Lou Salaün
email: lou.salaun@nokia.com

Chung Shue Chen
email: chungshue.chen@nokia.com

Wing Yi Chen
email: yichen@ie.cuhk.edu.hk

Shing Wong
email: wswong@ie.cuhk.edu.hk

Constant Delivery Delay Protocol Sequences for the Collision Channel Without Feedback

Keywords: Collision channel without feedback, protocol sequences, wireless ad-hoc networks, multiple access

We consider a collision channel model without feedback based on a time-slotted communication channel shared by K users. In this model, packets transmitted in the same time slot collide with each other and are unrecoverable. Each user accesses the channel according to an internal periodical pattern called protocol sequence. Due to the lack of feedback, users cannot synchronize their protocol sequences, leading to unavoidable collisions and varying throughput. Protocol sequences that provide constant throughput regardless of delay offsets between users are called shift-invariant (SI), they have been studied and characterized in previous work. We propose a new class of SI sequences: Constant Individual Delivery Delay (CIDD) sequences which ensure that the delay between two successfully delivered packets is constant for each user. We present a characterization of CIDD sequences. We also prove that CIDD sequences can achieve the lower bound of SI sequences period but not the optimal throughput.

I. INTRODUCTION

Many research activities have been conducted in the context of multiple access communications in which several users access a shared channel. Without synchronization, users might transmit packets at the same time causing collisions and the loss of packets. A possible solution is to use protocol that achieves collision-free transmission such as the well known time division multiple access (TDMA). However, in communication systems such as impulse radio [START_REF] Win | Impulse radio: how it works[END_REF], wireless sensor networks [START_REF] Li | Real-time QoS support in wireless sensor networks: a survey[END_REF] and ad hoc mobile networks [START_REF] Royer | A review of current routing protocols for ad hoc mobile wireless networks[END_REF], devices have constrained resource, limited communication power and need a flexible transmission scheme. Collision-free protocols such as TDMA may not be practical for these systems, since it requires stringent time synchronization. Other contention based random access protocols such as IEEE 802.11 CSMA/CA [START_REF]Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications[END_REF] can provide a more flexible transmission scheme. However, they still require some backoff algorithms and a feedback link which may not be practical for these low resource devices. These systems require simple multiple access protocol with no stringent time synchronization, frequent channel monitoring and feedback link.

Such a collision channel without feedback model was introduced by Massey and Mathys in [START_REF] Massey | The collision channel without feedback[END_REF]. In this model without feedback mechanism, senders cannot synchronize their transmission schedule, implying relative time offsets between users are unavoidable and unknown for them. They also proposed the idea of using protocol sequences as medium access control (MAC) schemes. Each user transmission schedule follows their protocol sequence. Many protocol sequences have been studied in the literature. One can consider slotted ALOHA [START_REF] Anantharam | The stability region of the finite-user slotted ALOHA protocol[END_REF] as a MAC using probabilistic protocol sequences, where each user transmits a packet with probability p independently. Other protocol sequences are deterministic, such as constantweight cyclically permutable codes [START_REF] Bitan | Constructions for optimal constant weight cyclically permutable codes and difference families[END_REF], [START_REF] Györfi | Constructions of binary constantweight cyclic codes and cyclically permutable codes[END_REF], linear congruence sequences [START_REF] Titlebaum | Time-frequency hop signals part I: Coding based upon the theory of linear congruences[END_REF], prime sequences [START_REF] Shaar | Collision resolution in contention access local area networks using concatenated prime sequences[END_REF], [START_REF] Shaar | Prime sequences: quasi-optimal sequences for or channel code division multiplexing[END_REF], wobbling sequences [START_REF] Wong | New protocol sequences for random-access channels without feedback[END_REF] and CRT sequences [START_REF] Shum | Construction and applications of CRT sequences[END_REF]. See [START_REF] Györfi | Coding for multiple access collision channel without feedback[END_REF] for a survey on these coding techniques.

Important features of protocol sequences are the sequence length (which should be as low as possible, e.g., for devices with memory limitations) and the resulting throughput. Some sequences have non-zero throughput regardless of relative delay offsets between users, this property was studied in [START_REF] Chen | User unsuppressible protocol sequences for collision channel without feedback[END_REF] as user unsuppressibility. Moreover, an important class of protocol sequences ensure constant throughput, independent of any relative delay offsets. This class is called shift-invariant protocol sequences (SI), they have been introduced in [START_REF] Massey | The collision channel without feedback[END_REF] and studied in [START_REF] Chen | Constructions of robust protocol sequences for wireless sensor and ad hoc networks[END_REF]- [START_REF] Zhang | Protocol sequences for the multiple-packet reception channel without feedback[END_REF]. Even if this class enjoys a zero-variance shift-invariant throughput, its main disadvantage is that its sequence length grows exponentially with the number of users.

In this paper, we propose a new class of SI sequences called Constant Individual Delivery Delay (CIDD) protocol sequences which ensure not only a constant throughput but also a constant delay between each successfully delivered packet of each user. After setting up the channel model and required notations in Section II, we show two important properties of SI sequences in Section III. In Section IV, we develop some properties of CIDD protocol sequences and prove its characterization theorem. We show that this class contains only a small number of instances. Furthermore, it does not achieve the optimal throughput of SI sequences except for the two users case. In Section V, we introduce Uniform Individual Delay (UID) protocol sequences, and we prove that they are equivalent to CIDD sequences. Finally, a brief conclusion is presented in Section VI.

II. CHANNEL MODEL AND NOTATIONS

We consider the collision channel without feedback model introduced in [START_REF] Massey | The collision channel without feedback[END_REF] in which K users share the same communication channel. Time is divided into discrete time slots and is denoted by t ∈ N. We suppose that each user i ∈ 1; K is aware of time slots boundaries and chooses to transmit following a periodic binary sequence s i (t), namely protocol sequence.

s i (t) =
1 , user i transmits a packet at time slot t 0 , user i keeps silent at time slot t Note that it is possible to remove the slot synchronized condition and have user transmitting in continuous time. Some approaches and discussions are presented in [START_REF] Massey | The collision channel without feedback[END_REF], [START_REF] Hui | Multiple accessing for the collision channel without feedback[END_REF], [START_REF] Thomas | Capacity of the wireless packet collision channel without feedback[END_REF]. However, we will not address it in this paper. At any time slot t, if more than one user transmit, a collision will occur and all packets transmitted in this time slot are lost. Since this model suppose no feedback mechanism, users are not aware of the packets lost.

Let s 1 , . . . , s K be binary sequences of least common period L. Since users cannot synchronize their protocol sequences, a realization will have delay offsets τ 1 , . . . , τ K ∈ 0; L -1 between users. We call cyclic shift of s i by τ i the sequence defined as

∀ t ∈ N, s (τi) i (t) = s i (t ⊕ τ i )
where ⊕ represents addition modulo L. The following example illustrates the above notations used in this paper.

Example 1. In this example, the protocol sequence s i has period L = 6, where s i (0) = s i (5) = 1, s i (3) = 0 and * denotes either 0 or 1 (value not specified, used for examples). s

(1) i and s

(2) i have τ i = 1 and τ i = 2 with respect to s

i .

s (0) i : 1 * * 0 * 1 s (1) i : 1 1 * * 0 * s (2) i : * 1 1 * * 0
Given the delay offsets τ 1 , . . . , τ K , we say that user i successfully delivers a packet at time t, if and only if

s (τi) i (t) j =i (1 -s (τj ) j (t)) = 1.
In the next section, we will define shift-invariant protocol sequences.

