Asian Journal of Mathematics & Statistics1994-54182077-2068Asian Network for Scientific Information10.3923/ajms.2008.57.62KhorshidianK.1200811We obtain a Strong Law of Large Numbers
(SLLN) for the reward process {Z(t), t≥0}, the cumulative reward gained
by operating a Semi-Markov system during the time interval [0, t]. The
important and striking point in this study is leaving the usual assumption
that rewards for each state are of constant rates. In most of applications
this frequently used assumption is not realistic, therefore we deal with
reward functions of general forms. The SLLN obtained is in the sense that
Lim_{t→∞}Z(t)/t = α, a.s. for some real value
α. Mild conditions for this SLLN are existence of sojourn times means
and integrability of reward functions with respect to sojourn time distributions.
As it has been shown, the parameter α coincides with the shift parameter
in asymptotic representation of E[Z(t)], t→∞.]]>Cinlar, E.,1969Cinlar, E.,1975Cinlar, E.,1975Khorshidian, K. and A.R. Soltani,2002Khorshidian, K. and A.R. Soltani,2003Masuda, Y. and U. Sumita,1991Pyke, R.,1961Pyke, R.,1961Pyke, R. and R. Schaufele,1964Soltani, A.R.,1996Soltani, A.R. and K. Khorshidian,1998Ushio, S. and M. Yasushi,1987Teugels, J.L.,1976