212 research outputs found
Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results
Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems
On deciding stability of multiclass queueing networks under buffer priority scheduling policies
One of the basic properties of a queueing network is stability. Roughly
speaking, it is the property that the total number of jobs in the network
remains bounded as a function of time. One of the key questions related to the
stability issue is how to determine the exact conditions under which a given
queueing network operating under a given scheduling policy remains stable.
While there was much initial progress in addressing this question, most of the
results obtained were partial at best and so the complete characterization of
stable queueing networks is still lacking. In this paper, we resolve this open
problem, albeit in a somewhat unexpected way. We show that characterizing
stable queueing networks is an algorithmically undecidable problem for the case
of nonpreemptive static buffer priority scheduling policies and deterministic
interarrival and service times. Thus, no constructive characterization of
stable queueing networks operating under this class of policies is possible.
The result is established for queueing networks with finite and infinite buffer
sizes and possibly zero service times, although we conjecture that it also
holds in the case of models with only infinite buffers and nonzero service
times. Our approach extends an earlier related work [Math. Oper. Res. 27 (2002)
272--293] and uses the so-called counter machine device as a reduction tool.Comment: Published in at http://dx.doi.org/10.1214/09-AAP597 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stability of Skorokhod problem is undecidable
Skorokhod problem arises in studying Reflected Brownian Motion (RBM) on an
non-negative orthant, specifically in the context of queueing networks in the
heavy traffic regime. One of the key problems is identifying conditions for
stability of a Skorokhod problem, defined as the property that trajectories are
attracted to the origin. The stability conditions are known in dimension up to
three, but not for general dimensions.
In this paper we explain the fundamental difficulties encountered in trying
to establish stability conditions for general dimensions. We prove that
stability of Skorokhod problem is an undecidable property when the starting
state is a part of the input. Namely, there does not exist an algorithm (a
constructive procedure) for identifying stable Skorokhod problem in general
dimensions
Instability in Stochastic and Fluid Queueing Networks
The fluid model has proven to be one of the most effective tools for the
analysis of stochastic queueing networks, specifically for the analysis of
stability. It is known that stability of a fluid model implies positive
(Harris) recurrence (stability) of a corresponding stochastic queueing network,
and weak stability implies rate stability of a corresponding stochastic
network. These results have been established both for cases of specific
scheduling policies and for the class of all work conserving policies.
However, only partial converse results have been established and in certain
cases converse statements do not hold. In this paper we close one of the
existing gaps. For the case of networks with two stations we prove that if the
fluid model is not weakly stable under the class of all work conserving
policies, then a corresponding queueing network is not rate stable under the
class of all work conserving policies. We establish the result by building a
particular work conserving scheduling policy which makes the associated
stochastic process transient. An important corollary of our result is that the
condition , which was proven in \cite{daivan97} to be the exact
condition for global weak stability of the fluid model, is also the exact
global rate stability condition for an associated queueing network. Here
is a certain computable parameter of the network involving virtual
station and push start conditions.Comment: 30 pages, To appear in Annals of Applied Probabilit
Algorithmic issues in queueing systems and combinatorial counting problems
Includes bibliographical references (leaves 111-118).Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2008.(cont.) However, these randomized algorithms can never provide proven upper or lower bounds on the number of objects they are counting, but can only give probabilistic estimates. We propose a set of deterministic algorithms for counting such objects for three classes of counting problems. They are interesting both because they give an alternative approach to solving these problems, and because unlike MCMC algorithms, they provide provable bounds on the number of objects. The algorithms we propose are for special cases of counting the number of matchings, colorings, or perfect matchings (permanent), of a graph.Multiclass queueing networks are used to model manufacturing, computer, supply chain, and other systems. Questions of performance and stability arise in these systems. There is a body of research on determining stability of a given queueing system, which contains algorithms for determining stability of queueing networks in some special cases, such as the case where there are only two stations. Yet previous attempts to find a general characterization of stability of queueing networks have not been successful.In the first part of the thesis, we contribute to the understanding of why such a general characterization could not be found. We prove that even under a relatively simple class of static buffer priority scheduling policies, stability of deterministic multiclass queueing network is, in general, an undecidable problem. Thus, there does not exist an algorithm for determining stability of queueing networks, even under those relatively simple assumptions. This explains why such an algorithm, despite significant efforts, has not been found to date. In the second part of the thesis, we address the problem of finding algorithms for approximately solving combinatorial graph counting problems. Counting problems are a wide and well studied class of algorithmic problems, that deal with counting certain objects, such as the number of independent sets, or matchings, or colorings, in a graph. The problems we address are known to be #P-hard, which implies that, unless P = #P, they can not be solved exactly in polynomial time. It is known that randomized approximation algorithms based on Monte Carlo Markov Chains (MCMC) solve these problems approximately, in polynomial time.by Dmitriy A. Katz-Rogozhnikov.Ph.D
Resource Management and Pricing in Networks
Resource management is important for network design and deployment. Resource management and allocation have been studied under a wide variety of scenarios --- routing in wired networks, scheduling in cellular networks, multiplexing, switching, and channel access in opportunistic networks are but a few examples. In this dissertation, we revisit resource management in the context of routing and scheduling in multihop wireless networks and pricing in single resource systems.
