Load Balancing in Arbitrary Network Topologies with Stochastic Adversarial Input

Abstract

We study the long-term (steady state) performance of a simple, randomized, local load balancing technique under a broad range of input conditions. We assume a system of n processors connected by an arbitrary network topology. Jobs are placed in the processors by a deterministic or randomized adversary. The adversary knows the current and past load distribution in the network and can use this information to place the new tasks in the processors. A node can execute one job per step, and can also participate in one load balancing operation in which it can move tasks to a direct neighbor in the network. In the protocol we analyze here, a node equalizes its load with a random neighbor in the graph. Our analysis of the protocol does not assume any particular input distribution. The input is generated by an arbitrary deterministic or probabilistic adversary subject only to some weak statistical properties. For stability and expected performance of the system we adopt the stochastic adversary model of [Borodin et al., J. ACM, 48 (2001), pp. 13--38]. For high-probability bounds we introduce a more restricted input model, the strongly bounded adversary. Assuming the stochastic adversarial input model, we show that if the adversary does not trivially overload the network (i.e., there is an integer $w\geq 1$ such that the expected number of new jobs in any interval of length w is bounded by $\lambda nw$ for some $\lambda < 1$), then the system is stable for any connected network topology, regardless of how the adversary allocates the new jobs between the processors. When the system is stable, the next performance parameter of interest is the waiting time of jobs. We develop expected and high probability bounds on the total load in the system and the waiting time of jobs in terms of the network topology. In particular, in the above stochastic adversary model, if the network is an expander graph, the expected wait of a task is O(w + log n), and in the strongly bounded adversary model the waiting time of a task is O(w + log n) with high probability. We contrast these results with the work stealing load balancing protocol, where we show that in sparse networks, the load in the system and the waiting time can be exponential in the network size.