In this note, we apply Stein's method to analyze the performance of general load balancing schemes in the many-server heavy-traffic regime. In particular, consider a load balancing system of $N$ servers and the distance of arrival rate to the capacity region is given by $N^{1-α}$ with $α> 1$. We are interested in the performance as $N$ goes to infinity under a large class of policies. We establish different asymptotics under different scalings and conditions. Specifically, (i) If the second moments linearly increase with $N$ with coefficients $σ_a^2$ and $ν_s^2$, then for any $α> 4$, the distribution of the sum queue length scaled by $N^{-α}$ converges to an exponential random variable with mean $\frac{σ_a^2 + ν_s^2}{2}$. (3) If the second moments quadratically increase with $N$ with coefficients $\tildeσ_a^2$ and $\tildeν_s^2$, then for any $α> 3$, the distribution of the sum queue length scaled by $N^{-α-1}$ converges to an exponential random variable with mean $\frac{\tildeσ_a^2 + \tildeν_s^2}{2}$. Both results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in \cite{zhou2020note}.
We study the load balancing system operating under Join the Shortest Queue (JSQ) in the many-server heavy-traffic regime. If $N$ is the number of servers, we let the difference between the total service rate and the total arrival rate be $N^{1-α}$ with $α>0$. We show that for $α>4$ the average queue length behaves similarly to the classical heavy-traffic regime. Specifically, we prove that the distribution of the average queue length multiplied by $N^{1-α}$ converges to an exponential random variable. Moreover, we show a result analogous to state space collapse. We provide two proofs for our r…
Efficiently exploiting servers in data centers requires performance analysis methods that account not only for the stochastic nature of demand but also for server heterogeneity. Although several recent works proved optimality results for heterogeneity-aware variants of classical load-balancing algorithms in the many-server regime, we still lack a fundamental understanding of the impact of heterogeneity on performance in finite-size systems. In this paper, we consider a load-balancing algorithm that leads to a product-form queueing model and can therefore be analyzed exactly even when the numbe…
In this paper we study dynamic averaging load balancing on general graphs. We consider infinite time and dynamic processes, where in every step new load items are assigned to randomly chosen nodes. A matching is chosen, and the load is averaged over the edges of that matching. We analyze the discrete case where load items are indivisible, moreover our results also carry over to the continuous case where load items can be split arbitrarily. For the choice of the matchings we consider three different models, random matchings of linear size, random matchings containing only single edges, and dete…
We analyze randomized dynamic load balancing schemes for multi-server processor sharing systems when the number of servers in the system is large and the servers have heterogeneous service rates. In particular, we focus on the classical power-of-two load balancing scheme and a variant of it in which a newly arrived job is assigned to the server having the least instantaneous Lagrange shadow cost among two randomly chosen servers. The instantaneous Lagrange shadow cost at a server is given by the ratio of the number of unfinished jobs at the server to the capacity of the server. Two different a…
We consider a system of N queues with decentralized load balancing such as power-of-D strategies(where D may depend on N) and generic scheduling disciplines. To measure the dependence of the queues, we use the clan of ancestors, a technique coming from interacting particle systems. Relying in that analysis we prove quantitative estimates on the queues correlations implying propagation of chaos for systems with Markovian arrivals and general service time distribution. This solves the conjecture posed by Bramsom et. al. in [*] concerning the asymptotic independence of the servers in the case of …
We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a classical load balancing policy for queueing systems with multiple parallel servers. Parallel server queueing systems are regularly analysed and dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling regime. However, when jobs must be dispatched to a server upon arrival, we advocate the Non-Degenerate Slowdown regime (NDS) to compare different load-balancing rules. In this paper we identify novel diffusion approximation and ti…
This paper considers the steady-state performance of load balancing algorithms in a many-server system with distributed queues. The system has $N$ servers, and each server maintains a local queue with buffer size $b-1,$ i.e. a server can hold at most one job in service and $b-1$ jobs in the queue. Jobs in the same queue are served according to the first-in-first-out (FIFO) order. The system is operated in a heavy-traffic regime such that the workload per server is $λ= 1 - N^{-α}$ for $0.5\leq α<1.$ We identify a set of algorithms such that the steady-state queues have the following universal s…
We consider the load balancing problem in large-scale heterogeneous systems with multiple dispatchers. We introduce a general framework called Local-Estimation-Driven (LED). Under this framework, each dispatcher keeps local (possibly outdated) estimates of queue lengths for all the servers, and the dispatching decision is made purely based on these local estimates. The local estimates are updated via infrequent communications between dispatchers and servers. We derive sufficient conditions for LED policies to achieve throughput optimality and delay optimality in heavy-traffic, respectively. Th…
It is well-known that the power-of-d choices routing algorithm maximizes throughput and is heavy-traffic optimal in load balancing systems with homogeneous servers. However, if the servers are heterogeneous, throughput optimality does not hold in general. We find necessary and sufficient conditions for throughput optimality of power-of-d choices when the servers are heterogeneous, and we prove that almost the same conditions are sufficient to show heavy-traffic optimality. Additionally, we generalize the sufficient condition for throughput optimality to a larger class of routing policies.
