Capacity of Wireless Networks with Heterogeneous Traffic

Transcription

1 Capacity of Wireless Networks with Heterogeeous Traffic Migyue Ji, Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata Cruz, 56 High Street, Sata Cruz, CA 95064, USA Palo Alto Research Ceter PARC, 3333 Coyote Hill Road, Palo Alto, CA 94304, USA {davidjmy, wzgold, hamid, Abstract We study the scalig laws for wireless ad hoc etwork i which the distributio of odes i the etwork is homogeeous but the traffic is heterogeeous. More specifically, we cosider the case i which a ode is the sik to k sources sedig differet iformatio, while the rest of the odes are part of uicast commuicatios with a uiform assigmet of source-destiatio pairs. We prove that the capacity of these heterogeeous etworks is Θ, where ad deote the maximum traffic for a cell ad the umber of odes i the etwork, respectively. Equivaletly, our derivatios reveal that, q whe k costat, the etwork capacity is equal to Θ for k O ad equal to Θ ` k for k Ω. Furthermore, the etwork capacity is Θ whe k costat. These results demostrate that the capacity of a heterogeeous etwork is domiated by the maximum cogestio i ay area of the etwork. I. INTRODUCTION The scalig laws of wireless ad hoc etworks with homogeeous traffic ad uiform distributio have bee extesively studies i the literature. The semial paper by Gupta ad Kumar [] evaluated the capacity of wireless ad hoc etwork with uiform traffic ad showed that the capacity scales as Θ uder the protocol model. The iformatio theoretic capacity of wireless ad hoc etworks with cooperatio amog odes was ivestigated by Xie ad Kumar [2], [3]. Zemliaov ad de Veciaa [4] ivestigated the throughput capacity with homogeeous traffic whe some odes are coected to the ifrastructure. Few prior works ivestigate heterogeeous traffic i the etwork. Keshavarz-Haddad et al. [5] itroduced the cocept of trasmissio area. Based o that defiitio, they itroduced a method to compute the upper boud of the capacity for differet traffic patters ad differet topologies of the etwork. However, the paper did ot itroduce ay closed-form scalig laws for the etwork capacity. Krishamurthy et al. [6] discussed differet heterogeeous traffic requiremets, which deped o the type of data such as audio ad video. Liu et al. [7] assumed a heterogeeous traffic for low-priority ad highpriority data with differet traffic models for them. Rodoplu et al. [8], [9] cosider a etwork with may sources selectig a sigle ode as destiatio. They itroduce the cocept of core capacity ad derived some aalytical results for capacity of this type of etwork ad compared it with uiform uicast core capacity. However, their derivatios did ot lead to a closed form scalig laws; istead, they showed simulatio results for the case i which there is a limited umber of odes i the etwork. To the best of our kowledge, this is the first paper that provides the scalig laws of such etwork with heterogeeous traffic as a fuctio of ad other etwork parameters. Iterestigly, we fid out that the capacity is domiated by the area i which the majority of traffic i the etwork passes. This result is ituitive whe we assume that all the traffic requiremet for each ode should be satisfied. Clearly, the ode with the highest traffic will domiate the capacity. The paper is orgaized as follows. Sectio II presets the assumptios ad defiitios eeded i our aalysis. Sectio III provides the routig scheme ad the lower boud throughput capacity for our etwork model. Sectio IV provides the upper boud. Some discussios are preseted i Sectio V ad the paper is cocluded i Sectio VI. II. WIRELESS NETWORK MODEL We cosider a etwork with odes uiformly distributed i a dese etwork, where the area of the etwork is a costat uit square. We assume heterogeeous traffic for the etwork, such that a sigle ode called the access poit is the destiatio for k sources i the etwork. For the rest of the k odes i the etwork we assume radom ad uiformly distributed source-destiatio pairs. Therefore, the source-destiatio pair selectio for uicast commuicatios is similar to that used by Gupta ad Kumar [] for the rest of k odes i the etwork. This etwork model is show i Figure. The trasmissio rage is assumed to be the same for all the odes ad the commuicatio betwee odes is poit-to-poit. A successful commuicatio betwee two odes is modeled accordig to the protocol model, which is defied below. Defiitio 2.: Protocol Model: Assume that there is a sigle commo commuicatio rage r for all odes. Node i at locatio X i ca successfully trasmit to ode ir at locatio X ir if X i X ir r ad for every ode k located at X k, k i that trasmits at the same time, X k X ir + r. The quatity guaratees a guard zoe aroud the receiver.

