Optimal Throughput Allocation in General Random-Access Networks

Optimal Throughput Allocation in General Random-Access Networks Piyush Gupta Alexander L. Stolyar Bell Laboratories, Lucent Technologies 600 Mountain Ave., 2C-374 Murray Hill, NJ 07974 Email: [email protected] Bell Laboratories, Lucent Technologies 600 Mountain Ave., 2C-322 Murray Hill, NJ 07974 Email: [email protected] Abstract— We consider a model for random-access communication in networks of arbitrary topology. We characterize the efficient (Pareto) boundary of the network throughput region as the family of solutions optimizing weighted proportional fairness objective, parameterized by link weights. Based on this characterization we propose a general distributed scheme that uses dynamic link weights to “move” the link-throughput allocation within the Pareto boundary to a desired point optimizing a specific objective. As a specific application of the general scheme, we propose an algorithm seeking to optimize weighted proportional fairness objective subject to minimum link-throughput constraints. We study asymptotic behavior of the algorithm and show that link throughputs converge to optimal values as long as link dynamic weights converge. Finally, we present simulation experiments that show good performance of the algorithm. I. I NTRODUCTION Next-generation wireless networks are likely to have a more decentralized architecture than the current cellular networks. For instance, in the emerging pico-cell architectures, the base station may not continue to perform the role of a central coordinating agent for the uplink access of user terminals. Such a decentralized architecture is already employed in 802.11-based wireless networks. An important issue in such networks is that of scheduling. Due to the decentralized control constraints, a natural approach to consider is random-access communication, as in the slotted Aloha wireless LAN and 802.11 systems. It is well known, however, that the multi-user contention for channel access, if not well regulated, can lead to significant throughput degradation even in wireless LAN random-access systems ([3] and references therein), where any two concurrent transmissions interfere with each other. The problem gets substantially more aggravated in the networks with more general interference structure (such as those arising in multi-hop communication), where there are pronounced hidden-node and exposed-node issues. Hence, it becomes important to determine the optimal throughput that can be achieved in such more general randomaccess systems and to devise distributed control schemes that can operate the network close to it. To address these problems, we consider a general model for random-access communication in networks of arbitrary topology. Briefly, the model is as follows (formal description in Section II). Consider a network with a finite set of nodes N . Each node n 2 N has one or several outgoing links (n; m) to a subset of other nodes m 2 Dn  N . In each time slot, node n accesses the channel with probability pn , and it chooses only one of its outgoing links to transmit on with probabilities pnm =pn. Each node transmits independently of other nodes. A transmission by node n interferes with and “erases” any simultaneous packet reception by any node k within a subset Nn  Dn . A transmission on a link (n; m) is successful if it is not erased by any other simultaneous transmission. The set of link throughputs  = fnm g is thus a function of the set of access probabilities p = fpnm g. This model is a generalization of the classical slotted Aloha system, proposed in [1] and analyzed in [8] (among others). It also generalizes recently studied models for general topology networks in [6], [12], where it is assumed that k 2 Nn if and only if n 2 Nk . The model of this paper is also related to another general interference model of [4] - the main difference is that in [4] erasures due to interference occur with certain probabilities, not necessarily equal to 1 or 0. As mentioned before, we seek to devise distributed control algorithms that achieve efficient throughput allocations . “Efficient” naturally means that allocation  is (or close to) Pareto optimal, i.e., it is such that it cannot be improved upon. Main Contributions  For the general random-access network model above, we characterize the Pareto boundary M  of the throughput region (under very mild additional condition), as a set of  optimizing the following Weighted Proportional Fairness (WPF) objective: max 2M Xw (n;m) nm log nm ; (1) for all possible sets of positive link “weights” fwnm g. This characterization generalizes that given in [8], [9] for the classical Slotted Aloha, and is similar to that obtained in [4] (for a different model).  