III. PROPERTIES OF SHIFT-INVARIANT PROTOCOL SEQUENCES

Definition 2 (Shift Invariant Protocol Sequences).

Let s 1 , . . . , s K be binary sequences of least common period L. {s 1 , . . . , s K } is said to be shift-invariant if each of its user's throughput is constant for every choice of offsets τ 1 , . . . , τ K .

∀ i ∈ 1; K , ∃ r i > 0, ∀ τ 1 , . . . , τ K , 1 L L-1 t=0 s (τi) i (t) j =i (1 -s (τj ) j (t)) = r i
where r i denotes user i's throughput.

Definition 3 (Duty Factor). Let {s 1 , . . . , s K } be a SI sequence set of least common period L. The duty factor f i is the fraction of time slots in which user i transmits.

f i = n i d i = 1 L L t=0 s i (t)
where

n i ≥ 0, d i ≥ 0, n i ≤ d j , so that f i ∈ [0; 1] ∩ Q.
In this paper, we will suppose n i ≥ 1, d i ≥ 2 and n i < d j to remove the trivial cases where either a user does not transmit at all or transmits in every time slot.

The following Theorem 4 and Theorem 5 are from [START_REF] Shum | Shift-invariant protocol sequences for the collision channel without feedback[END_REF], they describe the achievable throughput and the least common period of SI sequences. Their proofs can be found in [START_REF] Shum | Shift-invariant protocol sequences for the collision channel without feedback[END_REF].

Theorem 4 (Characterization of SI throughput).

Let {s 1 , . . . , s K } be a SI sequence set with duty factors f 1 , . . . , f K respectively. The throughput of user i is equal to

f i j =i (1 -f j ) . (1)
When all K users have the same duty factor f , the optimal (maximal) throughput is achieved by setting f = 1/K.

Theorem 5 (Characterization of SI period).

For any set of K SI sequences with duty factors

n 1 /d 1 , . . . , n K /d K , such that gcd(n i , d i ) = 1 for all i, the least common period L is divisible by K i=1 d i . In particular, the minimal period for given d 1 , . . . , d K is K i=1 d i .

IV. CONSTANT INDIVIDUAL DELIVERY DELAY SEQUENCES

We propose a new class of deterministic protocol sequences called Constant Individual Delivery Delay (CIDD) sequences. The main idea is to have not only a constant throughput like SI sequences, but also to ensure a constant delay between each packet delivered by a user, meaning that user i's packets are exactly successfully delivered every D i time slots, in a periodic manner of period D i .

Definition 6 (Constant Individual Delivery Delay Sequences).

A binary sequence set {s 1 , . . . , s K } of least common period L has CIDD if each user i's delay between two successive delivered packets is constant equal to a certain value

D i ∈ N * . ∀ i ∈ 1; K , ∀ τ 1 , . . . , τ K ∈ N, ∃ t ∈ N, ∀ t ∈ {t + qD i } q∈Z , s (τi) i (t ) j =i (1 -s (τj ) j (t )) = 1, ∀ t / ∈ {t + qD i } q∈Z , s (τi) i (t ) j =i (1 -s (τj ) j (t )) = 0,
where D i is the constant delivery delay of user i. Note that D i must divide L for all i.

Let s 1 , . . . , s K be binary sequences of least common period L. Suppose that {s 1 , . . . , s K } has constant individual delivery delays of D i for each user i ∈ 1; K . We first derive the following properties, then prove characterization Theorem 10. Lemma 7. {s 1 , . . . , s K } is SI with throughput of 1/D i for each user i ∈ 1; K . Proof: By definition of CIDD, each user i ∈ 1; K delivers one packet exactly one time per D i time slots. Thus their throughputs are respectively 1/D i and are shift invariant.

Since {s 1 , . . . , s K } is SI with throughput 1/D i > 0, in the following proofs we will use the fact that for each user i ∈ 1; K , there exists offsets τ 1 , . . . , τ K such that user i delivers a packet at time 0, i.e., s

(τi) i (0) = 1 and ∀ j = i, s (τj ) j (0) = 0. Lemma 8. Each s i is periodic with period D i .
Proof: Let i ∈ 1; K . Suppose that τ 1 , . . . , τ K are chosen such that user i delivers a packet at time 0. That is

s (τi) i (0) = 1, ∀ j = i, s (τj ) j (0) = 0.
Since s i is CIDD, we have, for all positive integer q s (τi) i

(qD i ) = 1, ∀ j = i, s (τj ) j (qD i ) = 0.
Suppose for the sake of contradiction that s i is not D iperiodic. Then

∃ τ i ∈ 0; L -1 , ∃ q ∈ 1; L/D i -1 , s (τ i ) i (0) = 1, s (τ i ) i (q D i ) = 0.
Under the offsets τ 1 , . . . , τ i-1 , τ i , τ i+1 , . . . , τ K , we have s

(τ i ) i (0) = 1 and ∀ j = i, s (τj ) j (0) = 0, (2)
s (τ i ) i (q D i ) = 0. (3)
User i delivers a packet at time 0 but fails to deliver a packet at time q D i , see Eqn. ( 2) and (3), respectively. This contradicts the fact that s i has constant delivery delay of D i . Thus, s i is D i -periodic.

The following example (4) shows a case of the proof by contradiction with 3 users i, j, k, where D i = 6 and the nonperiodic pattern is at τ i = 2 and q = 1.

Note that D i is not necessarily the fundamental period of s i . Its fundamental period is a divisor of D i . Lemma 9. Each D i is a common period of the sequence set {s 1 , . . . , s K }.

Proof: Let i, j ∈ 1; K , i = j. Suppose that τ 1 , . . . , τ K are chosen such that user i delivers a packet at time 0. Since s i has constant delivery delay of D i , we know that

∀ q ∈ 0; L/D i -1 , s (τi) i (qD i ) = 1, ∀ l = i, s (τ l ) l (qD i ) = 0.
Using the same argument as in Lemma 8, we will show that user j is also D i -periodic. Suppose for the sake of contradiction that s j is not D i -periodic. Then

∃ τ j ∈ 0; L -1 , ∃ q ∈ 1; L/D i -1 , s (τ j ) j (0) = 1, s (τ j ) j (q D i ) = 0.
Therefore, under the offsets τ 1 , . . . , τ j , . . . , τ K , user i delivers a packet at time q D i but not at time 0. This contradicts the constant delivery delay property of user i.

We prove that for all i, j ∈ 1; K , user j is D i -periodic. It follows that each D i is a common period of the sequence set {s 1 , . . . , s K }.

Theorem 10 (Characterization of CIDD Sequences).

Let s 1 , . . . , s K be binary sequences of least common period L. The following conditions are equivalent.

1) The sequence set {s 1 , . . . , s K } has constant individual delivery delay (CIDD) of D i for each user i ∈ 1; K . 2) The sequence set {s 1 , . . . ,

s K } is SI, such that L = 2 K and ∀ i ∈ 1; K , f i = 1/2, D i = L.
Proof: 1) ⇒ 2). Since L is the least common period of {s 1 , . . . , s K } and each D i is a common period (from Lemma 9), we know that L divides D i . We also know that, by definition, D i divides L. Hence, D i = L for all i.

Eqn. ( 5) follows from Lemma 7 and Eqn. (1) in Theorem 4.

