A Finite Queue Model Analysis of PMRC-based Wireless Sensor Networks

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1 A Fnte Queue Model Analyss of PMRC-based Wreless Sensor Networks Qaoqn L, Me Yang, Hongyan Wang, Yngtao Jang, Jazh Zeng College of Computer Scence and Technology, Unversty of Electronc Scence and Technology, Chengdu, Schuan, P. R. Chna Department of Electrcal and Computer Engneerng, Unversty of Nevada, Las Vegas, NV, 8954 Emals: Abstract In our prevous work, a hghly scalable and faulttolerant network archtecture, the Progressve Mult-hop Rotatonal Clustered (PMRC) structure, s proposed for constructng large-scale wreless sensor networks. Further, the overlapped scheme s proposed to solve the bottleneck problem n PMRC-based sensor networks. As buffer space s often scarce n sensor nodes, n ths paper, we focus on studyng the queung performance of cluster heads n PMRC-based sensor networks. We develop a fnte queung model to analyze the queung performance of cluster heads for both non-overlapped and overlapped PMRC-based sensor network. The average queue length and average queue delay of cluster head n dfferent layers are derved. To valdate the analyss results, smulatons have been conducted wth dfferent loads for both nonoverlapped and overlapped PMRC-based sensor networks. Smulaton results match wth the analyss results n general and confrm the advantage of selectng two cluster heads over selectng sngle cluster head n terms of the mproved queung performance. Key words: wreless sensor network, PMRC, overlap, fnte queue model. Introducton Due to ther benefts of low cost, rapd deployment, self-organzaton capablty, and cooperatve dataprocessng, wreless sensor networks have been proposed as a practcal soluton for a wde range of applcatons [], such as survellance and habt montorng, hazard detecton, ntellgent agrculture, automaton and control, ntellgent home, etc. Energy effcency s treated as the top desgn objectve for wreless sensor networks snce each sensor node has lmted power to consume. Prevous research shows that clustered structure [5] and mult-hop routng [3] acheve better energy effcency for large-scale sensor networks (as needed n many applcatons). In [5], a hghly scalable and fault-tolerant network archtecture, named as the Progressve Mult-hop Rotatonal Clustered (PMRC) structure s proposed, whch s sutable for the constructon of large-scale wreless sensor networks. In the PMRC structure, sensor nodes are parttoned nto layers accordng to ther dstances to the snk node. A cluster s composed of the nodes located n one layer and the cluster head n the upper layer closer to the snk node. The cluster head s responsble for forwardng data to ts upstream layers. A dstngushed feature of the PMRC structure s that two cluster heads (the prmary cluster head and the secondary cluster head) are selected for each cluster and the two cluster heads rotate to work. The study n [5] shows that by selectng two cluster heads, load balance s acheved and node lfe tme s prolonged. Smlar to other mult-hop structures, the PMRC structure also suffers from the bottleneck problem []. In the PMRC structure, the bottleneck problem s reflected on phenomenon that the network lfe tme s lmted by the node lfe tme of the cluster heads closer to the snk node. To solve ths problem, overlapped neghborng layers s proposed to balance the relay load at the cluster heads for all layers [4]. By overlappng layers, more cluster head canddates are avalable for each layer and the communcaton energy n each cluster can be reduced, whch ultmately helps prolongng network lfe tme. Smulaton results confrm that the overlapped scheme wth reasonable overlap ranges acheves sgnfcant mprovement n network lfe tme. Besdes energy effcency, qualty of servce (QoS) requrements [4][3], such as throughput, packet delay, packet loss, should also be satsfed n sensor networks. The buffer space of a sensor node s often lmted. Hence, properly choosng the buffer space s also mportant n desgnng sensor networks [8]. In the lterature, a number of research results on queung performance analyss for wreless sensor networks have been reported. A typcal approach (as adopted n [6][8][9][]) n queung analyss for wreless sensor networks s modelng a sensor node as a fnte FIFO queue usng a contnuous tme Markov chan assumng sensor nodes alternate between actve and sleep modes. The alternaton of actve/sleep mode may help reduce energy consumpton of a sensor node. However, the sleep mode wll cause extra delay and packet loss. In ths paper, we focus on the queung analyss of the PMRC-based wreless sensor networks. Generally all sensor nodes are assumed to be actve except the two cluster heads of one cluster may rotate to forward data n

