31 research outputs found
Rare Events of Transitory Queues
We study the rare event behavior of the workload process in a transitory
queue, where the arrival epochs (or points) of a finite number of jobs are
assumed to be the ordered statistics of independent and identically distributed
(i.i.d.) random variables. The service times (or marks) of the jobs are assumed
to be i.i.d. random variables with a general distribution, that are jointly
independent of the arrival epochs. Under the assumption that the service times
are strictly positive, we derive the large deviations principle (LDP) satisfied
by the workload process. The analysis leverages the connection between ordered
statistics and self-normalized sums of exponential random variables to
establish the LDP. This paper presents the first analysis of rare events in
transitory queueing models, supplementing prior work that has focused on fluid
and diffusion approximations
Transitory Queueing Networks
Queueing networks are notoriously difficult to analyze sans both Markovian
and stationarity assumptions. Much of the theoretical contribution towards
performance analysis of time-inhomogeneous single class queueing networks has
focused on Markovian networks, with the recent exception of work in Liu and
Whitt (2011) and Mandelbaum and Ramanan (2010). In this paper, we introduce
transitory queueing networks as a model of inhomogeneous queueing networks,
where a large, but finite, number of jobs arrive at queues in the network over
a fixed time horizon. The queues offer FIFO service, and we assume that the
service rate can be time-varying. The non-Markovian dynamics of this model
complicate the analysis of network performance metrics, necessitating
approximations. In this paper we develop fluid and diffusion approximations to
the number-in-system performance metric by scaling up the number of external
arrivals to each queue, following Honnappa et al. (2014). We also discuss the
implications for bottleneck detection in tandem queueing networks
Dominating Points of Gaussian Extremes
We quantify the large deviations of Gaussian extreme value statistics on
closed convex sets in d-dimensional Euclidean space. The asymptotics imply that
the extreme value distribution exhibits a rate function that is a simple
quadratic function of a unique "dominating point" located on the boundary of
the convex set. Furthermore, the dominating point is identified as the
optimizer of a certain convex quadratic programming problem, indicating a
"collusion" between the dependence structure of the Gaussian random vectors and
the geometry of the convex set in determining the asymptotics. We specialize
our main result to polyhedral sets which appear frequently in other contexts
involving logarithmic asymptotics. We also extend the main result to
characterize the large deviations of Gaussian-mixture extreme value statistics
on general convex sets. Our results have implications to contexts arising in
rare-event probability estimation and stochastic optimization, since the nature
of the dominating point and the rate function suggest importance sampling
measures
Asymptotically optimal appointment schedules with customer no-shows
We consider the problem of scheduling appointments for a finite customer
population to a service facility with customer no-shows, to minimize the sum of
customer waiting time and server overtime costs. Since appointments need to be
scheduled ahead of time we refer to this problem as an optimization problem
rather than a dynamic control one. We study this optimization problem in fluid
and diffusion scales and identify asymptotically optimal schedules in both
scales. In fluid scale, we show that it is optimal to schedule appointments so
that the system is in critical load; thus heavy-traffic conditions are obtained
as a result of optimization rather than as an assumption. In diffusion scale,
we solve this optimization problem in the large horizon limit. Our explicit
stationary solution of the corresponding Brownian Optimization Problem
translates the customer-delay versus server-overtime tradeoff to a tradeoff
between the state of a reflected Brownian motion in the half-line and its local
time at zero. Motivated by work on competitive ratios, we also consider a
reference model in which an oracle provides the decision maker with the
complete randomness information. The difference between the values of the
scheduling problem for the two models, to which we refer as the stochasticity
gap (SG), quantifies the degree to which it is harder to design a schedule
under uncertainty than when the stochastic primitives (i.e., the no-shows and
service times) are known in advance. In the fluid scale, the SG converges to
zero, but in the diffusion scale it converges to a positive constant that we
compute
Conditioned Functional Limits and Applications to Queues
We consider a renewal process that is conditioned on the number of events in
a fixed time horizon. We prove that a centered and scaled version of this
process converges to a Brownian bridge, as the number of events grows large,
which relies on first establishing a functional strong law of large numbers
result to determine the centering. These results are consistent with the
asymptotic behavior of a conditioned Poisson process. We prove the limit
theorems over triangular arrays of exchangeable random variables, obtained by
conditionning a sequence of independent and identically distributed renewal
processes. We construct martingale difference sequences with respect to these
triangular arrays, and use martingale convergence results in our proofs. To
illustrate how these results apply to performance analysis in queueing, we
prove that the workload process of a single server queue with conditioned
renewal arrival process can be approximated by a reflected diffusion having the
sum of a Brownian Bridge and Brownian motion as input to its regulator mapping.Comment: Submitted to Advances in/ Journal of Applied Probabilit
A Many-Server Functional Strong Law For A Non-Stationary Loss Model
The purpose of this note is to show that it is possible to establish a
many-server functional strong law of large numbers (FSLLN) for the fraction of
occupied servers (i.e., the scaled number-in-system) without explicitly
tracking either the age or the residual service times of the jobs in a
non-Markovian, non-stationary loss model. This considerable analytical
simplification is achieved by exploiting a semimartingale representation. The
fluid limit is shown to be the unique solution of a Volterra integral equation
An optimal control approach of day-to-day congestion pricing for stochastic transportation networks
Congestion pricing has become an effective instrument for traffic demand
management on road networks. This paper proposes an optimal control approach
for congestion pricing for day-to-day timescale that incorporates demand
uncertainty and elasticity. Travelers make the decision to travel or not based
on the experienced system travel time in the previous day and traffic managers
take tolling decisions in order to minimize the average system travel time over
a long time horizon. We formulate the problem as a Markov decision process
(MDP) and analyze the problem to see if it satisfies conditions for conducting
a satisfactory solution analysis. Such an analysis of MDPs is often dependent
on the type of state space as well as on the boundedness of travel time
functions. We do not constrain the travel time functions to be bounded and
present an analysis centered around weighted sup-norm contractions that also
holds for unbounded travel time functions. We find that the formulated MDP
satisfies a set of assumptions to ensure Bellman's optimality condition.