∀ i ∈ 1; K , n i d i j =i 1 - n j d j = 1 D i = 1 L . (5)
Using Theorem 5, we know that

K i=1 d i divides L. Therefore, (5) is equivalent to (6), with α i ∈ N * . ∀ i ∈ 1; K , n i j =i (d j -n j ) = 1 α i . (6)
The left hand side of ( 6) is a positive integer, so α i = 1 for each i ∈ 1; K . It follows that the unique solution is

n i = 1, d i = 2 for each i ∈ 1; K .
From this solution and Lemma 7, we conclude that {s 1 , . . . , s K } is SI with duty factor f i = n i /d i = 1/2 and delivery delay 2) ⇒ 1). From (1) the throughput of user i is f i j =i (1 -f j ) = 1/2 K . By definition of SI sequences, each user i will send exactly one packet per period L = 2 K regardless of the offsets. Since L is the least common period, user i's delivery delay is constant and D i = L.

D i = L = α i K i=1 d i = 2 K for each user i.
Theorem 10 shows that the CIDD sequences class is limited to a small number of instances. Indeed, for a K-users system with CIDD protocol sequences, each user throughput is 1/2 K , and their period (sequence length) is 2 K . According to Theorem 5, CIDD sequences achieve the minimal SI sequences period. However, according to Theorem 4, SI sequences optimal throughput is (K -1) K-1 /K K which is achieved by duty factor f = 1/K while CIDD sequences throughput is 1/2 K . Hence, CIDD sequences throughput is optimal only for K = 2 and its performance drop increases exponentially with K, as we can see on Fig. 1. In Section VI, we will discuss some special structures that allow to reduce the system complexity K in order to have better throughput performance.

In this paper, we have not dealt with explicitly constructing CIDD instances. However, [START_REF] Shum | Shift-invariant protocol sequences for the collision channel without feedback[END_REF] proposed simple methods to generate minimal period SI sequences for given duty factors and number of users. In particular, using duty factors f = 1/2, it follows from Theorem 10 that it is possible to generate every possible CIDD sequences.

V. UNIFORM INDIVIDUAL DELAY PROTOCOL SEQUENCES

User i's individual delay is the duration between t = 0 and its first delivered packet time [START_REF] Wu | Safety-message broadcast in vehicular ad hoc networks based on protocol sequences[END_REF]. Suppose that user offsets τ 1 , . . . , τ K are random variables following an independent discrete uniform distribution ∼ U (0, L -1), then individual delays follow a non-increasing distribution. In this section we study a particular class of protocol sequences that has uniform individual delay (UID) over all possible offsets. We also define UID sequences to be user unsuppressible, i.e., each user delivers at least one packet regardless of offsets. Thus, excluding trivial cases of all-ones or all-zeros sequences. We first introduce some notations and combinatorial properties, then prove that UID sequences are equivalent to CIDD sequences in Theorem 13.

The idea is to count the number of realizations of each individual delays over all the possible offsets. For that, we will reduce all τ 1 , . . . , τ K combinations to equivalence classes. Let N i (delay = t) be the number of realizations in which user i has its individual delay equal to t. We define the following equivalence relation

∀ (τ i ) i=1..K , (τ i ) i=1..K , (τ i ) i ∼ (τ i ) i ⇐⇒ ∃ t ∈ 0; L -1 , (τ i ) i = (τ i ⊕ t) i . (7)
The equivalence relation [START_REF] Bitan | Constructions for optimal constant weight cyclically permutable codes and difference families[END_REF] induces several equivalence classes

[(τ i ) i ] = {(τ i ) i | (τ i ) i ∼ (τ i ) i } = {(τ i ⊕ t) i , t ∈ 0; L -1 } (8)
which represents all offsets that are equal to (τ i ) i ignoring cyclic shift (cyclic shift of the same value t over all offsets: τ 1 ⊕t, . . . , τ K ⊕t). We then gather these equivalence classes (8) in a quotient set X/∼. Since τ i takes a finite number of values 0; L -1 , X/∼ is also finite. Hence, simplifying the calculation of delays counting with

N i (delay = t) = [(τi)i]∈X/∼ N i (delay = t | [(τ i ) i ])
where

N i (delay = t | [(τ i ) i ]
) is the number of realizations of delay = t over every offsets of [(τ i ) i ] equivalence class.

Definition 11 (Uniform Individual Delay Protocol Sequences).

A binary sequence set {s 1 , . . . , s K } has UID if, and only if, for all user i ∈ 1; K , there exists a threshold T > 0 such that

∀ t < T, N i (delay = t) = N i (delay = 0)
and

∀ t ≥ T, N i (delay = t) = 0.
Lemma 12. Let {s 1 , . . . , s K } be a set of protocol sequences with user unsuppressible property (i.e., each user delivers at least one packet regardless of offsets). Let D max (i, [(τ l ) l ]) and D min (i, [(τ l ) l ]) be respectively the maximum and minimum delivery delay (as defined in Section IV) of user i with offsets equivalence class [(τ l ) l ]. For simplicity, we use a shortened notation: D max , D min . They are well-defined since we suppose {s 1 , . . . , s K } is user unsuppressible and periodic. For all i ∈ 1; K , [(τ l ) l ] ∈ X/∼, we have the following properties: (b ):

(a) N i (delay = t | [(τ l ) l ]) is a non-increasing function of t. ∀ t, N i (delay = t + 1 | [(τ l ) l ]) ≤ N i (delay = t | [(τ l ) l ]) (
∀ t < D min , N i (delay = t | [(τ l ) l ]) = N i (delay = 0 | [(τ l ) l ]) (b ): N i (delay = D min | [(τ l ) l ]) < N i (delay = D min -1 | [(τ l ) l ]) (c
) Its values for t ≥ D max are equal to zero, expressible as (c ). For t < D max its values are not equal to zero, expressible as (c ).

(c ):

∀ t ≥ D max , N i (delay = t | [(τ l ) l ]) = 0 (c ): N i (delay = D max -1 | [(τ l ) l ]) = 0
Proof: Let L be the least common period of {s 1 , . . . ,

s K }. Let i ∈ 1; K , [(τ l ) l ] ∈ X/∼. Each element of [(τ l ) l ]
is a cyclic shift of (τ l ) l , so they all have the same delivery pattern ignoring cyclic shift. We define O

[(τ l ) l ] t
as the subset of [(τ l ) l ] that achieves individual delay of t for user i. As a consequence,

N i (delay = t | [(τ i ) i ]) = |O [(τ l ) l ] t |
where | . | denotes the cardinality of a finite set.

Proof of (a): Every offsets in [(τ l ) l ] give the same delivery pattern ignoring cyclic shift. Let (τ l + τ + 1) l ∈ X/∼ be such that user i's delay is t delay + 1 > 0. That is

s (τi+τ +1) i (t delay + 1) j =i (1 -s (τj +τ +1) j (t delay + 1)) = 1, ∀ t < t delay + 1, s (τi+τ +1) i (t) j =i (1 -s (τj +τ +1) j (t)) = 0. (9)
Then, if we consider offsets (τ l + τ ) l ∈ X/∼, we have

s (τi+τ ) i (t delay ) j =i (1 -s (τj +τ ) j (t delay )) = 1, ∀ t < t delay , s (τi+τ ) i (t) j =i (1 -s (τj +τ ) j (t)) = 0. (10)
Thus, user i's delay with offsets (τ l + τ ) l is t delay . In sum, we have proven that for every user i, for any (τ l + τ + 1) l ∈ X/∼ with delay t delay + 1 > 0, there is (τ l + τ ) l ∈ X/∼ with delay t delay (the converse (10) ⇒ (9) is not necessarily true due to the pattern's cyclic shift). Moreover, if we take two different offsets (τ l + τ + 1) l and (τ l + τ + 1) l in X/∼ which realize delay t delay + 1, then (τ l + τ ) l and (τ l + τ ) l are also different. We infer that there exists an injective function from O

