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... Theorem 2(ii) only.) 3.4. Preliminary observations We start with a characterization of optimal dual solutions, given below in Lemma 2, which follows from basic convex programming duality theory (cf. =-=[17]-=-, Sections 28 and 30). Suppose non-degeneracy condition (8) holds. By Kuhn-Tucker theorem, for any pair of optimal primal and dual solutions, v ∗ ∈ V ∗ and q ∗ ∈ Q ∗ , the complementary slackness cond...

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...we discuss in some detail in Sections 5.1 and 5.2, is to dynamic congestion control of communication networks. Recent advances in the theory of network congestion control originate from the discovery =-=[10]-=- that the congestion control mechanism employed by the Transport Control Protocol (TCP) in the Internet implicitly attempts to maximize the sum of traffic flow (concave) utilities, subject to the cons...

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...be asymptotically optimal as β ↓ 0in[3, 21]. Ifs404 STOLYAR we remove all commodity flows (so that we are not concerned about any utility), the GPD algorithm becomes a “MaxWeight”-type algorithm (cf. =-=[4,20,22,23]-=-), “greedily” pursuing minimization of Lyapunov function � N p(1/2)Q2n (t), and known to ensure stability of queueing networks if such is feasible at all. (Cf. [4] for recent results and review of pre...

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...function fn(t) iscontinuous, function hn(t) iscontinuous non-decreasing non-negative with hn(0) = 0, and, finally, time t cannot be a point of increase of hn(·) when qn(t) > 0. This implies (15) (cf. =-=[7]-=-, Proposition 2.2.3). Function f satisfying (11), (12) and the first part of (15), is unique because (according to these relations) it is uniquely determined by x and the initial condition f (0) = q(0...

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...k communication links are overloaded. (Note that the latter condition is necessary for stability of such networks.) This is a very large and active field, which we do not attempt to review here. (See =-=[11,14,16,18]-=- for recent reviews of the field.) We will only point out some aspects of congestion control modelling, to which we believe our results contribute, as illustrated by the application examples in Sectio...

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...y limited choices of instantaneously available transmission rates. This is the case when, for example, several network users inject traffic into a network via as406 STOLYAR shared wireless link. (Cf. =-=[1,8,15]-=- for basic features of transmission rate allocation over shared wireless links.) – Similarly, we allow traffic processing nodes in the network to be randomly timevarying, interdependent, with possibly...

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...k communication links are overloaded. (Note that the latter condition is necessary for stability of such networks.) This is a very large and active field, which we do not attempt to review here. (See =-=[11,14,16,18]-=- for recent reviews of the field.) We will only point out some aspects of congestion control modelling, to which we believe our results contribute, as illustrated by the application examples in Sectio...

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...y limited choices of instantaneously available transmission rates. This is the case when, for example, several network users inject traffic into a network via as406 STOLYAR shared wireless link. (Cf. =-=[1,8,15]-=- for basic features of transmission rate allocation over shared wireless links.) – Similarly, we allow traffic processing nodes in the network to be randomly timevarying, interdependent, with possibly...

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...veral examples in Section 5.3 that GPD algorithm provides a solution to a wide class of resource allocation problems in wireless networks, which include power usage and traffic rate constraints. (Cf. =-=[12,24,27]-=- for some problems with power constraints.) We show that GPD algorithm easily and naturally accommodates different system models and objectives by using, if necessary, virtual commodities and/or virtu...

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...e allow traffic processing nodes in the network to be randomly timevarying, interdependent, with possibly limited choices of instantaneously available service rates. (This aspect is also addressed in =-=[6]-=-, which is an independent and contemporaneous work with present paper. See remark in Section 5.1.) – Our models include routing and corresponding dynamics of the network queues. The latter means that ...

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...y limited choices of instantaneously available transmission rates. This is the case when, for example, several network users inject traffic into a network via as406 STOLYAR shared wireless link. (Cf. =-=[1,8,15]-=- for basic features of transmission rate allocation over shared wireless links.) – Similarly, we allow traffic processing nodes in the network to be randomly timevarying, interdependent, with possibly...

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...be asymptotically optimal as β ↓ 0in[3, 21]. Ifs404 STOLYAR we remove all commodity flows (so that we are not concerned about any utility), the GPD algorithm becomes a “MaxWeight”-type algorithm (cf. =-=[4,20,22,23]-=-), “greedily” pursuing minimization of Lyapunov function � N p(1/2)Q2n (t), and known to ensure stability of queueing networks if such is feasible at all. (Cf. [4] for recent results and review of pre...

