Product Forms for FCFS Queueing Models with Arbitrary Server-Job Compatibilities: An Overview

Gardner, Kristen; Righter, Rhonda

doi:10.48550/arxiv.2006.05979

Cited by 5 publications

(12 citation statements)

References 32 publications

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“…The patience of that supply is distributed as the mixture of two exponential random variables with rates θ s,1 and θ s,2 , which is no longer exponentially distributed. On the other hand, under the expanded state space considered in Adan et al [2018] which tracks the complete arrival sequence of supply of different types, one can derive a product-form solution by directly incorporating reneging rates (see Theorem 3.13 in Gardner and Righter [2020]). An interesting direction could be developing a new state space representation which preserves Markovian properties under type-dependent reneging rates.…”

Section: Discussionmentioning

confidence: 99%

“…We believe our ideas can be useful in the analysis more complex topologies. Note that a product form solution under general matching graphs with reneging can be obtained using the expanded state space representation in Adan et al [2018] and Gardner and Righter [2020] which tracks the entire arrival sequence. However, this state representation might not be tractable for computation.…”

Section: Discussionmentioning

confidence: 99%

“…Since the classic work of Jackson [1957,1963], researchers have pursued product form solutions for a diverse range of settings such as systems with redundant requests [Gardner et al, 2016], infinite bipartite matching system [Adan et al, 2017], and stochastic matching on general graphs [Moyal et al, 2017]. We refer readers to a recent overview by Gardner and Righter [2020]. Closest to our work are papers that study the parallel FCFS queues in the context of production and service systems.…”

Section: Related Workmentioning

confidence: 99%

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Matching Queues with Reneging: a Product Form Solution

Castro

Nazerzadeh

Yan

2020

Preprint

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Motivated by growing applications in two-sided markets, we study a parallel matching queue with reneging. Demand and supply units arrive to the system and are matched in an FCFS manner according to a compatibility graph specified by an N-system. If they cannot be matched upon arrival, they queue and may abandon the system as time goes by. We derive explicit product forms of the steady state distributions of this system by identifying a partial balance condition.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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Matching Queues with Reneging: a Product Form Solution

Castro

Nazerzadeh

Yan

2020

Preprint

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“…Over the last decade there has been interest in the performance analysis of policies that dispatch to the shortest among a set of queues, for instance, power-of-two-choices and join-the-idle-queue; see [40] for a survey. However, more recently there has been an increased focus on job replication and in the following, we review existing replication approaches; for further details, we refer the reader to the recent survey [25].…”

Section: Literature Reviewmentioning

confidence: 99%

Stability and Optimization of Speculative Queueing Networks

Anselmi¹,

Walton²

2021

Preprint

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We provide a queueing-theoretic framework for job replication schemes based on the principle "replicate a job as soon as the system detects it as a straggler". This is called job speculation. Recent works have analyzed replication on arrival, which we refer to as replication. Replication is motivated by its implementation in Google's BigTable. However, systems such as Apache Spark and Hadoop MapReduce implement speculative job execution. The performance and optimization of speculative job execution is not well understood. To this end, we propose a queueing network model for load balancing where each server can speculate on the execution time of a job. Specifically, each job is initially assigned to a single server by a frontend dispatcher. Then, when its execution begins, the server sets a timeout. If the job completes before the timeout, it leaves the network, otherwise the job is terminated and relaunched or resumed at another server where it will complete. We provide a necessary and sufficient condition for the stability of speculative queueing networks with heterogeneous servers, general job sizes and scheduling disciplines. We find that speculation can increase the stability region of the network when compared with standard load balancing models and replication schemes. We provide general conditions under which timeouts increase the size of the stability region and derive a formula for the optimal speculation time, i.e., the timeout that minimizes the load induced through speculation. We compare speculation with redundant-d and redundant-to-idle-queue-d rules under an S&X model. In light load, redundancy schemes provide better response times. However, for moderate to heavy loadings, redundancy schemes can lose capacity and have markedly worse response times when compared with a speculative scheme.

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“…Several recent works [1,2,3,15] pointed out the similarity between the stationary distribution of stochastic matching models under this first-come-first-matched policy and that of order-independent queues [5], a product-form queueing model that recently gained momentum in the queueing-theory community. Our first main contribution consists of formalizing the relation between stochastic non-bipartite matching models and order-independent loss queues, a loss variant of order-independent queues introduced in [6].…”

Section: Introductionmentioning

confidence: 99%

Stochastic Non-Bipartite Matching Models and Order-Independent Loss Queues

Comte

2020

Preprint

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The need for matching items with one another while meeting assignment constraints or preferences gave rise to several well-known problems like the stable marriage and roommate problems. While most of the literature on matching problems focuses on a static setting with a fixed number of items, several recent works incorporated time by considering stochastic models, in which items of different classes arrive according to independent Poisson processes and assignment constraints are described by an undirected non-bipartite graph on the classes. In this paper, we prove that the continuous-time Markov chain associated with this model has the same transition diagram as in a product-form queueing model called an order-independent loss queue. This allows us to adapt existing results on order-independent queues to stochastic matching models, and in particular to derive closed-form expressions for several performance metrics, like the waiting probability or the mean matching time, that can be implemented using dynamic programming. Both these formulas and the numerical results that they allow us to derive are used to gain insight into the impact of parameters on performance. In particular, we characterize performance in a so-called heavy-traffic regime in which the number of items of a subset of the classes goes to infinity while items of other classes become scarce.

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Product Forms for FCFS Queueing Models with Arbitrary Server-Job Compatibilities: An Overview

Cited by 5 publications

References 32 publications

Matching Queues with Reneging: a Product Form Solution

Matching Queues with Reneging: a Product Form Solution

Stability and Optimization of Speculative Queueing Networks

Stochastic Non-Bipartite Matching Models and Order-Independent Loss Queues

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