An Origin-Destination (OD) traffic matrix provides a major input to the design, planning and management of a telecommunications network. Since the Internet is being proposed as the principal delivery mechanism for telecommunications traffic at the present time, and this network is not owned or managed by a single entity, there are significant challenges for network planners and managers needing to determine equipment and topology configurations for the various sections of the Internet that are currently the responsibility of ISPs and traditional telcos. Planning of these sub-networks typically requires a traffic matrix of demands that is then used to infer the flows on the administrator's network. Unfortunately, computation of the traffic matrix from measurements of individual flows is extremely difficult due to the fact that the problem formulation generally leads to the need to solve an under-determined system of equations. Thus, there has been a major effort f rom among researchers to obtain the traffic matrix using various inference techniques. The major contribution of this thesis is the development of inference techniques for traffic matrix estimation problem according to three different approaches, viz: (1) deterministic, (2) statistical, and (3) dynamic approaches. Firstly, for the deterministic approach, the traffic matrix estimation problem is formulated as a nonlinear optimization problem based on the generalized Kruithof approach which uses the Kullback distance to measure the probabilistic distance between two traffic matrices. In addition, an algorithm using the Affine scaling method is developed to solve the constrained optimization problem. Secondly, for the statistical approach, a series of traffic matrices are obtained by applying a standard deterministic approach. The components of these matrices represent estimates of the volumes of flows being exchanged between all pairs of nodes at the respective measurement points and they form a stochastic counting process. Then, a Markovian Arrival Process of order two (MAP-2) is applied to model the counting processes formed from this series of estimated traffic matrices. Thirdly, for the dynamic approach, the dual problem of the multi-commodity flow problem is formulated to obtain a set of link weights. The new weight set enables flows to be rerouted along new paths, which create new constraints to overcome the under-determined nature of traffic matrix estimation. Since a weight change disturbs a network, the impact of weight changes on the network is investigated by using simulation based on the well-known ns2 simulator package. Finally, we introduce two network applications that make use of the deterministic and the statistical approaches to obtain a traffic matrix respectively and also describe a scenario for the use of the dynamic approach.

The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research.

The seminal paper by Gerber and Shiu (1998) gave a huge boost to the study of risk theory by not only unifying but also generalizing the treatment and the analysis of various risk-related quantities in one single mathematical function - the Gerber-Shiu expected discounted penalty function, or Gerber-Shiu function in short. The Gerber-Shiu function is known to possess many nice properties, at least in the case of the classical compound Poisson risk model. For example, upon the introduction of a dividend barrier strategy, it was shown by Lin et al. (2003) and Gerber et al. (2006) that the Gerber-Shiu function with a barrier can be expressed in terms of the Gerber-Shiu function without a barrier and the expected value of discounted dividend payments. This result is the so-called dividends-penalty identity, and it holds true when the surplus process belongs to a class of Markov processes which are skip-free upwards. However, one stringent assumption of the model considered by the above authors is that all the interclaim times and the claim sizes are independent, which is in general not true in reality. In this thesis, we propose to analyze the Gerber-Shiu functions under various dependent structures. The main focus of the thesis is the risk model where claims follow a Markovian arrival process (MAP) (see, e.g., Latouche and Ramaswami (1999) and Neuts (1979, 1989)) in which the interclaim times and the claim sizes form a chain of dependent variables. The first part of the thesis puts emphasis on certain dividend strategies. In Chapter 2, it is shown that a matrix form of the dividends-penalty identity holds true in a MAP risk model perturbed by diffusion with the use of integro-differential equations and their solutions. Chapter 3 considers