2 fxed tme ntervals. Each sensor node s modeled as a M/M//N queue model [0]. We then study the network performance n terms of average queue length and average queue delay, and explore the mpact of head rotaton and the overlapped scheme on these metrcs. Through analyss and smulaton results, we are gvng strong nsght nto the desgn parameters that affect the queung performance. The rest of the paper s organzed as follows. In secton, the network model and data flow model are descrbed. In Secton 3, we develop a fnte queue model of cluster heads and analyze the average queue length and the average queue delay of cluster heads at dfferent layers for both non-overlapped and overlapped PMRC structures. In Secton 4, smulaton results of the two performance metrcs are presented and compared wth the analyss results. Secton 5 concludes the paper.. Network Model and Data Flow Model. Network model Fg. llustrates an overlapped PMRC structure [4], Wthout loss of generalty, we assume that the sensor nodes are dstrbuted unformly wth densty ρ n a crcular area and the snk node s located at the center of the area. The crcular area can be parttoned nto a set of sub-areas, each one composed of the clusters formed n consecutve layers. Each sub-area can be represented as a fan shape wth angle θ (see Fg. ). Fgure Overlapped layers n a PMRC-based sensor network [4]. As shown n the fgure, layer occupes a crcular area and layer s shown n a rng shape. The grey area ndcates the overlapped area of layer and layer. Layer s consdered as the nner layer of layer and layer s consdered as the outer layer of layer. Note that the sensor nodes n the grey area stll belong to layer whle they are the canddate cluster heads for clusters n layer. Partcularly, the snk node s the cluster head of the sngle cluster formed n layer. The non-overlapped PMRC structure can be consdered as a specal case of the overlapped PMRC structure wth no overlapped area between adjacent layers.. Data flow model Assume that all nodes n the crcular area are actve n transmsson and a porton of these nodes are actve n sensng. We assume the sensed data are organzed nto packets wth a varable sze followng exponental dstrbuton. The data generatng process at each sensor node follows a Posson process, and all the sensng events n the wreless network occur ndependently. Each sensng node transmts ts sensed data to ts cluster head. Each cluster head then sends all sensed data wthn ts cluster and relays data comng from the cluster head n ts outer layer to the cluster head n ts nner layer. In the PMRC-based sensor network, sensor nodes belongng to the same cluster compete on the common channel for data transmsson. Accordng to [8], n a contenton-based sensor network, the relay packets from the outer layer can be also assumed as a Posson process as long the length of contenton wndow s much smaller than the nterval of packets. Hence, the arrval process at the cluster head of each layer can also be consdered as a Posson process followng the superposton property of Posson process. We also assume that there s no data aggregaton at all layers. 3. Analyss The followng notatons wll be used n our analyss. R: dameter of the crcular area of the wreless sensor network. n: maxmum number of layers n the crcular area. r: transmsson and sensng range of the sensor nodes. ρ: sensor node densty n the crcular area. α: rato of the number of sensng nodes to the total number of nodes, 0 α. θ: angle of the fan shape n overlapped PMRC structure. θ': angle of the fan shape n non-overlapped PMRC structure. r : range of the rng shape n layer for overlapped PMRC structure, where r =r. r ': range of the rng shape n layer for non-overlapped PMRC structure, where r '=r. x : range of the overlapped area of layer and layer +. λ 0 : data generaton rate of sensed data at each node. λ : data arrval rate at cluster head n layer for overlapped PMRC structure. λ ': data arrval rate at cluster head n layer for nonoverlapped PMRC structure. μ: data transmsson rate of a sensor node. μ : servce rate of cluster head n layer for overlapped PMRC structure. μ ': servce rate of cluster head n layer for nonoverlapped PMRC structure. K : buffer sze at cluster head n layer. N : number of neghborng sensor nodes of a cluster head n layer for overlapped PMRC structure.

