A service system with multiple types of customers, arriving as Poisson processes, is considered. The system has infinite number of servers, ranked by $1,2,3, \ldots$; a server rank is its ``location." Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to ``packing'' constraints. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered, such that the mean number of customers $r$ goes to infinity. We seek algorithms with the underlying objective of minimizing the location (rank) $U$ of the right-most (highest ranked) occupied (non-empty) server. Therefore, this objective seeks to minimize the total number $Q$ of occupied servers {\em and} keep the set of occupied servers as far at the ``left'' as possible, i.e., keep $U$ close to $Q$. In previous work, versions of {\em Greedy Random} (GRAND) algorithm have been shown to asymptotically minimize $Q/r$ as $r\to\infty$. In this paper we show that when these algorithms are combined with the First-Fit rule for ``taking'' empty servers, they asymptotically minimize $U/r$ as well.

翻译：本文研究一个多类型顾客的服务系统，顾客以泊松过程到达。系统具有无限数量的服务器，按$1,2,3, \ldots$排序；服务器序号即为其“位置”。每位顾客具有独立的指数分布服务时间，其均值由顾客类型决定。多位顾客（可能属于不同类型）可被安排至同一服务器接受服务，但需满足“装箱”约束。不同顾客的服务时间相互独立，即使在同一服务器中同时接受服务。我们考虑大规模渐近机制，即顾客平均数量$r$趋于无穷大。我们寻求算法的根本目标是最小化最右侧（序号最大）被占用（非空）服务器的位置（序号）$U$。因此，该目标旨在最小化被占用服务器总数$Q$，并尽可能使被占用服务器集合向“左侧”集中，即保持$U$接近$Q$。在先前研究中，已证明不同版本的贪心随机（GRAND）算法在$r\to\infty$时能渐近最小化$Q/r$。本文证明，当这些算法与“占用”空服务器的首次适应规则结合时，同样能渐近最小化$U/r$。