We use here a famous theorem often useful for numerical problems:
Theoreme 1 Let X ⊂Rn, n≥1 a closed set, and k · k a norm onR. Let F :X →X be a function such that there isα ∈[0,1[ such that
kF(x)−F(y)k ≤αkx−yk
for all x, y ∈X. Thus there is a single point P ∈X such that F(P) =P. Moreover, let x0 ∈X and let (xk)k∈N the sequence defined by induction by xk+1 =F(xk) for k∈N. Therefore limk→+∞xk =P and we have
kxk−Pk ≤αkkx0−Pk (CV) for allk∈N.
Remark: A priori, one does not know P, But thanks to the inequality (CV), we get a very good approximation because the convergence is very, very fast(α <1!!): the dream for a computer scientist. That’s why we try as soon as possible to reduce to a fixed point problem .
Step 1: nomenclature.. We denote ~τ = (τ0, ..., τ3) et~b = (b0, ..., b3), where for simplicity b0, ..., b3 denote the vector coordinates of b0,0,0,0. We denote
p(~τ)i = (1−pe)pi+pe
= (1−pe)(1−(1−Π3j=0(1−τj)Π3j=i+1(1−τj)) +pe
We have the relationship
τi= (1 +p(~τ)i+...+ (p(~τ)i)m+h)bi for alli, and thus,
bi= τi
1 +p(~τ)i+...+ (p(~τ)i)m+h.
Then we define the functionf :R4 →R4 with its coordinates:
(f(~τ))i = τi
It was reduced to a fixed point theorem problem: we will try now to apply the theoreme. Let fixed first of all a ”pleasant” norm:
kxk= sup{|xi|/ i= 0,1,2,3,4}. The problem is to evaluate the second term of the right side. We putA = p(~τ)i and B =p(~τ0)i. With a small calculation we get:
BecauseA, B >0, we get We assume henceforth thatM ≥2. Then we get
If we substitute (9.3) into (9.2), we get
|F(~τ)−F(~τ0)| ≤αk~τ −~τ0k
Thus,if we manage to get pe close to 1 (i.e. a big probabilty of error in a packet !), then we haveα <1.
remark: The problem is that in the formula ofα, if M is big, we are in a problem... We can maybe refine: Let be δ ∈ [0,1] and let us consider the domain we get back a small coefficient in the formula ofα.
Step 3: F is defined on [0,1]4, and it has to get its value in [0,1]4 too. Let Thus if we set that
k~bk ≤ 1 solution that converges fast. To summarize we have the following Theorem:
Theoreme 2 Assume that k~bk ≤ m+h+11 . Assume that α <1, where α is given by (9.4). Then there is ~τ ∈[0,1]4 such that f(~τ) =~b, and we can get
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Publications
Y. Haddad, D. Porrat. Femtocell SINR Performance Evaluation.In Proc. of International Conferences on Access Networks, Services and Tech-nologies. IARIA ACCESS 2010. Valencia Spain, September 2010.
Y. Haddad, D. Porrat. A Two-Tier Frequency Reuse Scheme in Proc. Of Second International Workshop on Indoor and Outdoor Femto cells (IOFC’10) in conjunction with IEEE PIMRC’10, Turkey, September 2010.
Y. Haddad, D. Porrat. Femtocell: Opportunities and Challenges of the Home Cellular Base Station for the 3G . In Proc. of IADIS International Multi Conference on Computer Science and Information Sys-tems (session on Wireless Applications and Computing), Algarve, Portugal, June 2009.
Y. Haddad, G. Le Grand. Throughput analysis of the IEEE 802.11e EDCA on a noisy channel in unsaturated modein Proc. of the 3rd ACM International Workshop on Wireless Multimedia Networking and Per-formance Modeling (WMuNeP) October 2007.
Y. Haddad, G. Le Grand. Performance Analysis of IEEE 802.11e EDCA under Finite Load in an Error Prone Channel. In Proc. of JDIR’2007, France, January 2007.