Solving probability transform functional equations for numerical inversion

Many methods for numerically inverting transforms require values of the transform at complex arguments. However, in some applications, the transforms are only characterized implicitly via functional equations. This is illustrated by the busy-period distribution in the M/G/1 queue. In this paper we provide conditions for iterative methods to converge for complex arguments. Moreover, we show that stochastic monotonicity properties can provide useful bounds.

[1]  Robert B. Cooper,et al.  An Introduction To Queueing Theory , 2016 .

[2]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[3]  B. T. Doshi.  An M/G/1 queue with a hybrid discipline , 1983, The Bell System Technical Journal.

[4]  Ward Whitt,et al.  Transient Behavior of the M/G/1 Workload Process , 1994, Oper. Res..

[5]  Marcel F. Neuts,et al.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1989 .

[6]  StehfestHarald.  Remark on algorithm 368: Numerical inversion of Laplace transforms , 1970 .

[7]  J. Keilson,et al.  On Time Dependent Queuing Processes , 1960 .

[8]  R. R. P. Jackson.  Introduction to Queueing Theory , 1943 .

[9]  R. R. P. Jackson,et al.  Introduction to the Theory of Queues , 1963 .

[10]  G. F. Newell,et al.  Introduction to the Theory of Queues. , 1963 .

[11]  B. Conolly.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1991 .

[12]  D. M. Lucantoni.  An algorithmic analysis of a communication model with retransmission of flawed messages , 1983 .

[13]  Ward Whitt,et al.  Numerical inversion of probability generating functions , 1992, Oper. Res. Lett..

[14]  Vincent Hodgson,et al.  The Single Server Queue. , 1972 .

[15]  D. P. Gaver,et al.  Observing Stochastic Processes, and Approximate Transform Inversion , 1966, Oper. Res..

[16]  N. L. Lawrie,et al.  Comparison Methods for Queues and Other Stochastic Models , 1984 .

[17]  Richard E. Nance,et al.  Busy Period Analysis of a Time-Sharing System: transform inversion , 1972, JACM.

[18]  Teunis J. Ott.  The covariance function of the virtual waiting-time process in an M/G/1 queue , 1977 .

[19]  Lajos Takcas.  Introduction to the Theory of Queues , 1962 .

[20]  Ward Whitt,et al.  Comparison methods for queues and other stochastic models , 1986 .

[21]  Louis A. Schmittroth,et al.  Numerical inversion of Laplace transforms , 1960, Commun. ACM.

[22]  Ward Whitt,et al.  The Fourier-series method for inverting transforms of probability distributions , 1992, Queueing Syst. Theory Appl..

[23]  C. Marshall.  The Single Server Queue, Revised Edition , 1983 .

[24]  Michael R. Middleton.  Transient effects in M/G/1 queues , 1979 .