Search Results (4)

An $MAP/PH/N$-type queuing system functioning within a finite-state Markovian random environment is studied. The random environment’s state impacts the number of available servers, the underlying processes of customer arrivals and service, and the impatience rate of customers. The impact on the state space of the underlying processes of customer arrivals and of the more general, as compared to exponential, service time distribution defines the novelty of the model. The behavior of the system is described by a multidimensional Markov chain that belongs to the classes of the level-independent quasi-birth-and-death processes or asymptotically quasi-Toeplitz Markov chains, depending on whether or not the customers are absolutely patient in all states of the random environment or are impatient in at least one state of the random environment. Using the tools of the corresponding processes or chains, a stationary analysis of the system is implemented. In particular, it is shown that the system is always ergodic if customers are impatient in at least one state of the random environment. Expressions for the computation of the basic performance measures of the system are presented. Examples of their computation for the system with three states of the random environment are presented as 3-D surfaces. The results can be useful for the analysis of a variety of real-world systems with parameters that may randomly change during system operation. In particular, they can be used for optimally matching the number of active servers and the bandwidth used by the transmission channels to the current rate of arrivals, and vice versa. Full article

This research examines a multi-server retrial queueing system with a batch Markov arrival process and a phase-type service time distribution. The system’s distinguishing feature is its ability to control the admission of retrial customers. An attempt by a customer to retry is successful only if the number of busy servers does not exceed certain threshold values, which may depend on the state of the fundamental process of the primary customer’s arrival. Impatient retrying customers may abandon the system without obtaining service. A group of primary customers that arrives while the number of available servers is fewer than the group size is either entirely rejected or occupies all available servers, while the remainder of the group transitions to the orbit. The system’s behavior, under a defined set of thresholds, is characterized by a multidimensional Markov chain classified as asymptotically quasi-Toeplitz. This enables the acquisition of the ergodicity condition and the computation of the steady-state distribution of the Markov chain and the system’s performance measures. The presented numerical examples demonstrate the impact of threshold value variation. An example of solving an optimization problem is presented. The importance of the account of the batch arrivals is shown. Full article

The operation of many queueing systems is adequately described by the structured multidimensional continuous-time Markov chains. The most well-studied classes of such chains are level-independent Quasi-Birth-and-Death processes, $GI/M/1$ type and $M/G/1$ type Markov chains, generators of which have the block tri-diagonal, lower- and upper-Hessenberg structure, respectively. All these classes assume that the matrices of transition rates are quasi-Toeplitz. This property greatly simplifies their analysis but makes them inappropriate for the study of many important systems, e.g., retrial queues with a retrial rate depending on the number of customers in orbit, queues with impatient customers, etc. The importance of such systems attracts significant interest to their analysis. However, in the literature, there is a methodological gap relating to the ergodicity condition of the corresponding Markov chains. To fulfill this gap and facilitate the analysis of a wide range of such systems, we show that under non-restrictive assumptions, the following hold true: (i) if the customers can balk or are impatient or non-persistent, then the Markov chain describing the behavior of the system belongs to the class of asymptotically quasi-Toeplitz Markov chains; (ii) this chain is ergodic; (iii) known algorithms can be applied for the calculation of the stationary distribution of the corresponding queueing system. Full article

In this paper, we analyze a multi-server queueing system with a marked Markov arrival process of two types of customers and a phase-type distribution of service time depending on the type of customer. Customers of both types are assumed to be impatient and renege from the buffers after an exponentially distributed number of times. The strategy of flexible provisioning of priorities is analyzed. It assumes a randomized choice of the customers from the buffers, with probabilities dependent on the relation between the number of customers in a priority finite buffer and the fixed threshold value. To simplify the construction of the underlying Markov chain and the derivation of the explicit form of its generator, we use the so-called generalized phase-type distribution. It is shown that the created Markov chain fits the category of asymptotically quasi-Toeplitz Markov chains. Using this fact, we show that the considered Markov chain is ergodic for any value of the system parameters and compute its stationary distribution. Expressions for key performance measures are presented. Numerical results that show how the parameters of the control strategy affect the system’s performance measurements are given. It is shown that the results can be used for managerial purposes and that it is crucial to take correlation in the arrival process into account. Full article