The Delay of Open Markovian Queueing Networks: Uniform Functional Bounds, Heavy Traffic Pole Multiplicities, and Stability

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 coe cients fCig of the expansion CM (1 )M + CM 1 (1 )M 1 + + C1 (1 ) + C0, where is the load factor, which are valid for all 2 [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 delay, valid for all arrival rates in the capacity region. The coe cients fCig can be optimized to provide the best bound at any desired value of the load factor, while still maintaining its validity for all 2 [0; 1). While the above LPs feature L(L+1)(M+1) 2 variables where L is the number of bu ers 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 tra c a bound obtainable from the Pollaczek-Khintchine formula, and can capture interactions between multiple bottleneck stations in heavy tra c. We Please address all correspondence to the third author. yThe research reported here has been supported by the U.S. Army Research O ce under Contract No. DAAH-04-95-1-0090, the National Science Foundation under Grant No. ECS-94-03571, the Joint Services Electronics Program under Contract No. N00014-90-J-1270, FAPESP grants 93/0603-1 and 94/1376-1, and by CNPq and IBM-Brasil. zDepartamento de Ciencia da Computacao, Instituto de Mathematica e Estatistica, University of Sao Paulo, Brazil. This work was done while the author was visiting the University of Illinois. xDepartment of Decision Sciences, Faculty of Business Administration, National University of Singapore, Singapore 0511. This work was done while the author was visiting the University of Illinois. {Department of Electrical and Computer Engineering, and the Coordinated Science Laboratory, University of Illinois, 1308 West Main St., Urbana, IL 61801. 1 also provide an explicit upper bound for all scheduling policies in acyclic networks, and for the FBFS policy in open re-entrant lines.