Cyclic Polling System Research Articles

An asymptotic property of branching-type overloaded polling networks

Abstract Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] studied the fluid asymptotics of the joint queue length process for an overloaded cyclic polling system with multigated service discipline by exploiting the connection with multi-type branching processes. In contrast to the heavy traffic behaviors, the cycle time of the overloaded polling system increases by a deterministic times over times under passage to the fluid dynamics and the fluid limit preserves some randomness. The present paper aims to extend the overloaded asymptotics in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] to the corresponding polling system with general branching-type service disciplines and customer re-routing policy. A unifying overloaded asymptotic property is derived. Due to the exhaustiveness, the property is a natural extension of the classical polling model with multigated service discipline in Remerova et al. [Random fluid limit of an overloaded polling model, Adv. Appl. Probab., 2014, 46, 76–101] and provides new exact results that have not been observed before for rerouting policy. Additionally, a stochastic simulation is undertaken for the validation of the fluid limit and the optimization of the gating indexes to minimize the total population is considered as an example to demonstrate the usefulness of the random fluid limit.

Optimizing product allocation in a polling-based milkrun picking system

E-commerce fulfillment competition evolves around cheap, speedy, and time-definite delivery. Milkrun order picking systems have proven to be very successful in providing handling speed for a large, but highly variable, number of orders. In this system, an order picker picks orders that arrive in real-time during the picking process; by dynamically changing the stops on the picker’s current picking route. The advantage of milkrun picking is that it reduces order picking set-up time and worker travel time compared with conventional batch picking systems. This article is the first to study order throughput times of multi-line orders in a milkrun picking system. We model this system as a cyclic polling system with simultaneous batch arrivals, and determine the mean order throughput time for three picking strategies: exhaustive, locally-gated, and globally-gated. These results allow us to study the effect of different product allocations in an optimization framework. We show that the picking strategy that achieves the shortest order throughput times depends on the ratio between pick times and travel times. In addition, for a real-world application, we show that milkrun order picking significantly reduces the order throughput time compared with conventional batch picking.

The analysis of batch sojourn-times in polling 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 steady-state 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 sojourn-times for different service disciplines in several numerical examples. Our results show that the best performing service discipline, in terms of minimizing the batch sojourn-time, depends on system characteristics.

Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines

The stability of a cyclic polling system, with a single server and two infinite-buffer queues, is considered. Customers arrive at the two queues according to independent batch Markovian arrival processes. The first queue is served according to the gated service discipline, and the second queue is served according to a state-dependent time-limited service discipline with the preemptive repeat-different property. The state dependence is that, during each cycle, the predetermined limited time of the server's visit to the second queue depends on the queue length of the first queue at the instant when the server last departed from the first queue. The mean of the predetermined limited time for the second queue either decreases or remains the same as the queue length of the first queue increases. Due to the two service disciplines, the customers in the first queue have higher service priority than the ones in the second queue, and the service fairness of the customers with different service priority levels is also considered. In addition, the switchover times for the server traveling between the two queues are considered, and their means are both positive as well as finite. First, based on two embedded Markov chains at the cycle beginning instants, the sufficient and necessary condition for the stability of the cyclic polling system is obtained. Then, the calculation methods for the variables related to the stability condition are given. Finally, the influence of some parameters on the stability condition of the cyclic polling system is analyzed. The results are useful for engineers not only checking whether the given cyclic polling system is stable, but also adjusting some parameters to make the system satisfy some requirements under the condition that the system is stable.

On fairness in polling systems

Which service discipline is more fair, exhaustive or gated? gated or globally-gated? How can we compare their fairness level? These questions are usually answered by handwaving as no fundamental research in this topic was yet conducted. Polling systems are widely used in many computer networks where several users compete for access to a common resource. Fairness is a fundamental aspect of polling systems; perhaps it serves as the motivation behind various polling schemes. Despite this fundamental role, fairness to customers in polling systems has not been extensively studied to date. This work is an attempt to model fairness in polling systems and study the relative fairness of various polling schemes. We focus, in the context of this work, on evaluating the system’s obedience to the FCFS policy as a measure of the fairness experienced by the individual customers. Using this metric we study the cyclic polling system under five different service disciplines: exhaustive, gated, binomial-gated, two-stage gated and globally-gated. For these systems we derive their “fairness” level both in a discrete system model and in a continuous model. We use the analysis as well as numerical examples to provide basic observations on fairness of polling systems.