3 N ': number of neghborng sensor nodes of a cluster head n layer for non-overlapped PMRC structure. 3. Node queue model For a cluster head n layer, wth the Posson arrval process of rate λ, the exponentally dstrbuted servce of rate μ, buffer sze K, t can be modeled as a M/M//K queue system, as shown n Fg.. λ μ K Fgure Queung model of cluster heads n layer. We observe that the arrval rate to the overlapped PMRC structure and non-overlapped PMRC structure are dfferent. In the followng, we wll show the dervaton of λ for overlapped PMRC structure followed by the dervaton of λ ' for non-overlapped PMRC structure. λ3( x, x, θ) = λ 0ραθ( n ( + r + r x x ) )/ (5) 3 Generally, we have, n rn( x, x,..., xn, θ ) = ( ( r x)) sn θ = n ( ( r x))( cos θ ) = (6) n λn( x, x,..., xn, θ) = λ 0 ραθ( n ( ( r x ) r n ) )/ + = (7) Accordng to the analyss n [4], θ s chosen at 7º for overlapped PMRC structure. The arrval rate at the cluster heads n each layer n nooverlapped PMRC structure can be derved as follows. Fgure 4 Top vew of three non-overlapped layers. Fgure 3 Top vew of three overlapped layers [4]. We frst consder the arrval rate of a cluster head n the overlapped area between layer and layer. Consder the fan shape wth angle θ whch corresponds to one cluster range n each layer. From Fg. 3, the arrval rate to the cluster head n layer can be derved as λ (θ)= λ 0 ρα(r r )θ/ =λ 0 ραθ(r r )/, where θ(r r )/ gves the area outsde layer. For smplcty, we normalze the value of r as. Assume R = n*r, then we get R=n. Thus λ (θ) can be derved as: λ( θ) = λ0ραθ( n )/ () Next, we derve λ as follows. To fnd out the overlapped area of layer and layer 3, we need calculate r, whch can be obtaned by geometry relatons as: r ( x, θ ) = ( x ) sn ( )( cos ) θ x θ () We then get λ ( x, θ) = λ 0 ραθ( n ( + r x ) )/ (3) We then derve r 3 and λ 3 as follows. r ( x, x, θ ) = ( + r x x ) sn θ 3 ( + r x x )( cos θ ) (4) Fgure 4 llustrates the layer relaton n non-overlapped PMRC structure. From Fg. 4, we can derve that λ '( θ ') = λ0ραθ '( n )/ (8) Accordng to geometry relatons, r ' s calculated as: θ ' r '( θ ') = ( sn ) (9) 4 We then get λ '( θ ') = λ0ραθ '( n ( + r ') ) / (0) Followng smlar way, we can derve r ' and λ ' for >. θ' s typcally set slghtly larger than θ as the cluster sze generated n the non-overlapped PMRC structure tends to be larger than that n the overlapped PMRC structure. For both overlapped and non-overlapped PMRC structure, the servce rate can be derved n the same way. Assume that sensor nodes n the same cluster compete on the common channel for data transmsson through random back-off scheme. The back-off tme s a random number wth a dscrete unform dstrbuted between 0 and CW-, where CW s the contenton wndow sze. Takng the example of overlapped PMRC structure, the probablty of the cluster head n layer wns the channel can be derved as follows. N P ( ) wn = () CW, where N s the number of neghborng nodes of the cluster head n layer competng for the channel.

4 Then, we have the rate of servce tme of cluster head n layer. N Pwn ( ) μ = μ = μ () CW 3. Queung performance metrcs In our study, we consder two queung performance metrcs, the average queue length at layer and the average queue delay at layer. To derve the average queue length at layer, we derve the steady-state queue length dstrbuton of cluster head n layer frst. Denote P j () as the probablty that there are j data packets n the buffer of a cluster head n layer. Then P j () can be derved as: ρ j Pj () = ρ, 0 K j K (3) j ρ + λ where ρ =. μ Then the average queue length (.e., the average number of data packets n the queue of the cluster head) at layer can be calculated as: K K + ρ ( K + ) ρ En [ ] = jpj( ) = K + j= ρ ρ (4) Accordng to Lttle s Law [0], the average queue delay at layer can be calculated as: E[ n ] Ed [ ] = (5) λ 4. Performance Evaluaton To valdate our analyss, smulatons of both overlapped and non-overlapped PMRC structure based on the smulaton models [4][5] developed on OPNET Modeler network smulator have been conducted. The wreless channel allocaton model provded n OPNET s used n our smulatons. 