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
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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)