Just as the characterization the Pareto boundary for the classical slotted Aloha model has provided many key insights utilized in the design of practical random-access systems, the above characterization of the Pareto boundary M  leads us the following distributed procedure for achieving a specific system objective. Link weights wnm are used as dynamic control parameters; nodes dynamically adjust their link weights wnm according to their own “satisfaction” with observed link throughputs; the link weights (in fact, only certain aggregates of them) are shared among neighboring nodes; nodes dynamically set their access probabilities to those optimal for the WPF objective with the current link weights. As a result, the set of link throughputs  “moves” to a desired point while staying within the Pareto boundary. In other words, although WPF (with fixed weights) in itself is a very common and useful resource allocation objective in communication networks (cf. [7]), it can also be used as a “tool” for efficient throughput allocation in randomaccess networks for perhaps very different objectives.  We apply the above general approach to the specific problem of achieving weighted proportional fairness subject to minimum link throughput constraints: max 2M X (n;m) nm log nm ; s.t. nm  rnm ; 8(n; m); (2) where nm > 0 and rnm  0 are fixed parameters. We propose an algorithm utilizing very simple token counter mechanism, similar to that in [2], [10], that dynamically increases the dynamic weight wnm of the links not achieving their required minimum throughput, so that eventually, in steady state, they do achieve it.  We study the dynamics of token counters and link throughputs under the above algorithm, and prove its optimality in the sense that, if the token counters converge, then the link throughputs converge to the unique solution of problem (2). Finally, we provide simulation result which show good performance of the algorithm. The rest of the paper is organized as follows. The formal model is described in Section II. The optimal solution to the WPF objective (with fixed weights) is in Section III. Section IV contains characterization of the throughput region, including smoothness properties of the Pareto boundary. In Section V we describe the distributed algorithm for problem (2). Section VI contains asymptotic analysis of the algorithm (as one of its parameters becomes small), and proves its optimality (in the sense described above). The simulation experiments are discussed in Section VII. Basic Notation. We use the notations R, R+ and R++ for the sets of real, real non-negative and real positive numbers, respectively. Corresponding I -times product spaces are denoted I . The space RI is viewed as a standard RI , R+I , and R++ vector-space, with elements x 2 RI being row-vectors x = (x1 ; : : : ; xI ), and with Eucleadian metric induced by the norm   kxk =: Pi x2i 1=2 : Vector equalities and inequalities are understood componentwise. II. T HE M ODEL Our model is as follows. (It is a generalization of the model of [6], [12].) The system consists of a finite set N = f1; 2; : : : ; N g of nodes, and operates in discrete time, with time slots indexed by t = 0; 1; 2; : : : . Let Dn  N n n denote the subset of nodes to which node n has data to send. A node n at any time t may attempt transmission of one unit of data (say, data packet) to one of the nodes m 2 Dn . When this happens, we say that node n makes transmission attempt on the link (n; m). We will denote by I =: f(n; m) j n 2 N ; m 2 Dn g the set of all system links, and by I its cardinality (i.e., the total number of links). We assume that a node cannot simultaneously (i.e., within the same slot) transmit on two or more different links. The interference between simultaneous transmissions in the network has the following structure. If a node transmits in a slot, any simultaneous attempt to transmit to this node will fail. If there are two or more simultaneous transmissions to a node, they all collide and fail. Any transmission attempt by node n will interfere with and “erase” any attempt to receive a message at any of the nodes within some subset of N , denoted by Nn . (The model of [6], [12] additionally assumes that m 2 Nn implies n 2 Nm .) Given the above assumptions, Dn  Nn . (In other words, a transmission attempt by node n may interfere with receiving at more nodes than it actually sends traffic