Flexible
Server Allocation and Customer Routing Policies for
Two Parallel
Queues when Service Rates are not Additive
Mark E. Lewis
Cornell University
School of Operations Research and Information Engineering
226 Rhodes Hall
Ithaca, New York 14853
Hyun-soo
Ahn
Operations and Management
Science, Ross School of Business
University of Michigan
701 Tappan Street
Ann Arbor, Michigan 48109-1234
We
consider the question of how routing and allocation can be coordinated to meet
the
challenge of demand variability in a
parallel queueing system serving two types of
customers.
A
decision-maker decides whether to keep customers at the station at which they
arrived or to
reroute them to the other station.
At the same time, the decision-maker has two servers and
must decide where to allocate
their effort. We analyze this joint decision-making scenario, but
add two important twists. First,
we allow the combined service rate (when the servers work
at the same station) to be super-additive or sub-additive. This captures positive or negative
externalities that arise during
collaboration. Second, routing costs are allowed to be strictly
positive. We seek an optimal control
policy under the discounted or long-run average cost
criteria.
Our
results show that in the super-additive case jobs should never be routed away
from
the lower cost queue. When jobs
are rerouted from the higher cost queue to the low cost
queue the optimal control is
monotone in the respective queue lengths. Moreover, we show
that the optimal allocation is a
non-idling priority rule based on the holding costs. In the
sub-additive case we find that the
optimal policy need not exhibit such a simple structure.
In
fact, the optimal allocation need not prioritize one station (it may split the
servers), and
the optimal routing need not be
monotone in the number of customers in each queue. We
characterize the optimal policy for a few
canonical cases, and discuss why intuitive policies
need not be optimal in the
general case. An extensive numerical study examines the benefit
of dynamically controlling both
routing and resource allocation; we discuss when using one of
the two levers – dynamic routing
and dynamic allocation – is sufficient and when using both
controls is warranted.