae for these starting conditions (which directly relate to the busy period in the m/m/ ∞  queue). y, a c-intercongestion period is defined as the period during which the number of customers is continuously below level c. Also for this situation a recursive scheme allows us to explicitly compute higher moments and covariances. Additionally we present the Laplace transform of a so-called intercongestion triple of the three performance quantities. It is also shown that expressions for the quantities of a c-intercongestion period can be used in an approximation for the c-congestion period. This is especially useful as the expressions for the c-intercongestion period are numerically more stable than those for the c-congestion period.