For open Markovian queueing networks, we study the functional dependence of the mean number in the system (and thus also the mean delay since it is proportional to it by Little's Theorem) on the arrival rate or load factor. We obtain linear programs (LPs) which provide bounds on the pole multiplicity M of the mean number in the system, and automatically obtain lower and upper bounds on the coefficients {Ci} of the expansion ρCM/(1 − ρ)M + ρCM−1/(1 − ρ)M−1 + . . . + ρC1/(1 − ρ) + ρC0, where ρ is the load factor, which are valid for all ρ ∈ [0, 1). Our LPs can thus establish the stability of open networks for all arrival rates within capacity, while providing uniformly bounding functional expansions for the mean delays, valid for all arrival rates in the capacity region. The coefficients {Ci} can be optimized to provide the best bound at any desired value of the load factor, while still maintaining its validity for all ρ ∈ [0, 1). While the above LPs feature L(L + 1)(M + 1)/2 variables where L is the number of buffers in the network, for balanced systems we further provide a lower dimensional LP featuring just S(S + 1)/2 variables, where S is the number of stations in the network. This bound asymptotically dominates in heavy traffic a bound obtainable from the Pollaczek–Khintchine formula, and can capture interactions between multiple bottleneck stations in heavy traffic. We provide an explicit upper bound for all scheduling policies in acyclic networks, and for the first batch first serve policy in open re-entrant lines.
