Stability of queueing networks

Bramson, Maury

doi:10.1214/08-ps137

Cited by 115 publications

(233 citation statements)

References 72 publications

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“…Related material for queueing networks is given in Bramson [3] and, for a general family of join the shortest queue networks, in Bramson [4]. Because of the similarity of these settings, we present a summary here and refer the reader to [4] for more detail.…”

Section: Markov Process Backgroundmentioning

confidence: 99%

Asymptotic independence of queues under randomized load balancing

2012

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Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-αN Poisson process, with α < 1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. [19], it was shown that when each arriving job is assigned to the shortest of D, D ≥ 2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N → ∞. This is a substantial improvement over the case D = 1, where queue sizes decay exponentially.The reasoning in [19] does not easily generalize to jobs with nonexponential service time distributions. A modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. [5]. The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N → ∞. This allows computation of queue size distributions and other performance measures of interest.In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult prob- lem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.

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Section: Markov Process Backgroundmentioning

confidence: 99%

Asymptotic independence of queues under randomized load balancing

2012

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“…In fact it is shown in Lemma 2 that Assumption (A2b) implies that all closed bounded sets A = {x : |x| ≤ m} are uniformly small. This implies that under (A2b) X is not only positive Harris recurrent, but is actually ergodic (see Theorem 4.3 in [5]). …”

Section: B(s)mentioning

confidence: 94%

“…If X(t) is positive Harris recurrent and if for some closed set A the recurrence time τ A satisfies P x (τ A < ∞) = 1 and A is uniformly small then X(t) is ergodic (see Theorem 4.3 in [5]). …”

Section: B(s)mentioning

confidence: 99%

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

Nazarathy

Weiss

2010

Performance Evaluation

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We consider a push pull queueing network with two servers and two types of jobs which are processed by the two servers in opposite order, with stochastic generally distributed processing times. This push pull network was introduced by Kopzon and Weiss, who assumed exponential processing times. It is similar to the KumarSeidman Rybko-Stolyar (KSRS) multi-class queueing network, with the distinction that instead of random arrivals, there is an infinite supply of jobs of both types. Unlike the KSRS network, we can find policies under which our push pull network works at full utilization, with both servers busy at all times, and without being congested. We perform fluid and diffusion scale analysis of this network under such policies, to show fluid stability, positive Harris recurrence, and to obtain a diffusion limit for the network. On the diffusion scale the network is empty, and the departures of the two types of jobs are highly negatively correlated Brownian motions. Using similar methods we also derive a diffusion limit of a re-entrant line with infinite supply of work.

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“…Queueing networks theory is used to model computer networks, telecommunication systems, distribution chains and warehouses, the manufacturing of products in factories, etc. (see [Bra08]). The real-time systems that are described in this thesis can be modeled as queueing networks where workers are tasks, jobs are packets, and resources are stations.…”

Section: Link With Queueing Networkmentioning

confidence: 96%

Stability of Adaptive Distributed Real-TimeSystems with Dynamic Resource Management

Rafiliu¹

2013

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T oday's embedded distributed real-time systems, are exposed to large variations in resource usage due to complex software applications, sophisticated hardware platforms, and the impact of their run-time environment. As efficiency becomes more important, the applications running on these systems are extended with on-line resource managers whose job is to adapt the system in the face of such variations. Distributed systems are often heterogeneous, meaning that the hardware platform consists of computing nodes with different performance, operating systems, and scheduling policies, linked through one or more networks using different protocols.In this thesis we explore whether resource managers used in such distributed embedded systems are stable, meaning that the system's resource usage is controlled under all possible run-time scenarios. Stability implies a bounded worst-case behavior of the system and can be linked with classic real-time systems' properties such as bounded response times for the software applications. In the case of distributed systems, the stability problem is particularly hard because software applications distributed over the different resources generate complex, cyclic dependencies between the resources, that need to be taken into account. In this thesis we develop a detailed mathematical model of an adaptive, distributed real-time system and we derive conditions that, if satisfied, guarantee its stability. vii viii Popupärvetenskaplig sammanfattning V iär omgivna av ett ständigtökande antal inbyggda datorsystem. De finns exempelvis i våra bilar, telefoner, fotokameror och tvättmaskiner. Vår förväntningär att dessa produkter ska vara säkra, effektiva samt hålla hög kvalitet på de tjänster de tillhandahåller. Detta innebär bland annat att de datorsystem som finns inbyggda i våra bilar inte får leda till att passagerare skadas eller i värsta fall mister sina liv. Av en telefon förväntar vi oss lång batteritidäven om vi använder dessa dagligen för att prata och skicka SMS, lyssna på musik, ta bilder, läsa nyheter och skicka e-post. Vi förväntar oss samtidigt att de tjänster och funktioner som finns i våra mobiltelefoner håller hög kvalitet. Alla våra krav och förväntningar kan endast uppfyllas om systemens inbyggda resurser används på ett effektivt sätt. Exempel på sådana resurserär processorer, minnen, batterier och kommunikationsmedier.I denna avhandling beskrivs hur resurser i inbyggda datorsystem bör användas och kontrolleras för att optimera dess tjänster. Vidare tas hänsyn till att ett modernt system har stora variationer i resursanvändning. Detta beror delvis på att det mångfald av tjänster som finns i dagens moderna datorsystem i allra högsta grad består av komplex programvara. Detta beroräven på den komplicerade och sofistikerade hårdvara som behövs för att stödja all inbyggd programvara ix x samt påverkan från systemets omgivning.För att kunna förlita oss på dessa inbyggda systemär det nödvändigt att studera huruvida dess resurshanterareär stabila. Denna avhandling presenterar ...

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Stability of queueing networks

Cited by 115 publications

References 72 publications

Asymptotic independence of queues under randomized load balancing

Asymptotic independence of queues under randomized load balancing

Positive Harris recurrence and diffusion scale analysis of a push pull queueing network

Stability of Adaptive Distributed Real-TimeSystems with Dynamic Resource Management

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