4. Smulaton settngs In the smulatons, we assume a 00mx00m geographcal area covered by a sensor network wth the snk node located at the center. All the sensor nodes are unformly dstrbuted n the network. Tab. lsts some basc parameters used n our smulatons. Table. Basc smulaton parameters. Parameter Value Sensor feld area 00m x 00m Node number (N) 400 Rado transmsson range (R ) t 40m Intal energy per node J Buffer sze 50 packets Mean packet sze 500 bytes Channel bandwdth Mbps Transmsson speed at each node (μ) 9600 bps Data generaton rate (λ 0 ) {0.3, 0.36, 0.4, 0.48, 0.54, 0.59} pkt/s Smulaton tme Untl the frst node death The followng performance metrcs are collected: Average queue length: the average queue length s averaged from the tme when the frst packet arrves at the queue untl the smulaton ends. Average queue delay: the queue delay of a packet s the dfference of the tme when the packet arrves at the queue and the tme when the packet leaves the queue. In the followng, we present the smulaton results of the two performance metrcs for two sets of dfferent confguraton scenaros: ) Non-overlapped PMRC, and transmsson range of 40m; ) Overlapped PMRC, and transmsson range of 40m. For each scenaro, we smulate two types of cluster head selecton strateges: sngle-head (represented as S n all fgures) and double-head (the prmary cluster head (PCH) s represented as DP and the secondary cluster (SCH) head s represented as DB n all fgures). In double-head selecton, the PCH and the SCH for a cluster rotates to work n 0 seconds nterval. The analyss results n Secton 3 (whch have been adjusted to ft for the square area) are represented as A n all fgures. For all smulatons, the same number of nodes evenly dstrbuted n the whole area are selected to sense the data and generate the packets. 4. Performance of non-overlapped PMRC structure Fgs. 5-6 present the averaged performance metrcs of cluster heads locatng n layers and for the nonoverlapped PMRC structure. The performance metrcs for cluster heads n other layers are omtted here for clearness reason. Fg. 5 shows that the average queue length of cluster heads at dfferent layers vs. the data generaton rate at each sendng node (.e., λ 0 ) wth transmsson range of 40m. It shows that the average queue length of cluster heads at one layer ncreases wth the ncrease of λ 0. Generally, the average queue length of one type of cluster heads n layer s larger than that of the same type of cluster heads n layer. Ths s consstent wth our ntuton that the average traffc load to cluster heads closer to the snk node s hgher than those farther from the snk node. Fg. 5 also shows that under the same set of sensng nodes, the average queue length of sngle cluster head s larger than that of the prmary/secondary cluster head at the same layer. Ths s because that wth dual cluster heads, the two heads rotate to receve data packets from ther outer layers. Ths reduces the queue length of each head. The average queue length of the PCH s larger than that of the SCH n the same layer. Ths s because the SCH may not cover all the nodes n ts cluster accordng to SCH selecton algorthm [5]. In Fg. 5, the analyss result of the average queue length for cluster heads n layer s close to that of the smulaton result wth sngle cluster head whle the analyzed result for cluster heads n layer s dfferent from the correspondng smulaton result. We expect that

5 ths dfference wll be mtgated by averagng the smulaton results of dfferent sets of sendng nodes. The performance metrcs for cluster heads n other layers are omtted here. Comparng Fgs. 5 and 7, the average queue length of sngle head at one layer for overlapped PMRC structure s lower than the correspondng result for non-overlapped PMRC structure. And t s apparent that the queue length of the prmary cluster head n one layer (for example DP_L) for overlapped PMRC s much smaller than that of ts correspondng result for nonoverlapped PMRC. Ths s the due to the fact that wth overlapped layers, more canddate cluster heads are avalable and the coverage of secondary cluster heads s mproved. Ths helps balance the traffc load between the prmary cluster head and the secondary cluster head at the same layer. Smlar results are reflected by comparng Fgs. 6 and 8. Fgure 5 Average queue length vs. λ 0 for non-overlapped PMRC structure. Fgure 7 Average queue length vs. λ 0 for overlapped PMRC structure. Fgure 6 Average queue delay vs. λ 0 for non-overlapped PMRC structure. Fg. 6 shows that the average queue delay of cluster heads at one layer ncreases wth the ncrease of λ 0. The trend s smlar to the queue length. Generally, the average queue length of one type of cluster heads n layer s larger than that of the same type of cluster heads n layer. Ths s because that the larger the queue length, the longer watng tme of a packet n the queue. Smlar to Fg. 5, under the same set of sensng nodes, the average queue delay of sngle cluster head s larger than that of the prmary/secondary cluster head at the same layer. In Fg. 6, the analyss result of the average queue delay s generally larger than the correspondng smulaton result. It s expected that the dfference between the smulaton results and analyzed results wll be mtgated by averagng the smulaton results of dfferent sets of sendng nodes. 