[(τ l ) l ] t+1 to O [(τ l ) l ] t . As a consequence, N i (delay = t delay + 1 | [(τ l ) l ]) ≤ N i (delay = t delay | [(τ l ) l ]) is proven.
Proof of (b ): Suppose that D min = 1, then (b ) is true. Now, suppose that D min > 1, we will show that the converse of the previous relation is true: (10) ⇒ (9) for t delay < D min -1. It is direct to see that [START_REF] Shaar | Collision resolution in contention access local area networks using concatenated prime sequences[END_REF] implies

s (τi+τ +1) i (t delay + 1) j =i (1 -s (τj +τ +1) j (t delay + 1)) = 1, ∀ 0 < t < t delay + 1, s (τi+τ +1) i (t) j =i (1 -s (τj +τ +1) j (t)) = 0. (11)
It remains to prove that [START_REF] Shaar | Prime sequences: quasi-optimal sequences for or channel code division multiplexing[END_REF] is also true for t = 0. Since the duration between t = t delay + 1 and t = 0 is less than D min , only one delivery in this duration is possible, i.e., s (τi+τ +1) i (0) j =i (1 -s (τj +τ +1) j (0) = 0. Hence, (10) ⇐⇒ (9) for t delay < D min -1. Using the same argument as previously, we infer there exists a bijection that maps O

[(τ l ) l ] t+1 to O [(τ l ) l ] t
for any t delay < D min -1. Thus, (b ) is proven. Proof of (b ): Proof of (a) shows that there exists an injective function

f : O [(τ l ) l ] t+1 → O [(τ l ) l ] t (τ l ⊕ 1) l → (τ l ) l Let (τ * l ) l ∈ [(τ l ) l ]
be such that user i delivers a packet at time t 1 = -1 (≡ L -1 (mod L)) and t 2 = D min -1. Since t 2 -t 1 = D min is the minimum delay, there is no delivery between t 1 and t 2 , so these offsets (τ * l ) l are a realization of

delay = D min -1. That is, (τ * l ) l ∈ O [(τ l ) l ] Dmin-1 . However, (τ * l ⊕ 1) l achieves delay = 0, that is (τ * l ⊕ 1) l ∈ O [(τ l ) l ] 0
. Hence, (τ * l ) l has no inverse image by f . We infer that

N i (delay = D min | [(τ l ) l ]) < N i (delay = D min -1 | [(τ l ) l ]).
Proof of (c ): Since user i's delivery delay is at most D max , its first delivered packet can only be sent at t < D max . Therefore, (c ) is proven.

Proof of (c ): Let's consider offsets (τ l ) l . By definition of D max , there exists two successive delivered packets at t 1 and t 2 such that t 2 -t 1 = D max . By cyclically shifting these offsets, it is possible to shift t 1 to position L -1 in the sequence and shift t 2 to position D max -1. Hence,

N i (delay = D max - 1 | [(τ l ) l ]) = 0.
Theorem 13 (Characterization of UID Sequences). Let s 1 , . . . , s K be binary sequences. The following conditions are equivalent.

1) The sequence set {s 1 , . . . , s K } has UID.

2) The sequence set {s 1 , . . . , s K } has CIDD.

Proof: Let {s 1 , . . . , s K } be a set of protocol sequences. By Definition 11, {s 1 , . . . , s K } has UID if, and only if, for all user i, there exists a threshold T > 0 such that

∀ t < T, N i (delay = t) = N i (delay = 0) and ∀ t ≥ T, N i (delay = t) = 0. (12)
Lemma 12(a) shows that for all

[(τ l ) l ] ∈ X/∼, N i (delay = t | [(τ l ) l ]
) is a non-increasing function of t, hence condition ( 12) is equivalent to: For all user i, for all

[(τ l ) l ] ∈ X/∼, ∀ t < T, N i (delay = t | [(τ l ) l ]) = N i (delay = 0 | [(τ l ) l ]) and ∀ t ≥ T, N i (delay = t | [(τ l ) l ]) = 0.
It follows from Lemma 12(b) and 12(c) that for each user i, D max (i) = D min (i) = T (note that this value does not depend on the choice of [(τ l ) l ] ∈ X/∼ in this case). That is, for each user i the delivery delay is constant regardless of any relative offsets between users.

VI. CONCLUSION

In this paper, we investigate constant individual delivery delay (CIDD) protocol sequences as a multiple access scheme for collision channel without feedback [START_REF] Massey | The collision channel without feedback[END_REF]. These sequences have strong properties of constant throughput and periodic delivery for each user in the system.

We first proved that they are equivalent to a small family of shift-invariant (SI) protocol sequences. Then, we infer that they achieve SI sequences minimal length. However, they do not achieve optimal SI throughput; indeed, Fig. 1 shows its performance drop increasing exponentially in the number of users. To overcome this issue of throughput performance while preserving their periodic delivery property, one possible solution would be to reduce the number of users communicating in the same channel. For example, in a graph network where nodes represent users and edges their communication links (or shared channels), the number of sequences needed to preserve SI and CIDD properties is the graph's chromatic number χ. In particular, the chromatic number of a planar graph is at most 4 (see the four color theorem in [START_REF] Cayley | On the colouring of maps[END_REF]- [START_REF] Appel | Every planar map is four colorable. part II: Reducibility[END_REF] for more information about planar graph coloring), therefore only 4 sequences are needed in this kind of structure. So, the performance drop would be approximately 41% (see Fig. 1). In an even more restricted grid structure with χ = 3, performance drop would be approximately 16%. Practical applications of this protocol to wireless sensor networks or ad hoc mobile networks is an interesting subject for future exploration.

Note that we also proved that CIDD sequences are equivalent to another class of protocol sequences called uniform individual delay (UID) protocol sequences.

Previous work shows that there is a tradeoff between the length of protocol sequences and the variance of throughput performance. SI sequences represent the extreme case where the throughput of each user is not affected by any relative delay offsets. In turn, the length of SI sequences must be exponential in the number of users. The CIDD property we developed in this paper, ensures very strong delivery stability and deterministic quality-of-service as a tradeoff for throughput performance drop which increases exponentially with the number of users.

: 1

 1 * * * 0 * 1 * * * 1 * s (0) j : 0 * * * * * 0 * * * * * s (0) k : 0 * * * * * 0 * * * * * s (τ i ) i : 1 * 1 * * * 0 * 1 * * * s (0) j : 0 * * * * * 0 * * * * * s (0) k : 0 * * * * * 0 * * * * *

Figure 1 .

 1 Figure 1. Optimal SI throughput and CIDD throughput comparison in a symmetric system (each user has the same duty factor) as a function of the number of users K.

  b) Its first D min values for t = 0 . . . D min -1 are equal, expressible as (b ), and its value at t = D min is different, expressible as (b ).

ACKNOWLEDGEMENT

The authors would like to thank Kenneth W. Shum for offering his software implementation of protocol sequences and valuable comments. A part of the work was carried out at Laboratory of Information, Networking and Communication Sciences (www.lincs.fr).


