A Note on Scheduling Identical Coupled Tasks in Constant Time

The coupled tasks problem consists in scheduling n jobs on a single machine. Each job i is made of two operations with processing times ai and bi and a fixed required delay Li between them. Operations cannot overlap in time but operations of dierent jobs can be interleaved. The objective is to minimize the makespan of the schedule. In this note we show that the problem with identical jobs (8i;ai = a;bi = b;Li = L) can be solved in constant time when a;b;L are fixed. This problem is motivated by radar scheduling applications where tasks corresponding to transmitting radiowaves and listening to potential echoes are coupled. A radar is a system using radiowaves to detect the presence of objects in a given domain. It can also compute the range as well as the relative radial velocity of these objects. Most radars consist of a transmitter, a single antenna and a receiver. The transmitter generates radiowaves which are sent out in a narrow beam by the antenna in a specific direction. Objects located in the beam intercept this signal and scatter back the energy in all directions. A portion of this energy is scattered back to the receiver of the radar listening to all potential echoes. See [7] and [12] for a detailed description of (airborne) radars. There are many interesting combinatorial optimization problems related to radar management. Barbaresco [3] as well as Winter and Baptiste [13] study real-time scheduling of airborne radars: Such radars have to search, track and identify potential targets. The waveforms of these tasks are most often incompatible and hence, cannot be processed simultaneously. Moreover, these tasks are repeated several times in a cyclic fashion. Altogether, this defines a