Theory And Applications Of T-QBD Process

It is well known that the quasi-birth-and-death process (abbreviated as QBD pro-cess) plays an important role in analysis of complicated stochastic models. For QBDprocess with finite phases, the systematical method is now available; for situationswith countably phases, it is still a challenging problem even though many researchworks have been done in various aspects. For example, the general method to compu-tation the exact stationary distribution remains unsolved.In the analysis of practical models, one important special QBD process withcountable phase is very common, that is, in the tri-diagonal infinitesimal generator,each sub-block is also a tri-diagonal matrix, we call such QBD process T-QBD pro-cess. In this dissertation, we will study T-QBD process and some stochastic modelsthat can be described using this special process. We bring forward the mathemati-cal description of T-QBD process starting from some practical queue models, someof them are level dependent, some are level independent, some with simple bound-aries, and some with complicated boundaries, these processes with various forms willbecome the main research object in this paper.In this paper, the author mainly studies the following problems:First, for the level independent T-QBD process, we discussed the tail characteris-tic of stationary distribution along level direction. After making further discussion onone of the tail analysis method in literature, we analyze two practical queue models,which are T-SPH/M/1 and M/T-SPH/1 queue(for the definition of T-SPH distribution,one can see Definition 2.4.2). The result indicates that under certain conditions, theirjoint stationary distributions have the characteristic of geometric decay in level direc-tion. Also, the result provides theoretical foundations for further analysis of these twomodels.Second, based on tail analysis of T-SPH/M/1 queue and M/T-SPH/1 queue, fur-ther analysis is made and the forms of rate operator of each model are given, so thatoperator-geometric solution of each model is reached. For the first model, we canget the joint stationary distribution, stationary length distribution and two marginaldistributions in close form; For the second model, we give the numerical solution of joint stationary solution based on the operator-geometric solution. Furthermore, forthis model, we can get the analytical solution of first several component of the jointstationary distribution using other methods. Though it is very difficult to get solutionof other component, it is possible to predict its'form.Third, we study the level-phase independence problem of joint stationary distri-bution of T-QBD process, and derive the necessary and sufficient conditions for leveland phase independence and both of the level and phase with geometric distributions.Taking two practical models as example, the practical method to verify these condi-tions is given. At the same time, one generalized independence is discussed, and onesufficient condition for independence problems of level and phase for general Markovprocess is given. Also, while the independence problem is discussed, we find an in-teresting fact of operator-geometric solution, that is, there exits some matrices whichhas properties as rate operator, for example, it fulfill the operator-geometric solutionbut is not rate operator, and the number of such matrix could be infinite.In the end, we bring forward the efficient algorithm to calculate the numericalsolution for joint stationary distribution of T-QBD process, which is called block rect-angle iterative(BRI) algorithm. This algorithm is applicable for all types of T-QBDprocess mentioned above, also, it is possible for it to be extended to higher dimen-sional and more generalized process, such as GI/M/1 type process. We studied theconvergence and computing complexity of this algorithm, furthermore, using a con-crete model, we give out the experience formula of terminating iteration times, thoughthe exact stopping condition under given error is not given. Based on numerical re-sults, by analyzing several practical models, we not only verify some conclusion inthe literature, but also find some new properties for such models. Therefore, the ef-fectiveness and applicability of our algorithm is proved.In the final part of this paper, we make conclusion to the whole thesis, and pointout some problems that are important in practical application and research but are notsolved in this paper, so that we can know what we will do in the future.