arXiv subjects

Burak Büke

Publications and source records attributed to Burak Büke.

4 recordsLinked to original sources

arXiv2026

Load Balancing with Individually Heterogeneous Servers

In this paper, we analyze the diffusion limit of Join-the-Shortest-Queue (JSQ) load balancing policies for a system with many parallel queues where each queue is being served by a server with a potentially different service rate. Prior asymptotic analyses of JSQ policies assumed servers to have service rates from a finite set which does not capture scenarios in modern data centers where service rates can vary at the level of individual servers. For systems with individually heterogeneous servers, tracking the empirical queue length distribution for each possible service rate becomes infeasible. To overcome this difficulty, we develop a framework based on measure-valued processes and provide a unified analysis of all JSQ-based load balancing policies under general tie-breaking rules. Our analysis identifies two key objects that distinguish the performance of different tie-breaking rules in the Halfin--Whitt regime, namely, the limiting fairness process which describes how idle servers are distributed across different service rates and the limiting routing measure which describes how arriving jobs are assigned across different service rates. In addition to characterizing the diffusion limits for different tie-breaking rules, we identify the tie-breaking rule which asymptotically minimizes the steady-state distributions of the diffusion-scaled total number of jobs and the diffusion-scaled number of waiting jobs. In proving these results, we develop crucial coupling-based sample-path comparisons which provide both policy-independent steady-state bounds and lower bounds to prove asymptotic optimality.

math.PR↗

arXiv2025

Many-Server Queueing Systems with Heterogeneous Strategic Servers in Heavy Traffic

In most service systems, the servers are humans who desire to experience a certain level of idleness. In call centers, this manifests itself as the call avoidance behavior, where servers strategically adjust their service rate to strike a balance between the idleness they receive and effort to work harder. Moreover, being humans, each server values this trade-off differently and has different capabilities. Drawing ideas on mean-field games we develop a novel framework relying on measure-valued processes to simultaneously address strategic server behavior and inherent server heterogeneity in service systems. This framework enables us to extend the recent literature on strategic servers in four new directions by: (i) incorporating individual choices of servers, (ii) incorporating individual abilities of servers, (iii) modeling the discomfort experienced by servers due to low levels of idleness, and (iv) considering more general routing policies. Using our framework, we are able to asymptotically characterize asymmetric Nash equilibria for many-server systems with strategic servers. In simpler cases, it has been shown that the purely quality-driven regime is asymptotically optimal. However, we show that if the discomfort increases fast enough as the idleness approaches zero, the quality-and-efficiency-driven regime and other quality driven regimes can be optimal. This is the first time this conclusion appears in the literature.

math.PR↗

arXiv2024

Asymptotic Optimality of the Speed-Aware Join-the-Shortest-Queue in the Halfin-Whitt Regime for Heterogeneous Systems

The Join-the-Shortest-Queue (JSQ) load balancing scheme is known to minimise the average response time of jobs in homogeneous systems with identical servers. However, for {\em heterogeneous} systems with servers having different processing speeds, finding an optimal load balancing scheme remains an open problem for finite system sizes. Recently, for systems with heterogeneous servers, a variant of the JSQ scheme, called the {\em Speed-Aware-Join-the-Shortest-Queue (SA-JSQ)} scheme, has been shown to achieve asymptotic optimality in the fluid-scaling regime where the number of servers $n$ tends to infinity but the normalised the arrival rate of jobs remains constant. {In this paper, we show that the SA-JSQ scheme is also asymptotically optimal for heterogeneous systems in the {\em Halfin-Whitt} traffic regime where the normalised arrival rate scales as $1-O(1/\sqrt{n})$.} Our analysis begins by establishing that an appropriately scaled and centered version of the Markov process describing system dynamics weakly converges to a two-dimensional reflected {\em Ornstein-Uhlenbeck (OU) process}. We then show using {\em Stein's method} that the stationary distribution of the underlying Markov process converges to that of the OU process as the system size increases by establishing the validity of interchange of limits. {Finally, through coupling with a suitably constructed system, we show that SA-JSQ asymptotically minimises the diffusion-scaled total number of jobs and the diffusion-scaled number of waiting jobs in the steady-state in the Halfin-Whitt regime among all policies which dispatch jobs based on queue lengths and server speeds.}

math.PR↗

arXiv2019

Many-Server Queues with Random Service Rates in the Halfin-Whitt Regime: A Measure-Valued Process Approach

We consider many-server queueing systems with heterogeneous exponential servers and renewal arrivals. The service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy traffic limit of these queues in random environment using probability measure-valued stochastic processes. We introduce the measure-valued fairness process which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these fairness processes do not converge in the usual Skorokhod-$J_1$ topology, hence we introduce a new notion of convergence based on shifted versions of these processes. We also introduce some useful martingales to identify limiting fairness processes under different routing policies.

math.PR↗