Moment-Based Approximations for the Moments of a Renewal-Reward Process with Asymmetric Triangular Distributed Interference of Chance

Abstract

This paper presents an effective approximation for the n^th order moments of the ergodic distribution of a renewal-reward process X(t) with asymmetric triangular distributed interference of chance. Many studies have investigated these processes and proposed asymptotic results in which they represented the remaining terms after the second or third term by big Oh (O) or small oh (o) notations. This method does not clarify how these terms convergence to zero. To overcome the ambiguity of the asymptotic expansion we propose approximation formulas to compute n^th order moments of the ergodic distribution of the considered process based only on the moments of random variables that produce the renewal process. The proposed approximation provides closed-form expression and can be used when the distribution functions of the random variables that produce the renewal process are unknown. Furthermore, a simulation study has been conducted to validate the proposed approach, demonstrating its high accuracy across a wide range of parameter configurations.

Keywords

• Renewal reward process
• Moment-based approximations
• Asymmetric triangular distribution
• Moments of ergodic distribution

References

1. [1] Anisimov, V., “Switching Processes in Queuing Models”, John Wiley & Sons: New York, (2013).
2. [2] Borovkov, A., “Stochastic Processes in Queueing Theory”, Springer Science & Business Media, (1976). DOI: https://doi.org/10.1007/978-1-4612-9890-5
3. [3] Brown, M., Solomon, H.A., “Second -order approximation for the variance of a renewal reward process”, Stochastic Processes and Their Applications, 3:301–314, (1975). DOI: https://doi.org/10.1016/0304-4149(75)90029-0
4. [4] Decreusefond, L., Moyal, P., “Stochastic Modeling and Analysis of Telecom Networks”, John Wiley & Sons, (2013).
5. [5] Finkelstein, M., Cha, J. H., “Stochastic Modelling for Reliability: Shocks, Burn-in and-Heterogeneous Populations”, Springer: London, (2013). DOI: https://doi.org/10.1007/978-1-4471-5028-2
6. [6] Gihman, I.I., Skorohod, A.V., “Theory of Stochastic Processes II”, Springer, (1975). DOI: https://doi.org/10.1007/978-3-642-61921-2
7. [7] Jiang, R., “Introduction to Quality and Reliability Engineering”, Springer:Berlin, (2015). DOI: https://doi.org/10.1007/978-3-662-47215-6
8. [8] Nakagawa, T., “Stochastic Processes: with Applications to Reliability Theory”, Springer: London, (2011). DOI: https://doi.org/10.1007/978-0-85729-274-2

9. [9] Smeitink, E., Dekker, R., “A simple approximation to the renewal function”, IEEE Transactions on Reliability, 39(1): 71-75, (1990). DOI: https://doi.org/10.1109/24.52614
10. [10] Smith, W. L., “On the cumulants of renewal process”, Biometrika, 46:1-29, (1959). DOI: https://doi.org/10.1093/biomet/46.1-2.1
11. [11] Hanalioglu, Z., Khaniyev, T., “Asymptotic results for an inventory model of type (s,S) with asymmetric triangular distributed interference of chance and delay”, Gazi University Journal of Science, 31(1): 174-187, (2018).
12. [12] Hanalioglu, Z., Khaniyev, T., “Limit theorem for a semi-Markovian stochastic model of type (s, S)”, Hacettepe Journal of Mathematics and Statistics, 48(2): 605-615, (2019). DOI: https://doi.org/10.15672/HJMS.2018.622
13. [13] Hanalioglu Z., Poladova A., Khaniyev T., “Approximation results for the moments of random walk with normally distributed interference of chance”, Turkish Journal of Mathematics, 47(1): 333-350, (2023). DOI: https://doi.org/10.55730/1300-0098.3363
14. [14] Hanalioglu, Z., Unver, F.N., Khaniyev T., “Asymptotic expansions for the moments of the renewal-reward process with a normal distributed interference of chance”, Applied and Computational Mathematics, 17(2):141-150, (2018).
15. [15] Khaniyev, T., Atalay, K.D., “On the weak convergence of the ergodic distribution for an inventory model of type (s, S)”, Hacettepe Journal of Mathematics and Statistics, 39(4): 599-611, (2010).
16. [16] Khaniyev, T., Kokangul A, Aliyev, R.T., “An asymptotic approach for a semi-Markovian inventory model of type (s, S)”, Applied Stochastic Models in Business and Industry, 29(5): 439-453, (2013). DOI: https://doi.org/10.1002/asmb.1918
17. [17] Khaniyev, T., Aksop, C., “Asymptotic Results for an inventory model of type (s,S) with a generalized Beta interference of chance”, TWMS Journal of Applied and Engineering Mathematics, 2(1):223-236, (2011).
18. [18] Aliyev, R., Ardıc, O., Khaniyev, T., “Asymptotic approach for a renewal-reward process with a general interference of chance”, Communication in Statistics: Theory and Methods, 45(14): 4237-4248, (2016). DOI: https://doi.org/10.1080/03610926.2014.917679
19. [19] Kamışlık, A.B., Alakoç, B., Kesemen, T., Khaniyev, T., “A semi-Markovian renewal reward process with Γ(g) distributed demand”, Turkish Journal of Mathematics, 44(4): 1250-1262, (2020). DOI: https://doi.org/10.3906/mat-2002-72
20. [20] Kamışlık, A.B., Kesemen, T., Khaniyev, T., “Inventory model of type (s, S) under heavy tailed demand with infinite variance”, Brazilian Journal of Probability and Statistics, 33(1): 39-56, (2019). DOI: https://doi.org/10.1214/17-BJPS376
21. [21] Kesemen, T., Kamışlık, A.B., Küçük, Z., Şenol, E., “Inventory model of type (s,S) with subexponential Weibull distributed demand”, Journal of the Turkish Statistical Association, 9(3): 81-92, (2016).
22. [22] Kamışlık, A.B., Baghezze, F. Kesemen, T., Khaniyev, T., “Moment-based approximations for stochastic control model of type (s, S)”, Communications in Statistics - Theory and Methods, 53(21): 7505-7516, (2024). DOI: https://doi.org/10.1080/03610926.2023.2268765
23. [23] Poladova, A., Tekin, S. Khaniyev, T., “Approximate formulas for renewal-reward process with dependent components and normally distributed interference of chance”, Statistics & Probability Letters. 224: 110493, (2025). DOI: https://doi.org/10.1016/j.spl.2025.110493
24. [24] Yazır, T., Kamışlık, A.B., Khaniyev,T., Hanalioglu, Z., “Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance”, Demonstratio Mathematica, 58(1): 20250153, (2025). DOI: https://doi.org/10.1515/dema-2025-0153
25. [25] Alakoç, B., Kamişlik, A.B., Yazır, T., Khaniyev, T., “Moment-based approximation for variance of semi-markovian random walk with gamma distributed interference of chance”, Turkish Journal of Mathematics, 48(6): 1037-1054, (2024). DOI: https://doi.org/10.55730/1300-0098.3559
26. [26] Başak,G., Khaniyev,T., “Rate of weak convergence of random walk with a generalized reflecting barrier”, Gazi University Journal of Science, 38(1): 474-90, (2025). DOI: https://doi.org/10.35378/gujs.1383377
27. [27] Kambo, N.S., Rangan, A., Hadji, E.M., “Moment based approximation to the renewal function”, Communication in Statistics-Theory and Methods, 41(5): 851-868, (2012). DOI: https://doi.org/10.1080/03610926.2010.533231
28. [28] Feller, W. “Introduction to Probability Theory and its Applications II”, John Wiley, (1971).