Abstract

In this paper, we study a Markovian queuing system consisting of two subsystems of an arbitrary structure. Each subsystem generates a multi-class Markovian arrival process of customers arriving to the other subsystem. We derive the necessary and sufficient conditions for the stationary distribution to be of product form and consider some particular cases of the subsystem interaction for which these conditions can be easily verified.

Highlights

  • The product form of the stationary distribution greatly simplifies the analysis of complex queueing systems

  • Naumov [21] obtained the necessary and sufficient conditions for a product-form solution for a Markov network consisting of two nodes, the first of which generates a Markovian arrival process (MAP) of customers arriving to the second node

  • Theorem 1 states that if the multi-class Markovian arrival processes (MMAP) arriving to each network node is replaced by a multi-class Poisson arrival process, the generators of the Markov processes representing each node in isolation are irreducible

Read more

Summary

Introduction

The product form of the stationary distribution greatly simplifies the analysis of complex queueing systems. These models, which include G-networks with signals [15], resets [16], and multiple customer classes [17], radically extend the class of Markovian queueing systems with the product-form stationary distributions The complexity of such systems continues to increase, with the introduction of new extensions of G-networks being an essential area of research [18,19]. Naumov [21] obtained the necessary and sufficient conditions for a product-form solution for a Markov network consisting of two nodes, the first of which generates a Markovian arrival process (MAP) of customers arriving to the second node.

Multi-Class Markovian Arrival Process
Interacting Markovian Queueing Systems
Examples
Conclusions

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.