  The PS-model treated in this paper is motivated by freelance job websites
where multiple freelancers compete for a single job. In the context of such
websites, multistage service of a job means collection of applications from
multiple freelancers. Under Markovian stochastic assumptions, we develop fluid
limit approximations for the PS-model in overload. Based on this approximation,
we estimate what proportion of freelancers get the jobs they apply for. In
addition, the PS-model studied here is an instant of PS with routing and
impatience, for which no Lyapunov function is known, and we suggest some
partial solutions.