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...these relations) it is uniquely determined by x and the initial condition f (0) = q(0). Estimate of the speed of convergence to a convex set The following lemma is a slight modification of Lemma 3 in =-=[21]-=-, and is presented here with a proof for completeness. Note that in the differential inclusion (105) in Lemma 20 it is only required that v(t) ∈ V . (There are no other conditions on v(t)). Lemma 20. ...

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...k communication links are overloaded. (Note that the latter condition is necessary for stability of such networks.) This is a very large and active field, which we do not attempt to review here. (See =-=[11,14,16,18]-=- for recent reviews of the field.) We will only point out some aspects of congestion control modelling, to which we believe our results contribute, as illustrated by the application examples in Sectio...

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...s not of any known primal-dual type. (Previously studied primal-dual dynamic systems include, for example, the classical ArrowHurwicz-Uzawa system (cf. [13]) and systems within passivity framework of =-=[26]-=-). 1.3. Discussion It is easy to observe that, if we remove all processing nodes from our model (so that there is no stability constraint), the GPD algorithm reduces to the Gradient algorithm [3, 21],...

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... of [26]). 1.3. Discussion It is easy to observe that, if we remove all processing nodes from our model (so that there is no stability constraint), the GPD algorithm reduces to the Gradient algorithm =-=[3, 21]-=-, always “greedily” choosing controls maximizing the drift of utility (“Lyapunov”) function H(X(t)), and shown to be asymptotically optimal as β ↓ 0in[3, 21]. Ifs404 STOLYAR we remove all commodity fl...

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...be asymptotically optimal as β ↓ 0in[3, 21]. Ifs404 STOLYAR we remove all commodity flows (so that we are not concerned about any utility), the GPD algorithm becomes a “MaxWeight”-type algorithm (cf. =-=[4,20,22,23]-=-), “greedily” pursuing minimization of Lyapunov function � N p(1/2)Q2n (t), and known to ensure stability of queueing networks if such is feasible at all. (Cf. [4] for recent results and review of pre...

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... the definition of an FSP, we obtain the fact that sample paths of z are FSPs w.p.1 (which completes the proof of the lemma). This can be done, for example, analogously to the proof of Theorem 7.1 in =-=[19]-=-. Alternatively, we can use Skorohod representation (cf. [5]), which allows us to construct all the processes z β (of the converging subsequence) and process z on a common probability space so that th...

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...ove network, consisting of a single generating switch, with no average power constraints, corresponds to a problem of utility based scheduling in wireless systems, subject to average rate constraints =-=[2, 15]-=-. (In fact, this single-switch model is more general than those in [2,15].) The specialized GPD algorithm for this case (which is the control rule for a generating switch, without the sums containing ...

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...veral examples in Section 5.3 that GPD algorithm provides a solution to a wide class of resource allocation problems in wireless networks, which include power usage and traffic rate constraints. (Cf. =-=[12,24,27]-=- for some problems with power constraints.) We show that GPD algorithm easily and naturally accommodates different system models and objectives by using, if necessary, virtual commodities and/or virtu...

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...k communication links are overloaded. (Note that the latter condition is necessary for stability of such networks.) This is a very large and active field, which we do not attempt to review here. (See =-=[11,14,16,18]-=- for recent reviews of the field.) We will only point out some aspects of congestion control modelling, to which we believe our results contribute, as illustrated by the application examples in Sectio...

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...be asymptotically optimal as β ↓ 0in[3, 21]. Ifs404 STOLYAR we remove all commodity flows (so that we are not concerned about any utility), the GPD algorithm becomes a “MaxWeight”-type algorithm (cf. =-=[4,20,22,23]-=-), “greedily” pursuing minimization of Lyapunov function � N p(1/2)Q2n (t), and known to ensure stability of queueing networks if such is feasible at all. (Cf. [4] for recent results and review of pre...

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...pace C R N [0;1) into the set of subsets of L R N (x(0)). From Lemmas 10 and 11 we see that operator A x(0) is closed, and its images A x(0) (f) are compact and convex. Thus, by Kakutani theorem (cf. =-=[9]-=-, Theorem XVI.5.1), operator A x(0) has asxed point. Statements (iii) and (iv) follow from the properties of operators A 1 and A 2 described in Lemmas 10 and 11. Proof of (v). Without loss of generali...

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...veral examples in Section 5.3 that GPD algorithm provides a solution to a wide class of resource allocation problems in wireless networks, which include power usage and traffic rate constraints. (Cf. =-=[12,24,27]-=- for some problems with power constraints.) We show that GPD algorithm easily and naturally accommodates different system models and objectives by using, if necessary, virtual commodities and/or virtu...