Abstract

Tools for performance evaluation often require techniques to match moments to continuous distributions or moments and correlation data to correlated processes. With respect to efficiency in applications, one is interested in low-dimensional (matrix) representations. For phase-type distributions (or matrix exponentials) of second order, analytic bounds could be derived earlier, which specify the space of permissible moments. In this paper, we add a correlation parameter to the first three moments of the marginal distribution to construct a Markovian arrival process of second order (MAP(2)). Exploiting the equivalence of correlated matrix-exponential sequences and MAPs in two dimensions, we present an algorithm that decides whether the correlation parameter is permissible with respect to the three moments and – if so – delivers a valid MAP(2) which matches the four parameters.

We also investigate the restrictions imposed on the correlation structure of MAP(2)s with hyperexponential marginals. Analytic bounds for the envelope correlation region (i.e., for arbitrary third moment) and for the specific correlation region (i.e., for fixed third moment) are given.

When there is no need for a MAP(2) representation (as in linear-algebraic queueing theory), the proposed procedure serves to check the validity of the constructed correlated matrix-exponential sequence.

Numerical examples indicate how these results can be used to efficiently decompose queueing networks.

Keywords: Markovian arrival processes; Second-order processes; Moment/correlation fitting; Correlation bounds; Queueing network decomposition

Article Outline

1. Introduction

2. Notation and preliminaries

2.1. The canonical form for correlated ME(2) sequences

3. The conversion algorithm

3.1. Uncorrelated processes: γ=0

3.2. Hyperexponential marginals and γ>0

3.3. The general situation γ≠0

3.3.1. Conversion algorithm

4. Experimental bounds of γ for Marie’s distribution

5. Analytic bounds for envelope and specific correlation regions

5.1. Bounds for the envelope correlation region

5.2. Specific correlation bounds for MAP(2)s with hyperexponential marginals

6. Application to decomposition of queueing networks

6.0.1. Mapping procedure

6.1. Dual tandem queue with MMPP(2) input

6.2. Dual tandem queue with MAP(2) input

6.3. Discussion

7. Conclusions

Acknowledgements

Appendix A. Appendix

References

Vitae