  QED-effects are known to occur in a strong laser pulse interaction with a
counter-propagating electron beam, among these effects being electron-positron
pair creation. We discuss the range of laser pulse intensities of J > 5*10^22
W/cm2 combined with electron beam energies of tens of GeV. In this regime
multiple pairs may be generated from a single beam electron, some of the
newborn particles being capable of further pair production. Radiation
back-reaction prevents avalanche development and limits pair creation. The
system of integro-differential kinetic equations for electrons, positrons and
\gamma-photons is derived and solved numerically.
