Abstract

A new queue, referred to here as the HetSigma queue, in the Markovian framework, is proposed in order to model nodes in modern telecommunication networks. The queue has many of the necessary ingredients, such as joint (or individual) Markov modulation of the arrival and service processes, superposition of K CPP (compound Poisson process) streams of (positive) customer arrivals, and a CPP of negative customer arrival stream in each of the modulating phases, a multiserver with c non-identical (can also be identical) servers, GE (generalised exponential) service times in each of the modulating phases and a buffer of finite or infinite capacity. Thus, the model can accommodate correlations of the inter-arrival times (of batches), and geometric as well as non-geometric batch size distributions of customers in both arrivals and services. The use of negative customers can facilitate modelling server failures, packet losses, load balancing, channel impairment in wireless networks, and in many other applications. An exact and computationally efficient solution of this new queue for steady state probabilities and performance measures is developed and presented.

A non-trivial application of the new queue to the performance evaluation of a wireless communication system is presented, along with numerical results, to illustrate the efficacy of the proposed method. The use of negative customers is also demonstrated. The new queue, perhaps with further evolution, has the potential to emerge as a generalised Markovian node model.

Keywords: MMCPP; Compound Poisson process; Generalised exponential; Unbounded batch-arrivals; Unbounded batch-services; Generalised Markovian node model; Spectral expansion; Mathematical transformations; Performance evaluation; Wireless networks

Article Outline

1. Introduction

2. Model description

2.1. Modulation

2.2. Customer arrival process

2.3. Arrival batch size distribution

2.4. The GE multi-server and the queueing capacity

2.5. Negative customer semantics

2.6. Condition for stability

3. The steady state balance equations

4. Transforming the balance equations

5. Spectral expansion solution of the balance equations

5.1. System with infinite queueing capacity

5.2. The case of c identical servers

5.3. No customer switching between servers

6. Application, numerical study and validation

6.1. Wireless communication scenario

6.2. Performance model

7. Conclusions

Acknowledgements

Appendix A. Obtaining the Q_l matrices

Computation

Appendix B. Automatic validation of equations

References

Vitae