Analysis of randomized join-the-shortest-queue (JSQ) schemes in large heterogeneous processor-sharing systems

In this paper, we investigate the stability and performance of randomized dynamic routing schemes for jobs based on the Join-the-Shortest Queue (JSQ) criterion in a heterogeneous system of many parallel servers. In particular, we consider servers that use processor sharing but with different server rates, and jobs are routed to the server with the smallest occupancy among a finite number of randomly sampled servers. We focus on the case of two servers that is often referred to as a Power-of-Two scheme. We first show that in the heterogeneous setting, uniform sampling of servers can cause a loss in the stability region and thus such randomized dynamic schemes need not outperform static randomized schemes in terms of mean delay in opposition to the homogeneous case of equal server speeds where the stability region is maximal and coincides with that of the static randomized routing. We explicitly characterize the stationary distributions of the server occupancies and show that the tail distribution of the server occupancy has a super-exponential behavior as in the homogeneous case as the number of servers goes to infinity. To overcome the stability issue, we show that it is possible to combine the static state-independent scheme with a randomized JSQ scheme that allows us to recover the maximal stability region combined with the benefits of JSQ, and such a scheme is preferable in terms of average delay. The techniques are based on a mean field analysis where we show that the stationary distributions coincide with those obtained under asymptotic independence of the servers and, moreover, the stationary distributions are insensitive to the job-size distribution

Hyper-Scalable JSQ with Sparse Feedback

Load balancing algorithms play a vital role in enhancing performance in data centers and cloud networks. Due to the massive size of these systems, scalability challenges, and especially the communication overhead associated with load balancing mechanisms, have emerged as major concerns. Motivated by these issues, we introduce and analyze a novel class of load balancing schemes where the various servers provide occasional queue updates to guide the load assignment. We show that the proposed schemes strongly outperform JSQ($d$) strategies with comparable communication overhead per job, and can achieve a vanishing waiting time in the many-server limit with just one message per job, just like the popular JIQ scheme. The proposed schemes are particularly geared however towards the sparse feedback regime with less than one message per job, where they outperform corresponding sparsified JIQ versions. We investigate fluid limits for synchronous updates as well as asynchronous exponential update intervals. The fixed point of the fluid limit is identified in the latter case, and used to derive the queue length distribution. We also demonstrate that in the ultra-low feedback regime the mean stationary waiting time tends to a constant in the synchronous case, but grows without bound in the asynchronous case

Load Balancing in the Non-Degenerate Slowdown Regime

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 timescale separation that provides insights into the performance of JSQ. We calculate the price of irrevocably dispatching jobs to servers and prove this to within 15% (in the NDS regime) of the rules that may manoeuvre jobs between servers. We also compare ours results for the JSQ policy with the NDS approximations of many modern load balancing policies such as Idle-Queue-First and Power-of-$d$-choices policies which act as low information proxies for the JSQ policy. Our analysis leads us to construct new rules that have identical performance to JSQ but require less communication overhead than power-of-2-choices.Comment: Revised journal submission versio

Randomized Assignment of Jobs to Servers in Heterogeneous Clusters of Shared Servers for Low Delay

We consider the job assignment problem in a multi-server system consisting of $N$ parallel processor sharing servers, categorized into $M$ ($\ll N$) different types according to their processing capacity or speed. Jobs of random sizes arrive at the system according to a Poisson process with rate $N \lambda$. Upon each arrival, a small number of servers from each type is sampled uniformly at random. The job is then assigned to one of the sampled servers based on a selection rule. We propose two schemes, each corresponding to a specific selection rule that aims at reducing the mean sojourn time of jobs in the system. We first show that both methods achieve the maximal stability region. We then analyze the system operating under the proposed schemes as $N \to \infty$ which corresponds to the mean field. Our results show that asymptotic independence among servers holds even when $M$ is finite and exchangeability holds only within servers of the same type. We further establish the existence and uniqueness of stationary solution of the mean field and show that the tail distribution of server occupancy decays doubly exponentially for each server type. When the estimates of arrival rates are not available, the proposed schemes offer simpler alternatives to achieving lower mean sojourn time of jobs, as shown by our numerical studies

Scalable Load Balancing Algorithms in Networked Systems

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.Comment: Ph.D. thesi