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Approximations of availability function using phase type distribution

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Abstract

The present work introduces simple approximations of one dimensional availability function using phase type distribution via moments, renewal function and Riemannian sum. A closed form solution for the availability function with phase type inter event times is not available in the literature. Availability function is expressed in terms of the renewal equations for number of failures and repairs. Examples are provided in two cases such as the combination of Exponential–Exponential and Erlang–Erlang alternating renewal processes. An Application of the problem to a communication system is used to test the approximations.

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References

  1. Arunachalam, V., Calvache, A., Tansu, A.: Some useful approximations for the availability function. Int. J. Rel. Quali. Saf. Eng. 22(2), 1550008 (2015)

    Article  Google Scholar 

  2. Asmussen, S., Rolski, T.: Computational methods in risk theory: a matrix—algorithmic approach. Ins. Math. Eco. 10, 257–274 (1991)

    Google Scholar 

  3. Bednara, L: Methods for approximating the availability functions. Thesis of science in Applied Mathematics, Delft University of Technology, Netherlands (2008)

  4. Buddhikot, M.M.: Understanding dynamic spectrum access: models, taxonomy and challenges. In Proceedings of 2nd IEEE Symposium on New Frontiers in Dynamic Spectrum Access Networks (IEEE DySPAN 2007), Ireland (2007)

  5. Bhat, U.N.: Elements of applied stochastic processes. John Wiley, New York (1972)

    Google Scholar 

  6. Frees, E.W., Nam, S.H.: Approximating expected warranty costs. Manag. Sci. 34, 1441–1449 (1988)

    Article  Google Scholar 

  7. Hadji, E.M., Kambo, N.S., Rangan, A.: Two-dimensional renewal function approximation. Comm. Stats. The. Meth. 44, 3107–3124 (2015)

    Article  Google Scholar 

  8. Nasreddine, J., Riihijarvi, J., Li, X., Petri M.: Exploring opportunistic access techniques using stochastic models: dynamic spectrum access without sensing. The 2011 military communications conference. Track 2- Network Protocols and Performance (2011)

  9. Kambo, N.S., Rangan, A., Hadji, E.M.: Moments based approximation to the renewal function. Comm. Stats. The. Meth. 41, 851–868 (2012)

    Article  Google Scholar 

  10. Kao, E.P.C., Smith, M.S.: Discounted and per unit net revenues and costs of product warranty: the case of phase type lifetime. Manag. Sci. 39, 1246–1255 (1993)

    Article  Google Scholar 

  11. Kao, E.P.C., Smith, M.S.: Computational approximations of renewal process relating to a warranty problem: the case of phase type lifetimes. Eur. J. Ope. Res. 90, 156–170 (1996)

    Article  Google Scholar 

  12. Kim, H., Shin, K.G.: Efficient discovery of spectrum opportunities with MAC- layer sensing in Cognitive radio networks. IEEE Trans. Mobile Comp. 7(5), 533–545 (2008)

    Article  Google Scholar 

  13. Mamer, J.W.: Discounted and per unit costs of product warranty. Manag. Sci. 33, 916–930 (1987)

    Article  Google Scholar 

  14. McHenry, M., McCloskey, D.: New York city spectrum occupancy measurements report September 2004. Shared Spectrum Company (http://www.sharedspectrum.com/), Tech. Rep., December (2004)

  15. Neuts, M. F.: Probability distributions of phase type. In Lib. Amicorum Prof. Emeritus H. Florin, Department of Mathematics, University of Louvain, Belgium (1975)

  16. Neuts, M.F.: Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. The John Hopkins University Press, Baltimore (1981)

    Google Scholar 

  17. Nguyen, D.G., Murthy, D.: Cost analysis of warranty policies. Nav. Res. Log. 31, 525–541 (1984)

    Article  Google Scholar 

  18. Pham Gia, T., Turkkan, N.: System availability in a gamma alternating renewal process. Nav. Res. Log. 46, 822–844 (1999)

    Article  Google Scholar 

  19. Ross, S.M.: Stochastic Processes, 2nd edn. John Wiley, New York (1996)

    Google Scholar 

  20. Wang, Z. H.: Reliability engineering theory and practice. 5th ed. Taipei Quality Control Society of Republic of China (1992)

  21. Wellens, M., Riihijarvi, J., Mahonen, P.: Empirical time and frequency domain models of spectrum use. Phy. Comm. 2(1–2), 10–32 (2009)

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to the Editor and referees for their comments and valuable suggestions which have improved the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, CEGC, Anna University, Chennai, Tamil Nadu, 600 025, India

    Y. Sarada

  2. Department of Mathematics and Actuarial Science, B. S. Abdur Rahman Crescent Institute of Science and Technology, Vandalur, Chennai, Tamil Nadu, 600 048, India

    R. Shenbagam

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  1. Y. Sarada
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  2. R. Shenbagam
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Correspondence to Y. Sarada.

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Appendix 1

Appendix 1

See Figures 1, 2, 3, 4, 5, 6.

Fig. 1
figure 1

t versus Availability function

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Fig. 2
figure 2

\({\overline{{\varvec{T}}} }_{{\varvec{O}}{\varvec{N}}}\) versus θ

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Fig. 3
figure 3

\(\overline{T }\) versus θ

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Fig. 4
figure 4

t versus \(E[T_{R} (t)]\)

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Fig. 5
figure 5

Different values of a versus \(E[T_{R} (t)]\)

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Fig. 6
figure 6

Different values of b versus \(E[T_{R} (t)]\)

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See Tables 1, 2, 3.

Table 1 Comparison between the values of the Availability function computed using the proposed models and the Exact value for an Erlang distribution case
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Table 2 Variation of a in θ for the proposed models
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Table 3 Variation of b in θ for the proposed models
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Sarada, Y., Shenbagam, R. Approximations of availability function using phase type distribution. OPSEARCH 59, 1337–1351 (2022). https://doi.org/10.1007/s12597-021-00560-2

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