Indagationes Mathematicae-New Series最新文献

Heavy-traffic limit theory is concerned with queues that operate close to criticality and face severe queueing times. Let $W$ denote the steady-state waiting time in the $\mathrm{GI}/G/1$ queue. Kingman (1961) showed that $W$, when appropriately scaled, converges in distribution to an exponential random variable as the system’s load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of $W$ (Pollaczek’s contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.

We consider the impulse control of Lévy processes under the infinite horizon, discounted cost criterion. Our motivating example is the cash management problem in which a controller is charged a fixed plus proportional cost for adding to or withdrawing from his/her reserve, plus an opportunity cost for keeping any cash on hand. Our main result is to provide a verification theorem for the optimality of control band policies in this scenario. We also analyze the transient and steady-state behavior of the controlled process under control band policies and explicitly solve for an optimal policy in the case in which the Lévy process to be controlled is the sum of a Brownian motion with drift and a compound Poisson process with exponentially distributed jump sizes.

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.

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.

We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.

The stationary reflected Brownian motion in a three-quarter plane has been rarely analyzed in the probabilistic literature, in comparison with the quarter plane analogue model. In this context, our main result is to prove that the stationary distribution can indeed be found by solving a boundary value problem of the same kind as the one encountered in the quarter plane, up to various dualities and symmetries. The main idea is to start from Fourier (and not Laplace) transforms, allowing to get a functional equation for a single function of two complex variables.

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions $(i,j)$ with $i+j\ge 0$ and small negative jumps, i.e., $i,j\ge -1$. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.

Two models involving a foreground and a background queue are studied in the steady state. Service is provided either by a single server whose speed depends on the total number of jobs present, or by several parallel servers whose number may be controlled dynamically. Job service times have a two-phase Coxian distribution. Incoming jobs join the foreground queue where they execute phase 1, and then possibly move to the background queue for the second phase at lower priority. The trade-offs between holding and energy consumption costs are examined by means of a suitable cost function. Two different two-dimensional Markov processes are solved exactly. The solutions are used in several numerical experiments, aimed at illustrating different aspects of system behaviour.

We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.

We introduce a queueing system that alternates between two modes, so-called working mode and vacation mode. During the working mode the system runs as an $M^X/G/1$ queue. Once the number of customers in the working mode drops to zero the vacation mode begins. During the vacation mode the system runs as a general queueing system (a service might be included) which is different from the one in the working mode. The vacation period ends in accordance with a given stopping rule, and then a random number of customers are transferred to the working mode. For this model we show that the number of customers given that the system is in the working mode is distributed as the sum of two independent random variables, one of them is the number of customers in an $M^X/G/1$ queue given that the server is busy. This decomposition result puts under the same umbrella some models that have already been introduced in the past as well as some new models.

Motivated by queueing applications, we consider a class of two-dimensional random walks, the invariant measure of which can be written as a linear combination of a finite number of product-form terms. In this work, we investigate under which conditions such an elegant solution can be derived by applying a finite compensation procedure. The conditions are formulated in terms of relations among the transition probabilities in the inner area, the boundaries as well as the origin. A discussion on the importance of these conditions is also given.