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 [[50.614291,3.13282],[50.604159,3.15239]], '500 m'); timeCoverage('30 min','1 H'); duplicate(1);

  location = requires('location'); var trace = requires('honeycomb'); var telephony = requires('gsm'); location.onLocationChange(function(event){ trace.sync({ lat : event.latitude, lng : event.longitude, signal : telephony.signalStrength(

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Nicolas Bouleau Cired

Was Anthropogenic Climate Change Falsifiable in 1925? Reflection on the Scientific Attitude

In 1925, a French astronomer J. Mascart achieved the gigantic work of studying more than three thousand publications in order to sum up the available knowledge about the climate variability. He published his analysis and conclusions in a book. We base on this work to tackle anew some epistemological problems. In particular we weigh up the relevance of the popperian doctrine in the case of climate change. As eventually seen, the strict framework of falsifiability does not practically operate and the most rational position seemed to stake on improving measurements and scientific networking. We qualify this position as "narrowly scientific". The case of climate change is, in fact, representative of any question of knowledge whose assessments and controls are postponed to the future, which is the case of every theory attempting to say more than available measurements.

This leads us to reconsider the narrowly scientific position, a wait and see policy which does not take into account the interests involved in contradictory representations. We discuss a more engaged attitude for research centres that is better placed to anticipate possible dangerous uses.

Le changement climatique anthropique était-il réfutable en 1925 ?

Réflexion sur l'attitude scientifique.

Nous étudions la pertinence de la doctrine popperienne dans le cas du changement climatique telle que cette connaissance était appréhendée en 1925 en nous fondant sur une analyse approfondie et quasi-exhaustive de J. Mascart des travaux et théories disponibles à cette époque. On s'aperçoit que le cadre strict de la réfutabilité n'est pas opérant et qu'à cette époque la position qui se voulait la plus rationnelle, apparentée à des points de vue qui seront, après guerre, ceux de Lakatos, Quine et Feyerabend, est finalement attentiste. Nous la qualifions de scientifique au sens étroit. Le cas du changement climatique est, en fait, générique de toute question de connaissance dont les sanctions et contrôles sont repoussés à plus tard, ce qui est le cas de toute théorie qui prétend dire plus que les mesures disponibles.

Ceci nous conduit à reconsidérer la position scientifique étroite qui ne tient pas compte des intérêts en jeu dans les représentations contradictoires. Nous discutons de la part des laboratoires une attitude plus engagée et plus anticipative d'usages possibles éventuellement dangereux.

I. Introduction. The Popperian doctrine and its critics.

The philosophy of Karl Popper is famous principally for the idea that certain intellectual constructions are easier than true science because they avoid verification by experiment. Among them are theories which present themselves as scientific, such as the dialectical materialism of Marx and Lenin, or which claim scientific status in a special sense like the sociological method explained by Max Weber; and collections of interpreted observations such as psychoanalysis, which are scientific according to some (Freud), or structurable knowledge according to others (Lacan).

In support of the theses of John Stuart Mill on the value of the open criticism of ideas, Karl Popper developed "a corpus of doctrine" able to define what is scientific and what is not, as well as the direction a more rigorous science should take: the real content of a theory is the sum of the statements which refute it, its potential falsifiers. A theory becomes more vulnerable as its content increases. When a theory is rejected by experiment, a new theory has to be found which fits in with what remains valid in the former while proposing an alternative for where the difficulty occurred. It is not sure that such a better theory can be found each time a problem occurs: this depends on the inventiveness of the researchers.

Since what scientists are going to discover is, by nature, unpredictable, the future of society is open and theories which claim to grasp the direction of History are non scientific. Popper's criticism of historicism should be considered as a polemical extension of his doctrine, with its sights firmly set on Marxism-Leninism, but it cannot easily be included in his epistemological thesis proper, since Popper does not deny the predictive value of scientific theories: on the contrary, Galle's discovery of Neptune using the calculations of Le Verrier is exemplary of the scientific approach. To avoid the contradiction, Popper is obliged to introduce his three-world theory, which is not easily acceptable, especially in the field of biology.

The failings of the Popperian doctrine have been set out by many authors, but they remain less well-known to the general public. School manuals only teach the idea of "falsifiability". The two most damaging critiques are those of Adolf Grünbaum, who pretends to believe in the doctrine in order to analyze the properties of the rationalism to which it leads1 , and of Paul Feyerabend, who does not believe in the doctrine at all, and whose main point is that things did not happen that way historically, not even in the case of Newtonian mechanics and relativity. To each of the questions: 1. Can we wish to live according to the rules of critical rationalism [the name given to the Popperian doctrine]? 2. Is it possible to keep both Science as we know it and these rules? Feyerabend answers no, basing his argument, and thus giving weight to it, on an analysis of real situations and the real human behaviour of scientists 2 , an analysis which continues through his whole work. Grünbaum and Feyerabend examine in detail the question of the replacement of one theory by another, and show that the matter is far from being as simple as Popper suggests.

The transition from Newtonian mechanics to relativist mechanics is a difficult case to hold up as the archetype of the truly scientific. The costume is too tight. One of the assets of Feyerabend's argument that "everything goes" is that it can receive in the field of knowledge all the issues in which we are involved today, and not only those of physics. The very notion of theory, with its logical and mathematical framework, such as the epistemologists used it during the three first quarters of the 20th century, is too restrictive. For instance, what is called chaos theory today is not a theory in this sense, but a phenomenological collection of properties of certain systems.

Two circumstances lead us to take a certain distance from Popper and from his better known critics alike. One, very sudden, was the development of modelling that accompanied the computer science boom during the last third of the 20th century, a new form of knowledge whose characteristics have to be studied. The other, more gradual, is the growing presence of environmental issues (waste, biodiversity, climate) and ethical issues (cloning, perinatal innovation, human genetics). Do Popper, Lakatos, and Feyerabend set themselves the right problem? When Feyerabend writes that theoretical anarchy is more apt to develop progress than doctrines based on law and order or again that the only principle which does not restrict progress is "everything goes", we have a right to wonder whether these positions are not showing their age, and whether the real problem today is not rather to know if technical innovation can be fostered with more respect for the values of civilization, in its most common sense as the opposite of barbarism, and with greater lucidity about the risks being generated3 .

In the following pages of this article, we carry through this reflection, basing ourselves on an analysis of the case of climate variability and its causes at the beginning of the 20th century. At that time, a synthesis of available scientific works was still possible, so that what was already a very complex issue has lessons for us when transposed to today.

II. How was the science of climate change to be taken forward in 1925?

The issue of climate change is peculiarly appropriate for putting Popper's ideas to the test. Central among contemporary debates in which science is expected to be an arbitrator, climate change is the province neither of metaphysics nor of psychological or social interpretation, but of the natural sciences, whose outlines it is bound to question. It is located in the very place where the purification of facts requires the conditions of observation to be made specific to the most anecdotal of detail and frustrates true understanding.

In this perspective, there are two principal advantages to basing our study on the work of Jean Mascart entitled Notes sur la variabilité des climats, 383p, Lyon, M. Audin 1925. Firstly, the author makes no attempt to defend one particular explanatory theory but rather strives with true scientific rigour to evaluate the various chains of argument as well as the quality of the measures. Secondly, his study presents a synthesis of the works available at that time that if not exhaustive is nonetheless outstandingly complete: he analyses almost 3100 bibliographical references involving more than 1000 authors.

The landscape, as we are going to see, is complex. We meet the most varied points of view, more or less well supported, which often defy comparison due to the multiplicity of explanatory fields they appeal to, and the specificity of the data they interpret. This great mass of documents and opinion creates a situation close to that we confront on most subjects today on a permanent basis with the Internet: information overload.

Jean Mascart shows a great concern for exactness. He was an astronomer, son of the physicist Eleuthère Mascart whose experiments prior to Michelson's against the existence of ether made him a pioneer in relativity theory 4 . But Jean Mascart is far from being interested only in astronomical factors. On the contrary, he attaches great value to all accounts and historical and economic testimonies, and in his conclusion, which betrays a certain epistemological unease, he stresses the importance of archival research and of the systematic collection of reliable data.

