In the classical setup used in process monitoring, the times between the collection of successive plotted samples are considered as nonrandom. However, in several real-life applications, it seems plausible to assume that the time needed to collect the necessary information for plotting a point in the control chart has a stochastic nature. Under this scenario, instead of focusing on the number of points plotted on the chart until an out-of-control signal is initiated, the appropriate statistic to look at is the total time until a signal is generated. If we denote by L the run length of a control chart and by Y_t,t = 1,2,…, the times between successive plotted points, then the compound random variable $urn:x-wiley:asmb:media:asmb2358:asmb2358-math-0001$ expresses the time to signal of a monitoring scheme, under a particular sampling policy. In this paper, we illustrate how S_L can be exploited to study various charts that are suitable for monitoring Poisson observations. We provide some results for the exact distribution of S_L that may facilitate the task of the performance assessment of a control chart with random plotting times; illustrations and several numerical comparisons that are useful for quality control experts who wish to practice them are presented, and finally, an illustrative example elucidating the implementation of the proposed model is also provided.

REFERENCES

1Montgomery DC. Introduction to Statistical Quality Control. 6th ed. New York, NY: John Wiley & Sons; 2009.
Google Scholar
2Celano G, Faraz A, Saniga E. Control charts monitoring product's loss to society. Qual Reliab Eng Int. 2014; 30: 1393-1407.
10.1002/qre.1562
Web of Science® Google Scholar
3Lucas JM, Davis DJ, Saniga EM. Detecting improvement using Shewhart attribute control charts when the lower control limit is zero. IIE Trans. 2006; 38: 699-709.
10.1080/07408170600692317
Web of Science® Google Scholar
4Wu S, Castagliola P, Khoo MBC. Run rules based phase II c and np charts when process parameters are unknown. Commun Stat Theory Methods. 2016; 45(4): 1182-1197.
10.1080/03610926.2013.861493
Web of Science® Google Scholar
5Lucas JM. Counted data CUSUM's. Technometrics. 1985; 27: 129-144.
10.1080/00401706.1985.10488030
Web of Science® Google Scholar
6Hawkins DM, Olwell DH. Cumulative Sum Charts and Charting for Quality Improvement. 1st ed. New York, NY: Springer-Verlag; 1998.
10.1007/978-1-4612-1686-5
Google Scholar
7Koutras MV, Eryilmaz S. Compound geometric distribution of order k. Methodol Comput Appl Probab. 2017; 19(2): 377-393.
10.1007/s11009-016-9482-y
Web of Science® Google Scholar
8Koutras VM, Koutras MV, Yalcin F. A simple compound scan statistic useful for modeling insurance and risk management problems. Insur Math Econ. 2016; 69: 202-209.
10.1016/j.insmatheco.2016.05.005
Web of Science® Google Scholar
9Jones-Farmer AL, Woodall WH, Steiner S, Champ CW. An overview of Phase I analysis for process improvement and monitoring. J Qual Technol. 2014; 46(3): 265-280.
10.1080/00224065.2014.11917969
Web of Science® Google Scholar
10Psarakis S, Vyniou AK, Castagliola P. Some recent developments on the effects of parameter estimation on control charts. Qual Reliab Eng Int. 2014; 30(8): 1113-1129.
10.1002/qre.1556
Web of Science® Google Scholar
11Yashchin E. Some aspects of the theory of statistical control schemes. IBM J Res Dev. 1987; 31(2): 199-205.
10.1147/rd.312.0199
Web of Science® Google Scholar
12Reynolds Jr MR, Amin RW, Arnold JC, Nachlas JA. $urn:x-wiley:asmb:media:asmb2358:asmb2358-math-0052$ charts with variable sampling intervals. Technometrics. 1998; 30: 181-192.
10.2307/1270164
Web of Science® Google Scholar
13Reynolds Jr MR, Amin RW, Arnold JC. CUSUM charts with variable sampling intervals. Technometrics. 1990; 32: 371-384.
10.1080/00401706.1990.10484721
Web of Science® Google Scholar
14Morais M, Natário I. Improving an upper one-sided c-chart. Commun Stat Theory Methods. 1998; 27: 353-364.
10.1080/03610929808832099
Web of Science® Google Scholar
15Morais MC, Pacheco A. Shewhart schemes with variable sampling intervals revisited. Seq Anal. 2007; 26: 265-282.
10.1080/07474940701404880
Google Scholar
16Chen Y-K, Chen C-Y, Chiou K-C. Cumulative conformance count chart with variable sampling intervals and control limits. Appl Stoch Model Bus Ind. 2011; 27(4): 410-420.
10.1002/asmb.848
Web of Science® Google Scholar
17Klein M. Two alternatives to the Shewhart $urn:x-wiley:asmb:media:asmb2358:asmb2358-math-0053$ control chart. J Qual Technol. 2000; 32: 427-431.
10.1080/00224065.2000.11980028
Web of Science® Google Scholar
18Khoo MBC. Design of runs rules schemes. Qual Eng. 2003; 16(1): 27-43.
