81 research outputs found
Scheduling control for queueing systems with many servers: asymptotic optimality in heavy traffic
A multiclass queueing system is considered, with heterogeneous service
stations, each consisting of many servers with identical capabilities. An
optimal control problem is formulated, where the control corresponds to
scheduling and routing, and the cost is a cumulative discounted functional of
the system's state. We examine two versions of the problem: ``nonpreemptive,''
where service is uninterruptible, and ``preemptive,'' where service to a
customer can be interrupted and then resumed, possibly at a different station.
We study the problem in the asymptotic heavy traffic regime proposed by Halfin
and Whitt, in which the arrival rates and the number of servers at each station
grow without bound. The two versions of the problem are not, in general,
asymptotically equivalent in this regime, with the preemptive version showing
an asymptotic behavior that is, in a sense, much simpler. Under appropriate
assumptions on the structure of the system we show: (i) The value function for
the preemptive problem converges to , the value of a related diffusion
control problem. (ii) The two versions of the problem are asymptotically
equivalent, and in particular nonpreemptive policies can be constructed that
asymptotically achieve the value . The construction of these policies is
based on a Hamilton--Jacobi--Bellman equation associated with .Comment: Published at http://dx.doi.org/10.1214/105051605000000601 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
A diffusion model of scheduling control in queueing systems with many servers
This paper studies a diffusion model that arises as the limit of a queueing
system scheduling problem in the asymptotic heavy traffic regime of Halfin and
Whitt. The queueing system consists of several customer classes and many
servers working in parallel, grouped in several stations. Servers in different
stations offer service to customers of each class at possibly different rates.
The control corresponds to selecting what customer class each server serves at
each time. The diffusion control problem does not seem to have explicit
solutions and therefore a characterization of optimal solutions via the
Hamilton-Jacobi-Bellman equation is addressed. Our main result is the existence
and uniqueness of solutions of the equation. Since the model is set on an
unbounded domain and the cost per unit time is unbounded, the analysis requires
estimates on the state process that are subexponential in the time variable. In
establishing these estimates, a key role is played by an integral formula that
relates queue length and idle time processes, which may be of independent
interest.Comment: Published at http://dx.doi.org/10.1214/105051604000000963 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Control of the multiclass queue in the moderate deviation regime
A multi-class single-server system with general service time distributions is
studied in a moderate deviation heavy traffic regime. In the scaling limit, an
optimal control problem associated with the model is shown to be governed by a
differential game that can be explicitly solved. While the characterization of
the limit by a differential game is akin to results at the large deviation
scale, the analysis of the problem is closely related to the much studied area
of control in heavy traffic at the diffusion scale.Comment: Published in at http://dx.doi.org/10.1214/13-AAP971 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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