Abstract
Performance measures of manufacturing systems have been intensively researched. However, little attention has been paid to the transient performance analysis of non-Markovian production systems. Therefore, this paper proposes a method to approximate the properties of a two-state non-Markovian system. In particular, an unreliable workstation with two states, operating and failed states, is considered. This system is a simplified version of an industrial manufacturing system. Moment-based approximations for the expected output quantity of the workstation at any arbitrary time is derived and discussed. In addition, an upper bound approximation for the variation of the produced amount is proposed. Failure and repair times are assumed to be arbitrarily distributed. The proposed approximations are compared with a simulated model using the ARENA 10 free version software to demonstrate the accuracy of the method. These approximations are nonparametric, easy to implement and depend only on the first three moments of the underlying distributions without recourse to the functional form of the distributions.
Similar content being viewed by others
References
Altiok T (1985) On the phase-type approximations of general distributions. IIE Trans 17:110–116
Altiok T (2007) Production lines with phase-type operation and repair times and finite buffers. Int J Prod Res 23:489–498. doi:10.1080/00207548508904723
Altiok T, Ranjan R (1989) Analysis of production lines with general service times and finite buffers: a two-node decomposition approach. Eng Costs Prod Econ 17:155–165. doi:10.1016/0167-188X(89)90065-7
Angius A, Horváth A, Colledani M (2014) Moments of accumulated reward and completion time in Markovian models with application to unreliable manufacturing systems. Perform Eval 75–76:69–88. doi:10.1016/j.peva.2014.02.005
Assaf R, Colledani M, Matta A (2014) Analytical evaluation of the output variability in production systems with general Markovian structure. OR Spectr 36:799–835. doi:10.1007/s00291-013-0343-6
Bobbio A, Telek M (1994) A benchmark for ph estimation algorithms: results for acyclic-ph. Commun Stat Stoch Model 10:661–677. doi:10.1080/15326349408807315
Buzacott JA, Shanthikumar JG (1993) Stochastic models of manufacturing systems. Prentice Hall, Prentice
Chen C-T, Yuan J (2004) Transient throughput analysis for a series type system of machines in terms of alternating renewal processes. Eur J Oper Res 155:178–197. doi:10.1016/S0377-2217(02)00838-X
Chen G, Wang C, Zhang L et al (2016) Transient performance analysis of serial production lines with geometric machines. IEEE Trans Autom Control 61:877–891. doi:10.1109/TAC.2015.2444071
Ciprut P, Hongler M-O, Salama Y (2000) Fluctuations of the production output of transfer lines. J Intell Manuf 11:183–189. doi:10.1023/A:1008942917166
Colledani M, Tolio T (2009) Performance evaluation of transfer lines with general repair times and multiple failure modes. Ann Oper Res 182:31–65. doi:10.1007/s10479-009-0595-3
Colledani M, Ekvall M, Lundholm T et al (2010a) Analytical methods to support continuous improvements at Scania. Int J Prod Res 48:1913–1945. doi:10.1080/00207540802538039
Colledani M, Matta A, Tolio T (2010b) Analysis of the production variability in multi-stage manufacturing systems. CIRP Ann Manuf Technol 59:449–452
Dallery Y, Gershwin SB (1992) Manufacturing flow line systems: a review of models and analytical results. Queueing Syst 12:3–94. doi:10.1007/BF01158636
Gershwin S (1993) Variance of the output of a tandem production system. In: Proceedings of the second international conference on queueing networks with finite capacity
Hendricks KB (1992) The output processes of serial production lines of exponential machines with finite buffers. Oper Res 40:1139–1147. doi:10.1287/opre.40.6.1139
Horvath A (2003) Approximating non-Markovian behaviour by Markovian models. Budapest University of Technology and Economics, Budapest
Inman R (1999) Empirical evaluation of exponential and independence assumptions in queueing models of manufacturing systems. Prod Oper Manag 8:409–432. doi:10.1111/j.1937-5956.1999.tb00316.x
Kambo NS, Rangan A, Hadji EM (2012) Moments-based approximation to the renewal function. Commun Stat Theory Methods 41:851–868. doi:10.1080/03610926.2010.533231
Lang A, Arthur J (1996) Parameter approximation for phase-type distributions. In: Alfa A, Chakravarty S (eds) Matrix-analytic methods in stochastic models. CRC Press, USA, pp 151–206
