Saddlepoint Approximation to Functional Equations in Queueing Theory and Insurance Mathematics |

Sunghoon Chung, PhD Candidate, Department of Statistics, Colorado State University

Monday, July 19, 2010

3:30 p.m., room 006, Statistics Bldg

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); g}, where G(s) is an MGF of G for which

a random sample can be obtained, and h is a smooth function. If G(s) estimates

G(s), then W(s) = h{G(s); g} estimates W(s). Generally, it can be shown that W(s)
converges to W(s) by the strong law of large numbers, which implies that F(t), the
cumulative distribution function (CDF) corresponding to W(s), converges to F(t), the
CDF of W, almost surely. If we set W*(s) = h{G*(s);g*}, where G*(s) and g* are
the empirical MGF and the estimator of g from bootstrapping, the corresponding CDF F*(t) can be used to construct the confidence band of F(t).

In this dissertation, we show that the saddlepoint inversion of W(s) is not only

fast, reliable, and 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 W(s) determined from the Pollaczek-Khinchin formula. Our saddlepoint estimation is compared with estimators based on some 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 g, 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 FB(t),
the CDF corresponding to B(s) and simulation results show that confidence bands of
FB(t) based on bootstrapping perform well.

Advisory Committee:

Dr. Ronald Butler, Advisor

Dr. Phil Chapman, Committee Member

Dr. Jennifer Hoeting, Committee Member

Dr. Louis Scharf, Electrical & Computer Engineering, Outside Member