Abstract
A single-server non-pre-emptive priority queueing system of a finite capacity with many types of customers is analyzed. Inter-arrival times can be correlated and batch arrivals are allowed. Possible unreliability of the server, implying the loss of a customer or the necessity of its service from the early beginning or some phase of the service, is taken into account. Initial priorities provided to various types of customers at the arrival moment can be varied (increased or decreased) after the random amount of time during the customer stay in the buffer. Such a type of queues arises in the modeling operation of various emergency care systems, information, and perishable goods delivering systems, etc. The stationary behavior of the system is described by the finite state multi-dimensional continuous-time Markov chain with the upper-Hessenberg block structure of the generator. The stationary distribution of the system states and some important characteristics of the system are calculated. The presented numerical examples illustrate opportunities to quantitatively evaluate the impact of the buffer capacity and customers’ mean arrival rate on the most important characteristics of the system. The possibility of solving optimization problems is briefly shown.
Highlights
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Customers arrival in the batch marked Markov arrival process (BMMAP) is defined by the irreducible continuous time Markov chain νt, t ≥ 0, having the finite state space {1, . . . , W }
The behavior of the system under study can be described by the regular irreducible continuous-time Markov chain
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The existing papers are classified according to the number of priority classes, arrival process, distribution of the service time, and time until the change of the priority and the obtained results. The model considered in paper [1] is the most general with respect to the pattern of the arrival process and the distribution of the service time and time until the change of the priority. After the exponentially distributed interval of time during the stay in the buffer, the established priority of a customer can increase.
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