On Queues with Markov Modulated Service Rates

Mahabhashyam, Sai Rajesh; Gautam, Natarajan

doi:10.1007/s11134-005-2158-x

Cited by 55 publications

(39 citation statements)

References 22 publications

(24 reference statements)

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“…In our case, the service speeds depend on an external environment that is governed by a Markov process. Analyses of single-server queueing models with Markov-modulated service speeds can be found in [17,27,29,30,37]. However, none of these papers concern themselves with the derivation of heavy-traffic asymptotics.…”

Section: Introductionmentioning

confidence: 99%

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

2014

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We study a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. The marginals of these limiting distributions are shown to be exponential with rates that can be computed by matrix-analytic methods. Moreover, we show how to numerically compute the joint distributions, by viewing the limit processes as multi-dimensional semi-martingale reflected Brownian motions in the non-negative orthant.

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Section: Introductionmentioning

confidence: 99%

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

2014

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“…The former [Gupta and Harchol-Balter, 2009;Rege and Sengupta, 1985;Zhang and Zwart, 2008] models the execution speed based on the current state of the system, i.e., the combination of the number of jobs and the multiprogramming levels. The latter [Mahabhashyam and Gautam, 2005;Casale and Tribastone, 2013;Zhang and Zwart, 2012;Dorsman et al, 2013] models the transition of execution speeds as Markov-modulated speed for single queue and multiple queues.…”

Section: ¾º¿º½ åó ð ò ì ð ê ×ôóò× ì ñ ×mentioning

confidence: 99%

“…Another related article is one by Massey [Massey, 2002], where the author focuses on deriving the queue process and the time-varying aspect, in particular an M t /M t /1 queueing system. In [Mahabhashyam and Gautam, 2005], the authors generalize the execution speed to any Markov process, and also the tail distribution of the workload in steady state. However, all the aforementioned analysis requires the arrival process to be Markovian.…”

Section: ¾º¿º½ åó ð ò ì ð ê ×ôóò× ì ñ ×mentioning

confidence: 99%

“…Considering varying service rates, the related work centers around two directions: (1) service rates depending on the state of the system [Rege and Sengupta, 1985;Gupta and Harchol-Balter, 2009] and (2) service rates changing due to external environmental processes [Mahabhashyam and Gautam, 2005;Boxma and Kurkova, 2001;Zhang and Zwart, 2012;Dorsman et al, 2013]. Motivated by the fact that the "speed" of a system increases with the number and variability of jobs, Gupta et al [Gupta and Harchol-Balter, 2009] developed the approximation for mean response times for M /G/PS − M P L and G I/G/PS − M P L queues with state-dependent service rate, where MPL denotes the multi-programming limit.…”

Section: º ëùññ öýmentioning

confidence: 99%

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Resource management of replicated service systems provisioned in the cloud

Björkqvist

Birke

Binder

2016

NOMS 2016 - 2016 IEEE/IFIP Network Operations and Management Symposium

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I certify that except where due acknowledgement has been given, the work presented in this thesis is that of the author alone; the work has not been submitted previously, in whole or in part, to qualify for any other academic award; and the content of the thesis is the result of work which has been carried out since the official commencement date of the approved research program. Lugano, 11 February 2015 ×ØÖ Ø Service providers seek scalable and cost-effective cloud solutions for hosting their applications. Despite significant recent advances facilitating the deployment and management of services on cloud platforms, a number of challenges still remain. Service providers are confronted with time-varying requests for the provided applications, inter-dependencies between different components, performance variability of the procured virtual resources, and cost structures that differ from conventional data centers. Moreover, fulfilling service level agreements, such as the throughput and response time percentiles, becomes of paramount importance for ensuring business advantages. Mathias BjörkqvistIn this thesis, we explore service provisioning in clouds from multiple points of view. The aim is to best provide service replicas in the form of VMs to various service applications, such that their tail throughput and tail response times, as well as resource utilization, meet the service level agreements in the most cost effective manner. In particular, we develop models, algorithms and replication strategies that consider multi-tier composed services provisioned in clouds. We also investigate how a service provider can opportunistically take advantage of observed performance variability in the cloud. Finally, we provide means of guaranteeing tail throughput and response times in the face of performance variability of VMs, using Markov chain modeling and large deviation theory. We employ methods from analytical modeling, event-driven simulations and experiments. Overall, this thesis provides not only a multi-faceted approach to exploring several crucial aspects of hosting services in clouds, i.e., cost, tail throughput, and tail response times, but our proposed resource management strategies are also rigorously validated via trace-driven simulation and extensive experiments.

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“…A MAP/MSP/1 queue is studied in [7] using matrix-analytical techniques in terms of the queue length, but waiting time results are not given. The Reference [21] studies a queue whose service speed changes according to an external environment governed by a Markov process.…”

Section: Introductionmentioning

confidence: 99%

System-theoretical algorithmic solution to waiting times in semi-Markov queues

Akar¹,

Sohraby²

2009

Performance Evaluation

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a b s t r a c tMarkov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples.

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On Queues with Markov Modulated Service Rates

Cited by 55 publications

References 22 publications

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

Heavy-traffic asymptotics for networks of parallel queues with Markov-modulated service speeds

Resource management of replicated service systems provisioned in the cloud

System-theoretical algorithmic solution to waiting times in semi-Markov queues

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