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The fundamental problem in the study of parallel-server systems is that of finding and analyzing routing policies of arriving jobs to the servers that efficiently balance the load on the servers. The most well-studied policies are (in decreasing order of efficiency) join the shortest workload (JSW), which assigns arrivals to the server with the least workload; join the shortest queue (JSQ), which assigns arrivals to the smallest queue; the power-of-[Formula: see text] (PW([Formula: see text])), which assigns arrivals to the shortest among [Formula: see text] queues that are sampled from the total of [Formula: see text] queues uniformly at random; and uniform routing, under which arrivals are routed to one of the [Formula: see text] queues uniformly at random. In this paper we study the stability problem of parallel-server systems, assuming that routing errors may occur, so that arrivals may be routed to the wrong queue (not the smallest among the relevant queues) with a positive probability. We treat this routing mechanism as a probabilistic routing policy, named a [Formula: see text]-allocation policy, that generalizes the PW([Formula: see text]) policy, and thus also the JSQ and uniform routing, where [Formula: see text] is an [Formula: see text]-dimensional vector whose components are the routing probabilities. Our goal is to study the (in)stability problem of the system under this routing mechanism, and under its “nonidling” version, which assigns new arrivals to an idle server, if such a server is available, and otherwise routes according to the [Formula: see text]-allocation rule. We characterize a sufficient condition for stability, and prove that the stability region, as a function of the system’s primitives and [Formula: see text], is in general smaller than the set [Formula: see text]. Our analyses build on representing the queue process as a continuous-time Markov chain in an ordered space of [Formula: see text]-dimensional real-valued vectors, and using a generalized form of the Schur-convex order.
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This content will become publicly available on June 30, 2026
Invariant States of Hydrodynamic Limits of Randomized Load-Balancing Networks
Randomized load-balancing algorithms play an important role in improving performance in large-scale networks at relatively low computational cost. A common model of such a system is a network of N parallel queues in which incoming jobs with independent and identically distributed service times are routed on arrival using the join-the-shortest-of-d-queues routing algorithm. Under fairly general conditions, it was shown by Aghajani and Ramanan that as the size of the system goes to infinity, the state dynamics converge to the unique solution of a countable system of coupled deterministic measure-valued equations called the hydrodynamic equations. In this article, a characterization of invariant states of these hydrodynamic equations is obtained and, when d=2, used to construct a numerical algorithm to compute the queue length distribution and mean virtual waiting time in the invariant state. Additionally, it is also shown that under a suitable tail condition on the service distribution, the queue length distribution of the invariant state exhibits a doubly exponential tail decay, thus demonstrating a vast improvement in performance over the case [Formula: see text], which corresponds to random routing, when the tail decay could even be polynomial. Furthermore, numerical evidence is provided to support the conjecture that the invariant state is the limit of the steady-state distributions of the N-server models. The proof methodology, which entails analysis of a coupled system of measure-valued equations, can potentially be applied to other many-server systems with general service distributions, where measure-valued representations are useful.
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- Award ID(s):
- 2246838
- PAR ID:
- 10627409
- Publisher / Repository:
- INFORMS
- Date Published:
- Journal Name:
- Mathematics of Operations Research
- ISSN:
- 0364-765X
- Subject(s) / Keyword(s):
- load balancing power of two choices stochastic network many-server queue fluid limit hydrodynamic limit measure-valued processes randomized algorithms invariant state equilibrium distribution cloud computing
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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