To situate his work, he first sets himself apart from the great narrative frescos such as that of Bertrand on the Révolutions du globe in which hypotheses on the formation of the earth's crust jostle together with Buffon's theory, Fourier's theory of heat, and Lyell's theory on the mass of waters, and so on. He criticizes the art of evocation without serious foundation, and he himself determines to draw on the experimental sciences: astronomy, geology, the physics of the globe, geography, oceanography, and meteorology. "It is necessary to retain in these matters", he writes, "a degree of scepticism, since it has been impossible until today to give an accurate definition of the very word climate that we use." For sure, the term climate sums up the invariability of the weather in a region once the daily and yearly variations have been abstracted, but the notion is vast in view of the physical parameters involved, and yet sins by omission in having no precise and clear marker as far as the long term is concerned. In other words, the signal is unknown as far as low frequencies are concerned. Mascart gradually adopts a position that we could qualify as positivist on this point, considering that it is rather useless to keep on trying to define this primitive word.

As recommended by Claude Bernard, Mascart looks for the efficient cause of the phenomena of glaciation by sifting the evidence of the plant fossil record. Ever the good astronomer, he finds it in the eccentricity of the earth's orbit, the precession of the equinoxes (variation of the earth's axis), and the variations of the activity of the sun. These causes being in any case indisputable and calculable, he deems it evident that the earth is not in a steady and constant state as far as its astronomical evolution is concerned, and that consequently climatic variations are currently taking place. His aim thus defines itself as "to search for the symptoms of climate change in the most recent human period". Naturally, he knows that "economic and political phenomena may intervene" -unbalanced irrigation, deforestation, and so on -and he cites Bourquin on that point: "the sunshine and desolation of lands where once bloomed great empires such as Tello, Babylon, Nineveh, Persepolis and Thebes were due, above all, to the lack of foresight or mistakes of men, and to political and economic events, rather than to natural phenomena."

After the astronomical causes, he approaches the physical causes and in particular the greenhouse effect. He credits the Belgian scholar W. Spring 5 with being the first to insist on the role played by carbon dioxide in the thermal balance of the atmosphere, ahead of Arrhenius (1895). Mascart expresses many reservations about Arrhenius' theory, which suggests that the carbon dioxide content variation hypothesis may explain variations at the geological scale, without astronomical cycles. He observes that if this were the case, the causes of the variations would remain obscure, and disputes the role of volcanoes and solubility in the ocean. He also disputes the orders of magnitude of the temperature changes connected with increases or decreases in carbon gas calculated by Arrhenius, which he finds excessive in both directions. But his principal disagreement is with Arrhenius' view that the recent and future increase of carbon dioxide content will be beneficial, by making winters milder and summers cooler, thanks to rain.

Mascart thinks that the proportion of carbon dioxide should be considered to be stable during historical time, with the exception of the recent period during which "the artificial combustion of coal has destroyed the balance, all the more as coal consumption has tremendously increased." Arrhenius' idea that the inescapable cooling of the Earth due to physical and astronomical causes is fortunately balanced by an increase in carbon dioxide of human origin (an increase which started in the 19 th century and which makes winter softer, summer more rainy, and plants grow better) is, to Mascart, a mistake, since the two phenomena are from different time scales. "This cooling [of our globe] is so very slow that for it to be demonstrated, very precise observations would have to be made, and continued over at least several centuriesperhaps even ten or twenty centuries." (p.113)

"Geographical causes" are to be understood as the analysis of air stream influence (influences on water and temperature transport, influences upon precipitation, erosion and modification of relief), the analysis of ocean currents (influences such as that of the Gulf Stream, or the melting of polar ice providing cold fresh water), and the influence of local circumstances (lakes softening the climate, volcanoes). It is clearly apparent in the works of certain authors studied that the causes do not take the logical form of a tree diagram, but that some are linked by feedback. For instance, the position of the Earth's axis has an obvious influence on the variation of the main air streams, which in turn may influence the axis of rotation by an inertia effect. We begin to see that the complexity of the phenomena makes it impossible to organise them hierarchically as the different terms of a convergent series. Confronted by the profusion of diverse theories, Mascart continues to defend the plurality of ideas. "The most important question is: what is a hypothesis?" he writes, and adds "A hypothesis is nothing but a momentary construction: it has no absolute or final value […] A hypothesis is good if it is fertile, if it stirs up research and criticism." Mascart would thus be closer to Lakatos' ideas than to Popper's. To him, the test of a hypothesis is not refutation by experiment. His position is more subtle: he has confidence in opinion that will prevail with time, and gives bonus points to what is likely to encourage future research. So, on a controversy which impassioned people at the time, he writes, "the very proof that this [the continental drift hypothesis] is right is the harshness of the criticism that it generates. And then, what is against it? Absolute facts, fundamental impossibilities? No: opposing it is only that its does not fit some other hypotheses, accepted today it is true, but what will be their own value tomorrow? […] for if the hypothesis is weak, time will carry it gently away …" Mascart comments that the authors hold to explanatory reasons that are overparticular, and concludes the chapters on the geographical causes with the observation that "All the successive theories are insufficient."

His approach to human causes, today called the anthropogenic effect, is a plea for a better management of old documents and archives. He explains how much the urban phenomenon changes the characteristics of the weather (temperature, rainfall) and that it has to be taken into account to maintain the comparability of measurements. He points out that it is the anomalies, the unusual phenomena of the climate, that should be put in correspondence with the economic facts, and that in this respect the averaging out of data often deletes the most interesting things. The poor quality of the measurements in 1925 had him write, "In these circumstances, given the brief duration of the few reliable observations, it seems a complete illusion to seek arguments for or against the hypothesis of climate variability among meteorological observations properly so called".

After several chapter devoted to detailing the astronomical causes that are his speciality, Mascart devotes a few pages to a discussion of the works relating sunspots to the price of wheat or other economic indicators (Herschel 1801, Arago 1825, Barral 1866, Chambers 1863, Hunter 1876, etc.). Whereas today we are concerned with the part played by the economy on the climate, Mascart considers it interesting that some causes of climate variation may shed light on economic science, which fits in with his permanent concern for scientific fertility.

The general conclusion of his book deals with the necessity of improving observations, the care to be given to measurements and their comparability, conservation, and indexation. The net result of his huge work is scarcely inspiring, and Mascart knows it. Conscious that his work could be considered confused, he recognizes that he hasn't drawn "the general lines at a stroke, broadly brushing in the steps", and that he still lacks a general point of view. He esteems that faced with "successive contradictory works" and "opposed hypotheses", the "spirit of association and collaboration" should be developed further.

What is outstanding in this book is the constant modesty of the author before the complexity of the subject. This attitude leads him to consider as an unyielding reality the fact it is impossible to choose among different theories, and that this situation is not bad in itself. "[These testimonies] are sometimes contradictory -and this is evidence of good faith on the part of the authors", he writes. (p.33) Mascart seems to take it for granted that scientists are disinterested. Yet knowledge in climate matters may be involved in active money-making schemes: this was the case with the plan to immerse the Sahara (at that time part of the French empire) by diverting 172 billion of cubic meters of water from the Mediterranean with a daily evaporation of 28 million cubic meters. Mascart presents the case and discusses both its feasibility and the stakes involved.