Transient analysis of cycle lengths in cyclic polling systems

We consider cyclic polling models with gated or globally gated service, and study the transient behavior of all cycle lengths. Our aim is to analyze the dependency structure between the different cycles, as this is an intrinsic property making polling models challenging to analyze. Moreover, the cycle structure is related to the output of a polling model and the current analysis may be useful to study networks of polling models. In addition, transient performance is of great interest in systems where disruptions or breakdowns may occur, leading to excessive cycle lengths. The time to recover from such events is a primary performance measure. For the analysis we assume that the distribution of the first cycle (globally gated) or N residence times (gated), where N is the number of queues, is known and that the arrivals are Poisson. The joint Laplace–Stieltjes transform (LST) of all x subsequent cycles (globally gated) or all x>N subsequent residence times (gated) is expressed in terms of the LST of the first cycle. From this joint LST, we derive first and second moments and correlation coefficients between different cycles. Finally, a heavy-tailed first cycle length or the heavy-traffic regime provides additional insights into the time-dependent behavior.

Stability of a cyclic polling system with an adaptive mechanism

We consider a single server cyclic polling system with multiple infinite-buffer queues where the server follows an adaptive mechanism: if a queue is empty at its polling moment the server will skip this queue in the next cycle. After being skipped, a queue is always visited in the next cycle. The service discipline in each queue is 1-limited. Using the fluid limit approach, we find the necessary and sufficient condition for the stability of such polling system.

Design and analysis of a frame-oriented dynamic bandwidth allocation scheme for triple-play services over EPONs

Ethernet Passive Optical Networks (EPONs) are regarded as an important technology for current access networks. However, EPONs perform poorly in differentiated service environments, and may therefore fail to meet the Quality of Service (QoS) demands of emerging triple-play services (i.e. Voice over IP, video and BE traffic). Accordingly, the present study proposes a novel Dynamic Bandwidth Allocation (DBA) scheme, designated as Frame-Oriented Interleaved Polling with Adaptive Cycle Time (FIPACT), to improve the differentiated QoS capability of EPON networks. In the proposed scheme, the Optical Line Terminal (OLT) divides the time domain of the upstream network bandwidth into a contiguous series of fixed-length frames. Within each frame, each Optical Network Unit (ONU) is guaranteed sufficient network resources to transmit a specific quota of each triple-play service. In this manner, the proposed DBA scheme can satisfy the differentiated QoS requirements of triple-play services. Therefore, based on the M/G/1 queueing analysis framework of multiuser cyclic polling systems, this study develops an analytical framework to derive the closed-form expressions for FIPACT mean packet delay. The simulation results validate the analytical results and confirm the efficiency and effectiveness of the proposed DBA scheme.

On the stability of a polling system with an adaptive service mechanism

We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.

Analysis of Mean Packet Delay for Dynamic Bandwidth Allocation Algorithms in EPONs

Closed-form mathematical expressions of network performances, such as the mean packet delay, are useful for evaluating a communication network during the design process. This paper provides derivations of closed-form expressions of the mean packet delay for the gated service and the limited service of dynamic bandwidth allocation in Ethernet passive optical networks (EPONs). Based on the M/G/1 queueing analysis framework of a multiuser cyclic polling system, we derive the mean packet delay expressions by modifying the expressions for the reservation time component of the total delay. Results from simulation experiments confirm that our analysis can accurately predict the mean packet delay. Finally, we extend the analysis to demonstrate how the limited service can protect packets transmitted by a light-load user from having excessive delays due to high traffic loads from other users in the same EPON. The analytical results indicate that, in selecting the maximum length of a scheduling cycle for the limited service, there is a tradeoff between the mean packet delay under uniform traffic and the guaranteed upper bound on the mean packet delay under nonuniform traffic.

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 at 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 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 the reneging of customers at polling instants. In more detail, whenever the server starts or ends a visit to a queue, some 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).

