A General Solution of the Integral Equation for Erlang Distribution

Lee Yoon Dong;Choi Hyemi;Lee Eun-kyung

  • 발행 : 2005.08.01


The mathematical properties of the sequentially operated systems are often described by integral equations. Reservoir system of a product and sequential probability ratio test (SPRT) are typical examples of sequentially operated systems. When the underlying random quantities follow Erlang distribution, a systematic method was developed to solve the integral equations. We extend the method to the cases having accrual functions of more general types. The solutions of the integral equations are represented as a linear combination of distribution functions, and the coefficients of the linear combination are obtained by solving linear system derived from the continuity condition of the solutions.


Erlang distribution;SPRT;Integral equation


  1. Chang, T. and Gan, F. (1995). A cumulative sum control chart for monitoring process variance, Journal of Quality Technology, 25, 109-119
  2. Choi, K and Kim, H. (1998). Bayesian computation for superposition of MUSA-OKUMOTO and Erlang (2) processes, Journal of Korean Applied Statistics, 11, 377-387
  3. Gan, F. and Choi, K. (1994). Computing Average Run Lengths for Exponential CUSUM Schemes, Journal of Quality Technology, 26, 134-139
  4. Khamis, S. H. (1961). Incomplete Gamma Functions Expansions of Statistical Distribution Functions, Statistique Mathematique, 385-396
  5. Kim, H. and Lee, S. (2000). Bayesian inference for mixture failure model of Rayleigh and Erlang pattern, Journal of Korean Applied Statistics, 13-2, 505-514
  6. Kohlruss, D. (1994). Exact formulas for the OC and the ASN functions of the SPRT for Erlang distributions, Sequential Analysis, 13, 53-62
  7. Lee, Y. D. (2004). Unified solutions of integral equations of SPRT for exponential random variables, Communications in Statistics, Series A, Theory and Method, 33, 65-74
  8. Vardeman, S. and Ray, D. (1985). Average Run Lengths for CUSUM Schemes When Observations Are Exponentially Distributed, Technometrics, 27, 145-150
  9. Stadje, W. (1987). On the SPRT for the mean of an exponential distribution, Statistics & Probability Letters, 5, 389-395
  10. Lee, E., Na, M.H. and Lee, YD. (2005). Solutions of integral equations related to SPRT for Erlang distribution, Journal of Korean Applied Statistics, 18, 57-66