We study online service systems in which requests arrive at locations in a network. A service may open a facility and connect customers, buy a subtree linking them to a root, or send a mobile server to visit them. Combining requests can save opening, network, or travel costs, but each request pays its own waiting time. The objective is the sum of these spatial costs and individual delays. We assume independent Poisson arrivals of known rates. The algorithms know how often requests occur, but observe them only upon arrival. We show that these rates suffice to make fixed spatial choices that compete with an offline optimum free to adapt to the realized input. We obtain long-run ratios of expectations below $27.249$ for facility location in finite metrics, $94.415$ for multi-root aggregation on trees, and $24.326$ for online service on trees. All three algorithms are deterministic; the multi-root algorithm also has polynomial-time preprocessing on rational inputs. The proofs connect rate-based spatial choices with lower bounds on unavoidable purchase, movement, and waiting costs. All main proofs are self-contained, and every guarantee includes the full cost of completing finite inputs.

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