A Service System with Packing Constraints: Greedy Randomized Algorithm Achieving Sublinear in Scale Optimality Gap

A service system with multiple types of arriving customers is considered. There is an infinite number of homogeneous servers. Multiple customers can be placed for simultaneous service into one server, subject to general packing constraints. The service times of different customers are independent even if they are served simultaneously by the same server; the service time distribution depends on the customer type. Each new arriving customer is placed for service immediately into either an occupied server, that is, one already serving other customers, as long as packing constraints are not violated or into an empty server. After service completion, each customer leaves its server and the system. The basic objective is to minimize the number of occupied servers in steady state. We study a greedy random (GRAND) placement (packing) algorithm, introduced in our previous work. This is a simple online algorithm that places each arriving customer uniformly at random into either one of the already occupied servers that can still fit the customer or one of the so-called zero servers, which are empty servers designated to be available to new arrivals. In our previous work, a version of the algorithm, labeled GRAND(aZ), is considered, in which the number of zero servers is aZ with Z being the current total number of customers in the system and positive a being an algorithm parameter. GRAND(aZ) is shown in our previous work to be asymptotically optimal in the following sense: (a) the steady-state optimality gap grows linearly in the system scale r (the mean total number of customers in service), that is, as c(a)r for some positive c(a), and (b) c(a) vanishes as a goes to zero. In this paper, we consider the GRAND(Z^p) algorithm, in which the number of zero servers is Z^p, where p < 1 is a fixed parameter, sufficiently close to 1. We prove the asymptotic optimality of GRAND(Z^p) in the sense that the steady-state optimality gap is sublinear in the system scale r. This is a stronger form of asymptotic optimality than that of GRAND(aZ).