# Control of Wireless Networks with Flow Level Dynamics under Constant Time Scheduling

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6 6 where we have ued Little law in deriving E [N x] t in the above equation. Specifically, the expected waiting time for a cla- uer i /µ x ), uing Little law we have E [N x] t = λ /µ x ) = ρ /x. Thu, we can rewrite 8) a max x g x)f x)d x 9) X Now, uppoe we wih to find optimal uer rate x [0, M ]. Let x be the maximum of g x ) = ρ /x U x ) in [0, M ] and the correponding optimum rate vector i x. Then, it i eay to ee that chooing f x) = δ x x ) will maximize 9) where δ.) i the delta function. Thu, the longterm utility maximization can be achieved by allowing uer of cla to tranmit at the optimal rate x. In ummary, when the traffic load atifie 6), the network i table under the propoed cheduling policy and no congetion control i needed. In addition, we can decouple the long-term utility maximization from tability under thi tability condition. Example: When the utility function i U x ) = lnx ) which correpond to proportional fair rate allocation among uer, we have g x ) = ρ /x lnx ). The global maximum of g x ) i x = e. Thu, if M > e, the optimal tranmiion rate to achieve maximum long-term utility i x = e. We will compare long-term utility under thi olution and for the cae where cro-layer congetion control algorithm i ued [2]. VI. CONGESTION CONTROL UNDER HEAVY LOAD In thi ection, we conider the heavy load regime where the bound tated in 5) i violated. Thi may be the cae when there are many long-lived flow in the network. A. Cro-Layer Congetion Control Algorithm In thi ubection, we preent a cro-layer congetion control algorithm which can tabilize the network under any offered load. We aume that the flow dynamic are low compared to the time cale of a congetion control algorithm. Hence, we aume that there are fixed number of uer of each cla in the network which will be denoted by N. Our propoed cro-layer congetion control algorithm work a follow. where U.) i the invere of derivative of utility function U.) and [x] + = max[x, 0]. The implicit cot are updated by q l t + ) = [ q l t) + α l S H k N x + κ )] + where κ > 0 i a mall number and α l i the tep-ize. Tranmiion cheduling: The regulated maximal cheduling policy i employed in each time lot. We aume that the utility function U x ) i increaing, trictly concave, twice differentiable. The propoed congetion control algorithm implicitly olve the following optimization problem. S maximize N U x ) ubject to S H k N x κ, l E 0) Note that contraint of thi optimization problem come from the contraining bound tated in 5). Here, the tranmiion rate of cla- uer i ued intead of it offered load. We introduce a mall κ > 0 on the right hand ide of thee contraint to enure that feaible olution of thi optimization problem trictly atify the contraining bound. By introducing Lagrange multiplier q l for the contraint in 0), we have the following Lagrangian L x, q) = S N U x ) l E q l [ S H k N x + κ Uing the tandard dual decompoition technique, we can calculate the uer rate from x t) = argmax N U x ) N x q l t) 0 x M l E :H k= which give the uer rate update rule preented in the algorithm. Alo, implicit cot update tep of the propoed algorithm come from the dual update of the underlying optimization problem. Now, let x be the optimal olution of the preented algorithm, we implement a x + kɛ)-regulator in the k-th hop of cla uer. We have the following reult on the tability of the preented algorithm. Propoition 4: If the tepize i choen to be mall enough, the preented algorithm can tabilize the network under any offered load. Proof: The proof i in Appendix C. It i expected that a congetion control algorithm hould tabilize the network under any offered load. In thi repect, the propoed congetion control algorithm doe a good job compared to thoe in [5], [2]. Note that in practice where the RAMM cheduling algorithm i implemented to approximate Uer rate i determined by the maximal matching policy RAMM intead of maximal x t) = min