Infinite-Server Resource Queueing Systems with Different Types of Markov-Modulated Poisson Process and Renewal Arrivals

Abstract

In this paper, we propose models that significantly expand the scope of practical applications, namely, queueing systems with various nodes for processing heterogeneous data that require arbitrary resource capacities for their service. When a customer arrives in the system, the customer typeis randomly selected according to a set of probabilities. Then the customer goes to the server of the corresponding device type, where its service is performed during a random time period with a distribution function depending on the type of customer. Moreover, each customer requires a random amount of resources, of which the distribution function also depends on the customer type, but is independent of its service time. The aim of this research was to develop a heterogeneous queueing resource system with an unlimited number of servers and an arrival process in the form of a Markov-modulated Poisson process or stationary renewal process, and with requests for a random number of heterogeneous resources. We have performed analysis under conditions of growing intensity of the arrival process. Here we formulate the theorems and prove that under high-load conditions, the joint asymptotic probability distribution of the n-dimensional process of the total amounts of the occupied resources in the system is a multidimensional Gaussian distribution with parameters that are dependent on the type of arrival process. As a result of numerical and simulation experiments, conclusions are drawn on the limits of the applicability of the obtained asymptotic results. The dependence of the convergence of experimental results on the type of distribution of the system parameters (including the distributions of the service time and of the customer capacity) are also studied. The results of the approximations may be applied to estimating the optimal total number of resources for a system with a limited amount of resources.

Application of Mass Service Theory to Economic Systems Optimization Problems—A Review

“…On the other hand, controversial points have been identified. For example, there is a contradiction in the approaches of some authors: E. Pankratova et al propose a mass service model with an unlimited number of servers [17], and Hao and Yifei are based on the idea that it is economically inexpedient to open a large number of servers [21]; in optimization issues, scientists rely on the development of alternative hypotheses and models [22], which is not always able to reflect the single best solution; and in the context of forecasting, a number of alternative prediction methods should be applied rather than limiting to comparing only two different models, as was conducted in May et al [41].…”

“…Scientists have revealed the following regularity: the optimal number of server resources has the highest value for the first case and the lowest for the third model [16]. In the example of heterogeneous resource-based mass service systems, Pankratova et al offer a method for determining the optimal level of resource provision for the k-th channel of the system [17]:…”

Ergodic Characteristics of Payment Points under the Slow Second Cashier Problem

“…Suppose that one chooses the departure epochs to study the customer process in view of the Poisson arrivals assumption employed and upon departures from both cashiers in view of varying work rates, then the PASTA principle holds good. By the arguments in [21][22], the customer process described above forms an irreducible and aperiodic Markov chain as time grows large. Again, the time process becomes a stationary process as time grows large [23][24].…”

Stalling in Queuing Systems with Heterogeneous Channels

“…Efrosinin and Sctrik [20,21] showed that, in an M/M/K queue, the optimal threshold levels may also depend on the states of slower servers; however, this influence is negligible. In [22], a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue were investigated. The authors of [23] provided an analysis of the dependence of the convergence of experimental results on the type of distribution used by the system parameters in heterogeneous queueing resource system with an unlimited number.…”