Fluid Models Driven By An M/M/1Vacation Queue

In traditional queueing theory the object of interest is usually a system of one or moreservers at which customers arrive, who want to receive some kind of service. Indeed, thecommon feature of this class of models is that individual customers arrive to receive somekind of service and leave after the service, and each customer arrival and customer departureare modeled as discrete events, reflecting the discrete nature of the modeled phenomena.With the development of the times and the emergence of high and new technology, the statespace of the existing discrete system is more and more huge. The various properties of thediscrete system gradually like that of the corresponding continuous one, which is so-calledfluid limit. The traditional discrete queue system becomes more and more intractable andinconvenient. Therefore, from the1950s onwards, people began to consider the fluid model.Based on the previous literatures on the subject of fluid queues, we introduce the va-cation policy(including work vacation policy) into the driven queueing system. The fluidmodels driven by various vacation M/M/1queue with certain fluid net input rate structureare studied. The analytic expression of joint distribution function (or its Laplace transform)of the fluid model is derived. Then the mean buffer content is also obtained. By means ofa large number of numerical examples, the mutative trend of the performance measures ofthe system is demonstrated, which verify the reasonability and accuracy of the conclusionsin the paper.Firstly, the fluid model driven by the M/M/1/N queue with multiple and single vacationis analyzed, where the fluid net input rate is controlled by the state of the server in the drivensystem. The finite dimensional matrix differential equation satisfied by the stationary jointdistribution function of the fluid model (i.e.3-dimensional Markov process) is constructed.Using a key lemma, the analytic solution is derived by means of spectral analysis method.Finally, the distribution function of the stable buffer content and the mean buffer content areobtained.Secondly, the fluid model driven by the classical M/M/1queue is discussed, where thenet input rate is regulated by the number of customers in the driven system. The model isproved to be a two-dimensional Markov process with mixed state space. The infinite di- mensional differential simultaneous equations satisfied by the stationary joint distributionfunction are constructed. Just as the stationary queue length follows the geometric distrib-ution, the Laplace transform of the joint distribution is of similar power function structure.Thus the geometric solution method is created, which can be used to deal with a class offluid model driven by birth-death processes. Based on this fact, the Laplace-Stieltjes trans-form(LST) and the concise formula of the mean buffer content are derived.Subsequently, as a generalization of the model in chapter3, fluid models driven byan M/M/1vacation queue are further considered. Two kind of vacation policy—multiplevacation policy and multiple vacation and N-policy are introduced into the M/M/1queuedriven system. The geometric solution method adopted in chapter3is generalized into highdimensional matrix geometric solution method. As for the model with multiple vacations,the net input rate is assumed to be regulated by the state of the driven system. Just as theLaplace transform of the stationary joint distribution function in M/M/1queue driven modelis of power function form, the Laplace transform is proved to be of certain matrix powerfunction form. While as for the model with multiple vacations and N-policy, the net inputrate is supposed to be regulated by the number of customers in the driven system. TheLaplace transform is shown to be of similar matrix power function form or matrix factorialform. Finally, the Laplace-Stieltjes transform of the steady-state distribution of the buffercontent in the above two fluid models are concisely expressed, as well as the performancemeasure—mean buffer content.Finally, using the matrix geometric solution method, the fluid model driven by anM/M/1queue with multiple work vacations is further studied. Given that the net input rateis controlled by the state of the driven system, the Laplace transform of the steady-state jointdistribution of the fluid queue is also proved to be of matrix geometric structure. Based onthis fact, the Laplace-Stieltjes transform of the steady-state buffer content distribution andthe mean buffer content are derived. Specially, two special cases are considered.(1) Whenthe vacation time is zero, the model is degenerated into the fluid model driven by an classi-cal M/M/1queue.(2) When the service rate during the period of the work vacation in thedriven system is zero, the model is reduced into the fluid driven by the M/M/1queue withmultiple general vacations. The conclusions in the two special models is the same with the previous ones.In a word, this paper studies the fluid model driven by an M/M/1queue with variousvacations(including work vacations). The analytic expression of joint distribution function(or its Laplace transform) of the fluid model is derived. Then the performance measure suchas the mean buffer content is also obtained. By introducing various vacation strategies tothe fluid model, we can provide greater variability and flexibility for the design and controlof input rate and output rate. Thus the fluid model can be adapted to the wider applicationbackground. In addition, comprehensive application of various methods such as spectralanalytic method, geometric solution method and matrix geometric solution method, canenrich the method of fluid model. We provide a unified theory frame and uniform methodto analyze the fluid model driven by various M/M/1vacation queue, enriching the researchresults in the field of fluid queue.