The first issue addressed is of delays in multihop wireless networks. The resource under contention is capacity which is allocated by a joint routing and scheduling algorithm. Delay in wireless networks is a key issue gaining interest with the growth of interactive applications and proliferation of wireless networks.
We start with an investigation of the back-pressure algorithm (BPA), an algorithm that activates the schedule with the largest sum of link weights in a timeslot. Though the BPA is throughput-optimal, it has poor end-to-end delays. Our investigation identifies poor routing decisions at low loads as one cause for it. We improve the delay performance of max-weight algorithms by proposing a general framework for routing and scheduling algorithms that allow directing packets towards the sink node dynamically. For a stationary environment, we explicitly formulate delay minimization as a static problem while maintaining stability. We see similar improved delay performance with the advantage of reduced per time-slot complexity.
Next, the issue of pricing for flow based models is studied. The increasing popularity of cloud computing and the ease of commerce over the Internet is making pricing a key issue requiring greater attention. Although pricing has been extensively studied in the context of maximizing revenue and fairness, we take a different perspective and investigate pricing with predictability. Prior work has studied resource allocations that link insensitivity and predictability. In this dissertation, we present a detailed analysis of pricing under insensitive allocations. We study three common pricing models --- fixed rate pricing, Vickrey-Clarke-Groves (VCG) auctions, and congestion-based pricing, and provide the expected operator revenue and user payments under them. A pre-payment scheme is also proposed where users pay on arrival a fee for their estimated service costs. Such a mechanism is shown to have lower variability in payments under fixed rate pricing and VCG auctions while generating the same long-term revenue as in a post-payment scheme, where users pay the exact charge accrued during their sojourn. Our formulation and techniques further the understanding of pricing mechanisms and decision-making for the operator
New Perspectives on Modelling and Control for Next Generation Intelligent Transport Systems
This PhD thesis contains 3 major application areas all within an Intelligent Transportation
System context.
The first problem we discuss considers models that make beneficial use of the large
amounts of data generated in the context of traffic systems. We use a Markov chain
model to do this, where important data can be taken into account in an aggregate form.
The Markovian model is simple and allows for fast computation, even on low end computers,
while at the same time allowing meaningful insight into a variety of traffic system
related issues. This allows us to both model and enable the control of aggregate, macroscopic
features of traffic networks. We then discuss three application areas for this model:
the modelling of congestion, emissions, and the dissipation of energy in electric vehicles.
The second problem we discuss is the control of pollution emissions in
eets of hybrid
vehicles. We consider parallel hybrids that have two power units, an internal combustion
engine and an electric motor. We propose a scheme in which we can in
uence the mix
of the two engines in each car based on simple broadcast signals from a central infrastructure.
The infrastructure monitors pollution levels and can thus make the vehicles
react to its changes. This leads to a context aware system that can be used to avoid pollution
peaks, yet does not restrict drivers unnecessarily. In this context we also discuss
technical constraints that have to be taken into account in the design of traffic control
algorithms that are of a microscopic nature, i.e. they affect the operation of individual
vehicles. We also investigate ideas on decentralised trading of emissions. The goal here
is to allocate the rights to pollute fairly among the
eet's vehicles.
Lastly we discuss the usage of decentralised stochastic assignment strategies in traffic
applications. Systems are considered in which reservation schemes can not reliably be
provided or enforced and there is a signifficant delay between decisions and their effect. In
particular, our approach facilitates taking into account the feedback induced into traffic
systems by providing forecasts to large groups of users. This feedback can invalidate the
predictions if not modelled carefully. At the same time our proposed strategies are simple
rules that are easy to follow, easy to accept, and significantly improve the performance
of the systems under study. We apply this approach to three application areas, the assignment
of electric vehicles to charging stations, the assignment of vehicles to parking
facilities, and the assignment of customers to bike sharing stations.
All discussed approaches are analysed using mathematical tools and validated through
extensive simulations
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