An infinite server system with packing constraints and ranked servers

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.