We consider a queuing system with multitype customers and
nonhomogeneous flexible servers, in the heavy traffic asymptotic regime
and under a complete resource pooling (CRP) condition. For the
input-queued (IQ) version of such a system (with customers being
queued at the system “entrance,” one queue per each type), it
was shown in the work of Mandelbaum and Stolyar that a simple parsimonious
Gcμ scheduling rule is optimal in that it asymptotically
minimizes the system customer workload and some strictly convex
queuing costs. In this article, we consider a
different—output-queued (OQ)—version of the model,
where each arriving customer must be assigned to one of the servers
immediately upon arrival. (This constraint can be interpreted as immediate
routing of each customer to one of the “output
queues,” one queue per each server.) Consequently, the space of
controls allowed for an OQ system is a subset of that for the
corresponding IQ system.We introduce the MinDrift routing rule for OQ systems (which
is as simple and parsimonious as Gcμ) and show that this
rule, in conjunction with arbitrary work-conserving disciplines at the
servers, has asymptotic optimality properties analogous to those
Gcμ rule has for IQ systems. A key element of the analysis is
the notion of system server workload, which, in particular,
majorizes customer workload. We show that (1) the MinDrift rule
asymptotically minimizes server workload process among all OQ-system
disciplines and (2) this minimal process matches the minimal possible
customer workload process in the corresponding IQ system. As a corollary,
MinDrift asymptotically minimizes customer workload among all disciplines
in either the OQ or IQ system.
Dynamic scheduling of a system with two parallel servers in heavy traffic with resource pooling: asymptotic optimality of a threshold policy
