24 research outputs found
A stochastic resource-sharing network for electric vehicle charging
We consider a distribution grid used to charge electric vehicles subject to voltage stability and various other constraints. We model this as a class of resource
Heavy-traffic approximations for a layered network with limited resources
Motivated by a web-server model, we present a queueing network consisting of two layers. The first layer incorporates the arrival of customers at a network of two single-server nodes. We assume that the inter-arrival and the service times have general distributions. Customers are served according to their arrival order at each node and after finishing their service they can re-enter at nodes several times (as new customers) for new services. At the second layer, active servers act as jobs which are served by a single server working at speed one in a Processor-Sharing fashion. We further assume that the degree of resource sharing is limited by choice, leading to a Limited Processor-Sharing discipline. Our main result is a diffusion approximation for the process describing the number of customers in the system. Assuming a single bottleneck node and studying the system as it approaches heavy traffic, we prove a state-space collapse property. The key to derive this property is to study the model at the second layer and to prove a diffusion limit theorem, which yields an explicit approximation for the customers in the system
Markovian polling systems with an application to wireless random-access networks
Motivated by an application in wireless random-access networks, we study a class of polling systems with
Markovian routing, in which the server visits the queues in an order governed by a discrete-time Markov chain.
Assuming that the service disciplines at each of the queues fall in the class of branching-type service disciplines,
we derive a functional equation for (the probability generating function of) the joint queue length distribution
conditioned on a point in time when the server visits a certain queue. From this functional equation, expressions
for the (cross-)moments of the queue lengths follow. We also derive a pseudo-conservation law for this class
of polling systems. Using these results, we compute expressions for certain system parameters that minimise
the total expected amount of work in systems that arise from the wireless random-access network setting. In
addition, we derive approximations for those parameters that minimise a weighted sum of mean waiting times in
these systems. Based on these expressions, we also present an adaptive control algorithm for finding the optimal
parameter values in a distributed fashion, which is particularly relevant in the context of wireless random-access
networks
Large fork-join queues with nearly deterministic arrival and service times
In this paper, we study an N server fork-join queue with nearly deterministic arrival and service times. Specifically, we present a fluid limit for the maximum queue length as N → ∞. This fluid limit depends on the initial number of tasks. In order to prove these results, we develop extreme value theory and diffusion approximations for the queue lengths
Asymptotic analysis of Emden–Fowler type equation with an application to power flow models
Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider
A L\'evy input fluid queue with input and workload regulation
We consider a queuing model with the workload evolving between consecutive
i.i.d.\ exponential timers according to a
spectrally positive L\'evy process that is reflected at zero, and
where the environment equals 0 or 1. When the exponential clock
ends, the workload, as well as the L\'evy input process, are modified; this
modification may depend on the current value of the workload, the maximum and
the minimum workload observed during the previous cycle, and the environment
of the L\'evy input process itself during the previous cycle. We analyse
the steady-state workload distribution for this model. The main theme of the
analysis is the systematic application of non-trivial functionals, derived
within the framework of fluctuation theory of L\'evy processes, to workload and
queuing models
Renewal processes with costs and rewards
We review the theory of renewal reward processes, which describes renewal processes that have some cost or reward associated with each cycle. We present a new simplified proof of the renewal reward theorem that mimics the proof of the Elementary Renewal Theorem and avoids the technicalities in the proof that is presented in most textbooks. Moreover, we mention briefly the extension of the theory to partial rewards, where it is assumed that rewards are accrued not only at renewal epochs but also during the renewal cycle. For this case, we present a counterexample which indicates that the standard conditions for the renewal reward theorem are not sufficient; additional regularity assumptions are necessary. We present a few examples to indicate the usefulness of this theory, where we prove the inspection paradox and Little's law through the renewal reward theorem
Exact solution to a Lindley-type equation on a bounded support
We derive the limiting waiting-time distribution FW of a model described by the Lindley-type equation W=max{0,B-A-W}, where B has a polynomial distribution. This exact solution is applied to derive approximations of FW when B is generally distributed on a finite support. We provide error bounds for these approximations