Extensions of the Queueing Relations L = λW and H = λG

P. W. Glynn and W. Whitt

Operations Research, Vol. 37, 634-644 (1989)

This paper extends the fundamental queueing relations L = λW and H = λG that relate customer averages (the customer average waiting time W or cost G) to associated time averages (the time-average queue length L or cost H) given an arrival process with arrival rate λ. These relations can be established by focusing on a two-dimensional cumulative input process that has the two one-dimensional cumulative input processes of interest as marginals. Relations between the marginal averages are established for cumulative input processes that may not be representable as integrals or sums. The general framework includes the continuous versions of L = λW and H = λG due to T. Rolski and S. Stidham as well as the standard version of H = λG, and can be extended to higher dimensions. Inequalities are also established when some of the conditions for equality do not hold. Moreover, central limit theorem versions of H = λG are established, extending our recent results for L = λW.