Atomic Hierarchical Routing Games in Communication Networks

Transcription

1 Atomic Hierarchica Routing Games in Communication etworks Vijay Kambe, Eitan Atman, Rachid E-Azouzi Vinod Sharma Dept. of Industria Engineering Management, IIT - Kharagpur, West Benga, India Maestro group, IRIA, 004 Route des Lucioes, Sophia Antipois, France LIA, University of Avignon, 339, chemin des Meinajaries, Avignon, France Dept. of Eectrica Communication Engineering, Indian Institute of Science, Bangaore, India Abstract Theoretica studies on routing games in networks have so far deat with reciproca congestion effects between routing entities. But, with the advent of technoogies ike Cognitive Radio, we have networks which support differentiation of fows. In a two priority eve mode a user can be high priority or ow priority there is a cost for such a cassification. The point of departure of this mode from the traditiona routing scenarios is the absence of reciprocity in the congestion effects: The ow priority fow faces congestion from both high priority as we as ow priority fow whie the high priority fow is immune to the congestion effects from the ow priority fow. This hierarchy is naturay present in contexts where there are primary (icensed users secondary (unicensed users who can sense their environment because there are equipped with a cognitive radio [9. We study such kind of routing scenarios for the cases of atomic users. We estabish the existence the uniqueness of ash equiibrium further we show the existence of a potentia function for inear congestion costs a certain priority cassification pricing scheme. atura appications of this mode to Cognitive Radio are aso pointed out. I. ITRODUCTIO on-cooperative routing has ong been studied both in the framework of road-traffic as we as in the framework of wireess networks. Such frameworks aow to mode the fow configuration that resuts in networks in which routing decisions are made in a non-cooperative distributed manner between the users. In the case of a finite (not very arge number of agents, the resuting fow configuration corresponds to the so caed ash equiibrium defined as a situation in which no agent has an incentive to deviate uniateray. In other words, the ash equiibrium does not state what can or cannot happen when more than one decision maker changes their strategy (route simutaneousy. When the decision makers in a ash game are discrete finite in number, a ash equiibrium can be achieved without the costs of a used routes being equa, contrary to Wardrop s equiibrium principe [11. In some cases, Wardrop s principe has been shown to represent a imiting case of the ash equiibrium principe, as the number of users becomes very arge [4, [5, [13. In the context of routing in networks, the ash equiibrium has been introduced by Orda et a [10 as we as [ in order to predict the traffic pattern that occurs when each of severa non-cooperative sources of traffic (say a service provider has traffic to send it shoud spit (or route its traffic among severa avaiabe paths in a network. Each service provider wishes to minimize its own cost, which is however infuenced by the routing decisions of other service providers. A reated mode was studied thirty years ago in [14. one of these mentioned works consider the cases where the congestion effects between the routed entities are not reciproca. To buid a theoretica framework for studying such probems to study the issues invoved is the main objective of this paper. We consider a different kind of Hierarchica Routing game in which the routed traffic can be of two types: High priority Low priority. The high priority traffic is unaffected by the routing profie of the Low priority traffic. On the other h the ow priority traffic faces congestion effects from both high priority ow priority traffic. The non-atomic version of the game was studied in [?. In this paper we consider the atomic version of the game. An atomic hierarchica game is payed between controers or service providers who can divide their tota traffic fow into these two types, cassifying it as high priority being more expensive than cassifying it as ow priority. After this division they have to make the decision of how to route their traffic on the network. Each service provider seeks to minimize its own cost which is infuenced by: 1. His cassification decision. His routing decision the routing decisions of a the other