Math Meth Oper Res (2017) 86:71–102
Scheduling for a processor sharing system with linear
· Yoni Nazarathy
Received: 21 July 2016 / Published online: 20 March 2017
© Springer-Verlag Berlin Heidelberg 2017
Abstract We consider the problem of scheduling arrivals to a congestion system with
a ﬁnite number of users having identical deterministic demand sizes. The congestion
is of the processor sharing type in the sense that all users in the system at any given
time are served simultaneously. However, in contrast to classical processor sharing
congestion models, the processing slowdown is proportional to the number of users
in the system at any time. That is, the rate of service experienced by all users is
linearly decreasing with the number of users. For each user there is an ideal departure
time (due date). A centralized scheduling goal is then to select arrival times so as to
minimize the total penalty due to deviations from ideal times weighted with sojourn
times. Each deviation penalty is assumed quadratic, or more generally convex. But
due to the dynamics of the system, the scheduling objective function is non-convex.
Speciﬁcally, the system objective function is a non-smooth piecewise convex function.
Nevertheless, we are able to leverage the structure of the problem to derive an algorithm
that ﬁnds the global optimum in a (large but) ﬁnite number of steps, each involving the
solution of a constrained convex program. Further, we put forward several heuristics.
The ﬁrst is the traversal of neighbouring constrained convex programming problems,
that is guaranteed to reach a local minimum of the centralized problem. This is a form
of a “local search”, where we use the problem structure in a novel manner. The second
is a one-coordinate “global search”, used in coordinate pivot iteration. We then merge
these two heuristics into a uniﬁed “local–global” heuristic, and numerically illustrate
the effectiveness of this heuristic.
Department of Statistics and the Federmann Center for the Study of Rationality,
The Hebrew University of Jerusalem, Jerusalem, Israel
School of Mathematics and Physics, The University of Queensland, Brisbane, Australia