Tail asymptotics for the queue size distribution in the MAP/G/1 retrial queue

We consider a MAP/G/1 retrial queue where the service time distribution has a finite exponential moment. We derive matrix differential equations for the vector probability generating functions of the stationary queue size distributions. Using these equations, Perron–Frobenius theory, and the Karamata Tauberian theorem, we obtain the tail asymptotics of the queue size distribution. The main result on light-tailed asymptotics is an extension of the result in Kim et al. (J. Appl. Probab. 44:1111–1118, 2007) on the M/G/1 retrial queue.     

Global and local asymptotics for the busy period of an M/G/1 queue

We consider an M/G/1 queue with subexponential service times. We give a simple derivation of the global and local asymptotics for the busy period. Our analysis relies on the explicit formula for the joint distribution for the number of customers and the length of the busy period of an M/G/1 queue.     

Tail asymptotics for the queue size distribution in a discrete-time Geo/G/1 retrial queue

We consider a discrete-time Geo/G/1 retrial queue where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically geometric. Remarkably, the result is inconsistent with the corresponding result in the continuous-time counterpart, the M/G/1 retrial queue, where the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function.     

The idle period of the finite G/M/1 queue with an interpretation in risk theory

We consider a G/M/1 queue with restricted accessibility in the sense that the maximal workload is bounded by 1. If the current workload V _t of the queue plus the service time of an arriving customer exceeds 1, only 1−V _t of the service requirement is accepted. We are interested in the distribution of the idle period, which can be interpreted as the deficit at ruin for a risk reserve process R _t in the compound Poisson risk model. For this risk process a special dividend strategy applies, where the insurance company pays out all the income whenever R _t reaches level 1. In the queueing context we further introduce a set-up time a∈[0,1]. At the end of every idle period, an arriving customer has to wait for a time units until the server is ready to serve it.     

Regularly varying tail of the waiting time distribution in M/G/1 retrial queue

We consider an M/G/1 retrial queue where the service time distribution has a regularly varying tail with index −β, β>1. The waiting time distribution is shown to have a regularly varying tail with index 1−β, and the pre-factor is determined explicitly. The result is obtained by comparing the waiting time in the M/G/1 retrial queue with the waiting time in the ordinary M/G/1 queue with random order service policy.     

Unreliable M/G/1 retrial queue: monotonicity and comparability

In this paper we investigate the monotonicity properties of an unreliable M/G/1 retrial queue using the general theory of stochastic ordering. We show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and increasing convex ordering. We obtain conditions of comparability of two transition operators and we obtain comparability conditions of the number of customers in the system. Inequalities are derived for the mean characteristics of the busy period, number of customers served during a busy period, number of orbit busy periods and waiting times. Inequalities are also obtained for some probabilities of the steady-state distribution of the server state. An illustrative numerical example is presented.     

Asymptotic behavior of the loss rate for Markov-modulated fluid queue with a finite buffer

We consider a Markov-modulated fluid queue with a finite buffer. It is assumed that the fluid flow is modulated by a background Markov chain which may have different transitions when the buffer content is empty or full. In Sakuma and Miyazawa (Asymptotic Behavior of Loss Rate for Feedback Finite Fluid Queue with Downward Jumps. Advances in Queueing Theory and Network Applications, pp. 195–211, Springer, Cambridge, 2009), we have studied asymptotic loss rate for this type of fluid queue when the mean drift of the fluid flow is negative. However, the null drift case is not studied. Our major interest is in asymptotic loss rate of the fluid queue with a finite buffer including the null drift case. We consider the density of the stationary buffer content distribution and derive it in matrix exponential forms from an occupation measure. This result is not only useful to get the asymptotic loss rate especially for the null drift case, but also it is interesting in its own light.     

Performance of an ATM multiplexer queue in the fluid approximation using the Beneš approach

This paper describes a new method for evaluating the queue length distribution in an ATM multiplexer assuming the cell arrival process can be assimilated to a variable rate fluid input. The method is based on a result due initially to Bene allowing the analysis of queues with general input. Its extension to fluid input systems is considered here in the case of a superposition of on/off sources. We derive an upper bound on the complementary queue length distribution. The method is most easily applied in the case of Poisson burst arrivals (infinite sources model). In this case, we derive analytic expressions for the tail of the queue length distribution. A corrective factor is deduced to convert the upper bounds to good approximations. Numerical results justify the accuracy of the method and demonstrate the impact of certain traffic characteristics on queue performance.     

