A Multi-Type Queueing Inventory System—A Model for Selection and Allocation of Spectra

Thulaseedharan Salini Sinu Lal,Dmitry Kozyrev,Achyutha Krishnamoorthy,Vladimir Vishnevsky,Varghese Chaukayil Joshua

doi:10.3390/math10050714

Abstract

The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network. The data packets (customers), belonging to different classes, arrive according to a batch marked the Markovian arrival process (BMMAP). The inventory considered is of multi-type (different types of channels becoming available) and are generated according to a marked Markovian arrival process (MMAP). The number of distinct types of inventory and that of the customers are the same. Arriving customers are allowed to wait in finite buffers of each category which are reserved for distinct classes of customers except for the most general class, which is provided with an infinite waiting space. The number of servers also equals the number of distinct types of inventory. When items of a particular type arrive in the inventory, the service starts, providing the buffer of customers of the corresponding class is non-empty. The service can be viewed as a selection process with Coxian distributed service times. The system is analyzed using the matrix analytic method and performance measures are obtained. The model is illustrated with suitable numerical examples.

Highlights

We propose a model for a multiclass queueing inventory system into which customers belonging to different classes and items of different types arrive
We assume that customers belonging to each class arrive in batches according to a batch marked the Markovian arrival process (BMMAP)
The model discussed in this paper provides an efficient mechanism for the selection and allocation of available limited spectra for transmission of heterogeneous data in a network

Summary

Introduction

We propose a model for a multiclass queueing inventory system into which customers belonging to different classes and items of different types arrive. Chakravarthy, and Krishnamoorthy [2] studied a single server queueing-inventory system with two commodities in which the system has finite storage space for each type. This model is analyzed using the matrix analytic method. Meng and Heragu [15] investigate a batch size modeling of a multi-item, discrete manufacturing system via an open queuing network These authors do not consider positive service time to each customer before the inventory is provided to them.

Mathematical Model

Stability Condition

Stationary Distribution

Profit Function

Numerical Illustration

Conclusions