2 B. The traffic caused by access ode Let us defie a traffic from ode i to ode j as commodity [9]. Clearly, the umber of commodities for access ode is k which is also equivalet to the umber of lies paths passig through the cell that cotais the access ode. For simplicity of the aalysis, we assume that the access ode is located at the ceter of the etwork. Now we compute the umber of commodities for a cell that has a distace of x from the access ode. From Fig. 2 ad by choosig X i C 2, the area of triagle is Fig.. The Network Model Defiitio 2.2: Feasible Throughput: A throughput of λ i bits per secod is said to be feasible for the i th source-destiatio pair if there is a commo trasmissio rage r, ad a scheme to schedule trasmissios ad there are routes betwee source ad destiatio, such that source i ca trasmit to its destiatio at such rate successfully. For heterogeeous traffic, the feasible throughput is defied for each source-destiatio pair. Defiitio 2.3: Order of Throughput Capacity: The total throughput capacity is said to be of order Θf bits per secod if there exist a costat c ad c such that lim Prλ λ i cf is feasible ; ad i limif Prλ λ i c f is feasible <. i III. THE LOWER BOUND OF THE CAPACITY We eed to emphasize that there are two types of traffic i our model. Oe traffic is associated to the k sources trasmittig packets to the access ode ad the other traffic stems from the rest of k odes i the etwork with uicast commuicatios. Therefore, we eed to defie the routig protocol ad schedulig uder this traffic model. A. The Routig Scheme ad the Schedulig Protocol The selectio of sources for the access ode i is based o the techique described i [0]. We radomly ad uiformly select k locatios i the etwork ad choose the closest odes to these k locatios as sources for the access ode. The routig trajectory is a straight lie L i from access ode to these k locatios. The the packets traverse from each source to destiatio i a multi-hop fashio passig through all the cells that cross L i. For the rest of j odes with uicast traffic where j k, both selectios of source-destiatio pairs ad routig is similar to the above techique. For the schedulig scheme, we utilize a TDMA scheme similar to [0] with some modificatios to take ito accout the heterogeeity of the traffic. S XiAB 2d 2 x + d 2 d 2 < 2d x, 2 d C is selected to guaratee the coectivity betwee adjacet cells i the etwork [] ad C is a costat factor. Fig. 2. d A geometric descriptio of traffic by the access ode i the etwork Theorem 3.: For ay cell with a distace of from the access ode, the upper boud for the umber of commodities caused by the traffic from the access ode is X i N j < 2 d k 3 whe k Ω. Proof: The average umber of lies passig through the cell E[N xj ] whose distace from access ode i is is less tha 2 d k sice k source odes are uiformly distributed i the etwork. Utlizig the Cheroff boud [], we have Pr N xj E[N xj ] > δ E[N xj ] < exp [ + δlog + δ δen xj ] 4 ad Pr N xj E[N xj ] < δ E[N xj ] ] < exp [ δ2 2 E[N ] 5

3 where 0 < δ <. Combiig the results ad cosiderig E[N xj ] < 2d k, we obtai Pr N xj E[N xj ] > δ E[N xj ] < [ exp + δlog + δ δ 2d ] ] k + exp [ δ2 2d k. 2 6 Thus, the probability that the values of the radom variables N xj for all j ca simultaeously be arbitrarily close to E[N xj ] is give by Pr N xj E[N xj ] < δ E[N xj ] j Pr N xj E[N xj ] > δ E[N xj ] j Pr [ N xj E[N xj ] > δ E[N xj ] ] j [ exp + δlog + δ δ 2d ] k > j ] + exp [ δ2 2d k. 7 2 Deote that if k Ω ad d Θ probability teds to whe. C. The traffic caused by uicast commuicatios, the this I