to.) Also, because a node n transmission makes simultaneous successful receiving impossible, n 2 Nn , for all n. Consider the following “Slotted Aloha-type” random access strategy. Each node n in each time slot transmits with probability pn , independently of other nodes and of the past history. And when node n does transmit, it chooses a particular link to transmit on, among the links (n; m); m 2 Dn , also randomly, with probabilities pnm =pn summing up to 1, that is X m2Dn pnm = pn : (3) Given this strategy, the average throughputs on the network links are given by nm = pnm k: m2Nk ; k6=n (1 pk ): (4) The dependence of the set (vector) of throughputs  = 2 I ) 2 RI on the set (vector) of access nm ; (n; m) ( Y + probabilities p 2 P =: f(pnm ; (n; m) 2 I ) 2 [0; 1℄I j (3) holdsg; given by (4), will be denoted by (p). Clearly, function (p) is continuous. III. O PTIMAL S OLUTION FOR THE W EIGHTED P ROPORTIONAL FAIRNESS O BJECTIVE The following Theorem 1 is a generalization of the corresponding result in [6], in that it applies to a more general model and optimization objective. (It also generalizes some of the results of [4].) The theorem shows that the problem of choosing access probabilities optimizing the weighted proportional fairness objective is relatively easy to solve, and it serves as a starting point for the development in this paper. n 2 N , let us denote by S =: f(`; k) j k 2 D ; k 2 N g the set of all links (`; k ) which either originate at n or are such that a transmission by node n interferes with that on (`; k ). Theorem 1: For arbitrary set of positive weights w = fw ; (n; m) 2 Ig 2 R++ , there exists a unique set of access probabilities p 2 P that maximizes the function w log  : F= (5) ( )2I The optimal p is given by: w : p = (6) ( )2Sn w For each ` n n I nm X nm nm n;m P nm nm `k `;k Remark 1. Expression (6) can be equivalently rewritten as p nm where W P = nm m = `: m (7) in m X : in m w ; 2Nn W w 2D` (8) `m is the sum of the weights of all links “incoming” to node m; in we will call Wm the incoming weight of node m. Proof of Theorem 1. Consider a fixed node n. Suppose first that the set S =: f(`; k) 2 S j ` 6= ng n n is non-empty. (In other words, node n’s transmissions interfere with transmissions on at least one link not originating at n.) In this case, any p maximizing F must be such that 0 < pnm  pn < 1 for all m 2 Dn . Then, if we substitute (4) and (3) into (5), we see that F p = nm w p X nm nm (`;k) which yields p Summing up (9) over whose solution is p n = P P 2Dn w m p = (1 nm n) m2D n P 2Dn w m nm + w 2Sn P `k 1 p =0 n (9) `k = 2Sn w `k (`;k) m p n , 2Dn w : )2Sn w (`;k nm `k Expressions (10) and (9) give (6). In the case when Sn = ;, it is easy to see that the access probabilities pnm ; m 2 Dn , must maximize m wnm log pnm subject to m pnm  1. The unique solution is P w P 2D : w means that f(n; k ) j k p nm = nm k n nk j 0  (p) for some p 2 Pg: (11) We denote by M  =: f 2 M j   0 2 M implies 0 =  g (12) the subset of maximal elements of M , which can be called the Pareto boundary of M . Characterizing boundary M  is the main focus of this section. We denote by  =: M  \ R M++ (13) ++ the subset of Pareto boundary M  consisting of vectors with I all strictly positive components. The following proposition describes basic properties of the throughput region M  . We omit the straightforward proof. Proposition 1: (i) Throughput region M is a compact set. (ii) Set M  is non-empty. For any  2 M  there exists p 2 P such that  = (p). I It follows from Theorem 1 that for any w 2 R++ ,  =   (p(w )) 2 M++. The natural question is whether or not  we can the converse is true, namely, that for any  2 M++    find w such that  = (p(w )). The answer is basically yes, under a mild additional condition, as we show below in Theorem 2. Let U denote the system log-throughput region. More precisely, (14)  g: U  =: flog x j x 2 M  \ R++  M++ (15) Lemma 1: The system log-throughput region U is a closed convex subset of the negative orthant of R . Proof. It is easy to observe that, for any link i 2 I , log  (p) I I (10) P I where here and below log applied to a vector is understood component-wise. The Pareto boundary of U is , we obtain an equation for nm M =: f0 2 [0; 1℄ I nm P P From this point on in the paper, for brevity, we sometimes denote links (n; m) 2 I by a single index i, j , etc. We define the system throughput region M as the set of all non-negative vectors, which can be majorized by vectors of the form (p), namely, U =: flog x j x 2 M \ R++ g; ; w : )2Sn w (`;k IV. S YSTEM T HROUGHPUT R EGION C HARACTERIZATION