Una
“ nueva ”
programación
para
el
proceso
de
modelización
del
cambio
químico El
programa
se
presenta
como
secuencias
de
enseñanza‐aprendizaje
(Méhéut
y
Psillos,


The result of Theorem 3.3 is for independent and memoryless erasure events. One might think that the result goes through for correlated erasure events across the links. However, this is not true. In fact, we can show that time-sharing is not capacity-achieving even for “degraded” channels.

Consider an (m, n)-erasure channels with correlated erasure events.3 _{We model}

the correlated event as follows. For each set A ⊆ {1, . . . , m}, Prj(A) denotes the probability that the transmitted signals from inputs i ∈ Ac _{are erased and signals} from i ∈ A are received successfully at receiver j. In the following Proposition, we look at the necessary and sufficient conditions for “degraded“-ness.

Proposition 3.1. Considering the previous notation, receiverj is a degraded version

of receiverk iff for any subset U of the power set of {1, . . . , m}, with the property that

B ∈ U,A ⊆ B =⇒ A ∈ U, we have X A∈U Prj(A)≤ X A∈U Prk(A). (3.20)

3_{Since the capacity region of broadcast channels with average probability of error constraint only} depends on the marginals [35], we assume that different users’ channels are independent from each other.

Proof. It is not hard to check that Prj(·) is a degraded version of Prk(·) if and only if there exists a stochastic matrix γ(·,·) of dimension of 2m_×2m _{indexed by the subsets} of {1, . . . , m} with γ(S,T) = 0 forS,T, S *T such that

Prj(A) =

X

B⊆A

γ(A,B)Prk(B)

for all A,B subsets of{1, . . . , m}.

With this, the necessity of (3.20) is clear. To prove the sufficiency, we use an argu- ment used in [44]. The idea is based on constructing a max-flow network. Apart from sourcesand destinationt, there are two nodesl(A) andr(A) for eachA ⊆ {1, . . . , m}.

s is connected tol(A) with a link with capacity Prk(A). r(A) is connected tot with a link with capacity Prj(A). Finally, l(A) and r(B) are connected with a link of unlimited capacity if and only if B ⊆ A. Consider a cut [Vs,Vt] of finite capacity in this network. Let Vt =Vtl∪ Vtr be the partition of Vt to the nodes in the l side and the r side. Now, because of the finiteness of the capacity of the cut, we should have

r(B)∈ Vr

t, B ⊆ A=⇒l(A)∈ Vrl.

Therefore, if we define V∗ _{={A|∃r(B)∈ V}r

t s.t. B ⊆ A}, we have

Vr _⊆r(V∗_{) and l(V}∗_{)⊆ V}l

t. (3.21)

The cut-capacity C(Vs) can be written as

C(Vs) = X l(A)∈Vl t Prk(A) + 1− X r(B)∈Vr t Prj(B) ≥ 1− X B∈V∗ Prj(B) + X A∈V∗ Prk(A) ≥ 1,

where the second line follows from (3.21), and the last line follows from (3.20). This suggests that the value of min-cut is equal to one, and the minimum cut is the one that isolates either s or t from the rest of nodes. In this case, using the max-flow min-cut theorem, the max-flow is also one. Therefore, the flow in each of the links between s (respectivelyt) and the intermediate nodes is equal to the capacity of the corresponding link. Defining the flow between l(A) andr(B) as γ(B,A)·Prk(A), we can easily see that γ(·,·) is the desired stochastic matrix.

Remark 3.6. As mentioned earlier, for the independent erasure case, different no-

tions of “degraded,” “more capable,” and “less noisy” are equivalent. However, it is not clear whether the same is true for the case when erasures are correlated. In Proposition 3.1 we identified the class of “degraded” channels. However, characteriz- ing the class of “more capable” channels is not an easy task. In order to see it, note

that channelj is “more capable” than channel k if and only if for any distribution on

the input X = (X1, . . . , Xm) we have

I(X;Yk)≤I(X;Yj).

