Queueing Systems Using Roots with Applications

Dear Colleagues,

Until the mid-1960s, a queueing or a related problem was considered solved if the solution was given in some form of a generating function or Laplace transformation. This is because the inversion was either considered not necessary or a trivial problem. However, the inversion of transforms arising in queueing and other stochastic models (except for simple cases) was not as easy as was once thought, and hence the results could not be used to solve practical problems. Many users expressed concerns that such solutions were inadequate. Well-known mathematicians such as Kendall and Kleinrock have made contributions [1–2]. Kendall (1964) made a famous remark that queueing theory wears the Laplacian curtain. Kleinrock (1975, p. 291) states “one of the most difficult parts of this method of spectrum factorization is to solve for the roots.” Neuts (see Stidham (2001) [3]) states that “In discussing matrix-analytic solutions, I had pointed out that when the Rouché roots coincide or are close together, the method of roots could be numerically inaccurate. When I finally got copies of Crommelin’s papers, I was elated to read that, for the same reasons as I, he was concerned about the clustering of roots. In 1932, Crommelin knew; in 1980, many of my colleagues did not ....”. In this connection, see also Neuts’ book (1981, pp.27-30 [4]). After this (even though the root-finding algorithm and QROOT developed by Chaudhry (see below) were available), several other researchers made similar remarks. The quotes of a few recent cases are as follows. In 2005, Mejia-Téllez made the following statement, “If the batch size is large, the determination of these roots is difficult….” [5]. In a recent paper, while analyzing a bulk-service queue, Bar-Lev et al. (2007) stated that “This general solution requires the calculation of the zeros of A^(M)(z) - z^M which in practice can result in numerical inaccuracies especially when the decision variable M assumes a large value….” [6]. Furthermore, Maity and Gupta (2015) [7] have stated several difficulties in obtaining results, even when using the spectral theory approach, an approach which may be simpler than the matrix-geometric approach used in several papers by Chakka, that is a more involved method of roots. For detailed comments, see also Chakka’s (1995) thesis [8].

At the early stages of the work (during the 1980s), Professor Chaudhry started where most other researchers have stopped, i.e., he concentrated on inverting transforms and thus lifting the Laplacian curtain. One of the methods, which has turned out to be a powerful one, involves the use of roots (even when the number involved is large) of transcendental or high-degree polynomial equations. When software packages such as MAPLE and MATHEMATICA could not find a large number of roots (they do now, probably built on Chaudhry’s algorithm), Professor Chaudhry developed and used a program called QROOT (Chaudhry (1991) [9]). An algorithm for finding a large number of roots is also available in Chaudhry (1991) [10]. When this method was used in one of the early papers, a referee made comments (for more details on this and other aspects, see the Appendix) on the paper (Brière and Chaudhry (1989) [11]), published in Advances in Applied Probability.

What has been outlined above is true even when we use Markovian arrival and/or Markovian service process, as well as for infinite- and finite-space models. This has been shown in several papers by Chaudhry and his collaborators. Again due to a lack of space, we quote here only three papers: one for Markovian arrival (Singh et al. (2016) [12], one for Markovian Service (Chaudhry et al. (2012) [13]), and one for finite-space (Chaudhry et al. (1991) [14]) that used the roots’ method. In the case of the queueing model BMAP/G/1 (see Singh et al. (2016) [12]), the method has been compared, and it has been shown that the root-finding method is superior to the matrix-analytic method.

In view of what has been outlined above, the purpose of this Special Issue is to showcase how the roots method works. The Special Issue welcomes submissions of all related research and reviews in theoretical analysis and their applications. Topics include, but are not limited to, analyzing the mathematical modeling of queues or waiting lines using the root-finding method.

1. Kendall, D.G. Some recent work and further problems in the theory of queues. Theor. Prob. Appl. 1964, 9, 1–13.
2. Kleinrock, L. Queueing Systems. Volume 1: Theory; John Wiley & Sons: New York, NY, USA, 1975.
3. Stidham, S., Jr. Applied Probability in Operations Research: A Retrospective; University of North Carolina, Department of Operations Research: Chapel Hill, NC, USA, 2001
4. Neuts, M.F. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach; John Hopkins University Press Baltimore, MD, USA, 1994.
5. Mejia-Téllez, Juan. Numerical Method for the Single-Server Bulk-Service Queueing System with Variable Service Capacity, M/G^Y/1, with Discretized Service Time Probability Distribution. In Proceedings of the Annual International Conference of the German Operations Research Society (GOR), Bremen, Germany, 7–9 September 2005, 811–816.
6. Bar-Lev, S. K., Parlar, M., Perry, D., Stadje, W., Van der Duyn Schouten, F. A. Applications of bulk queues to group testing models with incomplete identification, J. Oper.Res. 2007, 183: 226–237.
7. Maity, A., Gupta, U.C. A comparative numerical study of the spectral theory approach of Nishimura and the roots method based on the analysis of BDMMAP/G/1 queue. Int. J. Stoch. Anal. 2015; 2015; org/10.1155/2015/958730
8. Chakka, R. Performance and Reliability Modelling of Computing Systems Using Spectral Expansion. Ph.D. Thesis, Department of Computing Science, University of Newcastle, UK, 1995.
9. Chaudhry, M. L. QROOT Software Package; A&A Publications: Kingston, ON, Canada, 1991.
10. Chaudhry, M. L. Numerical issues in computing steady-state queueing-time distributions of single-server bulk-service queues: M/G ^b/1 and M/G ^d/1, ORSA J. Computing. 1991, 4, 300–310.
11. Brière G., Chaudhry, M. L. Computational analysis of single-server bulk-service queues M/GY/1. Applied. Probab. 1989, 21, 207–225
12. Singh, G., Gupta, U. C., Chaudhry, M. L. Detailed Computational Analysis of Queueing-time Distributions of BMAP/G/1 Using Roots, Appl. Probab. 2016, 53, 1078–1097.
13. Chaudhry, M. L., Samanta, S. K. and Pacheco, A. Analytically Explicit Results for the GI/C-MSP/1 queueing system using roots. Eng. Inf. Sci. 2012, 26, 221–244.
14. Chaudhry, M. L., Gupta, U. C., Agarwal, M. On exact computational analysis of distributions of numbers in system for M/G/1/N+1 and GI/M/1/N+1 queues using roots. Oper. Research. 1991,18, 679–694.

Prof. Dr. Mohan Chaudhry
Prof. Dr. Veena Goswami
Dr. Abhijit Datta Banik
Guest Editors

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• queueing systems
• queueing models
• Markovian arrival process
• Markovian service process
• Laplace transforms
• probability generating functions