U + matching i ued in tep 3 of thi algorithm), we can modify q l t), M the propoed congetion control algorithm by replacing on the right hand ide of the contraint in 0) by µ and modify l E :H k= the algorithm accordingly. B. Some Implementation Iue In practice, there i no imple way to know whether or not the contraining bound in 5) i atified. Therefore, it i unknown when the congetion control algorithm propoed in ection VI.A hould be activated. Moreover, it would be wie to avoid performing congetion control if it i unneceary. Alo, uer in the network hould tranmit at their optimal rate a calculated in ection V to maximize the long-term ]

11 for cla and for all uer clae at link l in time lot t, repectively. Let u denote by b l and a l the previou link and next link of link l on the route of cla- uer. Alo, let Cl t) and Dl t) be the number of packet tranmitted from regulator and tranmiion queue in time lot t, repectively. We have the following queue update equation Q l t + ) = Q l t) Dl t) + Cl t) 4) Pl t + ) = Pl t) Cl t) + Db t). l 5) Thu, we have Q l t+)+pa t+) = Q l l t)+pa t)+c l l t) Ca t). 6) l We will ue the following Lyapunov function for the ytem where V P, Q) = V Q) + ξv 2 P, Q) 7) V Q) = l V 2 P, Q) = l Q l t) Q k t) P a l + Q l ) 2. In fact, thi Lyapunov function wa alo ued in [9]. Now, let u conider V Qt + )) V Qt)) = 2 ) Q l t) Q k t + ) Q k t)) l + ) Ql t + ) Q l t)) l ) Q k t + ) Q k t)) = 2 l + l Q l t) ) Ck t) D k t)) C l t) D l t)) C k t) D k t)) ) 8).9) Since the number of packet tranmitted from regulator and tranmiion queue in each time lot are bounded, the econd term in 9) can be bounded by a contant B. Thu, we have [ E V Qt + )) V Qt)) P t), Qt) ] 2E [ l Q l t) 2E Q l t) l:q l t) C k t) D k t)) C k t) ) P t), Qt) ] ) D k t) + B P t), Qt) ] + B 2. Let L be the larget number of hop travered by any uer cla, we have E Q l t) Ck t) P R t), Qt) l:q l t) k Q l t) H k ρ + Lɛ. 20) R l:q l t) k Alo, due to the definition of maximal cheduling policy, we have Q l t) D k t) Q l t). 2) l:q l t) l:q l t) For traffic load atifying 5), we can find ɛ and δ mall enough uch that S H k ρ + Lɛ + δ, l E. 22) From 20), 2), 22), we have [ E V Qt + )) V Qt)) P t), Qt) ] Q l t) δ + B 2 δ l:q l t) l Q l t) + B 3. 23) Now, uing the procedure a in [9], we can obtain [ E V 2 P t + ), Qt + )) V 2 P t), Qt)) P t), Qt) ] Q l t) η Pa t) + B 4. 24) l l l Combining the reult in 23), 24), we have [ E V P t + ), Qt + )) V P t), Qt)) P t), Qt) ] [ ] δ ξ Q l t) ξη Pa R t) + B 5. 25) l l l l δ We can chooe ξ mall enough uch that ξ > 0. Thu, the drift will be negative if the regulator and/or tranmiion queue become large enough. Therefore, the tability reult follow by uing theorem 2 of [26]. Note that the choen Lyapunov function doe not take regulator queue on the firt hop of each uer cla into account. Thee regulator queue are, however, table becaue their output rate i ρ + ɛ which i larger than the average input load i.e., ρ ). APPENDIX II PROOF OF PROPOSITION 2 Conider any link AB between node A and B. We will find the probability that at leat one link in the interference et I AB i cheduled. Thi event will be denoted a M AB in the equel. A mentioned before, RAMM cheduling algorithm include BP-SIM cheduling algorithm for the one-hop node excluive) interference model propoed in [7] a a pecial cae. In the following we prove propoition 2 for k 2. For the pecial cae of k =, we refer the reader to [7] for the proof and the correponding analyi. Note that the proof for the cae k 2 i very challenging and completely different from that for the

