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The emergence of new communication services with anytime access to content, such as personalized time-shifted television (TV) programming, requires a combination of unicast and multicast content distribution, and possibly caching, for efficient support of user requests. We present a general framework for the analysis and design of such delivery systems that takes into account network resource constraints, time constraints, content features, user preferences, and service availability requirements. In this framework, the time-shifted content delivery system is modeled via the well-known probabilistic urn model with novel occupancy probabilities reflecting actual content popularity. New approximations to occupancy distributions are derived which enable fast computation of various system performance measures, such as likelihood of access to requested programs. We illustrate the utility of this analytical framework via a prototypical scenario in emerging personalized wireless video services.