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In this thesis, we are interested in the modelling of queueing networks with finite buffers and with general service time distributions. Queueing networks models have shown to be very useful tools to evaluate the performance of complex systems in many application fields (manufacturing, communication networks, traffic flow, etc.). In order to analyze such networks, the original distributions are most often transformed into tractable distributions, so that the Markov theory can then be applied. Our main originality lies in this step of the modelling process. We propose to discretize the original distributions by probability mass fitting (PMF). The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval. PMF has the advantage to be simple, refinable, and to conserve the shape of the distribution. Moreover, we show that it does not require more phases, and thus more computational effort, than concurrent methods. From the distributions transformed by PMF, the evolution of the system can then be modelled by a discrete Markov chain, and the performance of the system can be evaluated from the chain. This global modelling method leads to various interesting results. First, we propose two methodologies leading to bounds on the cycle time of the system. In particular, a tight lower bound on the cycle time can be computed. Second, probability mass fitting leads to accurate approximation of the performance measures (cycle time, work-in-progress, flow time, etc.). Together with the bounds, the approximations allow to get a good grasp on the exact measure with certainty. Third, the cycle time distribution can be computed in the discretized time and shows to be a good approximation of the original cycle time distribution. The distribution provides more information on the behavior of the system, compared to the isolated expectation (to which other methods are limited). Finally, in order to be able to analyze larger networks, the decomposition technique can be applied after PMF. We show that the accuracy of the performance evaluation is still good, and that the ability of PMF to accurately estimate the distributions brings an improvement in the application of the decomposition. In conclusion, we believe that probability mass fitting can be considered as a valuable alternative in order to build tractable distributions for the analytical modelling of queueing networks.

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We define a service factory to be a network of service-related-workstations, at which assigned workers process work-in-progress that flows through the workstations. Examples of service factory work include mail processing and sorting, check processing and telephoned order processing. Exogenous work may enter the factory at any workstation according to any time-of-day profile. Work-in-progress flows though the factory in discrete time according to Markovian routings. Workers, who in general are cross trained, may work part time or full time shifts, may start work only at designated shift starting times, and may change job assignments at midshift. In order to smooth the flow of work-in-progress through the service factory, work-in-progress may be temporarily inventoried (in buffers) at work stations. The objective is to schedule the workers (and correspondingly, the workflow) in a manner that minimizes labor costs subject to a variety of service-level, contractural and physical constraints. Motivated in part by analysis techniques of discrete time linear time-invariant (LTI) systems, an object-oriented linear programming (OOLP) model is developed. Using exogenous input work profiles typical of large U. S. mail processingfacilities, illustrative computational results are included.

Analyses are addressed for a number of problems in queueing systems and stochastic modeling that arose due to an investigation into techniques that could be used to approximate general closed networks. In Chapter II, a method is presented to calculate the system size distribution at an arbitrary point in time and at departures for a (n)/G/1/N queue. The analysis is carried out using an embedded Markov chain approach. An algorithm is also developed that combines our analysis with the recursive method of Gupta and Rao. This algorithm compares favorably with that of Gupta and Rao and will solve some situations when Gupta and Rao's method fails or becomes intractable. In Chapter III, an approach is developed for generating exact solutions of the time-dependent conditional joint probability distributions for a phase-type renewal process. Closed-form expressions are derived when a class of Coxian distributions are used for the inter-renewal distribution. The class of Coxian distributions was chosen so that solutions could be obtained for any mean and variance desired in the inter-renewal times. In Chapter IV, an algorithm is developed to generate numerical solutions for the steady-state system size probabilities and waiting time distribution functions of the SM/PH/1/N queue by using the matrix-analytic method. Closed form results are also obtained for particular situations of the preceding queue. In addition, it is demonstrated that the SM/PH/1/N model can be implemented to the analysis of a sequential two-queue system. This is an extension to the work by Neuts and Chakravarthy. In Chapter V, principal results developed in the preceding chapters are employed for approximate analysis of the closed network of queues with arbitrary service times. Specifically, the (n)/G/1/N queue is applied to closed networks of a general topology, and a sequential two-queue model consisting of the (n)/G/1/N and SM/PH/1/N queues is proposed for tandem queueing networks.

