Tail asymptotics for the fundamental period in the MAP/G/1 queue

This paper studies the tail behavior of the fundamental period in the MAP/G/1 queue. We prove that if the service time distribution has a regularly varying tail, then the fundamental period distribution in the MAP/G/1 queue has also regularly varying tail, and vice versa, by finding an explicit expression for the asymptotics of the tail of the fundamental period in terms of the tail of the service time distribution. Our main result with the matrix analytic proof is a natural extension of the result in (de Meyer and Teugels, J. Appl. Probab. 17: 802–813, 1980) on the M/G/1 queue where techniques rely heavily on analytic expressions of relevant functions. I.-S. Wee’s research was supported by the Korea Research Foundation Grant KRF 2003-070-00008.     

Moments of claims in a Markovian environment

This paper considers discounted aggregate claims when the claim rates and sizes fluctuate according to the state of the risk business. We provide a system of differential equations for the Laplace–Stieltjes transform of the distribution of discounted aggregate claims under this assumption. Using the differential equations, we present the first two moments of discounted aggregate claims in a Markovian environment. We also derive simple expressions for the moments of discounted aggregate claims when the Markovian environment has two states. Numerical examples are illustrated when the claim sizes are specified.     

Palladium-catalysed 5-endo-trig allylic (hetero)arylation

A palladium-catalysed intramolecular allylic (hetero)arylation strategy for the synthesis of fused cyclopentenes incorporated with all-carbon quaternary and spiro centres is described. The method is straightforward, shows broad scope, proceeds in synthetically useful yields, and provides a rare means to construct complex cyclopentanoids. The reaction is believed to involve a kinetically unfavourable 5-endo-trig carbocyclisation of the tethered (π-allyl)palladium system. Further, this method was successfully applied as the key step in the total synthesis of diterpene natural products taiwaniaquinone H and dichroanone.

A palladium-catalysed intramolecular allylic (hetero)arylation strategy for the synthesis of fused cyclopentenes incorporated with all-carbon quaternary and spiro centres is described.