Open queueing networks : optimization and performance evaluation models for discrete manufacturing systems

Includes bibliographical references (p. 41-45).Research supported by Fundac̦ão de Amparo a Pesquisa do Estado de São Paulo, Brazil.by Gabriel R. Bitran, Reinaldo Morabito

Modeling Conveyor Merges in Zone Picking Systems

In many order picking and sorting systems conveyors are used to transport products through the system and to merge multiple flows of products into one single flow. In practice, conveyor merges are potential points of congestion, and consequently can lead to a reduced throughput. In this paper, we study merges in a zone picking system. The performance of a zone picking system is, for a large part, determined by the performance of the merge locations. We model the system as a closed queueing network that describes the conveyor, the pick zones, and the merge locations. The resulting model does not have a product-form stationary queue-length distribution. This makes exact analysis practically infeasible. Therefore, we approximate the behavior of the model using the aggregation technique, where the resulting subnetworks are solved using matrix-geometric methods. We show that the approximation model allows us to determine very accurate estimates of the throughput when compared with simulation. Furthermore, our model is in particular well suited to evaluate many design alternatives, in terms of number of zones, zone buffer lengths, and maximum number of totes in the systems. It also can be used to determine the maximum throughput capability of the system and, if needed, modify the system in order to meet target performance levels

Performance analysis of networks on chips

Modules on a chip (such as processors and memories) are traditionally connected through a single link, called a bus. As chips become more complex and the number of modules on a chip increases, this connection method becomes inefficient because the bus can only be used by one module at a time. Networks on chips are an emerging technology for the connection of on-chip modules. In networks on chips, switches are used to transmit data from one module to another, which entails that multiple links can be used simultaneously so that communication is more efficient. Switches consist of a number of input ports to which data arrives and output ports from which data leaves. If data at multiple input ports has to be transmitted to the same output port, only one input port may actually transmit its data, which may lead to congestion. Queueing theory deals with the analysis of congestion phenomena caused by competition for service facilities with scarce resources. Such phenomena occur, for example, in traffic intersections, manufacturing systems, and communication networks like networks on chips. These congestion phenomena are typically analysed using stochastic models, which capture the uncertain and unpredictable nature of processes leading to congestion (such as irregular car arrivals to a traffic intersection). Stochastic models are useful tools for the analysis of networks on chips as well, due to the complexity of data traffic on these networks. In this thesis, we therefore study queueing models aimed at networks on chips. The thesis is centred around two key models: A model of a switch in isolation, the so-called single-switch model, and a model of a network of switches where all traffic has the same destination, the so-called network of polling stations. For both models we are interested in the throughput (the amount of data transmitted per time unit) and the mean delay (the time it takes data to travel across the network). Single-switch models are often studied under the assumption that the number of ports tends to infinity and that traffic is uniform (i.e., on average equally many packets arrive to all buffers, and all possible destinations are equally likely). In networks on chips, however, the number of buffers is typically small. We introduce a new approximation specifically aimed at small switches with (memoryless) Bernoulli arrivals. We show that, for such switches, this approximation is more accurate than currently known approximations. As traffic in networks on chips is usually non-uniform, we also extend our approximation to non-uniform switches. The key difference between uniform and nonuniform switches is that in non-uniform switches, all queues have a different maximum throughput. We obtain a very accurate approximation of this throughput, which allows us to extend the mean delay approximation. The extended approximation is derived for Bernoulli arrivals and correlated arrival processes. Its accuracy is verified through a comparison with simulation results. The second key model is that of concentrating tree networks of polling stations (polling stations are essentially switches where all traffic has the same output port as destination). Single polling stations have been studied extensively in literature, but only few attempts have been made to analyse networks of polling stations. We establish a reduction theorem that states that networks of polling stations can be reduced to single polling stations while preserving some information on mean waiting times. This reduction theorem holds under the assumption that the last node of the network uses a so-called HoL-based service discipline, which means that the choice to transmit data from a certain buffer may only depend on which buffers are empty, but not on the amount of data in the buffers. The reduction theorem is a key tool for the analysis of networks of polling stations. In addition to this, mean waiting times in single polling stations have to be calculated, either exactly or approximately. To this end, known results can be used, but we also devise a new single-station approximation that can be used for a large subclass of HoL-based service disciplines. Finally, networks on chips typically implement flow control, which is a mechanism that limits the amount of data in the network from one source. We analyse the division of throughput over several sources in a network of polling stations with flow control. Our results indicate that the throughput in such a network is determined by an interaction between buffer sizes, flow control limits, and service disciplines. This interaction is studied in more detail by means of a numerical analysis

