Comment to "Coverage by Randomly Deployed Wireless Sensor Networks"

Comment to “Cov erage b y Randoml y Deploy ed Wireless Se nsor Net work s ”. by Bhupendra Gupta 1 F aculty of Engineering and Sciences, Indian Institute of Information T ec hnolog y (DM)-Jabalpur-48 2011 , India. 1 In tro d uction. In the ab ov e pap er, Lemma (4), on pa g e 2659 play the key role for deriving the main results in the pape r . The statement a s well as the pro of of Lemma (4), pa ge 2 659 , [1] is not correct. Here mainly tw o serious error s in this Lemma, one is in the statemen t of lemma and the other one in the pro of of Lemma (4 ), on page 2666, [1]. In the Lemma (4), page 2659 , [1], W an and Yi, states the lemma as follo ws, Lemma 1.1 Assume Ω is the disk r e gion and let r n b e such that nπ r 2 n → ∞ and nπ r 3 n → 0 . Then for any p oint z ∈ δ Ω φ n,r n ( z ) ∼  nπ r 2 n 2  k k ! e − nπr 2 n 2 . (1.1) Firstly t he statement o f Lemm a (4), pag e 2659, [1], is wrong. Since authors has tak en nπ r 2 n → ∞ . This implies φ n,r n ( z ) → 0 , bec ause in the r ight hand side of (1.1), nπ r 2 n is in the negative power of ex p o nent ia l. 1 Corresp onding Author, email: bh up en@iii tdm.in, gupta.bh up endra@gmail.com Hence a utho r s should r emov e the condition nπr 2 n → ∞ , from the statement o f Lemma (1.2) (Lemma (4 ), page 2659, [1]). Now the second and more serious er ror is in the proo f of Lemma (4), on page 2 666, [1 ]. Here authors ha s derive the low e r b o und of φ ( z ) as follows φ ( z ) ≥ k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 >  nπ r 2 n 2  k k ! e − nπr 2 n 2 . (1.2) By using the second inequality , authors were getting lose low er b o und, while the ob jective of the lemma is to g et the exa c t expression for φ ( z ) . Also, authors has given the pro of for the upp er bound of φ ( z ) as follows φ ( z ) ≤ k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 + nr 2 ar csin √ πr 2 =    k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2    e nr 2 ar csin √ πr 2 ∼ k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 ∼  nπ r 2 n 2  k k ! e − nπr 2 n 2 The problem is with the la st ‘ ∼ ’ in the abov e expressio n. F rom (1.2), we hav e k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 >  nπ r 2 n 2  k k ! e − nπr 2 n 2 . It is clear that one can not replace ‘ a > b ’ by ‘ a ∼ b .’ Even in case o f clear convergence i.e ., a → b, here it is not allow ed, beca use the authors were g iv ing the deriv a- tion of upper bound for φ ( z ) and  nπr 2 n 2  k k ! e − nπr 2 n 2 is the 1 low er b ound of P k i =0  nπr 2 n 2  i i ! e − nπr 2 n 2 . F ollowing is the cor r ected version of Lemma (4). Lemma 1.2 Assume Ω is the disk r e gion and let r n b e such that nπ r 3 n → 0 . Then for any p oint z ∈ δ Ω k X i =1  nπ r 2 n 2  i i ! e − nπr 2 n 2 ≤ φ n,r n ( z ) ≤ (1+ C ) k X i =1  nπ r 2 n 2  i i ! e − nπr 2 n 2 , (1.3) wher e C > 0 is an arbitr ary c onst ant. Pro of. The idea o f the pro of is s ame, except the math- ematics for getting the b ounds of φ ( z ) , which leads to totally different b ounds. Her e w e are giving only those steps which are different from the o riginal pr o of. F or the low er bound φ ( z ) ≥ k X i =1  nπ r 2 n 2  i i ! e − nπr 2 n 2 . Now for the uppe r b ound φ ( z ) ≤ k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 + nr 2 ar csin √ πr 2 =    k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2    e nr 2 ar csin √ πr 2 ≤ (1 + C ) k X i =0  nπ r 2 n 2  i i ! e − nπr 2 n 2 , since n r 2 acrsi n √ π r 2 = 1 2 nπ 1 / 2 r 3 + 1 2 3 6 nπ 3 / 2 r 5 + . . . , and nπ r 3 n → 0 . This change in Lemma leads a drastic c hang e in a ll the result deriv ed in the article [1 ]. References [1] P .J. W an and C.W. Yi, “Cov erag e b y Ra ndomly De- ploy ed Wireles s Sensor Netw orks” , IEEE T r ansac- tion On Info r mation Theo r y , vol.52, No.6 , June 2006 . 2