this sectio, we derive the umber of lies passig through each cell because of uicast traffic i the etwork. Sice the uicast traffic is distributed uiformly i the etwork, this value is the same for all the cells i the etwork. Lemma 3.2: For ay cell S, the maximum umber of lies itersectig this cell caused by uicast traffic is give by PrMaximum umber of lies L i passig through S C 2 k, whe k costat. Proof: Our proof is similar to that of [0] except that we accout for k uicast pairs i the etwork. The probability that the destiatio ode j is x away from the source ode is C 3 πx+d [0] where C 3 is a costat. Thus, the probability p that there is a lie passig through the cell S which is with distace x from j is PrL i itersects S p < 2 d 2d x k C 3 πx + d dx C 4 where C 4 is a costat value. Each of k odes radomly ad uiformly selects ay other ode i the etwork as 8 destiatio. Defie i.i.d. radom variable I i as { If Li itersect S I i 0, Otherwise where i, 2,, k. It is clear from Eq. 8 that PrI i p < C 4. Deote Z k i I i as the umber of lies passig through the cell S. Thus for positive values of a ad m ad usig Cheroff Boud, we have Furhter, 9 PrZ > m EeaZ. 0 eam Ee az + e a p k Let s defie m C 2 k exp ke a p expc 4 ke a, the Eq. becomes PrZ > C 2 k exp k C 4e a C 2 a. 2 If we select C 2 such that C 2 a C 4 e a ǫ > 0, the PrZ > C 2 k exp ǫ k. 3 If the area for each cell is defied as s 2 Θ, the by utilizig the uio boud we arrive at PrSome cells have more tha k lies PrZ > k all the cells s 2 exp ǫ k exp ǫ k. 4 2C This probability goes to zero as teds to ifiity as log as k costat. D. The Lower Boud of the Capacity Case of k costat: From the previous two sectios, we deduce that the umber of lies passig through a cell with distace x from the access ode is upper bouded as 2dk x + C 2 k ad for the cell that cotais the access ode is k+c 2 k. I the traditioal aalysis of capacity with homogeeous traffic, the iverse of traffic for a cell usig a TDMA scheme provides the throughput capacity. Give that this value varies for differet cells i heterogeeous traffic, we assig a badwidth to the cell that is

4 proportioal to the umber of lies passig through a cell. This assigmet is based o the fact that each lik i the etwork has the same badwidth similar to the approach by Gupta ad Kumar but more allocatio of badwidth is give to a cell with higher traffic. Clearly, our results demostrate that the cell that cotais the access ode has the highest traffic. If we divide the etwork ito layers of cells startig from the access poit as show i Fig. 3, the traffic for cells i each layer is the same order. Let s assume the traffic for each layer is T i where i,..., Θ. The our badwidth requiremet for each layer is give by W o T o W T... W Θ c. 5 T Θ Note that W o W max, T o ad c is a pre-determied fuctio of. This assumptio basically meas that more badwidth is provided to a cell with higher traffic. 2 Case of k costat: Uder this coditio, clearly all the traffic is cotributed by the access ode ad sice each source is sedig differet packet to the access ode, the achivable capacity is Ω by allowig oe source at the time to trasmit its packet to the access ode. Combiig the above results, we state the followig theorem for the achivable lower boud. Theorem 3.3: The achievable lower boud for a heterogeeous traffic with maximum umber of traffic of for a cell ca be give as follows. Ω whe k C 5 C lower 7 Ω whe k C 5 Note that Theorem 3. is proved oly for k Ω. However whe k O, we ca still take advatage of the upper boud for because there is less traffic uder