IEEE TRANSACTIONS ON AUTOMATIC CONTROL 10 2)Impact of Heterogeneity:Second,we consider the impact E.Full Coverage Control and Sensing Energy Consumption of heterogeneity.We have the following technical lemma. Control Lemma 4.3:For arbitrary g that 0<q<1, From previous discussions,we can know that critical ESR is the critical condition for full coverage of the operational region. Therefore,we can control the ESR to promise the network 22) achieve full coverage under different random mobility patterns, as shown in Figure 6.As the total number of sensors increases. where cu,r,u are as defined in the system model,and r= 0.12 I.I.D mobility model Proof:Please refer to Appendix ● -1-dimensional random walk model 0.1 Then we can have the following results. 0.08 (a)Under I.I.D.Mobility Model: 0.06 E.i.d=日 logn+log log n (23) 0.04 (b)Under 1-Dimensional Random Walk Mobility Model: 0.02 E.w.>日 log n log log n (24) n 0 500 100015002000 2500300035004000 Total Number of Sensors n E.w.<日 (log n +log log n) (25) n是+9 Fig.6. Relationship between ESR and total number of sensors n. The lower bound in (b)comes from the Cauchy Inequality that the critical ESR decreases for II.D mobility model and 1- dimensional random walk mobility model.When we want to 三店小（宫 control the full coverage,we can make the network achieve the critical ESR under I.I.D mobility and 1-dimensional random walk mobility model as illustrated in the figure. And the upper bound results from Lemma 4.3. Hence,with the sensing energy model Eheterogene- Since the sensing energy consumption is positively correlated to the ESR as discussed previously,if we want to further ity does not make any difference to sensing energy in WSNs under ii.d.mobility model or stationary WSNs,which can be decrease the energy consumption,we have to decrease the ESR. This further reduction of ESR will make the network not be seen from (20)and (23).However,from (21)and (24)we full covered,thus sacrificing the coverage performance,which know that under 1-dimensional random walk mobility model, is actually a tradeoff control realized by the value of ESR. sensing energy consumption will increase due to heterogeneity. And there is a trade-off in mobile WSNs that designers must V.K-COVERAGE IN MOBILE HETEROGENEOUS WSNS face:on one hand,heterogeneous WSNs composed of high- UNDER POISSON DEPLOYMENT MODEL end sensors with large sensing range and low-end sensors with small sensing range can reduce the cost of WSNs and In this section,we study k-coverage at an instant and guarantee a satisfactory sensing performance;on the other hand, over a time interval in mobile heterogeneous WSNs.The heterogeneity will increase sensing energy consumption.From sensor locations within the operational region are modeled the upper bound in (25),the order of energy consumption as 2-dimensional Poisson process with density n.Thus,the in terms of n approaches the order (i.e.,n2)in the case coverage problem can be described by the frequently used without heterogeneity.Hence,the energy efficiency will not Poisson-Boolean model.Besides,the 2-dimensional random be dramatically deteriorated by the heterogeneity. walk mobility model is employed in this part. Remark 4.1:The sensing process in WSNs depends on the A.K-coverage at an Instant area covered by each sensor.And under the 1-dimensional random walk mobility model,the area covered by sensors is We start with the results regarding k-coverage at an instant. on the same order of sensing radius r.In stationary WSNs or Theorem 5.1:Given an instant t>0,the expectation of the WSNs with i.i.d.mobility model,however,the covered area is fraction of operational region that is k-covered at instant t is on the order ofr2.This discrepancy of dependence on sensing radius leads the impact of heterogeneity to be different under g}=1- (k,m∑y=1yr) the two models. (k-1)1IEEE TRANSACTIONS ON AUTOMATIC CONTROL 10 2) Impact of Heterogeneity: Second, we consider the impact of heterogeneity. We have the following technical lemma. Lemma 4.3: For arbitrary q that 0 <q< 1,  u y=1 cyr2 y  <  u y=1 cyry 3 2  1 nq 1 2 , (22) where cy, ry, u are as defined in the system model, and ry = Θ log n n . Proof: Please refer to Appendix. Then we can have the following results. (a) Under I.I.D. Mobility Model: Ei.i.d. = Θ log n + log log n n . (23) (b) Under 1-Dimensional Random Walk Mobility Model: Er.w. > Θ log n + log log n n 2  ; (24) Er.w. < Θ  (log n + log log n) 3 2 n 3 2 + 1 2 q  . (25) The lower bound in (b) comes from the Cauchy Inequality that  u y=1 cy   u y=1 cyr2 y  >  u y=1 cyry 2 . And the upper bound results from Lemma 4.3. Hence, with the sensing energy model Ey ∝ r2 y, heterogene￾ity does not make any difference to sensing energy in WSNs under i.i.d. mobility model or stationary WSNs, which can be seen from (20) and (23). However, from (21) and (24) we know that under 1-dimensional random walk mobility model, sensing energy consumption will increase due to heterogeneity. And there is a trade-off in mobile WSNs that designers must face: on one hand, heterogeneous WSNs composed of high￾end sensors with large sensing range and low-end sensors with small sensing range can reduce the cost of WSNs and guarantee a satisfactory sensing performance; on the other hand, heterogeneity will increase sensing energy consumption. From the upper bound in (25), the order of energy consumption in terms of n approaches the order (i.e., n2) in the case without heterogeneity. Hence, the energy efficiency will not be dramatically deteriorated by the heterogeneity. Remark 4.1: The sensing process in WSNs depends on the area covered by each sensor. And under the 1-dimensional random walk mobility model, the area covered by sensors is on the same order of sensing radius r. In stationary WSNs or WSNs with i.i.d. mobility model, however, the covered area is on the order of r2. This discrepancy of dependence on sensing radius leads the impact of heterogeneity to be different under the two models. E. Full Coverage Control and Sensing Energy Consumption Control From previous discussions, we can know that critical ESR is the critical condition for full coverage of the operational region. Therefore, we can control the ESR to promise the network achieve full coverage under different random mobility patterns, as shown in Figure 6. As the total number of sensors increases, 0 500 1000 1500 2000 2500 3000 3500 4000 0 0.02 0.04 0.06 0.08 0.1 0.12 Total Number of Sensors n Critical ESR I.I.D mobility model 1−dimensional random walk model Fig. 6. Relationship between ESR and total number of sensors n. the critical ESR decreases for I.I.D mobility model and 1- dimensional random walk mobility model. When we want to control the full coverage, we can make the network achieve the critical ESR under I.I.D mobility and 1-dimensional random walk mobility model as illustrated in the figure. Since the sensing energy consumption is positively correlated to the ESR as discussed previously, if we want to further decrease the energy consumption, we have to decrease the ESR. This further reduction of ESR will make the network not be full covered, thus sacrificing the coverage performance, which is actually a tradeoff control realized by the value of ESR. V. K-COVERAGE IN MOBILE HETEROGENEOUS WSNS UNDER POISSON DEPLOYMENT MODEL In this section, we study k-coverage at an instant and over a time interval in mobile heterogeneous WSNs. The sensor locations within the operational region are modeled as 2-dimensional Poisson process with density n. Thus, the coverage problem can be described by the frequently used Poisson-Boolean model. Besides, the 2-dimensional random walk mobility model is employed in this part. A. K-coverage at an Instant We start with the results regarding k-coverage at an instant. Theorem 5.1: Given an instant t > 0, the expectation of the fraction of operational region that is k-covered at instant t is E{η(t)} = 1 − Γ k, πnu y=1 cyr2 y (k − 1)! ,