  We analyze the Two Level Processor Sharing (TLPS) scheduling discipline with
the hyper-exponential job size distribution and with the Poisson arrival
process. TLPS is a convenient model to study the benefit of the file size based
differentiation in TCP/IP networks. In the case of the hyper-exponential job
size distribution with two phases, we find a closed form analytic expression
for the expected sojourn time and an approximation for the optimal value of the
threshold that minimizes the expected sojourn time. In the case of the
hyper-exponential job size distribution with more than two phases, we derive a
tight upper bound for the expected sojourn time conditioned on the job size. We
show that when the variance of the job size distribution increases, the gain in
system performance increases and the sensitivity to the choice of the threshold
near its optimal value decreases.