Li J, Meerkov SM (2008) Production systems engineering. Springer Science and Business Media, Berlin
Li J, Blumenfeld DE, Huang N, Alden J (2009) Throughput analysis of production systems: recent advances and future topics. Int J Prod Res 47:3823–3851. doi:10.1080/00207540701829752
Lindsay BG, Pilla RS, Basak P (2000) Moment-based approximations of distributions using mixtures: theory and applications. Ann Inst Stat Math 52:215–230. doi:10.1023/A:1004105603806
Medhi J (1994) Stochastic processes, 2nd edn. Wiley, University of Michigan, New York
Meerkov SM, Zhang L (2008) Transient behavior of serial production lines with Bernoulli machines. IIE Trans 40:297–312. doi:10.1080/07408170701488037
Miltenburg G (1987) Variance of the number of units produced on a transfer line with buffer inventories during a period of length T. Nav Res Logist 34:811–822
Muth EJ (1968) A method for predicting system downtime. IEEE Trans Reliab 17:97–102. doi:10.1109/TR.1968.5217522
Narahari Y, Viswanadham N (1994) Transient analysis of manufacturing systems performance. IEEE Trans Robot Autom 10:230–244. doi:10.1109/70.282547
Papadopoulos HT (1996) An analytic formula for the mean throughput of K-station production lines with no intermediate buffers. Eur J Oper Res 91:481–494. doi:10.1016/0377-2217(95)00113-1
Papadopoulos HT, Heavey C (1996) Queueing theory in manufacturing systems analysis and design: a classification of models for production and transfer lines. Eur J Oper Res 92:1–27. doi:10.1016/0377-2217(95)00378-9
Rausand M, Hoyland A (1994) System reliability theory: models, statistical methods, and applications, 2nd edn. John Wiley & Sons, Inc., Hoboken, NJ, USA, p 636
Rismanchian F, Hadji EM (2014) Transient analysis of single machine production line dynamics. Int J Oper Res 11:40–50
Tan B (1997) Variance of the throughput of an N-station production line with no intermediate buffers and time dependent failures. Eur J Oper Res 101:560–576. doi:10.1016/S0377-2217(96)00191-9
Tan B (1998a) Agile manufacturing and management of variability. Int Trans Oper Res 5:375–388
Tan B (1998b) An analytical formula for variance of output from a series-parallel production system with no interstation buffers and time-dependent failures. Math Comput Model 27:95–112. doi:10.1016/S0895-7177(98)00031-4
Tan B (1998c) Effects of variability on the due-time performance of a continuous materials flow production system in series. Int J Prod Econ 54:87–100. doi:10.1016/S0925-5273(97)00132-1
Tan B (2000) Asymptotic variance rate of the output in production lines with finite buffers. Ann Oper Res 93:385–403. doi:10.1023/A:1018992327521
Tan B (2003) State-space modeling and analysis of pull-controlled production systems. Analysis and modeling of manufacturing systems. Springer, Berlin, pp 363–398
Telek M, Pfening A (1996) Performance analysis of Markov regenerative reward models. Perform Eval 27–28:1–18. doi:10.1016/S0166-5316(96)90017-6
Whitt W (1982) Approximating a point process by a renewal process, I: two basic methods. Oper Res 30:125–147
Wu W, Tang Y, Yu M, Jiang Y (2015) Computation and transient analysis of a k-out-of-n: G repairable system with general repair times. Oper Res Int J 15:307–324. doi:10.1007/s12351-015-0181-1
Xia B, Xi L, Zhou B, Du S (2013) An efficient analytical method for performance evaluation of transfer lines with unreliable machines and finite transfer-delay buffers. Int J Prod Res 51:1799–1819
Acknowledgments
We are indebted to the editor and anonymous referees of the journal for their constructive comments that helped us to improve the quality of this paper. In addition, the first author would like to thank Professor Alagar Rangan for sharing his insights regarding this study and for his suggestions. This work was partially supported by National Research Foundation of Korea (NRF). Grant funded by the Korean government (MSIP) under Grant No. NRF-2014R1A2A2A03003874.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rismanchian, F., Lee, Y.H. Moment-based approximations for first- and second-order transient performance measures of an unreliable workstation. Oper Res Int J 18, 75–95 (2018). https://doi.org/10.1007/s12351-016-0254-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12351-016-0254-9
Keywords
- Stochastic models of production systems
- Two-state non-Markovian system
- Performance measure analysis
- Output variability
- Transient analysis