ISR and Zero Noise

By a simple manipulation, the SINR condition takes the following form:

ISR_P(S, i) =ci

X

j∈S

I_ji P_i ≤ 1

β, (2.3)

where c_i = _P ^Pi

i−βNi. We call the left part of (2.3) Interference to Signal Ratio (ISR)³. This form of the SINR condition is widely preferred to the original form. The reason for this is that in this form the left part of the formula is more comfortable to manipulate because of additivity⁴:

ISR_P(S, i) =X

j∈S

ISR_P(j, i).

Under this notation, we call a set of links S a γ-feasible (or γ-SINR-feasible) set if for each linki∈S,

ISR_P(S, i)≤ 1 γ.

Similarly, each partitioning of Γ into a number of disjoint γ-feasible subsets is called a γ-feasible schedule. Using the additivity ofISR, [HW09] proved the following result that is very useful in asymptotic computations involving the SINR model. The theorem shows, in particular, that if one increases the β threshold of the SINR condition by a constant factor then the length of the optimum schedule changes by at most a constant factor. It should be noted that the theorem does not use the particular form of ISR, but just the fact that it is additive. This means that it is equally applicable for different

3In some of the related literature this is called “affectedness” or “affectance”.

4We will usually not write the set braces in the ISRP formula when the first argument is an one-element set. Since the letters for sets and one-elements are different, hopefully, no confusion will happen.

setting e.g. for the bidirectional SINR model (Chapter 8), provided that we have the additivity of ISR. We present the algorithm and the corresponding theorem below.

The algorithm uses a subroutine that we callfirstfit.

Procedurefirstfit

1. Input: a power schemeP, a numberγ⁰ >0 and a set of linksi₁, i₂, . . . in this fixed order.

2. Take empty setsS1, S2, . . . in this fixed order.

3. Fork= 1,2, . . ., do:

3.1. SetS_t←S_t∪{i_k}, wheretis the smallest index such thatISR_P(S_t, i_k)≤ _2γ¹0. 4. Output: obtained non-empty sets S1, S2, . . . in their original order.

Algorithm 2.4 [HW09] Transforming a γ-feasible schedule into a γ⁰-feasible schedule.

1. Input: aγ-feasible scheduleS₁, S₂, . . . of a set of links Γ w.r.t. power assignment P and a numberγ⁰>0.

2. SetW ← ∅. // an empty collection of sets 3. Fort= 1,2, . . ., do:

3.1. Leti1, i2, . . . , i_k be the links of S_t in this fixed order.

3.2. Set {T₁, T₂, . . . , T_p} ←firstfit(γ⁰, P, i₁, i₂, . . . , i_k).

3.3. Forf = 1,2, . . ., do:

3.3.1. Let j1, j2, . . . , j_h be the links ofT_f in the opposite of their original order inS_t.

3.3.2. Set W ←W∪firstfit(γ⁰, P, j₁, j₂, . . . , j_h).

4. Output: γ⁰-feasible scheduleW.

Theorem 2.13. [HW09] For any γ⁰ > γ > 0 and any power assignment P, Algo-rithm 2.4, applied to a γ-feasible schedule, outputs a γ⁰-feasible schedule, increasing the length of the original schedule by a factor of at most d2γ⁰/γe².

Proof. Consider the set S_t for some index t. Let i ∈ T_p be any link in the last set Tp. Since i did not “fit” in the sets T1, T2. . . , Tp−1, we have ISR_P(T_f, i) > _2γ¹0 for f = 1,2, . . . , p−1. Using additivity we getISR_P(S_t, i) > ^p−1_2γ0 . On the other hand, we have assumed thatISR_P(S_t, i)≤ _γ¹. From these inequalities we get that one application of firstfit splits the input set into p ≤ d2γ⁰/γe subsets. Since we have two nested applications of firstfit, the resulting number of sets is bounded by d2γ⁰/γe². It

remains to show that the resulting collection of sets is γ-feasible. Let i1, . . . , i_k be the fixed order of links in S_t. Consider a link i_s which was assigned to a set T_f after the first application of firstfit. Then we have that ISR_P(T_f ∩ {i₁, . . . , is−1}, i_s) ≤

1

2γ⁰. Since we use the opposite of the mentioned order for the second application of firstfit, we also have that ifi_s was eventually assigned to a setT_f⁰ ⊆T_f,ISR_P(T_f⁰ ∩ {i_s+1, . . . , i_k}, i_s)≤ _2γ¹0. This imples that

ISR_P(T_f⁰, i_s) =ISR_P(T_f⁰ ∩ {i₁, . . . , i_s−1}, i_s) +ISR_P(T_f⁰ ∩ {i_s+1, . . . , i_k}, i_s)≤ 1 γ⁰.

The theorem above helps to get rid of the noise factors from the SINR condition. This comes with the price of an assumption on the power levels of the nodes. According to the SINR condition, the minimum receive power required for a link i to win just the environmental noise is P_i = βN_i. We assume that the power is higher than this minimum level by a small factor. This is a common assumption in the related literature, e.g. [HM11b,Kes11].

Assumption. From this point on we assume that for each linki in the given set Γ, Pi ≥cβNi

for a constant 1< c≤2.

Theorem 2.14. Let S be a set of links with a power assignmentP. IfS is feasible with the SINR condition

1. T₀^cβ/(c−1)∈Ω(T_N^β),

2. each S_t isβ-feasible under the original SINR condition (t= 1,2, . . . T₀^cβ/(c−1)).

Proof. First let us note that

T₀^cβ/(c−1)≤T_N^cβ/(c−1) ∈Ω(T_N^β),

where the first part holds because the presence of a noise can only increase the length of the optimum schedules and the second part follows from Theorem2.13. For the second part of the theorem, consider e.g. S1. Since S1 is 2β-feasible with zero noise, we have P_i≥ _c−1^cβ P

j∈SI_ji, so

P_i−P_i/c≥βX

j∈S

I_ji;

hence, Thus,S1 isβ-feasible with noise valuesNi.

Theorem2.14shows that we lose only a constant factor of approximation if we setNi= 0 for alliin our calculations. Since our interest in this thesis is in asymptotic bounds and relations, we, based on Theorem2.14, make the following assumption for the rest of the text.

Assumption. We will do our calculations based on a modified SINR condition, where we assume zero noise for all receivers.

The assumption above makes ISR even more flexible as a function. Now that the coefficients c_i are all equal to 1, we extend ISR additively for the second argument as well as for the first argument. We have:

ISR_P(j, i) = I_ji

Note that the definition above is consistent as soon as ci =cj for all linksi, j.

At last, we will make yet another modification in the formula of ISR_P. Consider the following:

It has been shown (see e.g. [KV10] and [HM11a]) thatISR⁰_P is analytically more mean-ingful than ISR_P. The reason for this is that having one pair of links i, jthat interfere each other very much will destroy the information about other individual interferences in the sumISR_P(S1, S2); hence, ISR_P⁰ expresses more information about the common interference by a set of links thanISR_P. This will be made more evident in next chapter.

Note also that replacing ISR_P by ISR⁰_P will not affect the SINR condition: since we assume β≥1, it is easy to see that

ISR⁰_P(S, i)≤ 1

β if and only ifISR_P(S, i)≤ 1

β. (2.4)

In order to not overload the notation we, based on (2.4), make the following notation in the rest of the text.

Notation. We setISR_P(S₁, S₂) = X

j∈S1

X

i∈S2

min

1,I_ji P_i

.