  We investigate statistical properties of galaxy clusters in the context of
hierarchical clustering scenario, taking account of their formation epoch
distribution, motivated by the recent finding by Fujita and Takahara that X-ray
clusters form a fundamental plane, where the mass and formation epoch are
regarded as two independent parameters. Using the formalism which discriminates
between major merger and accretion, the epoch of a cluster formation is
identified with that of the last major merger. Since tiny mass accretion after
the formation does not much affect the core structure of clusters, the
properties of X-ray emission from clusters are determined by the total mass and
density at their formation time. Under these assumptions, we calculate X-ray
luminosity and temperature functions of galaxy clusters. We find that the
behavior of luminosity function is different from the model which does not take
account of formation epoch distribution, while the behavior of temperature
function is not much changed. In our model, luminosity function is shifted to a
higher luminosity and shows no significant evolution up to $z \sim 1$,
independent of cosmological models. The clusters are populated on the
temperature-luminosity plane with a finite dispersion. Since the simple scaling
model in which the gas temperature is equal to the virial temperature fails to
reproduce the observed luminosity-temperature relation, we also consider a
model which takes the effects of preheating into account. The preheating model
reproduces the observations much better.
