The impact of production interruptions on kitting, an analytical study

Nowadays, customers request more variation in a company's product assortment leading to an increased amount of parts moving around on the shop floor. To cope with this tendency, a kitting process can be implemented. As it gathers the necessary parts into a container prior to assembly, kitting enables a more cost-efficient and qualitative production. However, the performance of this preparation technique in an assembly process has merely been investigated. Therefore, we study a kitting process with two parts as a continuous-time Markovian queueing model. In particular, our study assess the impact of production interruptions on kitting

Numerical methods for queues with shared service

A queueing system is a mathematical abstraction of a situation where elements, called customers, arrive in a system and wait until they receive some kind of service. Queueing systems are omnipresent in real life. Prime examples include people waiting at a counter to be served, airplanes waiting to take off, traffic jams during rush hour etc. Queueing theory is the mathematical study of queueing phenomena. As often neither the arrival instants of the customers nor their service times are known in advance, queueing theory most often assumes that these processes are random variables. The queueing process itself is then a stochastic process and most often also a Markov process, provided a proper description of the state of the queueing process is introduced. This dissertation investigates numerical methods for a particular type of Markovian queueing systems, namely queueing systems with shared service. These queueing systems differ from traditional queueing systems in that there is simultaneous service of the head-of-line customers of all queues and in that there is no service if there are no customers in one of the queues. The absence of service whenever one of the queues is empty yields particular dynamics which are not found in traditional queueing systems. These queueing systems with shared service are not only beautiful mathematical objects in their own right, but are also motivated by an extensive range of applications. The original motivation for studying queueing systems with shared service came from a particular process in inventory management called kitting. A kitting process collects the necessary parts for an end product in a box prior to sending it to the assembly area. The parts and their inventories being the customers and queues, we get ``shared service'' as kitting cannot proceed if some parts are absent. Still in the area of inventory management, the decoupling inventory of a hybrid make-to-stock/make-to-order system exhibits shared service. The production process prior to the decoupling inventory is make-to-stock and driven by demand forecasts. In contrast, the production process after the decoupling inventory is make-to-order and driven by actual demand as items from the decoupling inventory are customised according to customer specifications. At the decoupling point, the decoupling inventory is complemented with a queue of outstanding orders. As customisation only starts when the decoupling inventory is nonempty and there is at least one order, there is again shared service. Moving to applications in telecommunications, shared service applies to energy harvesting sensor nodes. Such a sensor node scavenges energy from its environment to meet its energy expenditure or to prolong its lifetime. A rechargeable battery operates very much like a queue, customers being discretised as chunks of energy. As a sensor node requires both sensed data and energy for transmission, shared service can again be identified. In the Markovian framework, "solving" a queueing system corresponds to finding the steady-state solution of the Markov process that describes the queueing system at hand. Indeed, most performance measures of interest of the queueing system can be expressed in terms of the steady-state solution of the underlying Markov process. For a finite ergodic Markov process, the steady-state solution is the unique solution of $N-1$ balance equations complemented with the normalisation condition, $N$ being the size of the state space. For the queueing systems with shared service, the size of the state space of the Markov processes grows exponentially with the number of queues involved. Hence, even if only a moderate number of queues are considered, the size of the state space is huge. This is the state-space explosion problem. As direct solution methods for such Markov processes are computationally infeasible, this dissertation aims at exploiting structural properties of the Markov processes, as to speed up computation of the steady-state solution. The first property that can be exploited is sparsity of the generator matrix of the Markov process. Indeed, the number of events that can occur in any state --- or equivalently, the number of transitions to other states --- is far smaller than the size of the state space. This means that the generator matrix of the Markov process is mainly filled with zeroes. Iterative methods for sparse linear systems --- in particular the Krylov subspace solver GMRES --- were found to be computationally efficient for studying kitting processes only if the number of queues is limited. For more queues (or a larger state space), the methods cannot calculate the steady-state performance measures sufficiently fast. The applications related to the decoupling inventory and the energy harvesting sensor node involve only two queues. In this case, the generator matrix exhibits a homogene block-tridiagonal structure. Such Markov processes can be solved efficiently by means of matrix-geometric methods, both in the case that the process has finite size and --- even more efficiently --- in the case that it has an infinite size and a finite block size. Neither of the former exact solution methods allows for investigating systems with many queues. Therefore we developed an approximate numerical solution method, based on Maclaurin series expansions. Rather than focussing on structural properties of the Markov process for any parameter setting, the series expansion technique exploits structural properties of the Markov process when some parameter is sent to zero. For the queues with shared exponential service and the service rate sent to zero, the resulting process has a single absorbing state and the states can be ordered such that the generator matrix is upper-diagonal. In this case, the solution at zero is trivial and the calculation of the higher order terms in the series expansion around zero has a computational complexity proportional to the size of the state space. This is a case of regular perturbation of the parameter and contrasts to singular perturbation which is applied when the service times of the kitting process are phase-type distributed. For singular perturbation, the Markov process has no unique steady-state solution when the parameter is sent to zero. However, similar techniques still apply, albeit at a higher computational cost. Finally we note that the numerical series expansion technique is not limited to evaluating queues with shared service. Resembling shared queueing systems in that a Markov process with multidimensional state space is considered, it is shown that the regular series expansion technique can be applied on an epidemic model for opinion propagation in a social network. Interestingly, we find that the series expansion technique complements the usual fluid approach of the epidemic literature

Performance analysis of a decoupling stock in a make-to-order system

In a Make-to-Order system, products are only manufactured when orders are placed. As this may lead to overly long delivery times, a stock of semi-finished products can be installed to reduce production time: the so-called decoupling stock. As performance of the decoupling stock is critical to the overall performance and cost of the production system, we propose and analyse a Markovian model of the decoupling stock. In particular, we focus on a queueing model with two buffers, thereby accounting for both the decoupling stock as well as for possible backlog of orders. By means of numerical examples, we then quantify the impact of production inefficiency on delivery times and overall cost

Stochastic modelling of energy harvesting for low power sensor nodes

Battery lifetime is a key impediment to long-lasting low power sensor nodes. Energy or power harvesting mitigates the ependency on battery power, by converting ambient energy into electrical energy. This energy can then be used by the device for data collection and transmission. This paper proposes and analyses a queueing model to assess performance of such an energy harvesting sensor node. Accounting for energy harvesting, data collection and data transmission opportunities, the sensor node is modelled as a paired queueing system. The system has two queues, one representing accumulated energy and the other being the data queue. By means of some numerical examples, we investigate the energy-information trade-off

Performance evaluation of a decoupling inventory for hybrid push-pull systems

Nowadays, companies that oer product variety while maintaining short lead times and competitive quality and cost, gain a competitive edge over their competitors. Therefore, hybrid push-pull systems allow for efficiently balancing lead times and production costs. Raw materials are `pushed' into the semi-finished good warehouse and customers `pull' products by placing orders. As performance of the decoupling stock is critical to the overall performance of the manufacturing system, we define and analyse a Markovian queueing model with two buers, thereby accounting for both the decoupling stock as well as for possible backlog of orders. In particular, our study assesses the eect of variability in the production process and the ordering process on the performance of the decoupling stock