Optimization of Multiclass Queueing Networks: Polyhedral and Nonlinear Characterizations of Achievable Performance

We consider open and closed multiclass queueing networks with Poisson arrivals (in open networks), exponentially distributed class dependent service times, and with class dependent deterministic or probabilistic routing. For open networks, the performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose variants of a method to derive polyhedral and nonlinear spaces which contain the entire set of achievable response times under stable and preemptive scheduling policies. By optimizing over these spaces, we obtain lower bounds on achievable performance. In particular, we obtain a sequence of progressively more complicated nonlinear approximations (relaxations) which are progressively closer to the exact achievable space. In the special case of single station networks (multiclass queues and Klimov's model) and homogenous multiclass networks, our characterization gives exactly the achievable region. Consequently, the proposed method can be viewed as the natural extension of conservation laws to multiclass queueing networks. For closed networks, the performance objective is to maximize throughput. We similarly find polyhedral and nonlinear spaces that include the performance space and by maximizing over these spaces we obtain an upper bound on the optimal throughput. We check the tightness of our bounds by simulating heuristic scheduling policies for simple open networks and we find that the first order approximation of our method is at least as good as simulation-based existing methods. In terms of computational complexity and in contrast to simulation-based existing methods, the calculation of our first order bounds consists of solving a linear programming problem with both the number of variables and constraints being polynomial (quadratic) in the number of classes in the network. The i-th order approximation involves solving a convex programming problem in dimension O(Ri+l), where R is the number of classes in the network, which can be solved efficiently using techniques from semi-definite programming

Performance Modelling and Optimisation of Multi-hop Networks

A major challenge in the design of large-scale networks is to predict and optimise the total time and energy consumption required to deliver a packet from a source node to a destination node. Examples of such complex networks include wireless ad hoc and sensor networks which need to deal with the effects of node mobility, routing inaccuracies, higher packet loss rates, limited or time-varying effective bandwidth, energy constraints, and the computational limitations of the nodes. They also include more reliable communication environments, such as wired networks, that are susceptible to random failures, security threats and malicious behaviours which compromise their quality of service (QoS) guarantees. In such networks, packets traverse a number of hops that cannot be determined in advance and encounter non-homogeneous network conditions that have been largely ignored in the literature. This thesis examines analytical properties of packet travel in large networks and investigates the implications of some packet coding techniques on both QoS and resource utilisation. Specifically, we use a mixed jump and diffusion model to represent packet traversal through large networks. The model accounts for network non-homogeneity regarding routing and the loss rate that a packet experiences as it passes successive segments of a source to destination route. A mixed analytical-numerical method is developed to compute the average packet travel time and the energy it consumes. The model is able to capture the effects of increased loss rate in areas remote from the source and destination, variable rate of advancement towards destination over the route, as well as of defending against malicious packets within a certain distance from the destination. We then consider sending multiple coded packets that follow independent paths to the destination node so as to mitigate the effects of losses and routing inaccuracies. We study a homogeneous medium and obtain the time-dependent properties of the packet’s travel process, allowing us to compare the merits and limitations of coding, both in terms of delivery times and energy efficiency. Finally, we propose models that can assist in the analysis and optimisation of the performance of inter-flow network coding (NC). We analyse two queueing models for a router that carries out NC, in addition to its standard packet routing function. The approach is extended to the study of multiple hops, which leads to an optimisation problem that characterises the optimal time that packets should be held back in a router, waiting for coding opportunities to arise, so that the total packet end-to-end delay is minimised

Optimization of multiclass queueing networks : polyhedral and nonlinear characterization of achievable performance

Includes bibliographical references (p. 48-50).Supported by the National Science Foundation. ECS-8552419 Supported by the Presidential Young Investigator Award. DDM-9158118 Supported by the Draper Laboratory and by the Leaders for Manufacturing Program at MIT. Supported by the ARO. DAAL03-92-G0309Dimitris Bertsimas, Ioannis Ch. Paschalidis, John N. Tsitsiklis