service providers. A. Our Contributions We first define the atomic hierarchica routing game for which the existence of ash equiibrium is estabished under assumptions on the ink costs. We then derive a pricing scheme for the ow priority cassification in a hierarchica game on parae inks with inear ink costs which aows for an existence of potentia under certain conditions. These conditions are naturay satisfied by symmetric games for which we are thus abe to prove the uniqueness of ash equiibrium we are aso abe to present certain distributed agorithms based on evoutionary dynamics which converge to this equiibrium. atura appications of these modes to cognitive radio are pointed out. Our resuts are finay iustrated by an exampe of muti-channe hierarchica rate aocation under unsotted ALOHA protoco. Digita Object Identifier: /ICST.CROWCOM

2 II. ATOMIC ROUTIG GAME OVER PARALLEL LIKS Consider a network topoogy given by G = (E,{s, d} where E is a set of E = K parae inks connecting source node s to destination node d. The inks are assumed to be directed from the source to the destination. We consider a mode where the ow priority fow is compressed when it is routed whie the high priority fow is unaffected. For exampe if the fow is in the form of a stream of packets, then the packet size is compressed by a factor γ (0, 1 if it is routed as ow priority. Consider casses of users where cass i has a tota fow dem r i to ship from a source s to a destination d. Let be the high priority fow be the ow priority fow that payer i ships over ink E, =1,,K. With some abuse of notation we write Aso, x H = = i=1, x L = x = x H K =1 We define the foowing vectors + x L, = i=1 K =1 =( 1,, K =( 1,,xi,L K x H =(x 1,H,,x,H x L =(x 1,L,,x,L Introduce for each user i the cost density function G H (x H associated with cassifying the traffic as High priority on ink the cost density function G L (xh for cassifying it as ow priority on ink both dependent on the tota high priority fow on the ink with G H (a >G L (a a 0. For now, we consider constant functions of the form G H (x H =C H G L (xh =C L where C L C H are costs per unit fow. For each user i introduce for each ink a cost function J i,h (, x H for the high priority traffic routed by him on that ink which is dependent ony on the high priority traffic fow profie on that ink. Further introduce the cost function J i,l (γ, x H,γ x L for the ow profie traffic which depends on the entire fow profie on that ink. ote that this function depends on the ow priority fow after compression. We ony consider cost functions of the form J i,h T (x H J i,l (γ (, x H =, x H,γ x L = xi,l T (x H + x L which depend on the tota fows on the ink rather than the profie of individua fows. The ink congestion cost density functions T (. are convex, continuousy differentiabe monotone increasing in their argument. So the optimization probem faced by payer i is to find a fow profie x i = (, which soves the foowing probem: min K [ =1 T (x H +γ T (x H + γx L +G H (x H + G L (x H or if we consider constant pricing functions, K [ min subject to =1 T (x H +γ T (x H + γx L +C H + C L + = r i i =1,,, 0 i =1,,, E Under the given assumptions on the cost functions, given that the cassification costs are constant, the existence of ash equiibrium foows from arguments simiar to those in [10. Consider the case where the cassification costs are not constant. Then the Kuhn-Tucker conditions for this optimization probem are given by: (T (x H + T (x H +γ T (x H + γx L + (x H + G H (x H λ i =0, E (1 T (x H + T (x H +γ T (x H + γx L + (x H + G H (x H λ i 0, E ( (γt (x H + γx L +γ T (x H + γx L + G L (x H λ i =0, E (3 γt (x H + γx L +γ T (x H + γx L + G L (x H λ i 0, E (4 A. Existence of a potentia for inear cost density functions under a certain pricing scheme We now consider congestion cost per unit fow functions T (. to be inear in their argument i.e. T (a =k a, a >0 for some positive constant k for each ink E. The Kuhn Tucker conditions then reduce to the foowing: (k x H + k + γk + (x H + G H (x H λ i =0, E (5 k x H + k + γk + (x H + G H (x H λ i 0, E (6 (γk (x H + γx L +γ k + G L (x H λ i =0, E (7 γk (x H + γx L +γ k + G L (x H λ i 0, E (8 Digita Object Identifier: /ICST.CROWCOM