Applications in cloud platforms motivate the study of efficient load balancing under job-server constraints and server heterogeneity. In this paper, we study load balancing on a bipartite graph where left nodes correspond to job types and right nodes correspond to servers, with each edge indicating that a job type can be served by a server. Thus edges represent locality constraints, i.e., each job can only be served at servers which contained certain data and/or machine learning (ML) models. Servers in this system can have heterogeneous service rates. In this setting, we investigate the perfor…
This paper studies a class of load balancing algorithms for many-server ($N$ servers) systems assuming finite buffer with size $b-1$ (i.e. a server can have at most one job in service and $b-1$ jobs in queue). We focus on steady-state performance of load balancing algorithms in the heavy traffic regime such that the load of system is $λ= 1 - N^{-α}$ for $0<α<0.5,$ which we call sub-Halfin-Whitt regime ($α=0.5$ is the so-called the Halfin-Whitt regime). We establish a sufficient condition under which the probability that an incoming job is routed to an idle server is one asymptotically. The cla…
A fundamental challenge in large-scale networked systems viz., data centers and cloud networks is to distribute tasks to a pool of servers, using minimal instantaneous state information, while providing excellent delay performance. In this thesis we design and analyze load balancing algorithms that aim to achieve a highly efficient distribution of tasks, optimize server utilization, and minimize communication overhead.
We consider a system of $N$ parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among $d$ randomly selected servers otherwise $(1 \leq d \leq N)$. This load balancing scheme subsumes the so-called Join-the-Idle Queue (JIQ) policy $(d = 1)$ and the celebrated Join-the-Shortest Queue (JSQ) policy $(d = N)$ as two crucial special cases. We develop a stochastic coupling construction to obtain the diff…
We consider load balancing in service systems with affinity relations between jobs and servers. Specifically, an arriving job can be allocated to a fast, primary server from a particular selection associated with this job or to a secondary server to be processed at a slower rate. Such job-server affinity relations can model network topologies based on geographical proximity, or data locality in cloud scenarios. We introduce load balancing schemes that allocate jobs to primary servers if available, and otherwise to secondary servers. A novel coupling construction is developed to obtain stabilit…
We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where th…
We consider general large-scale service systems with multiple customer classes and multiple server (agent) pools, mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a natural (load balancing) routing/scheduling rule, Longest-Queue Freest-Server (LQFS-LB), in the many-server asymptotic regime, such that the exogenous arrival rates of the customer classes, as well as the number of agents in …
This paper studies load balancing for many-server ($N$ servers) systems. Each server has a buffer of size $b-1,$ and can have at most one job in service and $b-1$ jobs in the buffer. The service time of a job follows the Coxian-2 distribution. We focus on steady-state performance of load balancing policies in the heavy traffic regime such that the normalized load of system is $λ= 1 - N^{-α}$ for $0<α<0.5.$ We identify a set of policies that achieve asymptotic zero waiting. The set of policies include several classical policies such as join-the-shortest-queue (JSQ), join-the-idle-queue (JIQ), i…
Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be moved from a vertex to another that is carrying less load. We analyze the properties of balanced allocations of load. We extend the concept of balancedness from finite hypergraphs to their local weak limits in the sense of Benjamini and Schramm (2001) and Aldous and Steele (2004). To do this, we define a notion of unimodularity for hypergraphs which could be…