13 3 matching requet to a right node which i not an one-hop neighbor of A or B. Hence, there are at mot I x left node which can collide with the matching requet from C. For the econd cae, if C i not one of x node i.e., x one-hop neighbor of A or B). Then, thi cae occur with probability I 0 x)/2) I 0. In thi cae, the matching requet from node C will be uccefully received and the correponding link will be cheduled with a probability which i lower bounded by F 3 = I x 2 i=0 I x 2 i ) ) I x 2 B B m= m ) i B where in calculating the lower bound for P m we aume that C tranmit it matching requet to a right node which i not an one-hop neighbor of A or B. Hence, there are at mot I x 2 left node which can collide with the matching requet from C. B. There are two or more left node in I 0 Suppoe node C in the et I 0 become left and win the contention. Then node C hould have the mallet backoff value among all the node whoe matching requet can collide with the matching requet from C. Alo, node C hould end the matching requet to a node which i a right one. For eae of reference, we will refer to thi right node a node D in the equel. In general, D can belong to et I 0 or not. However, to find the lower bound of P m, we aume that D belong to I 0 ; therefore, there are at mot I 0 2 other left node beide C and D in I 0. A before, we aume the wort-cae cenario where there are I node whoe tranmiion can collide with that of node C. Recall that all x node which are one-hop neighbor of A or B belong to the et I. Similar to the previou cae, we conider the following two ub-cae. For the firt cae, C i one of x node i.e., x one-hop neighbor of A or B). In thi cae the matching probability can be lower bounded a F 4 = x I 0 2 i= I x j=0 I0 2 i ) ) I0 I x 2 j ) I x 2 B B m= ) m ) i+j B where i i the number of left node beide C and D in the et I 0. And j i the number of left node which belong to I but are not one-hop neighbor of A or B i.e., there i no link between thee node and A or B). We will denote thi et a I \ x in the equel. In general, D can belong to I \ x or not; however, to find the lower bound for P m, we only allow j take value from 0 to I x. In addition, C can be any node among x node o we have a factor of x before the um. Alo, we require that all left node i left node belonging I 0 and j left node belonging to I \ x) achieve larger backoff value than that of node C. For the econd cae, C belong to the et I \ x. In thi cae, the matching probability can be lower bounded a F 5 = I 0 x) I 0 2 i= I x 2 j=0 I0 2 i ) ) I0 I x 2 2 j ) I x 2 B B m= ) m ) i+j B where C can be one of I 0 x node o we have the factor I 0 x) before the um. Alo, to calculate the lower bound for the matching probability, we aume D alway belong to I \ x, o j can be at mot I x 2. Subtitute reult of all conidered cae into 26), the matching probability i lower bounded by P m P 0 m = 2/2) I 0 F + x 2)/2) I 0 F 2 +I 0 x)/2) I 0 F 3 + F 4 + F 5 27) where F, F 2, F 3, F 4, and F 5 are defined above. From the lower bound of the matching probability P 0 m derived above, we can calculate the lower bound of P m for any link AB a p = min x, I 0 P 0 m where we find the minimum of P 0 m over all poible x and I 0. Note that poible value of x and I 0 will be in the range of [3, 2d ] and [3, I 0 ], repectively. Here, x and I 0 are at leat three for the network to be connected i.e., A and B hould have at leat one one-hop neighbor). It i oberved that p i independent of the network ize and depend only on B, d, I, I 0. Now, we can chooe K uch that at leat one link in I l of a backlogged link l i cheduled with probability greater than µ after K round a follow: { K = min k : p ) k µ }. 28) k Thu, we can chooe K which i independent of network ize and only depend on B, d, I, I 0, µ uch that performance guarantee arbitrarily cloe to the contraining bound can be achieved. Example: For the grid network and two-hop interference model, we have I = 22, I 0 = x can take value of 5, 6, 8. With maximum backoff value B = 0, by uing the analyi preented above, the minimum number of cheduling round to achieve µ = 0.9 i K = 60. In fact, thi calculation i quite conervative becaue it conider the wort cae cenario. In practice, the ize of interference et decreae quickly over cheduling round, o the required value of K i much maller. APPENDIX III PROOF OF PROPOSITION 4 The proof i imilar to that of Propoition i.e., we ue the ame Lyapunov function and proof procedure). In particular,

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