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

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now

Fairness and efficiency for polling models with the K-gated service discipline

We study a polling model where we want to achieve a balance between the fairness of the waiting times and the efficiency of the system. For this purpose, we introduce the k-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using Ki gated service phases at queue i before the server switches to the next queue. We derive the distributions and means of the waiting times, a pseudo conservation law for the weighted sum of the mean waiting times, and the fluid limits of the waiting times. Our goal is to optimize the Ki's so as to minimize the differences in the mean waiting times, i.e. to achieve maximal fairness, without giving up too much on the efficiency of the system. From the fluid limits we derive a heuristic rule for setting the Ki's. In a numerical study the heuristic is shown to perform well

Fairness and efficiency for polling models with the K-gated service discipline

We study a polling model where we want to achieve a balance between the fairness of the waiting times and the efficiency of the system. For this purpose, we introduce the k-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using Ki gated service phases at queue i before the server switches to the next queue. We derive the distributions and means of the waiting times, a pseudo conservation law for the weighted sum of the mean waiting times, and the fluid limits of the waiting times. Our goal is to optimize the Ki's so as to minimize the differences in the mean waiting times, i.e. to achieve maximal fairness, without giving up too much on the efficiency of the system. From the fluid limits we derive a heuristic rule for setting the Ki's. In a numerical study the heuristic is shown to perform well

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)