This paper considers trajectory planning for a mobile robot which persistently relays data between pairs of far-away communication nodes. Data accumulates stochastically at each source, and the robot must move to appropriate positions to enable data offload to the corresponding destination. The robot needs to minimize the average time that data waits at a source before being serviced. We are interested in finding optimal robotic routing policies consisting of 1) locations where the robot stops to relay (relay positions) and 2) conditional transition probabilities that determine the sequence in which the pairs are serviced. We first pose this problem as a non-convex problem that optimizes over both relay positions and transition probabilities. To find approximate solutions, we propose a novel algorithm which alternately optimizes relay positions and transition probabilities. For the former, we find efficient convex partitions of the non-convex relay regions, then formulate a mixed-integer second-order cone problem. For the latter, we find optimal transition probabilities via sequential least squares programming. We extensively analyze the proposed approach and mathematically characterize important system properties related to the robot's long-term energy consumption and service rate. Finally, through extensive simulation with real channel parameters, we verify the efficacy of our approach. Significant advances in robotics over the past several years have created new possibilities in the design of communication systems.

Stability analysis consists of identifying conditions under which the number of jobs in a system is guaranteed to remain bounded over time. To date, such long-run performance guarantees have not been available for periodic approaches to dynamic scheduling problems. However, stability has been extensively studied in queueing theory. In this paper, we introduce stability to the dynamic scheduling literature and demonstrate that stability guarantees can be obtained for methods that build the schedule for a dynamic problem by periodically solving static deterministic sub-problems. Specifically, we analyze the stability of two dynamic environments: a two-machine flow shop, which has received significant attention in scheduling research, and a polling system with a flow-shop server, an extension of systems typically considered in queueing. We demonstrate that, among stable policies, methods based on periodic optimization of static schedules may achieve better mean flow times than traditional queueing approaches.