10.1081/QEN-120020769
Google Scholar
19Acosta-Mejia CA. Two sets of runs rules for the chart. Qual Eng. 2007; 19(2): 129-136.
10.1080/17513470701263641
Google Scholar
20Riaz M, Mehmood R, Does RJMM. On the performance of different control charting rules. Qual Reliab Eng Int. 2011; 27: 1059-1067.
10.1002/qre.1195
Web of Science® Google Scholar
21Rakitzis AC, Antzoulakos DL. On the improvement of one-sided S control charts. J Appl Stat. 2011; 38(12): 2839-2858.
10.1080/02664763.2011.570320
Web of Science® Google Scholar
22Costa AFB, Machado MAG. A single chart with supplementary runs rules for monitoring the mean vector and the covariance matrix of multivariate processes. Comput Ind Eng. 2013; 66(2): 431-437.
10.1016/j.cie.2013.07.008
Web of Science® Google Scholar
23Castagliola P, Achouri A, Taleb H, Celano G, Psarakis S. Monitoring the coefficient of variation using control charts with run rules. Qual Technol Quant Manage. 2013; 10: 75-94.
10.1080/16843703.2013.11673309
Web of Science® Google Scholar
24Bersimis S, Koutras MV, Papadopoulos GK. Waiting time for an almost perfect run and applications in statistical process control. Methodol Comput Appl Probab. 2014; 16(1): 207-222.
10.1007/s11009-012-9307-6
Web of Science® Google Scholar
25Koutras MV, Bersimis S, Maravelakis PE. Statistical process control using Shewhart control charts with supplementary runs rules. Methodol Comput Appl Probab. 2007; 9: 207-224.
10.1007/s11009-007-9016-8
Web of Science® Google Scholar
26 Western Electric Company. Statistical Quality Control Handbook. Easton, PA: Mack Printing; 1956.
Google Scholar
27Balakrishnan N, Koutras MV. Runs and Scans with Applications. New York, NY: John Wiley & Sons; 2002.
Google Scholar
28Brook D, Evans DA. An approach to the probability distribution of CUSUM run length. Biometrika. 1972; 59: 539-549.
10.1093/biomet/59.3.539
Web of Science® Google Scholar
29Champ CW, Woodall WH. Exact results for Shewhart control charts with supplementary runs rules. Technometrics. 1987; 29: 393-399.
10.1080/00401706.1987.10488266
Web of Science® Google Scholar
30Fu JC, Spiring FA, Xie H. On the average run lengths of quality control schemes using a Markov chain approach. Stat Probab Lett. 2002; 56: 369-380.
10.1016/S0167-7152(01)00183-3
Web of Science® Google Scholar
31Antzoulakos DL, Rakitzis AC. The modified r out of m control chart. Commun Stat Simul Comput. 2008; 37: 396-408.
10.1080/03610910701501906
Web of Science® Google Scholar
32Zhang Y, Castagliola P. Runs rules $urn:x-wiley:asmb:media:asmb2358:asmb2358-math-0054$ charts when process parameters are unknown. Int J Reliab Qual Saf Eng. 2010; 17(4): 381-399.
10.1142/S0218539310003858
Google Scholar
33Lucas JM, Crosier RB. Fast initial response for CUSUM quality control schemes. Technometrics. 1982; 24: 199-205.
Web of Science® Google Scholar
34Neuts MF. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Mineola, NY: Dover Publications; 1981.
Google Scholar
35Latouche G, Ramaswami V. Introduction to Matrix Analytic Methods in Stochastic Modeling. Philadelphia, PA: American Statistical Association and Society for Industrial and Applied Mathematics; 1999.
10.1137/1.9780898719734
Google Scholar
36He Q-M. Fundamentals of Matrix-Analytic Methods. New York, NY: Springer Science+Business Media; 2014.
10.1007/978-1-4614-7330-5
Google Scholar
37Feller W. An Introduction to Probability Theory and Its Applications. Vol. 1. 3rd ed. New York, NY: John Wiley & Sons; 1968.
Google Scholar
38Eisele K-T. Recursions for compound phase distributions. Insur Math Econ. 2006; 38: 149-156.
10.1016/j.insmatheco.2005.08.002
Web of Science® Google Scholar
39Weiß CH, Testik MC. CUSUM monitoring of first-order integer-valued autoregressive processes of Poisson counts. J Qual Technol. 2009; 41(4): 389-400.
10.1080/00224065.2009.11917793
Web of Science® Google Scholar
40Gan FF. An optimal design of cumulative sum control chart based on median run length. Commun Stat Simul Comput. 1994; 23(2): 485-503.
10.1080/03610919408813183
Web of Science® Google Scholar
41Low CK, Khoo MBC, Teoh WL, Wu Z. The revised m-of-k runs rule based on median run length. Commun Stat Simul Comput. 2012; 41(8): 1463-1477.
10.1080/03610918.2011.606950
Web of Science® Google Scholar
42Gan FF. An optimal design of CUSUM control charts for binomial counts. J Appl Stat. 1993; 20(4): 445-460.
10.1080/02664769300000045
Google Scholar