3 3 We now make the foowing assumption: A1. There exists an equiibrium such that for each ink, > 0 for some i if ony if > 0 for a i. Simiary > 0 for some i if ony if > 0 for a i Under this condition, we can now sum the above variationa inequaities over a payers to obtain the foowing set of inequaities defined on tota fows on the inks: x H (( +1k x H + γk x L + (x H x H + G H (x H λ =0, E (9 ( +1k x H + γk x L + (x H x H + G H (x H λ 0, E (10 x L (γk (x H + γx L +γ k x L + G L (x H λ =0, E (11 γk (x H + γx L +γ k + G L (x H λ 0, E (1 Here is the number of payers λ = i λ i. These are the variationa inequaities that define the Wardrop equiibrium with ink costs for a High priority user given by c H =( +1k x H + γk x L + (x H x H + G H (x H the ink costs for Low priority user given by c L = γk (x H + γx L +γ k x L + G L (x H ote that the decision for a user invoves both: the choice of ink the choice of cass, high priority or ow priority. But for the existence of a potentia function the foowing externaity symmetry condition (see [1 must be satisfied: which eads to: (c H x L = (cl x H γk = γk + G L (x H G L = γk (1 x H + K L Where K L is an arbitrary constant of integration (we assume it to be the same for a the inks. Let us consider the cassification cost density function for High priority fow to be of the form G H (x H =C H where C H is a constant. ow this is a potentia popuation game with ink costs given by: c H =( +1k x H + γk x L + C H (13 c L = γ ( +1k x L + γk x H + K L (14 the potentia function given by: Ψ( x = E ( +1k (x H + ( +1k γ (x L +C H x H + K L x L + γk x H x L (15 ote that this function is stricty convex. From [1 its foows that the tota ink fows at Wardrop equiibrium can be computed as a soution to the goba minimization probem of the convex function Ψ( x over its domain. The feasibe domain is given by the foowing set of constraints: E(x H + x L = r i = r i x H,x L 0 E Remark.1: Seect K L C H such that the foowing conditions are vaid: min k [ γk (1 r+k L > 0 C H > K L. Then since >1we have the foowing characteristics of the function G L (xh : G L (xh is ineary decreasing with increasing amount of High priority fow in the network. The first condition ensures that G L (xh > 0 over the feasibe domain. The second condition ensures that the high priority cassification cost per unit is greater than the ow priority cassification cost per unit over the entire domain. B. Characterization of equiibrium for symmetric payers with inear congestion costs We now consider the Hierarchica routing game with symmetric casses. We say that casses are symmetric if they have the same dems ink cost functions. Let us denote the dem r i = r for each cass i. It is cear that if the casses are symmetric, then assumption A1 is vaid that the tota fow on a ink at a particuar priority at equiibrium is shared equay by a casses. We thus have the foowing theorem: Theorem.1: Consider a hierarchica routing game between symmetric casses with inear congestion cost density functions as described above. Consider a cassification cost density function for ow priority users to of the form G L (x H = γk (1 x H + K L that for the high priority users is constant C H. Then, The unique Wardrop equiibrium fow profies for the ink costs given by c H c L are obtained as a soution to the goba minimization probem of minimizing the stricty convex function Ψ( x over the feasibe domain. Under A1, there exists a ash equiibrium to the origina probem such that the tota ink fows are the unique Wardrop equiibrium with the transformed costs c H c L If the users are symmetric, A1 hods, the Wardrop equiibrium eads to a unique ash equiibrium for our origina probem where, each cass sends over the ink 1 under a particuar priority, of the tota prescribed fow at Wardrop equiibrium. Digita Object Identifier: /ICST.CROWCOM

4 4 C. Distributed Computation of ash equiibrium: Popuation dynamics The existence of a potentia function in a convex game ushers in a ot of nice properties. First of a, as mentioned before, the equiibrium can be computed as the goba optimum of a singe convex function: The potentia function a hoard of different agorithms for convex optimization can be used to compute the equiibrium. But there are another cass of agorithms based on popuation dynamics which have been shown to converge to the ash equiibrium for our case of potentia games, see [1. These popuation dynamics based agorithms can easiy be impemented in a distributed fashion. Let c H c L for each E be as given by equations (13 (14. Aso et P {H, L} Introduce the foowing distributed evoutionary dynamics: Repicator Dynamics: ( ẋ i,p = x i,p c P 1 r i E ( c H + c L Brown Von eumann (B Dynamics: Define [ ζ i,p 1 =max r i ( c H + c L c P, 0 Then ẋ i,p = ( r i ζ i,p E x i,p (ζ i,h E + ζ i,l A symmetric fow configuration is the one in which the individua fows of each user on any ink are the same. ote that the dynamics are defined in terms of the individua path fows rather than the tota fows x H x L of the potentia game, thus it is necessary to start from from a symmetric point so that the fina configuration is symmetric, which according to theorem (.1 is a necessary condition for it to be a ash equiibrium. Using [1 that proves the convergence of B for potentia popuation games, we see that B converges to the origina ash equiibrium for the symmetric case. Simiar resut hods for the repicator dynamics provided we start at an interior point, see [1, [15. We thus have the foowing theorem. Theorem.: Assume that the casses are symmetric. Then A1 hods The B dynamics converges to the ash equiibrium provided we start from a symmetric point. The Repicator dynamics converges to ash equiibrium provided we start from an interior symmetric point. III. APPLICATIOS A hierarchica routing game arises naturay in some context of practica interests. For exampe, this hierarchy is naturay present in contexts where there are primary (icensed users secondary (unicensed users who can sense their environment because there are equipped with a cognitive radio [9. In cognitive radio networks it requires that secondary users dynamicay opportunisticay utiize unused icensed spectrum without interfering with the primary users. Another situation where this mode finds a natura appication is the two power ALOHA system. We sha consider this scenario as an iustration of the appicabiity of our mode in the next section. A. Hierarchica non-cooperative rate contro over mutipe ALOHA channes Consider an infinite popuation of mobie terminas. We use the mode of unsotted aoha where the goba arriva of new packets from a the mobies foows a poisson process with rate λ. Consider M channes over which the packets can be routed under the ALOHA protoco. Aso two eves of power can be chosen by each mobie: High Low. The time required to send the packet is 1 time unit. So if there is a transmission in the period [t 1,t+ 1, then there is a coision. If a High power packet coides with another High power packet then both packets are ost, simiary if a ow power packet coides with a ow power packet. If a high power packet coides with a ow priority packet, then the high power packet transmission is successfu whie the ow power packet is ost. This is caed the capture phenomenon. A mobies are identica in the sense that the contribution to the goba rate λ of the arriva rate from each mobie, though negigibe, is the same for a mobies. Let η be the rate contribution of a singe mobie where η is very sma compared to λ. The popuation of mobies is divided into groups each being controed by a controer i where i =1,,. A these controers are symmetric each contros the same popuation size of mobies hence the same rate. They face the decision of spitting this rate amongst different channes as we as power eves. The rate being controed by each controer is given by λ. Let λi,h λ i,l be the high ow power rate of packets respectivey from controer i on channe. Let λ H λ L be the tota high ow power rates respectivey on a channe. λ H + λ L = λ The probabiity of successfu transmission for a packet choosing high power on a ink is P H (λ H =e λh that for a packet choosing ow power is P L (λ H + λ L =e (λh +λl The channe utiity density per unit rate is assumed to be an increasing function of the packet transmission probabiity. We choose the ogarithmic function, then the channe cost density (negative of the utiity density is rendered inear in the rates on that channe: c H = k n(e λh =k (λ H Digita Object Identifier: /ICST.CROWCOM