However, Sn = ; 2 Dn g = Sn , and thus expression (6) is still valid. The dependence of the set (vector) of access probabilities I p 2 P on the set (vector) of positive link weights w 2 R++ , given by (6), will be denoted by p(w). (Clearly, p(w) is invariant with respect to scaling of w by a positive constant.) i is a concave scalar function of the set of access probabilities I p. Then, for any (1) = (p(1) ) 2 M \ R++ and (2) = (2) I (p ) 2 M \ R++ , a convex combination of log (1) and log (2) is (1) + 2 log (2)  log ( 1 p(1) + 2 p(2)) 2 U: This means that u 2 U as well. Since any u 2 U is dominated by u0 = log (p0 ) 2 U for some p0 , the convexity of U follows. Region U is closed because M is closed, and log u= 1 log is a continuous mapping. Theorem 2, presented just below, characterizes the Pareto boundary of the throughput region. This result is analogous to the results of Sections 3.4-3.5 of [4], which apply to a closely related - but different - model. In particular, Theorem 2 generalizes Theorem 6 of [4]. Consider the directed graph with vertices being links i 2 I , and the edge from i = (n; m) to j existing if and only if j 2 Sn n i. We will call this graph a link dependence graph. (In the case when there is at most one link originating from each node, this graph could be called “interference graph” the term used in [4].) Since function p(w) is invariant with respect to scaling of I w, we can restrict the domain R++ of p(w) to the normalized : I w = 1 g : set B = fw 2 R++ j i i Theorem 2: Suppose, the link dependence graph is strongly connected. (There is a directed path from any vertex to any other.) Then, function (p(w)) defines a homeomorphism (mutually continuous one-to-one mapping) between B and  . Moreover, M  M++ ++ is a smooth (I 1)-dimensional surface. Proof of Theorem 2. The outline of the proof that (p(w)) is a homeomorphism is as follows. For any w 2 B , (p(w)) 2  . Then, we establish the following sequence of assertions, M++  . for a fixed  2 M++ I Assertion 1. There exists w 2 R++ such that  =  (p(w )). Assertion 2. Vector p(w ) is the unique vector p 2 P solving equation  = (p). Assertion 3. Vector w is the unique vector w 2 B solving equation  = (p(w )).  Assertion 4. If 0 !  , then w0 ! w , where 0 2 M++ 0 0 and  = (p(w )). Due to space limitation, we do not give the detailed proof of  . (All proofs these assertions, and of the smoothness of M++ are given in [5].) P V. P ROVIDING M INIMUM L INK T HROUGHPUT G UARANTEES : A D ISTRIBUTED A LGORITHM Suppose we want an efficient “distributed” random access algorithm that satisfies certain minimum link throughput requirements whenever this is feasible at all. It is also desirable that the “leftover” system capacity, after satisfying the minimum throughput constraints, is allocated in a “fair” fashion. We will combine and specify this two objectives as follows. Suppose, a weight i > 0 and a minimum throughput requirement ri  0 is given for each link i 2 I , that is, I there are two parameter vectors = f i ; i 2 Ig 2 R++ I and r = fri ; i 2 Ig 2 R+ . We want a distributed random access algorithm such that, in steady state, the link throughput allocation  solves the following optimization problem: max X x2M i log xi (16) i subject to x  r: (17) Equivalently, in terms of log-throughput region U , we seek  such that u = log  solves the problem: max u2U X i u i i (18) subject to u  log r: Note that if  X (19) is a solution to (16)-(17) and it has the form  2 arg max x2M  i log xi (20) i I I , then  2 R++ and this for some f i ; i 2 Ig 2 R++ solution is unique. This and Theorem 2 easily imply that when the link dependence graph is strongly connected, the solution  of (16)-(17) (if any) is unique. The algorithm we propose is as follows. As before, let t = 0; 1; 2; : : : denote a time slot. (A) Each node n, maintains a “token counter” (token queue length) Qnm (t) for each of its outgoing links (n; m), which is updated according to the following rule: Qnm (t + 1) = Qnm (t) + rnm hnm (t) ; t = 0; 1; 2; : : : ; (21) where hnm (t) = 1 if there was a successful transmission on link (n; m) in slot t, and hnm (t) = 0 otherwise. (We will use vector notation Q(t) = (Qi (t); i 2 I ).) (B) Each node n, for each of its outgoing links (n; m), calculates dynamic weight wnm (t) = nm + Qnm (t), where > 0 is some (typically small) parameter, and sets its access probabilities in slot t according the expression (7). Implementation considerations. Of course, the big question is: How can a node n “know” the incoming weights Wmin (defined in (8)) of the nodes m 2 Nn , i.e. nodes it