Probability, Markov chains, queues, and simulation

This book, with its four parts, represents a valuable reference to probability, Markov Chains, queuing systems and computer simulation. The first part begins with some useful concepts in probability, and discusses the elements of probability space and conditional probability. In addition, independence, the law of total probability and Bayes’ theorem are investigated. Principles of counting, permutations and combinations are also outlined. After that, the book describes discrete or continuous random variables, giving their statistical measures and many common distributions. Further, the common inequalities used in probability are stated. The second part discusses classes of discrete stochastic processes in both discreteand continuous-time. It gives various statistical characteristics for Markov chains (e.g. state classifications, probability distributions, ergodicity and reversibility), with some types of such chains and processes such as embedded Markov chains, random walks, semi-Markov processes and renewal processes. Computation of stationary and transient distributions of Markov chains is also investigated, with algorithms and software code, based on some computational methods that adapted the nature of Markov chains. The third part introduces several queuing models under various situations, such as the number of servers, state dependent and truncation, including the birth–death process for Markovian queues and measures of effectiveness of such models. Queuing systems with single server and phasetype distributions given in the first part are discussed for both arrival and service processes via the Neuts’ matrix-geometric approach. Markovian single server queues are also analyzed by the z-transform technique. Moreover, general arrival and service queues denoted by M/G/1 and G/M/1 are treated by the embedded Markov chain approach, involving some features such as residual service time, busy period and priority scheduling. This part is ended by treatment of some queuing networks. The final part deals with the simulation of uniformly and non-uniformly distributed random numbers, writing program simulation codes and the accuracy of simulation. The book is closed by an appendix that reviews the linear algebra used in the book. The book material is invaluable and presented with clarity. Chapters of the book are covered by various examples and exercises. A minor criticism of the book comes in not stating a hint about Markovian queues with features as balking and reneging. The book represents a valuable text for courses in statistics and stochastic processes, so it is strongly recommended to libraries.