For (possibly correlated) erasure channels, the above condition can be written as a linear inequality involving the joint entropies of subsets of inputs, i.e.,

0≤X A (Prj(A)−Prk(A)) | {z } α(A) H(Xi, i∈ A),

where P_Aα(A) = 0. Therefore, to identify the class of “more capable” channels, one

should be able to characterize the set of all valid linear inequalities over the entropies

of subsets of m random variables. However, the later problem has proved to be a very

challenging problem. Identifying the set of linear inequalities over the space of entropy

Now consider a 2 by 2 degraded erasure broadcast channel with probability of erasure events as shown in table . Consider the supporting hyperplane µ= (µ1, µ2),

and let us look at the boundary point corresponding to this point.

It is not hard to check that the boundary point of the time-sharing region corre- sponding to this hyperplane is

(R1, R2) = µ max i=1,2 ½ µi(Pri({1}) + Pri({1,2})) ¾ ,max i=1,2 ½ µi(Pri({2}) + Pri({1,2})) ¾¶ .

On the other hand, using the Bergman’s formula [40], we know that the rate vectors in the capacity region of this degraded channel should satisfy

R1 ≤I(Y1;X|U) = Pr1({1,2})H(X1, X2|U) + Pr1({1})H(X1|U) + Pr1({2})H(X2|U)

R2 ≤I(Y2;U) = Pr2({1,2})I(X1, X2;U) + Pr2({1})I(X1;U) + Pr2({2})I(X2;U),

for some joint distribution Pr (U, X = (X1, X2)).

To find the boundary point of the capacity region corresponding to hyperplane

µ= (µ1, µ2), we have to solve the following optimization problem

f(µ) = max

Pr (U,X)µ1R1+µ2R2.

One can further write

f(µ) = max

Pr (U,X)µ2Pr2({1,2})H(X1, X2) +µ2Pr2({1})H(X1) +µ2Pr2({2})H(X2) +

aH(X1, X2|U) +bH(X1|U) +cH(X2|U),

where a = (µ1Pr1({1,2})− µ2Pr2({1,2})), b = (µ1Pr1({1}) −µ2Pr2({1})), and

H(X2), and H(X1, X2) with 1, 1, and 2 respectively. This gives

f(µ) ≥ max

Pr (U,X)aH(X1, X2|U) +bH(X1|U) +cH(X2|U) (3.22)

+ 2µ2Pr2({1,2}) +µ2Pr2({1}) +µ2Pr2({2}).

The optimization problem in (3.22) can be equivalently viewed as a Linear Program (LP) over the set of entropy vectors h = (H(X1, X2|U), and H(X1|U), H(X2|U)),

subject to the following constraints

H(X1, X2|U) ≤ H(X1|U) +H(X2|U),

H(Xi|U) ≤ H(X1, X2|U), i= 1,2,

0 ≤ H(Xi|U)≤1, i= 1,2.

Solving this LP gives the following upper bound for f(µ):

f(µ)≥max{0, a+b, a+c, a+b+c,2a+b+c}+2µ2Pr2({1,2})+µ2Pr2({1})+µ2Pr2({2}).

Furthermore, the following upper bound can be always achieved by consideringX1, X2

to be i.i.d. uniform and letting U be a member of {∅, X1, X2,(X1, X2), X1⊕X2}. It

can be checked that the first four possibilities for U correspond to performing time- sharing. However U =X1⊕X2 does not correspond to time-sharing. Based on the

above argument we have the following result.

Proposition 3.2. The capacity region,C, of adegraded (2,2)-erasure broadcast chan- nel with correlated erasure events is

C = conv (RT S∪ {(1−Pr1(∅),Pr2({1,2}))}),

A ∅ {1} {2} {1,2} Pr1(·) 0 0.35 0.35 0.3

Pr2(·) 0.5 0.15 0.15 0.2

Table 3.1: Probability of erasure events for Example 3.2.

of set A.

Example 3.2. Consider a degraded (2,2)-erasure channel with probability of erasures shown in Table 3.1. Using Proposition 3.2, we have plotted the capacity region of this channel in Figure 3.4. The time-sharing region is also shown in the same figure. Note that for this channel the capacity region is strictly greater than the time-sharing region.