the dual MAP risk model which is a reflection of the ordinary MAP model. A threshold dividend strategy is applied to the model and various risk-related quantities are studied. Our methodology is based on an existing connection between the MAP risk model and a fluid queue (see, e.g., Asmussen et al. (2002), Badescu et al. (2005), Ramaswami (2006) and references therein). The use of fluid flow techniques to analyze risk processes opens the door for further research as to what types of risk model with dependency structure can be studied via probabilistic arguments. In Chapter 4, we propose to analyze the Gerber-Shiu function and some discounted joint densities in a risk model where each pair of the interclaim time and the resulting claim size is assumed to follow a bivariate phase-type distribution, with the pairs assumed to be independent and identically distributed (i.i.d.). To this end, a novel fluid flow process is constructed to ease the analysis. In the classical Gerber-Shiu function introduced by Gerber and Shiu (1998), the random variables incorporated into the analysis include the time of ruin, the surplus prior to ruin and the deficit at ruin. The later part of this thesis focuses on generalizing the classical Gerber-Shiu function by incorporating more random variables into the so-called penalty function. These include the surplus level immediately after the second last claim before ruin, the minimum surplus level before ruin and the maximum surplus level before ruin. In Chapter 5, the focus will be on the study of the generalized Gerber-Shiu function involving the first two new random variables in the context of a semi-Markovian risk model (see, e.g., Albrecher and Boxma (2005) and Janssen and Reinhard (1985)). It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation, and some discounted joint densities involving the new variables are derived. Chapter 6 revisits the MAP risk model in which the generalized Gerber-Shiu function involving the maximum surplus before ruin is examined. In this case, the Gerber-Shiu function no longer satisfies a defective renewal equation. Instead, the generalized Gerber-Shiu function can be expressed in terms of the classical Gerber-Shiu function and the Laplace transform of a first passage time that are both readily obtainable. In a MAP risk model, the interclaim time distribution must be phase-type distributed. This leads us to propose a generalization of the MAP risk model by allowing for the interclaim time to have an arbitrary distribution. This is the subject matter of Chapter 7. Chapter 8 is concerned with the generalized Sparre Andersen risk model with surplus-dependent premium rate, and some ordering properties of certain ruin-related quantities are studied. Chapter 9 ends the thesis by some concluding remarks and directions for future research.

No / Given only the traffic correlations of counts and intervals, a Batch Renewal Arrival Process (BRAP) is completely determined, as the least biased choice and thus, it provides the analytic means to construct suitable traffic models for the study of queueing systems independently of any other traffic characteristics. In this context, the BRAP and the Batch Markovian Arrival Process (BMAP) are employed in the continuous time domain towards the analysis of the stable BRAP/GE/1 and BMAP/GE/1 queues with infinite capacity, single servers and generalized exponential (GE) service times. Novel closed form expressions for the steady state probabilities of these queues are obtained, based on the embedded Markov chains (EMCs) technique and the matrix-geometric (M-G) method, respectively. Moreover, the stable GEsGGeo/GE/1 queue with GE-type service times and a GEsGGeo BRAP consisting of bursty GE-type batch interarrival times and a shifted generalized geometric (sGGeo) batch size distribution is adopted to assess analytically the combined adverse effects of varying degrees of correlation of intervals between individual arrivals and the burstiness of service times upon the typical quality of service (QoS) measure of the mean queue length (MQL). Moreover, a comprehensive experimental study is carried out to investigate numerically the relative impact of count and interval traffic correlations as well as other traffic characteristics upon the performance of stable BRAP/GE/1 and BMAP/GE/1 queues. It is suggested via a conjecture that the BRAP/GE/1 queue is likely to yield pessimistic performance metrics in comparison to those of the stable BMAP/GE/1 queues under the worst case scenario (i.e., a worst case scenario) of the same positive count and interval traffic correlations arising from long sojourn in each phase.