As far as climate variability and the question of human influence are concerned, the multiplicity of effects and causes is impressive, and the mechanisms which could explain them still more so, so that many authors make the mistake of proposing theories. "All the authors seem to become intoxicated with the hope of finding simple and single origins [of climate variations]" but "each time a cause is considered, it at once appears insufficient and is completed by mere hypotheses". The scientists seem to be at a loss faced with this complexity, as if they do not want simply to recognise it, or do not know how to examine it as it is, and can do nothing other than propose explanatory theories. This is the most common form of discourse in scientific publications. The theories are fairly rapidly contradicted by certain examples. But Mascart clearly shows that these refutations invalidate the numbers and the order of magnitude of the conclusions but not the theories themselves, which retain a part of possible truth. This is thus effectively not a situation where the Popperian approach is practicable.

Furthermore, on the problem of the anthropogenic effect, we do not know what would have happened if human beings had not existed. This remark is significant. There is no reason why the path we have not followed should be thought of in a more determinist manner than the path we are following. It could certainly have been warmer than it is at present: we can imagine, for instance that the oilfields not worked by human beings would have spread, naturally or after earthquakes, over the surface of the oceans, forming a film which would have obviously changed the evaporation and created consequences that are hard to assess6 . Reading Mascart's work, we realize that the epistemologists Popper, Kuhn, and Lakatos were mainly thinking of the case of physics, which they raised, consciously or not, to the rank of generic situation for scientific knowledge. For in the situation described to us, no direction clearly stands out as the rational path. The line adopted by our author is a sort of "wait and see": he places his trust in the scientific community as an institution that will abandon weak hypotheses over time and organize itself to make use of accessible data7 . This attitude has two failings, however: first its passivity, which is not the best way of tackling risks, and then the somewhat idyllic image of scientific society it relies upon. We shall return to that point later.

The epistemological landscape is here composed of many theories which sometimes oppose and sometimes complement one another. Mathematical formalisms are present through the astronomical and fluid thermodynamics calculations, and co-exist alongside geographical and economic explanations. The result is a vast entanglement of causal dynamics. It is noticeable that the first rational steps consisted in simplifying so as better to understand. For instance, in the 19 th century atmospheric circulations were calculated on an ideal sphere without continents or oceans where only physical laws applied. But since such models gave a poor explanation of the observations, towards the end of the century (Maurice de Tastes 1874), an approach more respectful of the observed particularities was tried, and followed from one complexity to another. The situation as Mascart felt it can be described epistemologically as a miscellany of the points of view of Lakatos, Quine and Feyerabend.

It is indeed above all necessary to promote conditions for more cooperative research programmes that have a higher concern for accuracy and comparability of data (Lakatos). It should also be recognised that chronological series of numbers are always finite, and are likely to receive several interpretations, each of which can be improved so as to adapt to new data (Quine's underdetermination), Finally, it is unwise to reject anything definitively on methodological grounds a priori: it is better to give all ideas a chance (Feyerabend).

III. A Generic Situation

The chronological dimension is the principal axis of the problem of climate change, and one might imagine that this is why the Popperian doctrine fails to apply. Indeed, how can we know today if a theory is falsifiable if its consequences appear only in a century or two? Yet we cannot call all predictive theories unscientific, since prediction is of the essence of science itself, nor can we accept all theories if they contradict one another.

The chronological dimension pushes the sanctions or possible adjustments into the future, but the situation is the same in any field for theories that correctly satisfy today's checks. The deductions furnished by these theories are in advance of investigations requiring higher performance equipment and can only be evaluated after a better understanding of the field concerned and of its relationship with other domains of knowledge. This is evidently not only a question of precision. The refinement of instruments goes hand in hand with that of theories, and as mathematics demonstrates very well, it is ideas that are the factors of advance in the reading of complexity 8 . Alexandre Koyré has shown that at the time of Tycho Brahe the Copernican system was less precise than the Ptolemaic. As for what makes ideas take hold -what gives them this enlightening power -it has something to do with simplification but takes the most varied forms, as Feyerabend has analysed. To go beyond the available data by means of an interpretation which provides more than is strictly to be gleaned by observation: this is what is expected not only of theories in the classical sense, but of all models.

In effect, the epistemological question, "what is true knowledge?", or what makes one set of representations better than another, should be posed today at the level of models. The notion of theory that was utilised so extensively by the philosophers of the 19 th and 20 th centuries is now somewhat dated and very academic in hue. Now that simple algorithms can engender configurations of astounding variety and unexpectedness that defy all theoretical a priori descriptions, we have too many artificial examples stemming from no theory9 not to think that natural things are also capable of astonishing, specific combinations and dynamics that the search for theoretical laws alone cannot reach.

Let us attempt to analyse the contemporary situation starting from the problem of climate change, similar to that analysed by Mascart, but thinking in the context of modelling.

First, it is useful to distinguish between the use of models and the modelling (or model creation) that we are considering here. Models can be used in classical scientific activity: the satellites of Jupiter discovered by Gallileo were a model of the Copernican solar system; the Ehrenfest model helps us understand that the reversibility of the laws of mechanics is compatible with the statistical laws of thermodynamics. Modelling is a larger and more common activity, consisting in representations, usually but not always aided by computer, which are generally intended not so much to offer greater objectivity and universality as to help decision and action.

Modelling presents features which distinguish it from other forms of knowledge or representation.

First of all, it takes place in a precise social setting. It is done by someone (a designer, a research laboratory, a design consultancy) for someone, the client (private company or public organisation), in an economic and geographical space, at a certain date. This obviously does not mean that modelling cannot use tried and tested knowledge that is considered as universal and objective, since such knowledge is available to all. Rather, it can in addition draw heavily upon data known only to certain actors, and its conclusions can make sense only for the action of certain economic agents or of certain institutions.

Next, modelling is a hybrid language, comprising natural language, the language of the sciences, and often, the language of the engineering sciences whose several special terms fashioned by usage do not achieve perfect rigour (decennial high water level, for example). This hybrid language provides semantic input for mathematical and computer formalisms and connects them to the signifieds shared by the actors concerned.

Finally, modelling is usually under-determined relative to what it is trying to represent. The point made by W. V. Quine that theories can be understood in several incompatible ways is much more apparent and general in modelling. This is equally evident in cases where only a finite quantity of observed data is used, as where a finite number of points can give rise either to a polynomial or to a linear combination of sine and cosine, or where other models are compatible with the data, and different families of models are indefinitely perfectible as the number of observations increases. The phenomenon is general: an act of modelling can be validated as much as one would like, but it remains one interpretation among others.

The result of these defining characteristics is that the quality of modelling work cannot be evaluated without taking into account the person or organisation the work was done for, the actor who will make use of it. Let us return to the problem of Jean Mascart, who wanted to move things forward, but who was confronted by complex phenomena (retroactive effects and symbolic representations) explained by theories (today acts of modelling) that were sometimes contradictory, and often imprecise and hypothetical.

IV. Partisan representations are neither unworthy nor dishonest.

Mascart thinks it appropriate to wait. Wait for better observations, and also for the scientific community to work in a more cooperative manner by exchanging information and criticism so that ad hoc hypotheses are ultimately abandoned.

Today it seems that a growing number of academics recognise that the Popperian norm is relevant only in highly idealised cases, and that in practice one is before a mixture bringing together among others the views of Lakatos, Quine, and Feyerabend, where the direction of the rationality is hard to follow. The most scientific attitude would thus be to wait before pronouncing upon this or that truth or such and such a consequence, while trying to contribute to the research in progress. It must be recognised that this "confined" research -"recherche confinée" as Michel Callon10 calls it -far from the stakes, interests and conflicts of actors, is highly agreeable: ideas are favourably received and disciplinary recognition is gratifying. This attitude may appear wise, but it leads to the permanent victory of faits accomplis. As Ulrich Beck pointed out in the 80s, technological innovation emerges essentially through faits accomplis, on the one hand via the market release of products prepared in secret, and on the other via research laboratories which invent new possibilities without controlling who is going to use them 11 . An epistemology which is based on collective knowledge tested within the framework of the world scientific community and which leads scientists to wait for progress before pronouncing on possible consequences, maintains a passive conformity which protects two pillars of contemporary positivism: freedom of initiative for market release of products, and freedom for the researcher to work on whatever he sees fit. It is perfectly understandable how faits accomplis such as the spread of GMOs or the delinquent feeding of cattle leading to BSE are unstoppable in such a system. This has led to reactions such as the precaution principle, wholly understandably in the context.

