  This paper presents a new research direction for online Multi-Level
Aggregation (MLA) with delays. In this problem, we are given an edge-weighted
rooted tree $T$, and we have to serve a sequence of requests arriving at its
vertices in an online manner. Each request $r$ is characterized by two
parameters: its arrival time $t(r)$ and location $l(r)$ (a vertex). Once a
request $r$ arrives, we can either serve it immediately or postpone this action
until any time $t > t(r)$. We can serve several pending requests at the same
time, and the service cost of a service corresponds to the weight of the
subtree that contains all the requests served and the root of $T$. Postponing
the service of a request $r$ to time $t > t(r)$ generates an additional delay
cost of $t - t(r)$. The goal is to serve all requests in an online manner such
that the total cost (i.e., the total sum of service and delay costs) is
minimized. The current best algorithm for this problem achieves a competitive
ratio of $O(d^2)$ (Azar and Touitou, FOCS'19), where $d$ denotes the depth of
the tree.
  Here, we consider a stochastic version of MLA where the requests follow a
Poisson arrival process. We present a deterministic online algorithm which
achieves a constant ratio of expectations, meaning that the ratio between the
expected costs of the solution generated by our algorithm and the optimal
offline solution is bounded by a constant. Our algorithm is obtained by
carefully combining two strategies. In the first one, we plan periodic
oblivious visits to the subset of frequent vertices, whereas in the second one,
we greedily serve the pending requests in the remaining vertices. This problem
is complex enough to demonstrate a very rare phenomenon that ``single-minded"
or ``sample-average" strategies are not enough in stochastic optimization.
