Piecewise linear test functions for stability of queueing networks

Down, Douglas G.; Meyn, Sean

doi:10.1109/cdc.1994.411432

Cited by 55 publications

(62 citation statements)

References 11 publications

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“…Here b,. denotes the bandwidth, in bits/s, allocated to route r. With this change of variables one finds that (9) from which it follows that b* (c~x) = b* (x) and since U~(x) =…”

Section: Wrn R P-------~-----__ Ar Re'rmentioning

confidence: 99%

“…It is easy to see that this gradient exists almost everywhere, and, when it exists, it equals ~e, for some g. In order to obtain an appropriate Lyapunov function in Lemma 3.2 below we follow the result of [9] which is based on showing the existence of a smoothened version W of the function V that satisfies a drift condition in the sense of (18) for all x E R~. The proof of this lemma follows [9] and can be found in [18]. Given this results we can show that the network is indeed positive recurrent as follows.…”

Section: Me_c 34¢0mentioning

confidence: 99%

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Stability and performance analysis of networks supporting services with rate control-could the Internet be unstable?

Veciana

Lee

Konstantopoulos³

1999

IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer A

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Abstract--We consider the stability and performance of a model for networks supporting services that adapt their transmission to the available bandwidth. Not unlike real networks, in our model, connection arrivals are stochastic, each has a random amount of data to send, and the number of ongoing connections in the system changes over time. Consequently, the bandwidth allocated to, or throughput achieved by, a given connection may change during its lifetime as feedback control mechanisms react to network loads. Ideally, if there were a fixed number of ongoing connections, such feedback mechanisms would reach an equilibrium bandwidth allocation typically characterized in terms of its "fairness" to users, e.g., max-min or proportionally fair. In this paper we prove the stability of such networks when the offered load on each link does not exceed its capacity. We use simulation to investigate performance, in terms of average connection delays, for various fairness criteria. Finally, we pose an architectural problem in TCP/IPs decoupling of the transport and network layer from the point of view of guaranteeing connection-level stability, which we claim may explain congestion phenomena on the Internet.Index Terms--ABR service, bandwidth allocation, Lyapunov functions, performance analysis, proportional fairness, rate control, stability, TCP/IP, weighted max-min fairness.

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“…Here b,. denotes the bandwidth, in bits/s, allocated to route r. With this change of variables one finds that (9) from which it follows that b* (c~x) = b* (x) and since U~(x) =…”

Section: Wrn R P-------~-----__ Ar Re'rmentioning

confidence: 99%

Section: Me_c 34¢0mentioning

confidence: 99%

Stability and performance analysis of networks supporting services with rate control-could the Internet be unstable?

Veciana

Lee

Konstantopoulos³

1999

IEEE INFOCOM '99. Conference on Computer Communications. Proceedings. Eighteenth Annual Joint Conference of the IEEE Computer A

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“…We formulate the problem of determining the coe -cients of the Lyapunov function as a linear programming problem, which has unbounded objective values only if the coe cients and hence the Lyapunov function exist. Our linear program arises directly from the piecewise linear Lyapunov function introduced in Dai and Weiss (1996), which generalizes that of Botvich and Zamyatin (1992) and is simpler than that independently formulated by Down and Meyn (1994).…”

Section: Introductionmentioning

confidence: 99%

The Stability of Two-Station Multitype Fluid Networks

Dai

Vate

2000

Operations Research

118

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This paper studies the uid models of two-station multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is deÿned by the nominal workload conditions and the "virtual workload conditions," and we introduce two intuitively appealing phenomena-virtual stations and push starts-that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a uid solution that cycles to inÿnity, showing that the uid network is unstable. When all the workload conditions are satisÿed, we solve a network ow problem to ÿnd the coe cients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the uid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the interarrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding uid network is stable. Thus, the workload conditions are su cient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.

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“…In all cases, it was established that a multiclass network is stable if certain linear programming problems are bounded. To the best of our knowledge the sharpest such conditions are those of [7] and [8] obtained through the use of piecewise linear convex potential functions. For some specific examples (for example in [3]), the conditions obtained are indeed sharp.…”

mentioning

confidence: 99%

“…Kumar and Meyn [10] used quadratic potential functions, while Botvich and Zamyatin [3], Dai and Weiss [7], and Down and Meyn [8] used piecewise linear convex potential functions. In all cases, it was established that a multiclass network is stable if certain linear programming problems are bounded.…”

mentioning

confidence: 99%

Stability conditions for multiclass fluid queueing networks

Bertsimas¹,

Gamarnik

Tsitsiklis

1996

IEEE Trans. Automat. Contr.

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We find necessary and sufficient conditions for the stability of all work-conserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class (monotone piecewise linear convex) potential functions. We show that for two-station systems it is not possible for this class of potential function to give the new (sharp) conditions.

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Piecewise linear test functions for stability of queueing networks

Cited by 55 publications

References 11 publications

Stability and performance analysis of networks supporting services with rate control-could the Internet be unstable?

Stability and performance analysis of networks supporting services with rate control-could the Internet be unstable?

The Stability of Two-Station Multitype Fluid Networks

Stability conditions for multiclass fluid queueing networks

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