"A distributed queueing Medium Access Control (MAC) protocol is used in Distributed Queue Dual Bus (DQDB) networks. A modified version of the MAC protocol was proposed by R.R. Pillai and U. Mukherji in an attempt to overcome some of the shortcomings of the DQDB MAC protocol. They analyzed the performance of the system for Bernoulli arrivals and for large propagation delays between the nodes. We extend the performance analysis of the modified MAC protocol for a DQDB type of Network. The parameter of interest to us is the bus access delay. This has two components, viz., the request bus access delay and the data bu6 access delay. We use the model at the request point at node and present methods to evaluate the delay experienced in such a model. The model is an n-priority ./D/l queue with D vacations (non-preemptive priority) where n is the number of nodes sending requests on the request bus for transmission on the data bus. The methods presented help to evaluate the request bus access delay when the arrivals at each node are Markovian Arrival Processes (MAPs). The algorithms for evaluating the mean request bus access delay are based on matrix geometric techniques. Thus, one can use the algorithms developed in the literature to solve for the finite buffers case too. This model, for the request bus access delay, holds irrespective of the propagation delay between the nodes. We also evaluate the inter-departure time of class 1 customers and virtual customers in a 2-priority M/G/l system with G vacations (non-preemptive priority). In the case of Poisson arrivals at all the nodes, we would have a 2-priority M/D/l system with D vacations (non-preemptive priority). We thus evaluate the inter-arrival time of the free slots on the data bus as seen by Node 2. Note that this is independent of the number of active nodes in the network We then develop methods to evaluate the mean data bus access delay experienced by the customers at Node 2 in a three-node network with 2 nodes communicating with the third when the propagation delay between the nodes is large. We consider the case of finite Local Queue buffers at the two nodes. Using this assumption we arrive at process of arrivals to the Combined Queue and the process of free slots on the data bus to be Markov Modulated Bernoulli processes. The model at the combined queue at Node 2 then has a Quasi Birth-Death evolution. Thus, this system is solved by using the Ramaswami-Latouche algorithm. The stationary probabilities are then used to evaluate the mean data bus access delay experienced at Node 2. The finite buffer case of this system can be solved by G.Wi Stewart's algorithm. The method in modelling the system and the results are presented in detail for Poisson arrivals. The extension of this to more complex processes is also explained. We encounter in the analysis an explosion of the state-space of the system. We try to counter this by considering approximations to the process of free slots on the data bus. The approximations considered are on the basis of what are known as Idealized Aggregates. The performance of the approximation is also detailed. It works very well under low and moderate load but underestimates the mean delay under heavy load. Thereafter, we discuss the performance of the system with reference to the mean of the access delay and the standard deviation of the access delay under varying traffic at the two nodes. For this part we use simulation results to discuss the performance. The comparison between the performance measures at both the nodes is also done. Then we develop methods/techniques to understand the performance of the system when we have finite propagation delays between the nodes. We concentrate on the 3-node problem and calculate performance bounds based on linear programs. This is illustrated in detail for Bernoulli arrivals for the case of 1 slot propagation delay between the nodes as well as for the case of 2 slots propagation delay. The performance of the bounds obtained is also detailed. The presence of an idling system at the combined queue of Node 2 makes the bounds somewhat loose. Finally, we discuss the performance of the system with reference to the mean access delay and the standard deviation of the access delay under varying load on the system. Again, we rely on simulation studies. Finally, we study the performance of the system as a multiplexer. For this, we re­strict the traffic to Markov Modulated Processes (or those which would satisfy the Gartner-Ellis Theorem requirements). The traffic is characterized by what are known as Envelope Processes - Lower and Upper. The class of processes which satisfy the conditions of the Gartner-Ellis theorem come under the category where both the Envelope Processes exist and the Minimum Envelope Rate and the Maximum Lower Envelope Rate are the same. We use the system evolution equations at the combined queue at any node to develop re­lations between the various input and output processes. First, this is done for a. system of this kind, in isolation. Then, we consider this system as a part of the modified protocol and present relations, among the various input and output processes, which are specific to the modified protocol. The possible use of all of the above to do Admission Control at the entry point to the Asynchronous Transfer Mode (ATM) network is also presented.