Closed-form waiting time approximations for polling systems (original) (raw)

Closed-Form Waiting Time Approximations for Polling

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.

Closed-form waiting time approximations for polling systems. EURANDOM report 2009-030, EURANDOM

2009

Scheduling in polling systems in heavy traffic

ACM SIGMETRICS Performance Evaluation Review, 2013

We consider the classical cyclic polling model with Poisson arrivals and with gated service at all queues, but where the local scheduling policies are not necessarily First-Come-First-Served (FCFS). More precisely, we study the waitingtime performance of polling models where the local service order is Last-Come-First-Served (LCFS), Random-Orderof-Service (ROS) or Processor Sharing (PS). Under heavytraffic conditions the waiting times turn out to converge to products of generalized trapezoidal distributions and a gamma distribution. * The research of J.L. Dorsman is funded in the framework of the STAR-project "Multilayered queueing systems" by the Netherlands Organization for Scientific Research (NWO).

Mean value analysis for polling systems in heavy traffic

Proceedings of the 1st international conference on Performance evaluation methodolgies and tools - valuetools '06, 2006

In this paper we present a new approach to derive heavy-traffic asymptotics for polling models. We consider the classical cyclic polling model with exhaustive service at each queue, and with general service-time and switch-over time distributions, and study its behavior when the load tends to one. For this model, we explore the recently proposed mean value analysis (MVA), which takes a new view on the dynamics of the system, and use this view to provide an alternative way to derive closed-from expressions for the expected asymptotic delay; the expressions were derived earlier in [32], but in a different way. Moreover, the MVA-based approach enables us to derive closed-form expressions for the heavy-traffic limits of the covariances between the successive visit periods, which are key performance metrics in many application areas. These results, which have not been obtained before, reveal a number of insensitivity properties of the covariances with respect to the system parameters under heavy-traffic assumptions, and moreover, lead to simple approximations for the covariances between the successive visit times for stable systems. Numerical examples demonstrate that the approximations are accurate when the load is close enough to one.

A probabilistic approach to estimate the mean waiting times in the earliest deadline first polling

Computers & Industrial Engineering, 2013

In queueing system, the mean waiting times of messages are important measures to characterize the quality of service (QoS) under various requirements. In a time-critical system, message transactions which cannot meet deadline constraints might lead to catastrophic consequences. Currently, the waiting time estimations using the First-Come-First-Served (FCFS) and priority (PRI) strategies are already well developed. However, in the case of multi-queue dynamic environments, these quantities are more difficult to analyze due to multiple classes of messages are considered. In this paper, we aim to consider a polling system consisting of a number of parallel infinite-capacity single-server queues. We propose a probabilistic approach to derive the waiting times for different classes of messages by using non-preemptive earliest deadline first (EDF) polling policy. The resulting formula can also lead to the FCFS polling and PRI polling by altering the relative deadlines. Moreover, the bounds of waiting times are discussed. The accuracy of the proposed algorithm is established by comparisons with simulation results. The runtime results are in very good convergence with the theoretical predictions made by our formulas, in terms of prediction accuracies of waiting times and untimely service ratios of messages under various scenarios and timing constraints.

Heavy-traffic analysis of k-limited polling systems

Probability in the Engineering and Informational Sciences

In this paper, we study a two-queue polling model with zero switchover times and k-limited service (serve at most k i customers during one visit period to queue i, i=1, 2) in each queue. The arrival processes at the two queues are Poisson, and the service times are exponentially distributed. By increasing the arrival intensities until one of the queues becomes critically loaded, we derive exact heavy-traffic limits for the joint queue-length distribution using a singular-perturbation technique. It turns out that the number of customers in the stable queue has the same distribution as the number of customers in a vacation system with Erlang-k 2 distributed vacations. The queue-length distribution of the critically loaded queue, after applying an appropriate scaling, is exponentially distributed. Finally, we show that the two queue-length processes are independent in heavy traffic.