4.3 Performance of overlapped PMRC structure Fgs. 7-8 present the averaged performance metrcs of cluster heads locatng n layers and for overlapped PMRC structure. In overlapped PMRC structure, wth the same transmsson range, more layers are created than non-overlapped PMRC structure. For clearness purpose, Fgure 8 Average queue delay vs. λ 0 for overlapped PMRC structure. An abnormty n these fgures s that the average queue length for cluster heads n nner layers (e.g., DP_L) may not always larger than that of the cluster heads n outer layers (e.g., DP_L). Ths can be explaned that the coverage of SCH n outer layers may not be as good as n nner layers. We can deduce that the average queue length of cluster heads n nner layers wll not ncrease defntely. The analyss results of the average queue length (from Eqn. (4)) and average queue delay (from Eqn. (5)) of cluster heads at layers and are also shown on Fgs. 7 and 8, respectvely. Compared wth Fgs. 5 and 6, we can

6 see that the analyzed results are closer to the smulaton results. Ths s because that the number of clusters generated n overlapped PMRC at each layer s typcally more than that n non-overlapped PMRC, whch helps averagng the smulaton result. 5. Concluson and Future Works In ths paper, we develop a fnte queung model for analyzng the queung performance for cluster heads n both non-overlapped and overlapped PMRC-based wreless sensor network. The smulaton results of the average queue length and the average queue delay match the analyss results n general trend. The smulaton results also confrm that the scenaros wth double cluster heads outperform the scenaros wth sngle head n terms of the two performance metrcs. Compared wth non-overlapped PMRC, overlapped PMRC mproves the queung performance of the scenaros of double cluster heads. The presented analyss provdes a gudelne n decdng the buffer sze of sensor nodes n PMRC-based sensor networks. Future work ncludes the analyss of other network performance metrcs and ther relatons wth the energy consumpton at sensor nodes. [3] S. Tang and W. L, QoS supportng and optmal energy allocaton for a cluster based wreless sensor network, Computer Communcaton, vol. 9, pp , 006. [4] H. Wang, M. Yang, Y. Jang, S. Wang, L. Gewal, Overlapped layers for prolongng network lfetme n mult-hop wreless sensor networks, to be presented on ITNG, 008, Las Vegas, NV. [5] M. Yang, S. Wang, A. Abdelal, Y. Jang, and Y. Km, An mproved mult-layered archtecture and ts rotatonal scheme for large-scale wreless sensor networks, Proc. IEEE CCNC, 007, pp References [] I. F. Akyldz, W. Su, Y. Sankarasubramanam, and E. Cayrc, A survey on sensor networks, IEEE Commu. Mag., pp. 0-4, Aug. 00. [] C. F. Chassern, M. Garetto, Modelng the performance of wreless sensor networks, IEEE INFOCOM, 004, vol., pp. 7-. [3] J. Dng, K. Svalngam, R. Kashyapa and L.J. Chuan, A multlayered archtecture and protocols for large-scale wreless sensor networks, Proc. IEEE SVTC, 003, pp [4] A. Fallah, E. Hossan, and A. S. Alfa, QoS and energy trade off n dstrbuted energy-lmted mesh/relay networks: a queung analyss, IEEE Trans. Parallel and Dstrbuted Systems, vol.7, no.6, pp , 006. [5] W. Henzelman, A. Chandrakasan, and H. Balakrshnan, An applcaton-specfc protocol archtecture for wreless mcrosensor networks, IEEE Trans. Wreless Communcatons, vol., no. 4, pp , 00. [6] J. M. Lu, T. Lee, A framework for performance modelng of wreless sensor networks, Proc. ICC, 005, vol., no., pp [7] C Lu, B.M. Blum, T.F. Abdelzaher, J.A. Stankovc and T. He, RAP: a real-tme communcaton archtecture for large-scale wreless sensor networks, Proc. 8 th IEEE RTAS, 00, pp [8] J. Luo, L. Jang, and C. He, Fnte queung model analyss for energy and QoS tradeoff n contenton-based wreless sensor networks, Proc. ICC, 007, pp [9] S. Moon, S. Lee and H. Cha, A congeston control technque for the near-snk nodes n wreless sensor networks, Proc. UIC, 006, LNCS 459, pp [0] T. G. Robertazz, Computer Networks and Systems: Queueng Theory and Performance Evaluaton, 3 rd Edton, Sprnger-Verlag New York, New York, NY, 000. [] T. Shu, M. Krunz, and S. Vrudhula, Power balanced coverage-tme optmzaton for clustered wreless sensor networks, Proc. Int l Symp. Moble Ad Hoc Networkng & Computng,005, pp. -0. [] S. Soro and W. Henzelman, Prolongng the lfetme of wreless sensor networks va unequal clusterng, Proc. 9th IEEE Int l Parallel and Dstrbuted Processng Symp. (IPDPS), 005.