A polling model with multiple priority levels

In this paper we consider a single-server cyclic polling system. Between visits to successive queues, the server is delayed by a random switch-over time. The order in which customers are served in each queue is determined by a priority level that is assigned to each customer at his arrival. For this situation the following service disciplines are considered: gated, exhaustive, and globally gated. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type.

A duplex cyclic polling system for mixed queues

We consider a new mathematical model that adequately represents the workings of a cyclic polling system in high-speed wireless MESH-networks. The queues are serviced by two processing units (servers) in a cyclic fashion. Part of the queues are available for cyclic polling for both servers; each of the remaining queues is attached to its "own" processing unit in the servicing cycle. To study this system, we have applied the mean value approach and have obtained analytic expressions for average waiting times of claims in the queues and other characteristics. The paper also presents numerical examples.

Analysis and Derivation of Mean Packet Delay for Gated Service in EPONs

Closed form mathematical expressions of network parameters such as the mean packet delay are useful for evaluating the communication network performance in the design process. This paper provides a simple derivation of closed form expression of the mean packet delay for the gated service dynamic bandwidth allocation algorithm (DBAA) in an Ethernet Passive Optical Network (EPON). Using the queuing analysis framework of a multi-user cyclic polling system with reservation, we derive the mean packet delay expression by modifying the expression for the reservation time component of the total packet delay. The derivation can also be extended for a different polling algorithm in which the gating time is at the beginning of each data interval. We verify our analysis with results from simulation experiments.

A Two-Queue Polling Model with Two Priority Levels in the First Queue

In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type.

POLLING SYSTEMS WITH TWO-PHASE GATED SERVICE

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of “fairness” among the different customer classes. For this model, we use the classical theory of multitype branching processes to derive closed-form expressions for the Laplace–Stieltjes transform of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance and the trade-off between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.

Polling models with renewal arrivals: A new method to derive heavy-traffic asymptotics

We consider asymmetric cyclic polling systems with an arbitrary number of queues, general service-time distributions, zero switch-over times, gated service at each queue, and with general renewal arrival processes at each of the queues. For this classical model, we propose a new method to derive closed-form expressions for the expected delay at each of the queues when the load tends to 1, under proper heavy-traffic (HT) scalings. In the literature on polling models, rigorous proofs of HT limits have only been obtained for polling models with Poisson-type arrival processes, whereas for renewal arrivals HT limits are based on conjectures [E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems with zero switch-over times: A heavy-traffic principle, Ann. Appl. Probab. 5 (1995) 681–719; E.G. Coffman, A.A. Puhalskii, M.I. Reiman, Polling systems in heavy-traffic: A Bessel process limit, Math. Oper. Res. 23 (1998) 257–304; T.L. Olsen, R.D. van der Mei, Periodic polling systems in heavy-traffic: Renewal arrivals, OR Lett. 33 (2005) 17–25]. Therefore, the main contribution of this paper lies in the fact that we propose a new method to rigorously prove HT limits for a class of non-Poisson-type arrivals. The results are remarkably simple and provide new fundamental insight and reveal explicitly how the expected delay at each of the queues depends on the system parameters, and in particular on the interarrival-time distributions at each of the queues. The results also suggest simple approximations for the expected delay in stable polling systems. Numerical results show that the approximations are highly accurate when the system load is roughly 90% or more.

A model of the polling system for studying the broadband wireless networks

A study of the model of a cyclic (polling) system adequately describing the broadband wireless WiFi and WiMax centralized-control networks was presented. The server was assumed to have full information about the current system state. The queues are serviced by the exhaustive threshold discipline, that is, a queue is serviced if its length exceeds the given threshold. If the lengths of all queues are insufficient to start servicing, then the server stops polling the queue until any of them accumulates the required number of customers. Relying on the stationary probability distribution of the polling system states, the main performance characteristics such as the mean queue length, failure probability, and mean waiting time were established.

A Two‐Queue Polling Model with Regularly Varying Service and/or Switchover Times

We consider the cyclic polling system with two queues. One queue is severed according to the exhaustive discipline, and the other queue is served according to the 1‐limited discipline. At least one of the service and/or switchover times has a regularly varying tail. We obtain the tail behavior of the waiting time distributions. When one of the service and/or switchover times has an infinite second moment, we derive the heavy‐traffic behavior of the waiting time distribution at the 1‐limited queue.