Stability conditions for multiclass fluid queueing networks

Bertsimas, Dimitris; Gamarnik, David; Tsitsiklis, John N.

doi:10.1109/9.543999

Cited by 97 publications

(37 citation statements)

References 17 publications

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“…In these unstable examples, the total number of jobs in the system goes to inÿnity with time. Other recent work on the stability of queueing networks and uid networks includes Harrison and Nguyen (1995), Bramson (1997Bramson ( , 1999, Bertsimas et al (1996), Dumas (1996Dumas ( , 1997, Dai and Weiss (1996), Foss and Rybko (1995), Winograd and Kumar (1996), , Chen andZhang (1997, 1998), Morrison and Kumar (1998), and Hasenbein (1997).…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Bertsimas et al (1996) showed that a twostation uid network is globally stable if and only if a certain linear program has bounded objective value. In this paper we extend the results of Bertsimas et al by stating explicitly, in terms of the arrival rates and service times, necessary and su cient conditions for a two-station uid network to be globally stable.…”

Section: Introductionmentioning

confidence: 99%

“…The explicit description of necessary and su cient conditions for the global stability of two-station uid networks provides a number of corollaries not immediately available from the linear programming characterization of Bertsimas et al (1996). Most important among these is a complete understanding of global stability in two-station uid networks via virtual stations and push starts.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Stability of Two-Station Multitype Fluid Networks

Dai

Vate

2000

Operations Research

118

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“…The inequality (19) is a Lyapunov drift condition of the form developed in [41,18], and also similar to the bounds used in [11,4,28,26,43] to obtain performance bounds for networks. Under natural assumptions on the model the bound (19) implies that the controlled network is geometrically ergodic, so that the mean E[c(Q(t))] converges to its steady-state value geometrically fast from each initial condition [41,42,40,26,36,19].…”

Section: Work-stoppage Under a Myopic Policymentioning

confidence: 99%

Stability and Asymptotic Optimality of Generalized MaxWeight Policies

Meyn¹

2009

SIAM J. Control Optim.

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It is shown that stability of the celebrated MaxWeight or back pressure policies is a consequence of the following interpretation: either policy is myopic with respect to a surrogate value function of a very special form, in which the "marginal disutility" at a buffer vanishes for vanishingly small buffer population. This observation motivates the h-MaxWeight policy, defined for a wide class of functions h. These policies share many of the attractive properties of the MaxWeight policy:(i) Arrival rate data is not required in the policy.(ii) Under a variety of general conditions, the policy is stabilizing when h is a perturbation of a monotone linear function, a monotone quadratic, or a monotone Lyapunov function for the fluid model.(iii) A perturbation of the relative value function for a workload relaxation gives rise to a myopic policy that is approximately average-cost optimal in heavy traffic, with logarithmic regret.The first results are obtained for a general Markovian network model. Asymptotic optimality is established for a general Markovian scheduling model with a single bottleneck, and homogeneous servers.

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“…(ii) Optimized solutions for problems formulated on the fluid model can lead to efficient solutions for the corresponding problems formulated on the original stochastic network [8,9,10,11]. On the other hand, many of the flow control problems defined in the context of the abstracting fluid networks can be quite challenging themselves, giving rise to some very interesting optimal control problems [12,13]. The recent publication of [14] offers an excellent treatment of the role of fluid models in the control of complex stochastic networks, and it defines the state-of-art in this area.…”

mentioning

confidence: 99%

Optimal flow control in acyclic networks with uncontrollable routings and precedence constraints

Reveliotis

Bountourelis

2011

Discrete Event Dyn Syst

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This paper introduces a novel optimal flow control problem that seeks to convey a specified amount of fluid to each of the nodes of an acyclic digraph with a single source node, while minimizing the total amount of fluid inducted into the network. Two factors complicating the aforementioned task are (i) the presence of nodes with uncontrollable routing of the traversing flow and (ii) a set of precedence constraints regarding the satisfaction of the nodal fluid requirements. It is shown that the considered problem can be naturally formulated as a continuous-time optimal control problem that can be reduced to a hybrid optimal control problem with controlled switching. This property subsequently enables the solution of the considered problem through a Mixed Integer Programming formulation. Additional results in the paper establish the NP-hardness of the considered problem, highlight its affinity to some well known scheduling problems, and offer guidelines that can alleviate the increased problem complexity.

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Stability conditions for multiclass fluid queueing networks

Cited by 97 publications

References 17 publications

The Stability of Two-Station Multitype Fluid Networks

The Stability of Two-Station Multitype Fluid Networks

Stability and Asymptotic Optimality of Generalized MaxWeight Policies

Optimal flow control in acyclic networks with uncontrollable routings and precedence constraints

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