# Saddlepoint approximation to functional equations in queueing theory and insurance mathematics

## Date

2010

## Authors

Chung, Sunghoon, author

Butler, Ronald W., advisor

Scharf, Louis L., committee member

Chapman, Phillip L., committee member

Hoeting, Jennifer A. (Jennifer Ann), 1966-, committee member

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## Abstract

We study the application of saddlepoint approximations to statistical inference when the moment generating function (MGF) of the distribution of interest is an explicit or an implicit function of the MGF of another random variable which is assumed to be observed. In other words, let W (s) be the MGF of the random variable W of interest. We study the case when W (s) = h{G (s) ; λ}, where G (s) is an MGF of G for which a random sample can be obtained, and h is a smooth function. If Ĝ (s) estimates G (s), then Ŵ (s) = h{Ĝ (s) ; λ̂} estimates W (s). Generally, it can be shown that Ŵ (s) converges to W (s) by the strong law of large numbers, which implies that F̂ (t), the cumulative distribution function (CDF) corresponding to Ŵ (s), converges to F (t), the CDF of W, almost surely. If we set Ŵ* (s) = h{Ĝ* (s) ; λ̂}, where Ĝ* (s) and λ̂* are the empirical MGF and the estimator of λ from bootstrapping, the corresponding CDF F̂* (t) can be used to construct the confidence band of F(t). In this dissertation, we show that the saddlepoint inversion of Ŵ (s) is not only fast, reliable, stable, and accurate enough for a general statistical inference, but also easy to use without deep knowledge of the probability theory regarding the stochastic process of interest. For the first part, we consider nonparametric estimation of the density and the CDF of the stationary waiting times W and Wq of an M/G/1 queue. These estimates are computed using saddlepoint inversion of Ŵ (s) determined from the Pollaczek-Khinchin formula. Our saddlepoint estimation is compared with estimators based on other approximations, including the Cramér-Lundberg approximation. For the second part, we consider the saddlepoint approximation for the busy period distribution FB (t) in a M/G/1 queue. The busy period B is the first passage time for the queueing system to pass from an initial arrival (1 in the system) to 0 in the system. If B (s) is the MGF of B, then B (s) is an implicitly defined function of G (s) and λ, the inter-arrival rate, through the well-known Kendall-Takács functional equation. As in the first part, we show that the saddlepoint approximation can be used to obtain F̂B (t), the CDF corresponding to B̂(s) and simulation results show that confidence bands of FB (t) based on bootstrapping perform well.

## Description

## Rights Access

## Subject

waiting time distribution

Method of steepest descent (Numerical analysis)

saddlepoint approximation

Queuing theory

queueing theory

Bootstrap (Statistics)

nonparametric estimation

Insurance -- Statistical methods

busy period

Insurance -- Mathematics