interferes with? The form of (7) allows (at least in principle) the following natural procedure. Each transmission on any link (`; k ) contains the piggy-backed current dynamic weight w`k (t) of the link. Thus, each node m can maintain an estiin mate of its incoming weight Wm . Each node m periodically in “broadcasts” its Wm , so it can be “heard” by all the nodes whose transmissions may interfere with reception by node m. in (In particular, Wm can be piggy-backed into transmissions from node m.) Each node n listens to the broadcast messages described above, which allows it to estimate the sum in the denominator of (7). VI. A SYMPTOTIC BEHAVIOR OF THE ALGORITHM WITH SMALL PARAMETER In this section we study the dynamics of user throughputs and token counters, under the algorithm described in Section V, when parameter > 0 is small. Namely, we consider an asymptotic regime such that converges to 0. We study the dynamics of fluid sample paths (FSP), which are (roughly speaking) possible trajectories q (t) of a random process that is a limit of the process Q(t= ) as ! 0. (In other words, trajectories q (t) “approximate” the process Q(t) scaled down by factor , and with 1= time speed-up.) The main result of this section (Theorem 3) basically says that if FSP is such that q(t) converges to some finite vector q as t ! 1, then link throughputs converge to the unique solution to the problem (16)-(17). Remark 2. A stronger result would be to prove that the convergence of q (t) in fact holds as long as problem (16)-(17) is feasible; this would prove the asymptotic optimality of the algorithm, as it is done, for example, for the Greedy PrimalDual (GPD) [10] algorithm, for a different model. Proving asymptotic optimality of the algorithm of this paper may be a subject of future work. We note that, since problem (18)-(19) is convex, the GPD algorithm can in principle be applied (and be provably optimal), if we would “work” with logarithms of the throughputs; this however would require that we measure throughputs over longer time intervals (and thus updates token counters less frequently), which would result in a much “slower” algorithm. Remark 3. It is shown in [11] that the algorithm of this paper, with any fixed parameter , ensures that the users will receive the desired minimum throughputs as long as the constraint (17) is feasible. However, the results of [11] do not address the utility maximization objective (16). We now define the asymptotic regime and an FSP. Let us denote by Fi (t) = t 1 X s=0 hi (s); t = 0; 1; 2; : : : ; i 2 I ; the total number of successful transmissions on link i by (and excluding) time t, and denote F (t) = (Fi (t); i 2 I ). We extend the time domain of functions F (t) and Q(t) to all real t  0 by adopting the convention that they are constant within each time slot [t; t + 1) for all integer t  0. Consider a sequence f g of positive values of , converging to 0. For each , let (F (
); Q (
)) be a realization of the corresponding random process, with some fixed initial Q (0). Assume that the sequence of realizations is such that a functional law of large numbers (FLLN) condition holds for the process governing transmission attempt decisions by node n at different times t, given its “current” (depending on t) set of access probabilities. (The precise condition is given in [5].) Consider the following rescaled trajectory for each : f ( =( f (t); t  0); q q (t); t  0)); =( where f (t) = F (t= ) and q (t) = Q (t= ). Definition: A pair of vector-functions (f = (f (t); t  0); q = (q (t); t  0)) is called a fluid sample path (FSP), if the uniform on compact sets (u.o.c.) convergence (f ; q ) ! (f; q ) holds for at least one sequence f g and the corresponding sequence (f ; q ) of scaled trajectories, as defined above. Theorem 3: Suppose an FSP (f; q ) is such that q(t) ! q 2 R+I as t ! 1: Then, the problem (16)-(17) is feasible, with  = (p( + q )) being its unique optimal solution, and (d=dt)f (t) !  . To prove Theorem 3, we will first describe the basic FSP properties in Lemma 2. Lemma 2: The family of fluid sample paths has the following properties. (i) All component functions fi (t); t  0, and qi (t); t  0, are Lipschitz continuous, uniformly across all FSPs. Consequently, any FSP is such that proper derivatives fi0 (t) and qi0 (t) exist for almost all t  0 (with respect to Lebesgue measure). (ii) “Shift property.” If (f; q ) is an FSP, then for any d  0, (d f; d q ) is also an FSP, where f (t + d) f (d); [dq℄(t) = q(t + d); t  0: (iii) “Compactness.” If a sequence of FSPs (f (j ) ; q (j ) ) ! (f; q ) uniformly on compact sets as j ! 