Through this result, the existence of the optimal average cost of the MDP is
shown. A method based on approximate dynamic programming is proposed to resolve
the implementation and computational issues of solving the control problem.
Numerical results suggest that the proposed method efficiently solves the
problem and produces accurate solutions
Approximating Systems Fed by Poisson Processes with Rapidly Changing Arrival Rates
This paper introduces a new asymptotic regime for simplifying stochastic
models having non-stationary effects, such as those that arise in the presence
of time-of-day effects. This regime describes an operating environment within
which the arrival process to a service system has an arrival intensity that is
fluctuating rapidly. We show that such a service system is well approximated by
the corresponding model in which the arrival process is Poisson with a constant
arrival rate. In addition to the basic weak convergence theorem, we also
establish a first order correction for the distribution of the cumulative
number of arrivals over , as well as the number-in-system process for an
infinite-server queue fed by an arrival process having a rapidly changing
arrival rate. This new asymptotic regime provides a second regime within which
non-stationary stochastic models can be reasonably approximated by a process
with stationary dynamics, thereby complementing the previously studied setting
within which rates vary slowly in time
A queueing model with independent arrivals, and its fluid and diffusion limits
We introduce the {\Delta}(i)/GI/1 queue, a new queueing model. In this model,
customers from a given population independently sample a time to arrive from
some given distribution F. Thus, the arrival times are an ordered statistics,
and the inter-arrival times are differences of consecutive ordered statistics.
They are served by a single server which provides service according to a
general distribution G, with independent service times. The exact model is
analytically intractable. Thus, we develop fluid and diffusion limits for the
various stochastic processes, and performance metrics. The fluid limit of the
queue length is observed to be a reflected process, while the diffusion limit
is observed to be a function of a Brownian motion and a Brownian bridge
process, and is given by a 'netput' process and a directional derivative of the
Skorokhod reflected fluid netput in the direction of a diffusion refinement of
the netput process. We also observe what may be interpreted as a transient
Little's law. Sample path analysis reveals various operating regimes where the
diffusion limit switches between a free diffusion, a reflected diffusion
process and the zero process, with possible discontinuities during regime
switches. The weak convergence is established in the M1 topology, and it is
also shown that this is not possible in the J1 topology.Comment: Accepted, Queueing System
Risk-Sensitive Variational Bayes: Formulations and Bounds
We study data-driven decision-making problems in a parametrized Bayesian
framework. We adopt a risk-sensitive approach to modeling the interplay between
statistical estimation of parameters and optimization, by computing a risk
measure over a loss/disutility function with respect to the posterior
distribution over the parameters. While this forms the standard Bayesian
decision-theoretic approach, we focus on problems where calculating the
posterior distribution is intractable, a typical situation in modern
applications with %high-dimensional parameter space large datasets,
heterogeneity due to observed covariates and latent group structure. The key
methodological innovation we introduce in this paper is to leverage a dual
representation of the risk measure to introduce an optimization-based framework
for approximately computing the posterior risk-sensitive objective, as opposed
to using standard sampling based methods such as Markov Chain Monte Carlo. Our
analytical contributions include rigorously proving finite sample bounds on the
`optimality gap' of optimizers obtained using the computational methods in this
paper, from the `true' optimizers of a given decision-making problem. We
illustrate our results by comparing the theoretical bounds with simulations of
a newsvendor problem on two methods extracted from our computational framework