{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T12:41:56Z","timestamp":1771332116824,"version":"3.50.1"},"reference-count":25,"publisher":"Association for Computing Machinery (ACM)","issue":"3","license":[{"start":{"date-parts":[[2006,8,1]],"date-time":"2006-08-01T00:00:00Z","timestamp":1154390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Sen. Netw."],"published-print":{"date-parts":[[2006,8]]},"abstract":"<jats:p>\n            We study the problem of\n            <jats:italic>coverage<\/jats:italic>\n            in planar heterogeneous sensor networks. Coverage is a performance metric that quantifies how well a field of interest is monitored by the sensor deployment. To derive analytical expressions of coverage for heterogeneous sensor networks, we formulate the coverage problem as a set intersection problem, a problem studied in integral geometry. Compared to previous analytical results, our formulation allows us to consider a network model where sensors are deployed according to an arbitrary stochastic distribution; sensing areas of sensors need not follow the unit disk model but can have any arbitrary shape; sensors need not have an identical sensing capability. Furthermore, our formulation does not assume deployment of sensors over an infinite plane and, hence, our derivations do not suffer from the border effect problem arising in a bounded field of interest. We compare our theoretical results with the spatial Poisson approximation that is widely used in modeling coverage. By computing the Kullback-Leibler and total variation distance between the probability density functions derived via our theoretical results, the Poisson approximation, and the simulation, we show that our formulas provide a more accurate representation of the coverage in sensor networks. Finally, we provide examples of calculating network parameters such as the network size and sensing range in order to achieve a desired degree of coverage.\n          <\/jats:p>","DOI":"10.1145\/1167935.1167937","type":"journal-article","created":{"date-parts":[[2007,1,16]],"date-time":"2007-01-16T19:38:29Z","timestamp":1168976309000},"page":"325-358","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":120,"title":["Stochastic coverage in heterogeneous sensor networks"],"prefix":"10.1145","volume":"2","author":[{"given":"Loukas","family":"Lazos","sequence":"first","affiliation":[{"name":"Network Security Lab, University of Washington, Seattle, WA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Radha","family":"Poovendran","sequence":"additional","affiliation":[{"name":"Network Security Lab, University of Washington, Seattle, WA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2006,8]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCOM.2002.1024422"},{"key":"e_1_2_1_2_1","volume-title":"Proceedings of the IEEE MWCN '01","author":"Bettstetter C.","unstructured":"Bettstetter , C. and Krause , O . 2001. On border effects in modeling and simulation of wireless ad hoc networks . In Proceedings of the IEEE MWCN '01 . Bettstetter, C. and Krause, O. 2001. On border effects in modeling and simulation of wireless ad hoc networks. In Proceedings of the IEEE MWCN '01."},{"key":"e_1_2_1_3_1","volume-title":"Proceedings of the WCNC '02","author":"Bettstetter C.","unstructured":"Bettstetter , C. and Zangl , J . 2002. How to achieve a connected ad hoc network with homogeneous range assignment: An analytical study with consideration of border effects . In Proceedings of the WCNC '02 . 125--129. Bettstetter, C. and Zangl, J. 2002. How to achieve a connected ad hoc network with homogeneous range assignment: An analytical study with consideration of border effects. In Proceedings of the WCNC '02. 125--129."},{"key":"e_1_2_1_4_1","volume-title":"Vorlesungen uber Integralgeometrie 9","author":"Blaschke W.","unstructured":"Blaschke , W. 1955. Vorlesungen uber Integralgeometrie 9 , 3 rd Ed. Deutsch Verlag Wiss , Berlin, Germany . Blaschke, W. 1955. Vorlesungen uber Integralgeometrie 9, 3rd Ed. Deutsch Verlag Wiss, Berlin, Germany.","edition":"3"},{"key":"e_1_2_1_5_1","doi-asserted-by":"crossref","unstructured":"Bonnesen T. and Fenchel W. 1934. Theorie de convexen korper. Ergeb. Math. Springer Berlin Germany 18.  Bonnesen T. and Fenchel W. 1934. Theorie de convexen korper. Ergeb. Math. Springer Berlin Germany 18.","DOI":"10.1007\/978-3-642-93014-0_5"},{"key":"e_1_2_1_6_1","doi-asserted-by":"crossref","unstructured":"Cover T. M. and Thomas A. 1991. Elements of Information Theory. John Wiley and Sons New York NY.   Cover T. M. and Thomas A. 1991. Elements of Information Theory. John Wiley and Sons New York NY.","DOI":"10.1002\/0471200611"},{"key":"e_1_2_1_7_1","first-page":"693","article-title":"On some integral formulas relative to convex figures in the euclidean space e2","volume":"23","author":"Filipescu D.","year":"1971","unstructured":"Filipescu , D. 1971 . On some integral formulas relative to convex figures in the euclidean space e2 . Stud. Cerc. Mat. 23 , 693 -- 709 . Filipescu, D. 1971. On some integral formulas relative to convex figures in the euclidean space e2. Stud. Cerc. Mat. 23, 693--709.","journal-title":"Stud. Cerc. Mat."