  We consider two-stage tandem queueing systems with one dedicated server in
each station and a flexible server that can serve both stations. We assume
exponential service times, linear holding costs accrued by jobs present in the
system, and a collaborative work discipline. We seek optimal server allocation
strategies for systems without external arrivals (clearing systems). When the
combined rate of collaborating servers is less than the sum of their individual
rates (partial collaboration), we identify conditions under which the optimal
server allocation strategy is non-idling and has a threshold-type structure.
Our results extend previous work on systems with additive service rates. When
the aforementioned conditions are not satisfied we show by examples that the
optimal policy may have counterintuitive properties, which is not the case when
a fully collaborative service discipline is assumed. We also obtain novel
results for any type of collaboration when idling policies may be optimal.