1, then (f; q ) is f t [ d ℄( ) = also an FSP. (iv) For any FSP, for almost all t  0 we have: f 0 (t) = (p( + q(t))); (22) i 2 I, if qi (t) > 0; ri fi0 (t) qi0 (t) = (23) maxfri fi0 (t); 0g if qi (t) = 0: (Consequently, since f (t) is Lipschitz and (p(w)) is continuous, the derivative f 0 (t) exists for all t > 0.) and, for each  Proof. See [5]. Proof of Theorem 3. Using shift and compactness properties, from the FSP as in the theorem statement, it is easy to see that the “stationary” trajectory, given by q(t)  q ; f 0 (t)   = (p( q )) (24) is also an FSP. We immediately see that vector r cannot lie outside region M , for otherwise ri > i for at least one i, implying (by (23)) that qi (t) ! 1, which contradicts (24). If we use notation u = log  , then, by the definition of function (p(
)) (see (5)), X u 2 arg max ( i + qi )ui : (25) u2U i + If we specialize (23) for the stationary FSP defined above, we see that qi > 0 implies i = ri , or, equivalently, qi (ui r log i ) = 0 for all i: (26) If we view qi as Lagrange multipliers for the constrained optimization problem (18)-(19), we see that, by Kuhn-Tucker theorem, (25) and (26) imply that u is a solution to that problem. The uniqueness of solution u follows from the representation (25), as noted in the remark containing (20). VII. S IMULATION RESULTS We now present simulation results for the algorithm introduced in Section V. Throughout this section we always assume that weights i = 1 (see (16)) for all links, and so the system objective is to maximize the sum of the logarithms of link throughputs, subject to minimum link throughput constraints (17). The parameter = 0:001 is same throughout all experiments. We start with a simple 3-node network, shown in Figure 1. A bi-directional link on the figure, say between nodes 1 and 2, means that both (1; 2) and (2; 1) are communication (data 10 transmission) links of the system. There is no interference between nodes 2 and 3. (That is, 3 62 N2 and 2 62 N3 .) For this network, expressions (8) and (7) specialize to: W1in = w21 + w31 ; W2in = w12 ; W3in = w13 ; w1m w p1m = in ; pm1 = in m1 in ; m = 2; 3: in in W1 + W2 + W3 W1 + Wm Table I shows steady state link throughputs (after they converge) for two cases. The first is the “baseline” case when we do not impose any minimum throughput requirements, i.e., rnm = 0 for all links. In the second case, we introduce minimum throughput requirement r2;1 = 1=7 for link (2; 1), and leave rnm = 0 for all other links. We see that, in the second case, the algorithm indeed “lifts” the throughput of link (2; 1) to the desired minimum level (at the expense of the throughputs on the other links, of course.) The token counters (and therefore the dynamic weights) in the second scenario indeed “converge” (see [5]), up to some inevitable “jitter” since is finite; this guarantees (by Theorem 3) that the throughput allocation is indeed optimal. 1 1 2 9 3 4 Fig. 2. 1;10 10;1 2;10 10;2 3;10 10;3 4;10 10;4 5;9 9;5 6;9 9;6 7;8 8;7 all 8 5 6 7 10-node system. rn;m = 0 r5;9 = 0:1 0.0569 0.0901 0.0564 0.0894 0.0359 0.0689 0.0380 0.0670 0.0412 0.0691 0.0876 0.0716 0.1245 0.1817 0.0492 0.0896 0.0490 0.0891 0.0287 0.0744 0.0290 0.0742 0.0986 0.0492 0.0687 0.0668 0.1048 0.2042 TABLE II 10- NODE SYSTEM . S TEADY STATE LINK THROUGHPUTS . 2 3 Fig. 1.  1; 2  2; 1  1; 3  3; 1 all 3-node system. rn;m = 0 r2;1 = 1=7(= 0:1429) 0.1649 0.1115 0.1677 0.1114 0.1437 0.1432 0.1609 0.1023 TABLE I 3- NODE SYSTEM . S TEADY STATE LINK THROUGHPUTS . Next, we consider a more complex 10-node network, shown in Figure 2. As before, a solid bi-directional link means presence of a communication link in both directions. A bidirectional dashed link means mutual interference between nodes. (For example, 4 2 N9 and 9 2 N4 . Recall that also, if a communication link (n; m) exists, then, by our definitions, n causes interference to m; for example, 10 2 N4 .) As for the 3-node network, we consider two cases: “baseline,” with rnm = 0 for all links, and the second case where we have minimum throughput requirement for one of the links, namely r5;9 = 0:1 for link (5; 9). Steady state link throughputs for both cases are shown in Table II. We see that the throughput of link (5; 9) is indeed “lifted” to approximately the required level. (An interesting - although not very surprising - observation is that, in systems of general topology, introducing minimum throughput requirement on one of the links does not necessarily result in the decrease of the throughputs on all other links.) 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