A polling model with reneging at polling instants

In this paper we consider a single-server, cyclic polling system with switch-over times and Poisson arrivals. The service disciplines that are discussed, are exhaustive and gated service. The novel contribution of the present paper is that we consider reneging of customers at polling instants. In more detail, whenever the server starts or ends a visit to a queue, part of the customers waiting in each queue leave the system before having received service. The probability that a certain customer leaves the queue, depends on the queue in which the customer is waiting, and on the location of the server. We show that this system can be analysed by introducing customer subtypes, depending on their arrival periods, and keeping track of the moment when they abandon the system. In order to determine waiting time distributions, we regard the system as a polling model with varying arrival rates, and apply a generalised version of the distributional form of Little’s law. The marginal queue length distribution can be found by conditioning on the state of the system (position of the server, and whether it is serving or switching)

Heavy-traffic single-server queues and the transform method

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 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.</p

A polling model with smart customers

International audienceIn this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at the server's departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little's law is applied to the joint queue length distribution at customer's departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples

A polling model with smart customers

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little's law is applied to the joint queue length distribution at departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples

Closed-form waiting time approximations for polling systems

A typical polling system consists of a number of queues, attended by a single server in a fixed order. The present study derives closed-form approximations for the mean waiting times and mean marginal queue lengths of polling systems with renewal arrival processes, which can be computed by simple calculations. The results of the present research may be very suitable for the design and optimisation phase in many application areas, such as telecommunication, maintenance, manufacturing and transportation

Congestion analysis of unsignalized intersections: The impact of impatience and Markov platooning

This paper considers an unsignalized intersection used by two traffic streams. The first stream of cars is using a primary road, and has priority over the other stream. Cars belonging to the latter stream cross the primary road if the gaps between two subsequent cars on the primary road are larger than their critical headways. A question that naturally arises relates to the capacity of the secondary road: given the arrival pattern of cars on the primary road, what is the maximum arrival rate of low-priority cars that can be sustained? This paper addresses this issue by considering a compact model that sheds light on the dynamics of the considered unsignalized intersection. The model, which is of a queueing-theoretic nature, reveals interesting insights into the impact of the user behavior on the capacity.The contributions of this paper are threefold. First, we introduce a new way to analyze the capacity of the minor road. By representing the unsignalized intersection by an appropriately chosen Markovian model, the capacity can be expressed in terms of the solution of an elementary system of linear equations. The setup chosen is so flexible that it allows us to include a new form of bunching on the main road that allows for dependence between successive gaps, which we refer to as Markov platooning; this is the second contribution. The tractability of this model facilitates studying the impact that driver impatience and various platoon formations on the main road have on the capacity of the minor road. Finally, in numerical experiments we observe various surprising features of the aforementioned model.</p