Supermarket model on graphs

Budhiraja, Amarjit; Mukherjee, Debankur; Wu, Ruoyu

doi:10.1214/18-aap1437

Cited by 41 publications

(52 citation statements)

References 35 publications

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“…Specific results for power of d choices policies when h N =N are presented in Graham [12], see also Luczak and McDiarmid [21]. Budhiraja et al [3] considers also such a local interaction for a queueing network with an underlying graph which is, possibly, random, with jumps of size 1. In this setting when the graph is deterministic, the mean-field analysis can be carried out by using standard arguments.…”

Section: Resultsmentioning

confidence: 99%

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Analysis of large urn models with local mean-field interactions

Sun¹,

Pierpoint²

2019

Electron. J. Probab.

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The stochastic models investigated in this paper describe the evolution of a set of F N identical balls scattered into N urns connected by an underlying symmetrical graph with constant degree h N . After some random amount of time all the balls of any urn are redistributed locally, among the h N urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality h N of the range of interaction is not necessarily linear with respect to N as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results.When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of d choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of d choices policies.

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Section: Resultsmentioning

confidence: 99%

“…In particular H N (1)=H N , one denotes by h N the cardinality of H N . Relation (3) gives the property that the associated graph is symmetrical that is, if i∈H N (j) then j∈H N (i). We give some simple situations of graphs satisfying these assumptions.…”

Section: The Stochastic Modelmentioning

confidence: 99%

Analysis of large urn models with local mean-field interactions

Sun¹,

Pierpoint²

2019

Electron. J. Probab.

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“…Suppose that A(t−) = ∅. By (14) and the fact that G N is connected, we know that either |S(t−)| = 0 or |S(t−)| = N , contradicting (11). Thus, A(t−) = ∅.…”

Section: Proof Of Lemma A1mentioning

confidence: 97%

“…In the context of load balancing, Mukherjee et al [27] studied the join-the-shortest queue (JSQ) policy on general graphs, and showed that this policy is optimal in some sense, as long as the expected degree diverges fast enough as a function of the number of servers. Similarly, Budhiraja et al [10] studied the local power-of-d scheme on graphs, and showed that as long as the minimum degree of the graph diverges, and the degree-weighted adjacency matrix of the graph is asymptotically doubly-stochastic, the occupancy process converges to the same system of ODEs as that on the complete graph.…”

Section: Related Workmentioning

confidence: 99%

“…• Propagation of chaos: In general, a powerful stepping stone to establishing mean-field approximation results when the network/graph is not symmetric, is the asymptotic independence of the local state of the agents in the graph (this is also called "propagation of chaos"). This has been done in the context of weakly interacting particles [6], queueing systems [10,8,40], and in other general systems [19,37].…”

Section: Related Workmentioning

confidence: 99%

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Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory

SuLili

ZubeldiaMartin

LynchNancy

2019

Proc. ACM Meas. Anal. Comput. Syst.

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We consider multi-armed bandit problems in social groups wherein each individual has bounded memory and shares the common goal of learning the best arm/option. We say an individual learns the best option if eventually (as t → ∞) it pulls only the arm with the highest expected reward. While this goal is provably impossible for an isolated individual due to bounded memory, we show that, in social groups, this goal can be achieved easily with the aid of social persuasion (i.e., communication) as long as the communication networks/graphs satisfy some mild conditions. In this work, we model and analyze a type of learning dynamics which are well-observed in social groups. Specifically, under the learning dynamics of interest, an individual sequentially decides on which arm to pull next based on not only its private reward feedback but also the suggestion provided by a randomly chosen neighbor. To deal with the interplay between the randomness in the rewards and in the social interaction, we employ the mean-field approximation method. Considering the possibility that the individuals in the networks may not be exchangeable when the communication networks are not cliques, we go beyond the classic mean-field techniques and apply a refined version of mean-field approximation:• Using coupling we show that, if the communication graph is connected and is either regular or has doubly-stochastic degree-weighted adjacency matrix, with probability → 1 as the social group size N → ∞, every individual in the social group learns the best option.• If the minimum degree of the graph diverges as N → ∞, over an arbitrary but given finite time horizon, the sample paths describing the opinion evolutions of the individuals are asymptotically independent. In addition, the proportions of the population with different opinions converge to the unique solution of a system of ODEs. Interestingly, the obtained system of ODEs are invariant to the structures of the communication graphs. In the solution of the obtained ODEs, the proportion of the population holding the correct opinion converges to 1 exponentially fast in time.Notably, our results hold even if the communication graphs are highly sparse.

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Stability of JSQ in queues with general server-job class compatibilities

2020

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We consider Poisson streams of exponentially distributed jobs arriving at each edge of a hypergraph of queues. Upon arrival, an incoming job is rooted to the shortest queue among the corresponding vertices. This generalizes many known models such as power-of-d load balancing and JSQ (join the shortest queue) on generic graphs. We provide a generic condition for stability of this model. We show that some graph topologies lead to a loss of capacity, implying more restrictive stability conditions than in, e.g., complete graphs.

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Supermarket model on graphs

Cited by 41 publications

References 35 publications

Analysis of large urn models with local mean-field interactions

Analysis of large urn models with local mean-field interactions

Collaboratively Learning the Best Option on Graphs, Using Bounded Local Memory

Stability of JSQ in queues with general server-job class compatibilities

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