Transform Methods for Heavy-Traffic Analysis

Hurtado-Lange, Daniela; Maguluri, Siva Theja

doi:10.1287/stsy.2019.0056

Cited by 26 publications

(34 citation statements)

References 63 publications

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“…We present two different proof techniques to study the two regimes. To establish the result in the quality-driven regime, we generalize the characteristic function method, which was first developed in [25]. To study the critical regime, we develop the inverse Fourier transform method.…”

Section: Main Contributionsmentioning

confidence: 99%

“…One of these methods is the Drift method introduced in [11] which analyzes the single server queue. This method was further generalized to analyze switch [29], flexible load balancing [41], generalized switch [23] systems include transform method [25], basic adjoint relationship (BAR) method [7], and Stein's method [16]. In addition, a single-server queue with state-dependent control was studied in [26].…”

Section: Literature Reviewmentioning

confidence: 99%

“…In addition, a single-server queue with state-dependent control was studied in [26]. Among these methodologically, the closest to the current work is the transform method presented in [25].…”

Section: Literature Reviewmentioning

confidence: 99%

“…Step 2: The key idea in the proof is to use as the test function for ∈ R and set its drift to zero in steady state. While this step is similar to the transform method in [25], the key challenge is that when we let ↑ ∞, we do not get an explicit expression for the characteristic function of ¯ ∞ . We instead get that the limit of every convergent sub sequence satisfies the following implicit equation:…”

Section: Sketch Of Proofsmentioning

confidence: 99%

“…Moreover, the mean of the exponential depends only on the variance of the inter-arrival and service distributions, but not on the whole distribution. Recent work [11,15,25] has also characterized the rate of convergence to the exponential, thus enabling us to approximate the stationary queue length when is not small. Recent developments in online platforms and matching markets such as ride-hailing, food delivery services, etc have led to an interest in the study of two-sided queues.…”

Section: Introductionmentioning

confidence: 99%

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A Heavy Traffic Theory of Matching Queues

Varma¹,

Maguluri²

2021

Preprint

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Motivated by emerging applications in online matching platforms and marketplaces, we study a two-sided queue. Customers and servers that arrive into a two-sided queue depart as soon as they are matched. It is known that a state-dependent control is needed to ensure the stability of a two-sided queue. However, analytically studying the steady-state behaviour of a two-sided queue, in general, is challenging. Therefore, inspired by the heavy-traffic regime in classical queueing theory, we study a two-sided queue in an asymptotic regime where the control decreases to zero. It turns out that there are two different ways the control can be sent to zero, and we model these using two parameters viz., that goes to zero and that goes to infinity. Intuitively, modulates the magnitude of the control and is the threshold after which we modulate the control.We show that depending on the relative rates of and , there is a phase transition in the limiting regime. We christen the regime when → 0, the quality-driven regime, and the limiting behaviour is a Laplace distribution. The phase transition starts in the regime when, goes to a nonzero constant when the limiting distribution is a Gibbs distribution, and so we call it the critical regime. When → ∞, we conjecture that the limiting distribution is uniform and prove that in a special case. We call this the profit-driven regime. These results are established using two related proof techniques. The first one is a generalization of the characteristic function method, and the second is a novel inverse Fourier transform method.

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Section: Main Contributionsmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Sketch Of Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Heavy Traffic Theory of Matching Queues

Varma¹,

Maguluri²

2021

Preprint

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A Mathematical Model for Cloud-Based Scheduling Using Heavy Traffic Limit Theorem in Queuing Process

Rashidifar

Chen

Bouzary

et al. 2022

Lecture Notes in Mechanical Engineering

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Cloud manufacturing (CMfg) is a service-oriented manufacturing paradigm that distributes resources in an on-demand business model. In the cloud manufacturing environment, scheduling is considered as an effective tool for satisfying customer requirements which has attracted attention from researchers. In this case, quality of service (QoS) in the scheduling plays a vital role in assessing the impacts of the distributed resources in operation on the performance of scheduling functions. In this paper, a queuing system is employed to model the scheduling problem with multiple servers and then scheduling in cloud manufacturing is classified based on various QoS requirements. Moreover, a set of heavy traffic limit theorems is introduced as a new approach to solving this scheduling problem in which different heavy traffic limits are provided for each of QoS-based scheduling classes. Finally, the number of operational resources in the scheduling is determined by considering the results obtained in the numerical analysis of the heavy traffic limit with different queue disciplines. The results show that different numbers of active machines in various QoS requirements classes play a vital role in that the required QoS metrics such as the expected waiting time and the expected completion time which are critical performance indicators of the cloud’s service are intimately related.

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Job assignment in large-scale service systems with affinity relations

2019

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We consider load balancing in service systems with affinity relations between jobs and servers. Specifically, an arriving job can be allocated to a fast, primary server from a particular selection associated with this job or to a secondary server to be processed at a slower rate. Such job-server affinity relations can model network topologies based on geographical proximity, or data locality in cloud scenarios. We introduce load balancing schemes that allocate jobs to primary servers if available, and otherwise to secondary servers. A novel coupling construction is developed to obtain stability conditions and performance bounds using a coupling technique. We also conduct a fluid limit analysis for symmetric model instances, which reveals a delicate interplay between the model parameters and load balancing performance.

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Transform Methods for Heavy-Traffic Analysis

Cited by 26 publications

References 63 publications

A Heavy Traffic Theory of Matching Queues

A Heavy Traffic Theory of Matching Queues

A Mathematical Model for Cloud-Based Scheduling Using Heavy Traffic Limit Theorem in Queuing Process

Job assignment in large-scale service systems with affinity relations

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