The analysis of batch sojourn-times in polling systems

Queueing Systems

We consider a cyclic polling system with general service times, general switch-over times, and simultaneous batch arrivals. This means that at an arrival epoch, a batch of customers may arrive simultaneously at the different queues of the system. For the exhaustive service discipline, we study the batch sojourn-time, which is defined as the time from an arrival epoch until service completion of the last customer in the batch. We obtain exact expressions for the Laplace-Stieltjes transform of the steadystate batch sojourn-time distribution, which can be used to determine the moments of the batch sojourn-time and, in particular, its mean. However, we also provide an alternative, more efficient way to determine the mean batch sojourn-time, using mean value analysis. We briefly show how our framework can be applied to other service disciplines: locally gated and globally gated. Finally, we compare the batch sojourntimes for different service disciplines in several numerical examples. Our results show that the best performing service discipline, in terms of minimizing the batch sojourntime, depends on system characteristics.

A transient analysis of polling systems operating under exponential time-limited service disciplines

2009

In the present article, we analyze a class of time-limited polling systems. In particular, we will derive a direct relation for the evolution of the joint queue-length during the course of a server visit. This will be done both for the pure and the exhaustive exponential time-limited discipline for general service time requirements and preemptive service. More specifically, service of individual customers is according to the preemptive-repeat-random strategy, i.e., if a service is interrupted, then at the next server visit a new service time will be drawn from the original service-time distribution. Moreover, we incorporate customer routing in our analysis, such that it may be applied to a large variety of queueing networks with a single server operating under one of the before-mentioned time-limited service disciplines. We study the time-limited disciplines by performing a transient analysis for the queue length at the served queue. The analysis of the pure time-limited discipline builds on several known results for the transient analysis of the M/G/1 queue. Besides, for the analysis of the exhaustive discipline, we will derive several new results for the transient analysis of an M/G/1 during a busy period. The final expressions (both for the exhaustive and pure case) that we obtain for the key relations generalize previous results by incorporating customer routing or by relaxing the exponentiality assumption on the service times. Finally, based on the interpretation of these key relations, we formulate a conjecture for the key relation for any branching-type service discipline operating under an exponential time-limit.

A novel approach to queue stability analysis of polling models

Performance Evaluation, 2000

Previous work in the stability analysis of polling models concentrated mainly on stability of the whole system. This system stability analysis, however, fails to model many real-world systems for which some queues may continue to operate under an unstable system. In this paper we address this problem by considering queue stability problem that concerns stability of an individual queue in a polling model. We present a novel approach to the problem which is based on a new concept of queue stability orderings, dominant systems, and Loynes' theorem. The polling model under consideration employs an m-limited service policy, with or without prior service reservation; moreover, it admits state-dependent set-up time and walk time. Our stability results generalize many previous results of system stability. Furthermore, we show that stabilities of any two queues in the system can be compared solely based on their (λ/m)'s, where λ is the customer arrival rate to a queue.

Heavy-traffic limits for polling models with exhaustive service and non-FCFS service order policies

47, 2015

We study cyclic polling models with exhaustive service at each queue under a variety of non-FCFS local service orders, namely Last-Come-First-Served (LCFS) with and without preemption, Random-Order-of-Service (ROS), Processor Sharing (PS), the multi-class priority scheduling with and without preemption, Shortest-Job-First (SJF) and the Shortest Remaining Processing Time (SRPT) policy. For each of these policies, we first express the waiting-time distributions in terms of intervisit-time distributions. Next, we use these expressions to derive the asymptotic waiting-time distributions under heavy-traffic assumptions, i.e., when the system tends to saturate. The results show that in all cases the asymptotic waiting-time distribution at queue i is fully characterized and of the form ΓΘ i , with Γ and Θ i independent, and where Γ is gamma distributed with known parameters (and the same for all scheduling policies). We derive the distribution of the random variable Θ i which explicitly expresses the impact of the local service order on the asymptotic waiting-time distribution. The results provide new fundamental insight in the impact of the local scheduling policy on the performance of a general class of polling models. The asymptotic results suggest simple closed-form approximations for the complete waiting-time distributions for stable systems with arbitrary load values. The accuracy of the approximations is evaluated by simulations.

Closed-form waiting time approximations for polling systems (original) (raw)

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