We consider a perimeter defense problem in a planar conical environment comprising a turret that has a finite range and non-zero startup time. The turret seeks to defend a concentric perimeter against N ≥ 2 intruders. Upon release, each intruder moves radially towards the perimeter with a fixed speed. To capture an intruder, the turret's angle must be aligned with that of the intruder's angle and must spend a specified startup time at that orientation. We address offline and online versions of this optimization problem. Specifically, in the offline version, we establish that in general parameter regimes, this problem is equivalent to solving a Travelling Repairperson Problem with Time Windows (TRP-TW). We then identify specific parameter regimes in which there is a polynomial time algorithm that maximizes the number of intruders captured. In the online version, we present a competitive analysis technique in which we establish a fundamental guarantee on the existence of at best (N – 1)-competitive algorithms. We also design two online algorithms that are provably 1 and 2-competitive in specific parameter regimes.