We study online network services that combine pending requests into batches. Each batch pays for its spatial service and for the waiting time of its oldest request. The objective is the sum of these costs over all batches. Thus requests can share both a spatial purchase and one maximum-delay charge. We assume independent Poisson arrivals of known rates, while actual requests are revealed only upon arrival. For stateless services, suppose that each type has an attainable minimum cost when served alone, that every batch costs at least the stand-alone minimum of each type it contains, and that one finite-cost action can serve any batch, regardless of its request multiplicities. We give a deterministic algorithm with long-run ratio of expectations $e/(e-1)$ against an unrestricted offline optimum. This factor is optimal, even allowing randomization, already at one request location. The key comparison measures the union of time intervals associated with requests. The theorem applies directly to multi-root and pair aggregation on trees. We also study facility location, where each request pays its own connection distance and the bounded-action assumption can fail. A deterministic policy gives a ratio $3$ for arbitrary nonnegative opening costs. It immediately serves a finite initial set of requests, computes an assignment from their observed arrival gaps, and then keeps that assignment fixed. Its analysis combines Poisson insertion with a time-slice packing bound on the offline optimum. Every guarantee includes the full cost of completing finite inputs without an end-of-input signal.

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