System occupancy of a two-class batch-service queue with class-dependent variable server capacity

Due to their wide area of applications, queueing models with batch service, where the server can process several customers simultaneously, have been studied frequently. An important characteristic of such batch-service systems is the size of a batch, that is the number of customers that are processed simultaneously. In this paper, we analyse a two-class batch-service queueing model with variable server capacity, where all customers are accommodated in a common first-come-first served single-server queue. The server can only process customers that belong to the same class, so that the size of a batch is determined by the number of consecutive same-class customers. After establishing the system equations that govern the system behaviour, we deduce an expression for the steady-state probability generating function of the system occupancy at random slot boundaries. Also, some numerical examples are given that provide further insight in the impact of the different parameters on the system performance

Dynamic server assignment in an extended machine-repair model

We consider an extension of the classical machine-repair problem. The machines, apart from receiving service from a single repairman, now also supply service themselves to queues of products. The extended model can be viewed as a two-layered queueing network, in which the queues of products in the ¿rst layer are generally correlated, due to the fact that the machines have to share the repairman’s capacity in the second layer. We are concerned with the dynamic control problem how the repairman should allocate his capacity to the machines at any point in time so that the long-term average (weighted) sum of the queue lengths of the ¿rst-layer queues is minimised. Since the optimal policy for the repairman cannot be found analytically due to the correlations in the queue lengths, we propose a near-optimal policy. We do this by combining intuition and results from queueing theory with techniques from Markov decision theory. Speci¿cally, we study the relative value functions for several policies for which the model can be decomposed in less complicated subsystems, and we combine the results with the classical one-step policy improvement algorithm. The resulting policy is easy to apply, scalable in the number of machines and is shown to be highly accurate over a wide range of parameter settings

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale reflected Brownian motions in the non-negative orthant. Keywords: Functional central limit theorem; Layered queueing networks; Machine-repair model; Semi-martingale reflected Brownian motio

Los espacios ideológicos en Carmen Baroja, escritora del 98

En las memorias escritas por Carmen Baraja en 1946, tituladas, Recuerdos de una mujer de la generación del 98, se aprecian claramente las diferentes etapas que ella misma establece para su vida: la niñez, la juventud, la madurez y la vejez, y cómo siempre en todas ellas exceptuando en la primera y en la última hay uná dialéctica entre la realidad y la voluntad o el deseo. Esta oposición se vehicula a través del espacio: el privado, representado en las distintas residencias de la familia Baroja, y el público, que se materializa en el exterior del hogar. En este trabajo me propongo analizar cómo el yo autobiográfico percibe ambos modelos espaciales y las restricciones y los vuelos de libertad que le ofrecen estos dos ámbitos

Polling systems with batch service

Motivated by applications in production and computer-communication systems, we study an N-queue polling system, consisting of an inner part and an outer part, and where products receive service in batches. Type-i products arrive at the outer system according to a renewal process and accumulate into a type-i batch. As soon as Di products have accumulated, the batch is forwarded to the inner system where the batch is processed. The service requirement of a type-i batch is independent of its size Di. For this model we study the problem of determining the combination of batch sizes D(opt) that minimizes a weighted sum of the mean waiting times. This model does not allow for an exact analysis. Therefore, we propose a simple closed-form approximation for D(opt), and present a numerical approach, based on the recently-proposed mean waiting-time approximation in [1]. Extensive numerical experimentation shows that the numerical approach is slightly more accurate than the closed-form solution, while the latter provides explicit insights into the dependence of the optimal batch sizes on the system parameters and into the behavior of the system. As a by-product, we observe near-insensitivity properties of D(opt), e.g. to higher moments of the interarrival and switch-over time distributions. Keywords: Polling systems, renewal arrivals, batch arrivals, optimal batch size

On two-queue Markovian polling systems with exhaustive service

We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace-Stieltjes transforms of the waiting-time distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traf¿c behaviour of properly scaled versions of these distributions, which results in compact and closed-form expressions for the distribution functions themselves. The heavy-traf¿c behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution. Keywords: Markovian routing, waiting-time distribution, queue length distribution, descendant set approach, heavy-traf¿c behaviou

Analysis of a two-layered network by means of the power-series algorithm

We consider an extension of the classical machine-repair model, also known as the computer-terminal model or time-sharing model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, supply service themselves to queues of products. The extended model can be viewed as a two-layered queueing network, of which the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The marginal and joint queue length distributions of the first-layer queues are hard to analyse in an exact fashion. Therefore, we apply the power-series algorithm to this model to obtain the light-traffic behaviour of the queue lengths symbolically. This leads to two accurate approximations for the marginal mean queue length. The first approximation, based on the light-traffic behaviour, is in closed form. The second approximation is based on an interpolation between the light-traffic behaviour and heavy-traffic results for the mean queue length. The obtained approximations are shown to work well for arbitrary loaded systems. The proposed numerical algorithm and approximations may prove to be very useful for system design and optimisation purposes in application areas such as manufacturing, computer systems and telecommunications. Keywords: Layered queueing networks; Light-traffic behaviour; Machine-repair model; Queue-length approximation

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 same 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. Keywords: Queue lengths; Binomial service disciplines; Markovian routing; Random routing; Wireless random-access network

The impact of scheduling policies on the waiting-time distributions in polling systems

We consider polling models consisting of a single server that visits the queues in a cyclic order. In the vast majority of papers that have appeared on polling models, it is assumed that at each of the individual queues, the customers are served on a first-come-first-served (FCFS) basis. In this paper, we study polling models where the local scheduling policy is not FCFS but instead is varied as last-come-first-served (LCFS), random order of service (ROS), processor sharing (PS), and shortest-job-first (SJF). The service policies are assumed to be either gated or globally gated. The main result of the paper is the derivation of asymptotic closed-form expressions for the Laplace–Stieltjes transform of the scaled waiting-time and sojourn-time distributions under heavy-traffic assumptions. For FCFS service, the asymptotic sojourn-time distribution is known to be of the form UG , where U and G are uniformly and gamma distributed with known parameters. In this paper, we show that the asymptotic sojourn-time distribution (1) for LCFS is also of the form UG , (2) for ROS is of the form U~G , where U~ has a trapezoidal distribution, and (3) for PS and SJF is of the form U~*G , where U~* has a generalized trapezoidal distribution. These results are rather intriguing and lead to new fundamental insight into the impact of the local scheduling policy on the performance of polling models. As a by-product, the heavy-traffic results suggest simple closed-form approximations for the complete waiting-time and sojourn-time distributions for stable systems with arbitrary load values. The accuracy of the approximations is evaluated by simulations. Keywords: Polling systems; Local scheduling policies; Waiting times; Heavy traffic; Approximation