143 research outputs found
Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic
We consider the problem of scheduling a queueing system in which many
statistically identical servers cater to several classes of impatient
customers. Service times and impatience clocks are exponential while arrival
processes are renewal. Our cost is an expected cumulative discounted function,
linear or nonlinear, of appropriately normalized performance measures. As a
special case, the cost per unit time can be a function of the number of
customers waiting to be served in each class, the number actually being served,
the abandonment rate, the delay experienced by customers, the number of idling
servers, as well as certain combinations thereof. We study the system in an
asymptotic heavy-traffic regime where the number of servers n and the offered
load r are simultaneously scaled up and carefully balanced: n\approx r+\beta
\sqrtr for some scalar \beta. This yields an operation that enjoys the benefits
of both heavy traffic (high server utilization) and light traffic (high service
levels.
Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic
A result of Ward and Glynn (2005) asserts that the sequence of scaled offered
waiting time processes of the queue converges weakly to a
reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the
traffic intensity approaches one. As a consequence, the stationary distribution
of a ROU process, which is a truncated normal, should approximate the scaled
stationary distribution of the offered waiting time in a queue;
however, no such result has been proved. We prove the aforementioned
convergence, and the convergence of the moments, in heavy traffic, thus
resolving a question left open in Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (1961) showing that an exponential
distribution approximates the scaled stationary offered waiting time
distribution in a queue in heavy traffic, our result confirms that
the addition of customer abandonment has a non-trivial effect on the queue
stationary behavior.Comment: 29 page
Asymptotic optimality of maximum pressure policies in stochastic processing networks
We consider a class of stochastic processing networks. Assume that the
networks satisfy a complete resource pooling condition. We prove that each
maximum pressure policy asymptotically minimizes the workload process in a
stochastic processing network in heavy traffic. We also show that, under each
quadratic holding cost structure, there is a maximum pressure policy that
asymptotically minimizes the holding cost. A key to the optimality proofs is to
prove a state space collapse result and a heavy traffic limit theorem for the
network processes under a maximum pressure policy. We extend a framework of
Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams
[Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing
network setting to the stochastic processing network setting to prove the state
space collapse result and the heavy traffic limit theorem. The extension can be
adapted to other studies of stochastic processing networks.Comment: Published in at http://dx.doi.org/10.1214/08-AAP522 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Redundancy Scheduling with Locally Stable Compatibility Graphs
Redundancy scheduling is a popular concept to improve performance in
parallel-server systems. In the baseline scenario any job can be handled
equally well by any server, and is replicated to a fixed number of servers
selected uniformly at random. Quite often however, there may be heterogeneity
in job characteristics or server capabilities, and jobs can only be replicated
to specific servers because of affinity relations or compatibility constraints.
In order to capture such situations, we consider a scenario where jobs of
various types are replicated to different subsets of servers as prescribed by a
general compatibility graph. We exploit a product-form stationary distribution
and weak local stability conditions to establish a state space collapse in
heavy traffic. In this limiting regime, the parallel-server system with
graph-based redundancy scheduling operates as a multi-class single-server
system, achieving full resource pooling and exhibiting strong insensitivity to
the underlying compatibility constraints.Comment: 28 pages, 4 figure
Queueing systems with many servers: Null controllability in heavy traffic
A queueing model has heterogeneous service stations, each consisting
of many independent servers with identical capabilities. Customers of
classes can be served at these stations at different rates, that depend on both
the class and the station. A system administrator dynamically controls
scheduling and routing. We study this model in the central limit theorem (or
heavy traffic) regime proposed by Halfin and Whitt. We derive a diffusion model
on with a singular control term that describes the scaling
limit of the queueing model. The singular term may be used to constrain the
diffusion to lie in certain subsets of at all times . We
say that the diffusion is null-controllable if it can be constrained to
, the minimal closed subset of containing all
states of the prelimit queueing model for which all queues are empty. We give
sufficient conditions for null controllability of the diffusion. Under these
conditions we also show that an analogous, asymptotic result holds for the
queueing model, by constructing control policies under which, for any given
, all queues in the system are kept empty on the time
interval , with probability approaching one. This introduces a
new, unusual heavy traffic ``behavior'': On one hand, the system is critically
loaded, in the sense that an increase in any of the external arrival rates at
the ``fluid level'' results with an overloaded system. On the other hand, as
far as queue lengths are concerned, the system behaves as if it is underloaded.Comment: Published at http://dx.doi.org/10.1214/105051606000000358 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
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