A note on insensitivity in stochastic networks

Journal of Applied Probability (Impact Factor: 0.59). 12/2006; 44(1). DOI: 10.1239/jap/1175267175
Source: arXiv


We give a simple and direct treatment of insensitivity in stochastic networks which is quite general and which provides probabilistic insight into the phenomenon. In the case of multi-class networks, the results generalise those of Bonald and Proutiere (2002, 2003).

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    • "Kang et al. (2007), Kang et al. (2009), Kelly et al. (2009) and Ye and Yao (2012) discussed product form resource pooling properties associated with the proportional fairness in heavy traffic and large deviations regimes. Influenced by Whittle (1985), the quasi-reversibility and insensitivity property in Massoulié networks was studied by Bonald andProutì ere (2002) (2003) (2004) and Zachary (2007). For connections between proportional fairness and the queueing networks of Kelly (1975) and Baskett et al. (1975), see, e.g., Schweitzer (1979); Kelly (1989); Massoulié and Roberts (1999); Walton (2009), and Anselmi et al. (2013). "
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    ABSTRACT: We consider a family of discrete time multihop switched queueing networks where each packet moves along a fixed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy has the virtues of possessing a maximal stability region and not requiring explicit knowledge of traffic arrival rates. BackPressure has certain structural weaknesses because implementation requires information about each route, and queueing delays can grow super-linearly with route length. For large networks, where packets over many routes are processed by a queue, or where packets over a route are processed by many queues, these limitations can be prohibitive. In this article, we introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which is based on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler has a maximal stability region and does not require explicit knowledge of traffic arrival rates. The Proportional Scheduler has the advantage that information about the network's route structure is not required for scheduling, which substantially improves the policy's performance for large networks. For instance, packets can be routed with only next-hop information and new nodes can be added to the network with only knowledge of the scheduling constraints.
    Full-text · Article · Dec 2014
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    • "More general systems – with infinitely many servers and/or with more involved disciplines of serving – were studied further in [16], [17], [22], [31], [33], [37], [43], [51], [52], et al. Even quite recently, results in this direction were still under investigation under the name of " insensitivity " of a stationary regime (i.e., where there are some general invariants of a stationary distribution, which depend on the service time distribution only through its mean value) for advanced versions of Erlang type models in [1], [5], [36], [58], [60]. Note that most of these papers – with the exception of [22] and [52] – do not cite two other pioneering publications [33]–[34] and none of them except [16] tackles convergence rates; in the latter paper, the result about convergence rate bounds could be called partial in comparison to our Theorem 1 below. "
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    ABSTRACT: Polynomial convergence rates in total variation are established in Erlang--Sevastyanov's type problem with an infinite number of servers and a general distribution of service under assumptions on the intensity of serving.
    Preview · Article · Oct 2013 · Queueing Systems
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    • "Bonald and Proutiére [4] established that it induces product-form stationary distributions and is insensitive with respect to phase-type distributions. This policy is shown to be insensitive for general service time distributions, including the deterministic service considered here, by Zachary [37]. The relation between this policy, the proportionally fair allocation, and multiclass queueing networks is discussed in depth by Walton [34] and Kelly et al. [16]. "
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    ABSTRACT: We consider a switched (queueing) network in which there are constraints on which queues may be served simultaneously; such networks have been used to effectively model input-queued switches and wireless networks. The scheduling policy for such a network specifies which queues to serve at any point in time, based on the current state or past history of the system. In the main result of this paper, we provide a new class of online scheduling policies that achieve optimal average queue-size scaling for a class of switched networks including input-queued switches. In particular, it establishes the validity of a conjecture about optimal queue-size scaling for input-queued switches.
    Preview · Article · Oct 2011 · The Annals of Applied Probability
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