Matrix Queueing Theory

Traditional queueing theory deals mainly with one-dimensional stochastic processes like queue or orbit length, virtual or actual waiting or sojourn time, busy period, etc. Discrete or continuous time one-dimensional Markov chains and renewal theory are the main tools for their investigation. However, the increasing complexity of the modern telecommunication networks (heterogeneous bursty correlated flows of information with the different requirements to Quality of Service, possibility of dynamic resources distribution, etc.) gave raise to investigating the queues whose behavior can be modelled in terms of multi-dimensional Markov chains. The important particular case of such Markov chains assumes that one component of the chain is denumerable (often this component represents queue or orbit length) while the other components are finite (the state of some external random environment, the states of processes which control the rates of input, service, breakdowns, disasters, vacations, searches, etc.). Generally speaking, such a multi-dimensional Markov chain can be easily reduced to the usual one-dimensional Markov chain (e.g., just by enumeration the states in lexicographic order). However, following this way we lose the special structure (if it exists) of the one-step transition probability matrix and consequently lose the chance to investigate the chain analytically. The alternative way for investigating the multi-dimensional Markov chain consists of dividing the state space of the chain to the infinite set of so called levels (each level corresponds to a fixed state of the denumerable component) with the finite number of states belonging to the level. In such a way, the stationary probabilities of the states belonging to the same level constitute a finite state vector while the transition probability matrix consists of matrices defining transitions between the levels together with transition of finite components. So, the equilibrium equations for original Markov chain can be rewritten in the form of system of linear equations for the stationary probability vectors. If the structure of this system is appropriate for deriving some analytic results, these results have a form which is a rather transparent matrix form of results that are valid for the corresponding queueing system with one-dimensional state space. This way is related with the name of Marcel Neuts. In his book [12], he introduced into consideration and comprehensively investigated a special kind of two-dimensional continuous-time Markov chains (so called generalized Birth-and-Death processes and GI/M/1 type Markov chains). It allowed to get the stationary state distribution for a wide class of Markovian queues in the …