this coditio ad the upper boud holds. X i l l l l l log IV. THE UPPER BOUND OF THE CAPACITY Fig. 3. The layers aroud X i We first compute the capacity for the case whe k costat. The capacity ca be defied as The average umber of odes i each cell is proportioal to Θ, the the lower boud capacity is Θ log 8lW l C lower + W 0 Θ, T l MW max MW max l Θ l0 Θ MW max + T 0 8lc + c Θ, Θ c, Ω c W max Ω, 6 where M is the TDMA parameter that is required to separate cells i order to satisfy the protocol model. Note that the capacity defied i this paper is the total capacity sice the traffic for each ode is differet ad per ode capacity may ot be meaigful. C upper the sum of capacity for all cells the average umber of hops for source-destiatio pairs maximum badwidth expasio TDMA parameter. First, we cosider the case whe k Ω. It is easy to show that x 2l 2d 2 where l varies from a costat value up to Θ depedig o the locatio of cell from the access ode. From this lower boud for x, we ca derive the upper boud for T l. 2d k T l < 2l + C 2 k 2d 2 k + C 2 k l 0 l 0 8

5 The the capacity ca be derived as C upper Θ MW max l 8lW l L o r + W 0 L o r W max ML o rc 2 2k a W max ML o rc Θ 2d k 8l l 2l + C 2 k 2d 2 + k + C 2 k Θ + k + C 2 k Θ l W max ML o rc 2 2k k+ 2l C 2 k Θ l 8l + l 8l + Θ l 8l 2l W max ML o rc 2 2kΘ + log + k + C 2 k Θ W max ML o rc 2 2kΘ + C 2 kθ b W max ML o cθ 2 2kΘ + C 2 kθ W max ML o c 2 2k + C 2 k O c W max O 9 a is derived by replacig W l T l c ad b is derived by replacig r with Θ. L i this derivatio is the average legth of each uicast or the average legth over all 4 distaces betwee k sources ad the access ode. Secod, we cosider the case whe k O. From 8, we ca see that the maximum traffic i the etwork still satisfies this coditio. Thus, we ca derive the same result as 9. The case of k costat is straightforward sice we ca at most have oe data set to the access ode whe all the commuicatios ivolve the access ode. Fially, from the aalysis above, we derive a tight boud for the capacity. Theorem 4.: I a radom ad hoc etwork, uder the heterogeeous traffic patter with oe ode performig as the destiatio for k source odes ad other odes have uicast commuicatios, the overall capacity is Θ, k C 5, k O C Θ, k C 5, k Ω k, Θ. whe k C 5 20 Proof: We kow that the capacity of this etwork is Θ, where k + C 2 k. The it is straightforward to see that for differet values of k, eq. 20 ca be derived. V. DISCUSSION Fig. 4 shows the throughput capacity of a wireless etwork obtaied from 20 as a fuctio of the umber of sources for the access ode. As the umber of the sources for this access ode k icreases from to Θ, the capacity of the etwork is Θ which is the well kow result computed by Gupta ad Kumar for homogeeous traffic model. We call this regio as Homogeeous Traffic regio. It is clear that the capacity of the etwork i this regio is domiated by the uiform uicast traffic. Oce the value of k passes this threshold of Θ, the capacity of the etwork is Θ k which is smaller tha the capacity of the Homogeeous Traffic regio. The capacity of the etwork is domiated by the access ode which is the bottleeck i the etwork ad we call this capacity regio as Heterogeeous Traffic regio. This result implies that for the cells ear the access ode, we should assig more resources badwidth or time to guaratee the data rate for each traffic. Fially if the umber of sources for the access ode is such that k C 5, the the capacity is Θ which is the same as broadcast trasport capacity [2]. Sice the umber of sources is relatively large i this case, we call this capacity regio as All to Oe Traffic regio. We ca see that almost all of the odes have traffic for the access ode, thus, for the extreme case that all the odes have traffic to the access ode, at each time, oly oe ode ca trasmit. Furthermore, the capacity we calculated is a ormalized capacity by the maximum badwidth. We ca see without this ormalizatio, the capacity of the etwork is c which is ot related to k see Eqs. 6 ad 9. However, to achieve the same capacity for all odes ad for differet

