R.D. van der Mei

Applications of polling systems

Since the first paper on polling systems, written by Mack in 1957, a huge number of papers on this topic has been written. A typical polling system consists of a number of queues, attended by a single server. In several surveys, the most notable ones written by Takagi, detailed and comprehensive descriptions of the mathematical analysis of polling systems are provided. The goal of the present survey paper is to complement these papers by putting the emphasis on applications of polling models. We discuss not only the capabilities, but also the limitations of polling models in representing various applications. The present survey is directed at both academicians and practitioners.

A New Method for Deriving Waiting-Time Approximations in Polling Systems with Renewal Arrivals

We study the waiting-time distributions in cyclic polling models with renewal arrivals, general service and switch-over times, and exhaustive service at each of the queues. The assumption of renewal arrivals prohibits an exact analysis and reduces the available analytic results to heavy-traffic asymptotics, limiting results for large switch-over times and large numbers of queues, and some numerical algorithms. Motivated by this, the goal of this paper is to propose a new method for deriving simple closed-form approximations for the complete waiting-time distributions that work well for arbitrary load values. Extensive simulation results show that the approximations are highly accurate over a wide range of parameter settings.

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.

Heavy traffic analysis of polling models by mean value analysis

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 or gated 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-form expressions for the expected asymptotic delay; the expressions were derived earlier in [R.D. van der Mei, H. Levy, Expected delay in polling systems in heavy traffic, Adv. Appl. Probab. 30 (1998) 586-602], 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.

Fluid approximation of a call center model with redials and reconnects

In many call centers, callers may call multiple times. Some of the calls are re-attempts after abandonments (redials), and some are re-attempts after connected calls (reconnects). The combination of redials and reconnects has not been considered when making staffing decisions, while not distinguishing them from the total calls will inevitably lead to under- or overestimation of call volumes, which results in improper and hence costly staffing decisions.Motivated by this, in this paper we study call centers where customers can abandon, and abandoned customers may redial, and when a customer finishes his conversation with an agent, he may reconnect. We use a fluid model to derive first order approximations for the number of customers in the redial and reconnect orbits in the heavy traffic. We show that the fluid limit of such a model is the unique solution to a system of three differential equations. Furthermore, we use the fluid limit to calculate the expected total arrival rate, which is then given as an input to the Erlang A formula for the purpose of calculating the service levels and abandonment probabilities. The performance of such a procedure is validated numerically in the case of both single intervals with constant parameters and multiple intervals with time-dependent parameters. The results demonstrate that this approximation method leads to accurate estimations for the service levels and the abandonment probabilities. We model the customer redial and reconnect behaviors in call centers.We approximate the service levels and abandonment percentages of such a model.A fluid model is proposed, and the corresponding fluid limit is derived.The performance of our approximation is evaluated numerically.

Analysis of a two-layered network by means of the power-series algorithm

We consider an extension of the classical machine-repair model, also known as the computer-terminal model or time-sharing model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, supply service themselves to queues of products. The extended model can be viewed as a two-layered queueing network, of which the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The marginal and joint queue length distributions of the first-layer queues are hard to analyse in an exact fashion. Therefore, we apply the power-series algorithm to this model to obtain the light-traffic behaviour of the queue lengths symbolically. This leads to two accurate approximations for the marginal mean queue length. The first approximation, based on the light-traffic behaviour, is in closed form. The second approximation is based on an interpolation between the light-traffic behaviour and heavy-traffic results for the mean queue length. The obtained approximations are shown to work well for arbitrary loaded systems. The proposed numerical algorithm and approximations may prove to be very useful for system design and optimisation purposes in application areas such as manufacturing, computer systems and telecommunications.

A simple index rule for efficient traffic splitting over parallel wireless networks with partial information

Multi-path communication solutions provide a promising means to improve the network performance in areas covered by multiple wireless access networks. Today, little is known about how to effectively exploit this potential. We study a model where flows are transferred over multiple parallel networks, each of which is modeled as a processor sharing node. The goal is to minimize the expected transfer time of elastic data traffic by smartly dispatching the flows to the networks, based on partial information about the numbers of foreground and background flows in each of the nodes. In the case of full state information, the optimal policy can be derived via standard MDP-techniques, but for models with partial information an optimal solution is hard to obtain. An important requirement is that the splitting algorithm is efficient, yet simple, easy-to-implement, scalable in the number of parallel networks and robust against changes in the parameter settings. We propose a simple index rule for splitting traffic streams based on partial information, and benchmark the results against the optimal solution in the case of full state information. Extensive simulations with real networks show that this method performs extremely well under practical circumstances for a wide range of realistic parameter settings.

Compliance tables for an EMS system with two types of medical response units

In this paper, we consider an Emergency Medical Services (EMS) system with two types of medical response units: Rapid Responder Ambulances (RRAs) and Regular Transport Ambulances (RTAs). The key difference between both is that RRAs are faster, but they lack the ability to transport a patient to the hospital. To maintain the ability to respond to emergency requests timely when ambulances get busy, we consider compliance tables, which indicate the desired locations of the available ambulances. Our system brings forth additional complexity to the problem of computing optimal compliance tables, as we have two kinds of ambulances. We propose an Integer Linear Program (ILP) computing compliance tables for such a system, which uses outcomes of a Hypercube model as input parameters. Moreover, we include nestedness constraints and we set bounds on the relocation times in the ILP. To obtain more credible results, we simulate the computed compliance tables for different input parameters. Results show that bounding the time a relocation may last is beneficial in certain settings. Besides, including the nestedness constraints ensures that the number of relocations and the relocation time is reduced, while the performance stays unaffected.

Product-form results for two-station networks with shared resources

Queueing networks are studied with two stations: either in tandem or in parallel, and with a common service resource shared among the two stations. First, a necessary and sufficient criterion, called adjoint reversibility, is provided to decide whether the system possesses a product form or not. This criterion unifies both the parallel (a reversible) and the tandem (a non-reversible) system in one product-form theorem. Next, the criterion is applied separately for the parallel and tandem system to obtain a number of new product-form examples which also includes non-balanced capacity sharing. Despite, but also due to, the different parallel and tandem mechanisms we observe that for certain examples the product form has the same structure, while for others there are essential differences. In addition, it is also proven that several models cannot have a product-form result. The results provide new insights and a step forward in understanding the behavior of multi-layered queueing networks in which resources are shared among stations.

Queueing networks with a single shared server: light and heavy traffic

We study a queueing network with a single shared server, that serves the queues in a cyclic order according to the gated service discipline. 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 and tandem queues. We derive exact limits of the mean delays under both heavy-traffic and light-traffic conditions. By interpolating between these asymptotic regimes, we develop simple closed-form approximations for the mean delays for arbitrary loads.