Numerical approximations for the steady‐state waiting times in a GI/G/1 queue

This paper focuses on easily computable numerical approximations for the distribution and moments of the steadystate waiting times in a stable GI/G/1 queue. The approximation methodology is based on the theory of Fredholm integral equations and involves solving a linear system of equations. Numerical experimentation for various M/G/1 and GI/M/1 queues reveals that the methodology results in estimates for the mean and variance of waiting times within ±1% of the corresponding exact values. Comparisons with competing approaches establish that our methodology is not only more accurate, but also more amenable to obtaining waiting time approximations from the operational data. Approximations are also obtained for the distributions of steadystate idle times and interdeparture times. The approximations presented in this paper are intended to be useful in roughcut analysis and design of manufacturing, telecommunications, and computer systems as well as in the verification of the accuracies of inequalities, bounds, and approximations.     

A D’Alembert Formula for Flat Surfaces in the 3-Sphere

We find all the flat surfaces in the unit 3-sphere $\mathbb{S}^{3}We find all the flat surfaces in the unit 3-sphere that pass through a given regular curve of with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat M?bius strips of  .      

A PλM-policy for an M/G/1 queueing system

We introduce P_λ^M-service policy for an M/G/1 queueing system. The stationary distribution of the workload under this policy is explicitly obtained through a decomposition technique, renewal reward theorem, and level crossing argument.     

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

This paper studies the geometric decay property of the joint queue-length distribution {p(n ₁,n ₂)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c ₁,c ₂,d ₁ and d ₂, an upper bound of the decay rate is derived in the sense

Erratum to: Stability of Diffusion Coefficients in an Inverse Problem for the Lotka-Volterra Competition System

First we establish a Carleman estimate for Lotka-Volterra competition-diffusion system of three equations with variable coefficients. Then the internal observations with two measurements are allowed to obtain the stability result for the inverse problem consisting of retrieving two smooth diffusion coefficients in the given parabolic system for the dimension n≤3. The proof of the results rely on Carleman estimates and certain energy estimates for parabolic system.     

Optimal sensor placement for modal identification of structures using genetic algorithms—a case study: the olympic stadium in Cali,Colombia

Adequate sensor placement plays a key role in such fields as system identification, structural control, damage detection and structural health monitoring of flexible structures. In recent years, interest has increased in the development of methods for determining an arrangement of sensors suitable for characterizing the dynamic behavior of a given structure. This paper describes the implementation of genetic algorithms as a strategy for optimal placement of a predefined number of sensors. The method is based on the maximization of a fitness function that evaluates sensor positions in terms of natural frequency identification effectiveness and mode shape independence under various occupation and excitation scenarios using a custom genetic algorithm. A finite element model of the stadium was used to evaluate modal parameters used in the fitness function, and to simulate different occupation and excitation scenarios. The results obtained with the genetic algorithm strategy are compared with those obtained from applying the Effective Independence and Modal Kinetic Energy sensor placement techniques. The sensor distribution obtained from the proposed strategy will be used in a structural health monitoring system to be installed in the stadium.     

On the Cauchy Problem for D t 2-D xa(t,x)D x in the Gevrey Class of Order s > 2

We study the Cauchy problem for second order hyperbolic equations with non negative characteristic form of two independent variables. We show that for such equations in divergence-free form, the Cauchy problem is well posed in the Gevrey class of order less than 5/2.     

On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

We consider the Cauchy problem for the one dimensional cubic nonlinear Schrödinger equation \(iu_t+u_{xx}-|u|^2u=0\). As the first step local well-posedness in the modulation space \(M_{2,p}\) (\(2\le p<\infty \)) is derived (see Theorem 1.4), which covers all the subcritical cases. Afterwards in order to approach the endpoint case, we will prove the almost global well-posedness in some Orlicz type space (see Theorem 1.8), which is a natural generalization of \(M_{2,p}\), and is almost critical from the viewpoint of scaling. The new ingredient is an endpoint version of the two dimensional restriction estimate (see Lemma 3.7).     

Higher order regularity results in 3D for the Landau–Lifshitz equation with an exchange field

We derive and prove higher order regularity results for the Landau–Lifshitz (LL) equation describing the evolution of spin fields in continuum ferromagnets. These results are based on the weight κ(s)={min1,s}$\kappa \left(s\right)=\{min1,s\}$ which helps to get nonlinear parts under control.     

Continuous approximation for a 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid

A modification of a 1D analog of the Gol’dshtik mathematical model for separated flows of an incompressible fluid is considered. The model is a nonlinear differential equation with a boundary condition. Nonlinearity in the equation is continuous and depends on a small parameter. When this parameter tends to zero, we have a discontinuous nonlinearity. The results of the solutions are in agreement with the results obtained for the 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid.     

A subjective Bayesian approach to the theory of queues II — Inference and information in M/M/1 queues

This is a sequel to Part I of A Subjective Bayesian Approach to the Theory of Queues. The focus here is on inference and a use of Shannon's measure of information for assessing the amount of information conveyed by the various types of data from queues. The notation and terminology used here is established in Part I.