This paper considers modeling and operational analysis of a generic asymmetric service-system situation in which (a) Red agents, potentially threatening, but possibly requiring assistance, arrive according to some partially known and possibly changing pattern in time and space; and (b) Reds are impatient: have effectively limited unknown deadlines or times of availability for Blue service, i.e., detection, classification, and attack in a military setting, or emergency assistance in other settings, such as medical care. This paper discusses various service options by Blue service agents and devises several approximations allowing one to compute efficiently proportions of tasks of different types that are successfully served or, more generally, if different rewards are associated with different types of tasks, the percentage of the possible reward gained. The paper suggests a heuristic policy for a Blue server to select the next task to perform and to decide how much time to allocate to that service. The paper also discusses this and related policies for a number of specific examples.
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