A unifying onservation law for single-server queues

Urtzi Ayesta LAAS-CNRS 7, Avenue Colonel Ro he 31077, Toulouse e-mail: urtzi laas.fr Abstra t We develop a onservation law for a multilass GI/GI/1 queue operating under a general workonserving s heduling dis ipline. For singlelass single-server queues, onservation laws have been obtained for both non-anti ipating and anti ipating dis iplines with general servi e time distributions. For multilass single-server queues, onservation laws have been obtained for (i) non-anti ipating dis iplines with exponential servi e time distributions and (ii) non-preemptive non-anti ipating dis iplines with general servi e time distributions. The unifying onservation law we develop generalizes already existing onservation laws. In addition it overs popular non-anti ipating multilass time-sharing dis iplines su h as Dis riminatory Pro essor Sharing (DPS) and Generalized Pro essor Sharing (GPS) with general servi e time distributions. As an appli ation we show that the unifying onservation law an be used to ompare the expe ted un onditional response time under two s heduling dis iplines. 1 Introdu tion The soalled workonserving property is fundamental to single-server (multilass) systems. Let us onsider a single-server queue with M job lasses. Let Uj(t) be the un nished work at time t of lass-j jobs, j = 1, . . . ,M , and let U(t) = ∑Mj=1 Uj(t) denote the total un nished work in the system. The un nished work in the system, U(t), is a fun tion that has verti al jumps at arrival epo hs equal in size to the orresponding servi e requirements of the job and remains onstant when it hits the horizontal axis. We say that the s heduling dis ipline is workonserving if U(t) de reases at rate 1(sec/sec) whenever U(t) > 0. A sample path argument shows that the un nished work in the system, U(t), is the same regardless of the workonserving s heduling dis ipline being deployed. In this paper we fo us on onservation laws for the time average un nished work. We refer to Green and Stidham [11℄ and Sigman [23℄ for the derivation and appli ation of sample-path onservation laws. Let U j denote the time average un nished work of lass j, j = 1, . . . ,M . The workonservation property implies that the total time average un nished work, U = M ∑