The average cost of operating a queueing network depends on several factors such as the complexity of the network and the service policy used. Approximate linear programming (ALP) is a method that can be used to compute an accurate lower bound on the optimal average cost as well as generate policies to be used in operating the network. These average cost solutions and policies are dependent on the type of basis function used in the ALP. In this paper, the ALP average cost solutions and policies are analyzed for twelve networks with four different types of basis functions (quadratic, linear, pure exponential, and mixed exponential). An approximate bound on the optimality gap between the ALP average cost solution and the optimal average cost solution is computed for each system, and the size of this bound is determined relative to the ALP average cost solution. Using the same set of networks, the performance of ALP generated policies are compared to the performance of the heuristic policies first-buffer-first-served (FBFS), last-buffer-first-served (LBFS), highest-queue-first-served (HQFS), and random-queue-first-served (RQFS). In general, ALP generated average cost solutions are considerably smaller than the simulated average cost under the corresponding policy, and therefore the approximate bounds on the optimality gaps are quite large. This bound increases with the complexity of the queueing network. Some ALP generated policies are not stabilizing policies for their corresponding networks, especially those produced using pure exponential and mixed exponential basis functions. For almost all systems, at least one of the heuristic policies results in mean average cost less than or nearly equal to the smallest mean average cost of all ALP generated policies in simulation runs. This means that generally there exists a heuristic policy which can perform as well as or better than any ALP generated policy. In conclusion, a useful bound on the optimality gap between the ALP average cost solution and the optimal average cost solution cannot be computed with this method. Further, heuristic policies, which are more computationally tractable than ALP generated policies, can generally match or exceed the performance of ALP generated policies, and thus computing such policies is often unnecessary for realizing cost benefits in queueing networks. / text

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No description available.

The study of stochastic queueing networks is quite important due to the many applications including transportation, telecommunication, and manufacturing industries. Since there is often no explicit solution to these types of control problems, numerical methods are needed. Following the method of Boué-Dupuis, we use a Dynamic Programming approach of optimization on a controlled Markov Chain that simulates the behavior of a fluid limit of the original process. The search for an optimal control in this case involves a Skorokhod problem to describe the dynamics on the boundary of closed, convex domain. Using relaxed stochastic controls we show that the approximating numerical solution converges to the actual solution as the size of the mesh in the discretized state space goes to zero, and illustrate with an example. / Ph. D.

The Internet has been more and more intensively used in recent years. Although network infrastructure has been regularly upgraded, and the ability to manage heavy traffic greatly increased, especially on the core networks, congestion never ceases to appear, as the amount of traffic that flow on the Internet seems to be increasing at an even faster rate. Thus, congestion control mechanisms play a vital role in the functioning of the Internet. Active Queue Management (AQM) is a popular type of congestion control mechanism that is implemented on gateways (most notably routers), which can predict and avoid the congestion before it happens. When properly configured, AQMs can effectively reduce the congestion, and alleviate some of the problems such as global synchronisation and unfairness to bursty traffic. However, there are still many problems regarding AQMs. Most of the AQM schemes are quite sensitive to their parameters setting, and these parameters may be heavily dependent on the network traffic profile, which the administrator may not have intensive knowledge of, and is likely to change over time. When poorly configured, many AQMs perform no better than the basic drop-tail queue. There is currently no effective method to compare the performance of these AQM algorithms, caused by the parameter configuration problem. In this research, the aim is to propose a new analytical model, which mainly uses discrete-time queueing theory. A novel transient modification to the conventional equilibrium-based method is proposed, and it is utilised to further develop a dynamic interactive model of responsive traffic and bottleneck queues. Using step-by-step analysis, it represents the bursty traffic and oscillating queue length behaviour in practical network more accurately. It also provides an effective way of predicting the behaviour of a TCP-AQM system, allowing easier parameter optimisation for AQM schemes. Numerical solution using MATLAB and software simulation using NS-2 are used to extensively validate the proposed models, theories and conclusions.