6 C log log k log k Fig. 4. The capacity result values of k, we eed to allocate more badwidth to the more cogested areas of the etwork. Fig. 5 demostrates that i the Homogeous Traffic regio, the maximum badwidth eeded is ot related to k. However, i the Heterogeous Traffic regio, the badwidth grows liearly with k, which is the price for keepig the overall capacity the same. Fially, i the All to Oe Traffic regio, the order of the maximum badwidth does ot chage. W max c REFERENCES [] P. Gupta ad P.R.Kumar, The capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 46, o. 2, pp , March [2] L.-L. Xie ad P.R.Kumar, A etwork iformatio theory for wireless commuicatio: Scalig laws ad optimal operatio, IEEE Trasactios o Iformatio Theory, vol. 50, o. 5, pp , May [3], O the path-loss atteuatio regime for positive cost ad liear scalig of trasport capacity i wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 52, o. 6, pp , Jue [4] A. Zemliaov ad G. de Veciaa, Capacity of ad hoc wireless etworks with ifrastructure support, IEEE Joural o Selected Areas i Commuicatios, vol. 23, o. 3, pp , March [5] A. Keshavarz-Haddad ad R. Riedi, Bouds for the capacity of wireless multihop etworks imposed by topology ad demad, i MobiHoc, September 2007, pp [6] F. Yu, V. Krishamurthy, ad V. C.M.Leug, Cross-layer optimal coectio admissio cotrol for varible bit rate multimedia traffic i packet wireless cdma etworks, IEEE Trasactios o Sigal Processig, vol. 54, o. 2, pp , February [7] W. Liu, X. Che, Y. Fag, ad J. M. Shea, Courtesy piggybackig: Supportig differetiated services i multiop mobile ad hoc etworks, IEEE Trasactios o Mobile Computig, vol. 3, o. 4, pp , [8] V. Rodoplu ad T. H.Meg, Core capacity of wireless ad hoc etworks, i The 5th Iteratioal Symposium o Wireless Persoal Multimedia Commuicatios, September 2002, pp [9] M. Kyoug ad V. Rodoplu, Core capacity regio of portable wireless etworks, i Globecom, September 2004, pp [0] F. Xue ad P. R. Kumar, Scalig Laws for Ad Hoc Wireless Networks: A Iformatio Theoretic Approach. NOW Publishers, [] B. Motwai ad P. Raghava, Radomized Algorithms. Cambridge Uiversity Press, 995. [2] D. Marco, E. J. Duarte-Melo, M. Liu, ad D. L. Neuhoff, O the may-to-oe trasport capacity of a dese wireless sesor etwork ad the compressibility of its data, i IPSN, kc c log k Fig. 5. The maximum badwidth required correspodig to differet k VI. CONCLUSION This paper preseted the first closed-form scalig laws for the capacity of wireless ad hoc etworks with heterogeeous traffic. More specifically, we assumed a access ode with k sources choosig this ode as destiatio ad the rest of odes i the etwork, havig uicast commuicatios. It was show that the capacity of such heterogeeous etworks is Θ. Equivaletly, our derivatios reveal that, whe k costat, the the capacity is equal to Θ for k O ad equal to Θ k for k Ω log. Furthermore, whe k costat, the the capacity is Θ. The results demostrate that, as it should be expected, the capacity of a heterogeeous etwork is domiated by the maximum traffic cogestio i ay area of the etwork.