Pooling and polling : creation of pooling in inventory and queueing models

The subject of the present monograph is the ‘Creation of Pooling in Inventory and Queueing Models’. This research consists of the study of sharing a scarce resource (such as inventory, server capacity, or production capacity) between multiple customer classes. This is called pooling, where the goal is to achieve cost or waiting time reductions. For the queueing and inventory models studied, both theoretical, scientific insights, are generated, as well as strategies which are applicable in practice. This monograph consists of two parts: pooling and polling. In both research streams, a scarce resource (inventory or server capacity, respectively production capacity) has to be shared between multiple users. In the first part of the thesis, pooling is applied to multi-location inventory models. It is studied how cost reduction can be achieved by the use of stock transfers between local warehouses, so-called lateral transshipments. In this way, stock is pooled between the warehouses. The setting is motivated by a spare parts inventory network, where critical components of technically advanced machines are kept on stock, to reduce down time durations. We create insights into the question when lateral transshipments lead to cost reductions, by studying several models. Firstly, a system with two stock points is studied, for which we completely characterize the structure of the optimal policy, using dynamic programming. For this, we formulate the model as a Markov decision process. We also derived conditions under which simple, easy to implement, policies are always optimal, such as a hold back policy and a complete pooling policy. Furthermore, we identified the parameter settings under which cost savings can be achieved. Secondly, we characterize the optimal policy structure for a multi-location model where only one stock point issues lateral transshipments, a so-called quick response warehouse. Thirdly, we apply the insights generated to the general multi-location model with lateral transshipments. We propose the use of a hold back policy, and construct a new approximation algorithm for deriving the performance characteristics. It is based on the use of interrupted Poisson processes. The algorithm is shown to be very accurate, and can be used for the optimization of the hold back levels, the parameters of this class of policies. Also, we study related inventory models, where a single stock point servers multiple customers classes. Furthermore, the pooling of server capacity is studied. For a two queue model where the head-of-line processor sharing discipline is applied, we derive the optimal control policy for dividing the servers attention, as well as for accepting customers. Also, a server farm with an infinite number of servers is studied, where servers can be turned off after a service completion in order to save costs. We characterize the optimal policy for this model. In the second part of the thesis polling models are studied, which are queueing systems where multiple queues are served by a single server. An application is the production of multiple types of products on a single machine. In this way, the production capacity is pooled between the product types. For the classical polling model, we derive a closedform approximation for the mean waiting time at each of the queues. The approximation is based on the interpolation of light and heavy traffic results. Also, we study a system with so-called smart customers, where the arrival rate at a queue depends on the position of the server. Finally, we invent two new service disciplines (the gated/exhaustive and the ??-gated discipline) for polling models, designed to yield ’fairness and efficiency’ in the mean waiting times. That is, they result in almost equal mean waiting times at each of the queues, without increasing the weighted sum of the mean waiting times too much

A polling model with smart customers

International audienceIn 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 the 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

Fairness and efficiency for polling models with the K-gated service discipline

We study a polling model where we want to achieve a balance between the fairness of the waiting times and the efficiency of the system. For this purpose, we introduce the k-gated service discipline. It is a hybrid of the classical gated and exhausted disciplines, and consists of using Ki gated service phases at queue i before the server switches to the next queue. We derive the distributions and means of the waiting times, a pseudo conservation law for the weighted sum of the mean waiting times, and the fluid limits of the waiting times. Our goal is to optimize the Ki's so as to minimize the differences in the mean waiting times, i.e. to achieve maximal fairness, without giving up too much on the efficiency of the system. From the fluid limits we derive a heuristic rule for setting the Ki's. In a numerical study the heuristic is shown to perform well

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 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 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

Closed-form waiting time approximations for polling systems

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

Position estimating in peer-to-peer networks

We present two algorithms for indoor positioning estimation in peer-to-peer networks. The setup is a network of two types of devices: reference devices with a known location and blindfolded devices that can determine distances to reference devices and each other. From this information the blindfolded devices try to estimate their positions. A typical scenario is navigation inside a shopping mall where devices in the parking lot can make contact with GPS satellites, whereas devices inside the building make contact with each other, devices on the parking lot, and devices fixed to the building. The devices can measure their in-between distances, with some measurement error, and exchange positioning information. However, other devices might only know their position with some error. We present two algorithms for positioning estimation in such a peer-to-peer network. The first one is purely geometric and is based on Euclidean geometry and intersecting spheres. We rewrite the information to a linear system, which is typically overdetermined. We use least squares to ??nd the best estimate for a device its position. The second approach can be considered as a probabilistic version of the geometric approach. We estimate the probability density function that a device is located at a position given a probability density function for the positions of the other devices in the network, and a probability density function of the measured distances. First we study the case with a distance measurement to a single other user, then we focus on multiple other users. We give an approximation algorithm that is the probabilistic analogue of the intersecting spheres method. We show some simulated results where ambiguous data lead to well defined probability distributions for the position of a device. We conclude with some open questions

Node counting in wireless ad-hoc networks

We study wireless ad-hoc networks consisting of small microprocessors with limited memory, where the wireless communication between the processors can be highly unreliable. For this setting, we propose a number of algorithms to estimate the number of nodes in the network, and the number of direct neighbors of each node. The algorithms are simulated, allowing comparison of their performance

Increasing Detection Performance of Surveillance Sensor Networks

We study a surveillance wireless sensor network (SWSN) comprised of small and low-cost sensors deployed in a region in order to detect objects crossing the field of interest. In the present paper, we address two problems concerning the design and performance of an SWSN: optimal sensor placement and algorithms for object detection in the presence of false alarms. For both problems, we propose explicit decision rules and efficient algorithmic solutions. Further, we provide several numerical examples and present a simulation model that combines our placement and detection methods