Load Balancing Under Strict Compatibility Constraints

Abstract

Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select [Formula: see text] random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor [Formula: see text] compared with the complete bipartite compatibility graph.

Utility Maximizing Load Balancing Policies

“…Similar models are considered in [29,41]. The former of these two papers broadens the class of graph sequences for which the steady-state fluid limit in [28] holds. For instance, [29] considers certain sequences of spatial graphs that do not satisfy the strong connectivity conditions stated in [28]; yet the number of servers compatible with any given task class goes to infinity.…”

“…For the model with strict compatibility constraints, general stability conditions are provided in [7,11]. In addition, [28] assumes that every new task joins the least busy of d compatible servers chosen uniformly at random, and provides connectivity conditions such that the occupancy process has the same process-level and steady-state fluid limit as in the case where the graph is complete bipartite. Similar models are considered in [29,41].…”

“…The former of these two papers broadens the class of graph sequences for which the steady-state fluid limit in [28] holds. For instance, [29] considers certain sequences of spatial graphs that do not satisfy the strong connectivity conditions stated in [28]; yet the number of servers compatible with any given task class goes to infinity. On the other hand, [41] extends the model considered in [28] by allowing for heterogeneous service rates and proves process-level and steady-state fluid limits in this setting.…”

“…For instance, [29] considers certain sequences of spatial graphs that do not satisfy the strong connectivity conditions stated in [28]; yet the number of servers compatible with any given task class goes to infinity. On the other hand, [41] extends the model considered in [28] by allowing for heterogeneous service rates and proves process-level and steady-state fluid limits in this setting. The model studied in [36,37] also allows for heterogeneous service rates.…”

Learning to Schedule Multi-Server Jobs with Fluctuated Processing Speeds

“…Load balancing policies are usually designed based on Continuous-Time Markov Chains (CTMC) and Lyapunov Stability theories. They assume that jobs arrive according to Poisson process and service rates of computing instances are exponentially distributed [17], [18], [36], [37], [38]. As an example, the most classic policy JSQ [16] dispatches each new arrvied job to the shortest queue available.…”