We describe a family of processes which are interesting generalizations of QBD processes. We first discuss a simple example and then consider the general case. These processes were introduced by Yeung and Sengupta [119] and Takine, Sengupta, and Yeung [114], who consider a more general class of processes than we do here. Our simplified presentation is intended to highlight the essential elements in their structure. The M/PH/1 LIFO QueueIn order to follow more easily the discussion in this chapter, it is helpful to have in mind a simple example of a process with a tree structure. One such example is provided by the M/PH/1 queue with LIFO service discipline. Thus, we consider a single server queue with an infinite buffer where customers are piled in a stack. Each newly arriving customer joins the top of the stack, preempting the customer being served, if there is one. When an interrupted customer is again on top of the stack, its service is resumed and proceeds from where it was interrupted. Arrivals form a Poisson process, and the service time distribution is PH with representation (r, T) of order m.Where customers are present, we represent the system state by ktuples of phase indices: the state (i l , i2i ... , ik ) with 1 < il , ... , ik < m indicates that there are k customers in the system; the customer at the top of the stack is active and currently executing the phase ik ;

The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution contains functional iterations, doubling methods, logarithmic reduction, cyclic reduction, and subspace iteration, all are described and analysed in detail. They are also adapted to interesting specific queueing models coming from applications. The book also offers a comprehensive and self-contained treatment of the structured matrix tools which are at the basis of the fastest algorithmic techniques for structured Markov chains. Results about Toeplitz matrices, displacement operators, and Wiener-Hopf factorizations are reported to the extent that they are useful for the numerical treatment of Markov chains. Every and all solution methods are reported in detailed algorithmic form so that they can be coded in a high-level language with minimum effort.

Quasi-birth-death processes are commonly used Markov chain models in queueing theory, computer performance, teletraffic modeling and other areas. We provide a new, simple algorithm for the matrix-geometric rate matrix. We demonstrate that it has quadratic convergence. We show theoretically and through numerical examples that it converges very fast and provides extremely accurate results even for almost unstable models.

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.

This paper presents the Laplace transform of the time until ruin for a fairly general risk model. The model includes both the classical and most Sparre-Andersen risk models with phase-distributed claim amounts as special cases. It also allows for correlated arrival processes, and claim sizes that depend upon environmental factors such as periods of contagion. The paper exploits the relationship between the surplus process and fluid queues, where a number of recent developments have provided the basis for our analysis.