},{"key":"e_1_2_1_8_1","volume-title":"Differential Forms with Applications to the Physical Sciences","author":"Flanders H.","unstructured":"Flanders , H. 1963. Differential Forms with Applications to the Physical Sciences . Academic Press , New York, NY . Flanders, H. 1963. Differential Forms with Applications to the Physical Sciences. Academic Press, New York, NY."},{"key":"e_1_2_1_9_1","unstructured":"Flanders H. 1967. Differential Forms. Prentice Hall Englewood Cliffs NJ.  Flanders H. 1967. Differential Forms. Prentice Hall Englewood Cliffs NJ."},{"key":"e_1_2_1_10_1","volume-title":"Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc '03)","author":"Gupta H.","unstructured":"Gupta , H. , Das , S. R. , and Gu , Q . 2003. Connected sensor cover: Self-organization of sensor networks for efficient query execution . In Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc '03) . 189--200. 10.1145\/778415.778438 Gupta, H., Das, S. R., and Gu, Q. 2003. Connected sensor cover: Self-organization of sensor networks for efficient query execution. In Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking & Computing (MobiHoc '03). 189--200. 10.1145\/778415.778438"},{"key":"e_1_2_1_11_1","volume-title":"Proceedings of WiOpt '03","author":"Kar K.","unstructured":"Kar , K. and Banerjee , S . 2003. Node placement for connected coverage in sensor networks . In Proceedings of WiOpt '03 . Kar, K. and Banerjee, S. 2003. Node placement for connected coverage in sensor networks. In Proceedings of WiOpt '03."},{"key":"e_1_2_1_12_1","volume-title":"Proceedings of IEEE INFOCOM '01","author":"Koushanfar F.","unstructured":"Koushanfar , F. , Meguerdichian , S. , Potkonjak , M. , and Srivastava , M . 2001. Coverage problems in wireless ad-hoc sensor networks . In Proceedings of IEEE INFOCOM '01 . 1380--1387. Koushanfar, F., Meguerdichian, S., Potkonjak, M., and Srivastava, M. 2001. Coverage problems in wireless ad-hoc sensor networks. In Proceedings of IEEE INFOCOM '01. 1380--1387."},{"key":"e_1_2_1_13_1","volume-title":"Proceedings of the 22nd International Conference on Distributed Computing Systems (ICDCSW '02)","author":"Krishnamachari B.","unstructured":"Krishnamachari , B. , Estrin , D. , and Wicker , S. B . 2002. The impact of data aggregation in wireless sensor networks . In Proceedings of the 22nd International Conference on Distributed Computing Systems (ICDCSW '02) . 575--578. Krishnamachari, B., Estrin, D., and Wicker, S. B. 2002. The impact of data aggregation in wireless sensor networks. In Proceedings of the 22nd International Conference on Distributed Computing Systems (ICDCSW '02). 575--578."},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/TC.2003.1204831"},{"key":"e_1_2_1_15_1","volume-title":"Proceedings of MASS '04","author":"Liu B.","unstructured":"Liu , B. and Towsley , D . 2004. A study of the coverage of large-scale sensor networks . In Proceedings of MASS '04 . Liu, B. and Towsley, D. 2004. A study of the coverage of large-scale sensor networks. 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In Proceedings of MobiCom '01 . 139--150. 10.1145\/381677.381691 Meguerdichian, S., Koushanfar, F., Qu, G., and Potkonjak, M. 2001. Exposure in wireless ad hoc sensor networks. In Proceedings of MobiCom '01. 139--150. 10.1145\/381677.381691"},{"key":"e_1_2_1_18_1","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1093\/biomet\/56.3.661","article-title":"The assymptotic values of certain coverage probabilities","volume":"56","author":"Miles R.","year":"1969","unstructured":"Miles , R. 1969 . The assymptotic values of certain coverage probabilities . Biometrika 56 , 661 -- 680 . Miles, R. 1969. The assymptotic values of certain coverage probabilities. Biometrika 56, 661--680.","journal-title":"Biometrika"},{"key":"e_1_2_1_19_1","volume-title":"Proceedings of IEEE INFOCOM '05","author":"Miorandi D.","unstructured":"Miorandi , D. and Altman , E . 2005. Coverage and connectivity of ad hoc networks in presence of channel randomness . In Proceedings of IEEE INFOCOM '05 . 491--502. Miorandi, D. and Altman, E. 2005. Coverage and connectivity of ad hoc networks in presence of channel randomness. In Proceedings of IEEE INFOCOM '05. 491--502."},{"key":"e_1_2_1_20_1","volume-title":"Proceedings of the IEEE International Conference on Robotics and Automation '04","author":"Poduri S.","unstructured":"Poduri , S. and Sukhatme , G. S . 2004. Constrained coverage for mobile sensor networks . In Proceedings of the IEEE International Conference on Robotics and Automation '04 . 165--172. Poduri, S. and Sukhatme, G. S. 2004. Constrained coverage for mobile sensor networks. In Proceedings of the IEEE International Conference on Robotics and Automation '04. 165--172."},{"key":"e_1_2_1_21_1","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1007\/BF02940725","article-title":"Geometrica intregral 4: Sobre la medida cinematica en el plano","volume":"11","author":"Santalo L.","year":"1936","unstructured":"Santalo , L. 1936 . 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