A polling model with an autonomous server

Queueing Systems, Jul 2009

This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only.

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A polling model with an autonomous server

Roland de Haan Richard J. Boucherie Jan-Kees van Ommeren This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only. Polling models are multi-queue systems with a single server. Typically, the server visits a queue, offers service to (a part of) the customers present at this queue, and then moves to a next queue. The specific details of the system may lead to quite distinct polling models. Polling models are typically characterized by: (i) the arrival process of the customers to the system (Poisson or more general), (ii) the service requirements - of the customers, (iii) the servicing policy of the server (exhaustive, gated, k-limited, etc.), (iv) the visit order of the server, and (v) the switch-over times of the server between visits to the queues. Excellent surveys on a broad class of polling models can be found in, e.g., [13]. Applications of polling models are ubiquitous. For instance, traffic light systems, multiple-access protocols for communication networks (e.g., IEEE 802.11) and product-assembly systems can be modeled as a polling model. In most of the (applications of) polling models, the server is assumed controllable. The goal is then to limit the time a server spends idle at a queue while there is still work in the system. To the contrary, in this article we assume that the server behaves autonomously (and thus is uncontrollable). More precisely, we assume that the server spends an exponentially distributed period of time at a queue irrespective of the number of customers present at each queue. A consequence of the autonomous server is that the services are subject to preemption. Applications of such specific polling models arise for instance in the context of wireless ad hoc networks in which cars, pedestrians or other moving objects that carry wireless equipment are used as communication hops. The class of polling models that is most closely related to our model is the class of so-called time-limited polling models [47]. Leung [4] analyzes a time-limited model in which the server remains an exponential time at a queue but service is nonpreemptive. Preemption is considered for a deterministic time-limited model by De Souza e Silva et al. [6] for Poisson arrivals and by Frigui and Alfa [5] for Markovian Arrival Processes. Eliazar and Yechiali [8, 9] studied a model with an exponential time limit and preemptive service. Observing that upon successful service completion at a queue the busy period in fact regenerates, the authors could obtain a direct, closed-form relation between the joint queue length at the end and the start of a server visit. In each of these models, the server is impatient and leaves a queue as soon as it becomes empty. A specific application of such a time-limited model to a timed token protocol can be found in [7]. A common assumption in polling model analysis is that the server moves to a next queue once the queue becomes empty. However, there also exists analytical work on models with a server that remains at a queue even when it becomes empty. These models are often referred to as patient server or stopping server models. The works of Eisenberg [10] and Borst [11] analyze several strategies for the server once the complete system becomes empty as to optimize some system performance measure. More recently, Boxma et al. [12, 13] considered a single-queue vacation model and a two-queue polling model in which the server upon arriving at an empty queue waits patiently for a certain duration before leaving again. We note that in the latter twoqueue polling model (contrary to the models in [10] and [11]) there is no notion of work conservation anymore, since the server may wait patiently at one queue while the other queue is nonempty. The only work we know of that includes both a given (random) visit time and a patient server that does not leave before the end of the visit time is [14]. This work considers the workload process for the autonomous server model with deterministic visit times. Due to the deterministic nature of the model, the queue lengths at the different queues can be decoupled and each queue is modeled (...truncated)


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Roland de Haan, Richard J. Boucherie, Jan-Kees van Ommeren. A polling model with an autonomous server, Queueing Systems, 2009, pp. 279-308, Volume 62, Issue 3, DOI: 10.1007/s11134-009-9131-z