Unreliable Retrial Queues in a Random Environment

This dissertation investigates stability conditions and approximate steady-state performance measures for unreliable, single-server retrial queues operating in a randomly evolving environment. In such systems, arriving customers that find the server busy or failed join a retrial queue from which they attempt to regain access to the server at random intervals. Such models are useful for the performance evaluation of communications and computer networks which are characterized by time-varying arrival, service and failure rates. To model this time-varying behavior, we study systems whose parameters are modulated by a finite Markov process. Two distinct cases are analyzed. The first considers systems with Markov-modulated arrival, service, retrial, failure and repair rates assuming all interevent and service times are exponentially distributed. The joint process of the orbit size, environment state, and server status is shown to be a tri-layered, level-dependent quasi-birth-and-death (LDQBD) process, and we provide a necessary and sufficient condition for the positive recurrence of LDQBDs using classical techniques. Moreover, we apply efficient numerical algorithms, designed to exploit the matrix-geometric structure of the model, to compute the approximate steady-state orbit size distribution and mean congestion and delay measures. The second case assumes that customers bring generally distributed service requirements while all other processes are identical to the first case. We show that the joint process of orbit size, environment state and server status is a level-dependent, M/G/1-type stochastic process. By employing regenerative theory, and exploiting the M/G/1-type structure, we derive a necessary and sufficient condition for stability of the system. Finally, for the exponential model, we illustrate how the main results may be used to simultaneously select mean time customers spend in orbit, subject to bound and stability constraints

Approximate performance analysis of production lines with continuous material flows and finite buffers

In this paper, we analyze production lines consisting of a number of machines or servers in series with a finite buffer between each pair of machines. The flow of products through the machines is continuous. Each machine suffers from breakdowns, because of, for example, failures, cleaning and changeover. The up- and downtimes are independent and generally distributed. We develop a new method to efficiently and accurately estimate the throughput and the mean buffer content of the production line. This method relies on decomposition of the production line into two-stage, one-buffer subsystems aggregating the up- and downstream part of the line. For each subsystem, the parameters of the aggregate up- and downtimes are determined iteratively by employing matrix-analytic techniques. The proposed method performs very well on a large test set consisting of over 49,000 cases. Remarkably, the performance of the method does not deteriorate in case of highly unpredictable up- and downtimes, as often seen in practice. We apply the method to a bottling line at brewery Heineken Den Bosch and an assembly line at NXP Semiconductors

Performance analysis of production lines with continuous material flows and finite buffers

This paper deals with the approximative analysis of production lines with continuous material flow consisting of a number of machines or servers in series and finite buffers in between. Each server suffers from operational dependent breakdowns, characterized by exponentially distributed up- and down-times. We construct an iterative method to efficiently and accurately estimate performance characteristics such as throughput and mean total buffer content. The method is based on decomposition of the production line into single-buffer subsystems. Novel features of the method are (i) modeling of the aggregate servers in each subsystem, (ii) equations to iteratively determine the processing behavior of these servers, and (iii) use of modern matrix-analytic techniques to analyze each subsystem. The proposed method performs very well on a large test set, including long and imbalanced production lines. For production lines with imbalance in mean down-times, we show that a more refined modeling of the servers in each subsystem performs significantly better. Lastly, we apply the iterative method to predict the throughput of a bottle line at brewery Heineken Den Bosch yielding errors of less than two percent

Fluid flow models in performance analysis

We review several developments in fluid flow models: feedback fluid models, linear stochastic fluid networks and bandwidth sharing networks. We also mention some promising new research directions

Two-dimensional fluid queues with temporary assistance

We consider a two-dimensional stochastic fluid model with $N$ ON-OFF inputs and temporary assistance, which is an extension of the same model with $N = 1$ in Mahabhashyam et al. (2008). The rates of change of both buffers are piecewise constant and dependent on the underlying Markovian phase of the model, and the rates of change for Buffer 2 are also dependent on the specific level of Buffer 1. This is because both buffers share a fixed output capacity, the precise proportion of which depends on Buffer 1. The generalization of the number of ON-OFF inputs necessitates modifications in the original rules of output-capacity sharing from Mahabhashyam et al. (2008) and considerably complicates both the theoretical analysis and the numerical computation of various performance measures

A feedback fluid queue with two congestion control thresholds

Feedback fluid queues play an important role in modeling congestion control mechanisms for packet networks. In this paper we present and analyze a fluid queue with a feedback-based traffic rate adaptation scheme which uses two thresholds. The higher threshold $B_{1}$ is used to signal the beginning of congestion while the lower threshold $B_{2}$ signals the end of congestion. These two parameters together allow to make the trade--off between maximizing throughput performance and minimizing delay. The difference between the two thresholds helps to control the amount of feedback signals sent to the traffic source. In our model the input source can behave like either of two Markov fluid processes. The first applies as long as the upper threshold $B_{1}$ has not been hit from below. As soon as that happens, the traffic source adapts and switches to the second process, until $B_{2}$ (smaller than $B_1$) is hit from above. We analyze the model by setting up the Kolmogorov forward equations, then solving the corresponding balance equations using a spectral expansion, and finally identifying sufficient constraints to solve for the unknowns in the solution. In particular, our analysis yields expressions for the stationary distribution of the buffer occupancy, the buffer delay distribution, and the throughput