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 ;

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.

Markovian fluid flow models are used extensively in performance analysis of communication networks. They are also instances of Markov reward models that find applications in several areas like storage theory, insurance risk and financial models, and inventory control. This paper deals with the transient (time dependent) analysis of such models. Given a Markovian fluid flow, we construct on the same probability space a sequence of queues that are stochastically coupled to the fluid flow in the sense that at certain selected random epochs, the distribution of the fluid level and the phase (the state of the modulating Markov chain) is identical to that of the work in the queue and the phase. The fluid flow is realized as a stochastic process limit of the processes of work in the system for the queues, and the latter are analyzed using the matrix-geometric method. These in turn provide the needed characterization of transient results for the fluid model. ORDER REPRINTS Ahn and RamaswamiAt a high-level, this article is but an implementation of the above simple ideas. As noted, the details of the construction of the needed queues is given in Sec. 3. In Sec. 4, we establish stochastic coupling at the spatial uniformization epochs and also show that each queue is of the QBD type; see Theorem 3 and Theorem 5. In Sec. 5, Theorem 8 we establish the stochastic process limit result that forms the basis of our approach. To let the main ideas flow uninterrupted, the technical details of its proof are moved to a later section, viz., Sec. 9.Lest the reader should miss a major subtlety of our construction, we note that to compare the sample path increments of the spatial discretizations, it is necessary to make their construction such that all of them are modulated by a common phase process and are nested. Our construction in Sec. 3 satisfies this important condition, and that is exploited many times, and particularly so to establish the stronger (pathwise) limit theorems.In Sec. 6, we develop the transient analysis of the queues using matrix-geometric techniques. These are then used in Sec. 7 to obtain the transient results for the fluid flow model through a limit process. Specifically, Sec. 7 introduces three fundamental kernels for the fluid model that hold the key to its transient analysis. In Sec. 8, numerical computations are performed for a set of examples, and the results are compared to those of Sericola. 14 Finally, in Sec. 10, we provide some concluding remarks.

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We discuss a single-server queue whose input is the versatile Markovian point process recently introduced by Neuts [22] herein to be called the N-process. Special cases of the N-process discussed earlier in the literature include a number of complex models such as the Markov-modulated Poisson process, the superposition of a Poisson process and a phase-type renewal process, etc. This queueing model has great appeal in its applicability to real world situations especially such as those involving inhibition or stimulation of arrivals by certain renewals. The paper presents formulas in forms which are computationally tractable and provides a unified treatment of many models which were discussed earlier by several authors and which turn out to be special cases. Among the topics discussed are busy-period characteristics, queue-length distributions, moments of the queue length and virtual waiting time. We draw particular attention to our generalization of the Pollaczek–Khinchin formula for the Laplace–Stieltjes transform of the virtual waiting time of the M/G/1 queue to the present model and the resulting Volterra system of integral equations. The analysis presented here serves as an example of the power of Markov renewal theory.