5 5 c L = k n(e (λh +λl =k (λ H + λ L where c H is the channe cost density for a controer routing on channe using high power, c L is the channe cost density for a controer routing on channe using ow power k is a constant for each channe. The cassification cost density for transmitting a unit rate with high power is given by C H it is a constant. According to the pricing scheme derived in the previous section, the cassification cost density for transmitting a unit rate as ow power on a channe is given by G L (λh = k (1 λ H + K L such that K L satisfies the properties mentioned before. The tota cost density functions (channe cassification are then given by : c H = k (λ H +C H c L = k (λ H + λ L + k (1 λ H + K L The equiibrium tota channe fows at high or ow power can then be found as the ones which minimize the potentia function: Ψ= ( +1k (λ H + ( +1k (λ L +C H λ H + K L λ L + k λ H λ L (16 1 Exampe: Consider a hierarchica ALOHA game between 4 symmetric controers. There are three channes 1, 3 with k 1 =1, k = k 3 =4. Choose λ =1, K L =3.5 C H =4. We minimize the potentia function above to get the foowing vaues of tota high power ow power rates on each channe at ash equiibrium: λ H = ( λ H 1, λ H, λ H 3 = (0.0358, , λ L = ( λ L 1, λ L, λ L 3 = (0.5357, 0.679, thus the ash equiibrium rates for each controer i are given by: λi,h = ( λ i,h i,h i,h 1, λ, λ 3 = ( , , λ i,l =( λ i,l i,l i,l 1, λ, λ 3 = ( , , The cost at ash equiibrium to each payer is the tota cost incurred by a the payers is We aso compute the gobay optima soution. At this soution, the tota high power ow power rates on each channe are: ˆλ H =(ˆλ H 1, ˆλ H, ˆλ H 3 =(0.149, , ˆλ L = (ˆλ L 1, ˆλ L, ˆλ L 3 =(0.487, 0.144, The tota gobay optima cost is which is esser than the tota cost at ash equiibrium. Thus even though the ash equiibrium is computed as a goba optima of the potentia function, minimizing this potentia does not minimize the tota cost incurred in the system. IV. COCLUSIOS We introduced a new cass of atomic routing games which aows for a hierarchy amongst the routed entities with nonreciproca congestion effects, appications of which arise frequenty in practica scenarios especiay with the advent of cognitive radio technoogy. Through our resuts we were abe to highight the issues that arise in modeing routing scenarios in the absence of reciproca congestion effects between casses of users. Finay a feasibe pricing scheme was derived which induces a potentia function for the atomic game on parae inks with inear ink costs symmetric payers. Our resuts were iustrated by an appication: mutichanne hierarchica rate aocation under unsotted ALOHA for which we expicity computed the ash equiibrium for the atomic case. REFERECES [1 E. Atman, T. Başar, T. Jiménez. Shimkin, Routing into two parae inks: game-theoretic distributed agorithms, Specia Issue of Journa of Parae Distributed Computing on Routing in Computer Communication etworks, pp , Vo. 61, o. 9, September 1, 001. [ E. Atman H. Kameda, Equiibria for muticass routing probems in muti-agent networks, In 40th IEEE CDC, Orondo, Forida, USA, December, 001 [3 D. Bertsekas R. Gaager, Data etworks, Prentice Ha, Engewood Ciffs, ew Jersey, [4 E. Atman L. Wynter, Equiibrium, games, pricing in transportation teecommunication, etwork spatia Economics, Specia Issue on Crossovers Between Transportation Teecommunication Modeing, E. Atman L. Wynter, Guest Eds, 4(1, March 004 [5 A. Haurie P. Marcotte, On the reationship between ash-cournot Wardrop equiibria, eworks, 15:95-308, 1985 [6 A. eyman, Correated equiibrium potentia games, Int. J. of Game Theory, 6:3-7, 1997 [7 P. Gupta P. R. Kumar, A system traffic dependent adaptive routing agorithm for ad hoc networks, Proceedings of the 36th IEEE Conference on Decision Contro, pp , San Diego, Dec [8 Y. A. Koriis A. Orda, Incentive-Compatibe Pricing Strategies for QoS Routing, in Proceedings of IEEE IFOCOM 99, ew York, Y, USA, March 1999 [9 Jin Zhang Qian Zhang Hong Kong, Stackeberg game for utiitybased cooperative cognitive radio networks, in the Proceedings of the tenth ACM internationa symposium on Mobie ad hoc networking computing, ew Oreans, LA, USA, 009. [10 A. Orda,. Rom. Shimkin, Competitive routing in muti-user communication networks, IEEE/ACM Transaction on etworking, Vo 1, pp. Pages , [11 J. G. Wardrop, Some theoretica aspects of road traffic research, Engineers, Part II, 195, pp [1 W. H. Shom, (001. Potentia Games with Continuous Payer Sets, J. Econ. Theory 97, [13 B. W. Wie R. L. Tobin, On the reationship between dynamic ash instantaneous user equiibria, etworks, 6:313-37, 1998 [14 R. Rosentha. A cass of games possessing pure-strategy nash equiibria. Internationa Journa of Game Theory, :65 67, [15 S. Shakkottai, E. Atman A. Kumar, The Case for oncooperative Mutihoming of Users to Access Points in IEEE WLAs, invited paper, IEEE Journa of Seected Areas in Communications, Vo 5 o 6, , Aug 007. Digita Object Identifier: /ICST.CROWCOM