But we can approach things differently. The narrowly scientific attitude, when the wait is over and science has progressed, leads to a situation that is just as confused as before. Simply, during this time, a certain number of faits accomplis have emerged, accomplished not by angels but by men of every stripe, including drug traffickers, unscrupulous egoists, terrorists, and the agencies of totalitarian regimes. And not less important are those accomplished anonymously, by everyone and no-one, due to the underlying evolution of mores just as the sociologists of modernity have described 12 .

The narrowly scientific attitude, which relies upon scientific progress to eliminate the least relevant representations, appears to be indissolubly connected to the belief that men are good and that groups, nations, organisations, companies, and networks are harmless.

More precisely, this attitude implicitly admits two theses. Firstly, that knowledge creation will escape logical contradictions if we work more in a "spirit of cooperation", with greater synergy between the modelling teams. In other words, the complexity encountered will dissipate if conflicts of interpretation are gradually reduced, for example by coming to agreements on experimental protocols which become incontestable standards; by eliminating all abstract or vague terms from competing theories so that only the strictly descriptive and factual skeleton is retained following the ideas of the neo-positivists of the Vienna circle; or by harmonising points of view by regular colloquia producing consensual syntheses. This irenism, which constantly risks becoming argument by authority, ignores the fact that differing interpretations are the principal means at our disposal for advancing in complexity. They are our guiding lights in the forest. Their mutual consistency is not a necessary condition for comprehension, which can happily accommodate "local maps", as René Thom puts it. In the context of discovery, scientific irenism is an illusion which can scarcely win the approval of researchers.

The second implicit thesis is that the purification necessary for scientific detachment is a process that can unfold over time, reality remaining unchanged. This was in fact the case from antiquity until, say, the 18 th Century: then the influence of science on the world through the intermediary of technology was relatively slow, this slowness in turn most probably favouring in us the emergence and the durability of philosophical categories of division (between the mind and the body, between facts and values, between knowledge and interest). But the appropriation of material power or the economic advantage that an innovation is likely to give is today extremely rapid. This surprises even the research milieu, where knowledge circulates at the rhythm of congresses and the reading of publications. The new link with possible social uses has modified the historic significance of laboratory work upstream of innovation 13 . In addition, the intentions of actors (other scientists, developers, etc.), have today become a permanent preoccupation, and a possible risk.

In front of the dangers currently run by our planet and its human and non human inhabitants, and projects to manipulate human nature itself 14 , a more active attitude on the part of scientists is indispensable. If he holds to the quest for objectivity, in which society tends to maintain him, the scientist will inevitably revert to the narrowly scientific attitude. But the moment the lobbies act to promote their strategies, academics who thus remain confined in their research (pure or applied) are transformed into the mere hired help of a logic of development that is insatiable, blind, and without fear.

We must therefore look to break out of this cycle of pure/applied researchdevelopment -expansion, which is the credo defended best by economic agents on the one hand and by the media on the other. Scientists have certainly a role to play in this change, for they bear the vague responsibility for all the absurdities that this motor can lead to. They have however only limited means at their disposal: they are ordinary citizens, with no financial or institutional power whatsoever. Their only advantage is that they know knowledge production well, and one of their most efficient instruments is their possession of a language able to deepen both preciseness and interpretations: modelling.

The way that suggests itself consists not only in criticising the results or methods of colleagues in order to improve them, but also in criticising the experiments and modelling, taking into consideration the interests these favour, and shouldering the imaginative task of reviewing their possible consequences.

Those who engage in producing counter-models, in partisan cautions in the good sense of the term, who contribute to the biodiversity of ideas and representations by accepting their own cultural roots, are the true defenders of the collective good as this is properly understood. I consider that the most appropriate context for the production of these critical models is likely that of the research centre, and that it would be normal in this respect for the cultural values of the teams to be present and made explicit, the base from which the work of interpretation and anticipation can be conducted. It is in "grey literature" that this self-questioning and commitment take place, for the major reviews will probably remain politically correct for a considerable time to come. 13 Examples are numerous concerning patents and the community of biologists: cf. notably D. Pestre Science, argent et politique, INRA ed. 2003. But the purest science is not free of such surprises: when the International Mathematical Union presented the Gauss prize in September 2006 to Kyosi Ito, 92, the best known mathematician in the banking world today, he confessed his astonishment when the importance of the Ito calculus in trading rooms was explained to him in the 1970s. The work he began after the war was at the heart of a large research movement, stochastic calculus, in which the Soviet and the French schools were particularly distinguished (the members of the latter being relatively left wing or extremely left wing politically). Since 1973 stochastic calculus has become one of the principal capitalist tools on organised markets. 14 See for example the trans-humanist movement, which brings together brilliant academics in the USA with the declared aim of genetically improving man, cf. J. Brockman The New Humanists Barnes & Noble 2003. This is the best real work for mankind: expose as much as can be exposed the risks that the technological adventure poses for it 15 . There is a loss to be accepted here: the era of knowledge gathering is over.

With regard to the greenhouse effect, as in other domains, there is no shortage of Mr. Homais16 to be found, the sort of person who has had some higher education and who will intone to anyone who will listen that "temperature is such a stochastic fractal that one can say everything and anything on the subject". Attempting to make such interlocutors understand that the anthropogenic effect has been proved cannot be the only aim of scientists. The problem is how to uncover the interests behind the actions and the words that hide them, in the very place where the processes occur. Concretely, this poses a large number of questions, most evidently that of the right of collective scientific authorities to interfere in public and private research. But there is also the question of research management, which to me is philosophically fundamental: in France we speak a great deal of making universities financially autonomous, while implicitly maintaining everywhere the universality of the same positivist classical science. Is it not the precise opposite that should be done, namely make the universities autonomous in the values they defend, while maintaining public credits for critical rationalism as such ?

The cartoonist Christophe evokes the astronomer Jean Mascart under the pseudonym Jean Scarmat, and credits him with special rhetorical powers, L'idée fixe du savant Cosinus, librairie Armand Colin 1899.

1925

  Climate map (after E. de Martonne), A. Chevalier and L. Cuenot, Traité de géographie physique, Paris, Armand Colin, 1925

A. Grünbaum, "Is the method of bold conjectures and attempted refutations justifiably the method of science?" Brit. Jour. Philosophical Sci.

(1976) 105-326. 2 P. Feyerabend, Against Method, New Left Books, 1975, Chapter 15 in particular.

Actually, Feyerabend approaches the legitimacy of non-dominant knowledge and the democratization of its production in other works, cf. Farewell to Reason Verso 1987, Wissenschaft als Kunst Suhrkamp 1984.

Cf. M.-A. Tonnelat Histoire du principede relativité, Flammarion 1971 and R. Mossery Léon Brillouin, Belin 1999, Pages 11-30.

W. Spring "Recherches sur les proportions d'acide carbonique contenues dans l'air" Mém. de l'Acad. de Belg. 1885.

In this respect, the imagination of climatologists is also quite productive "what will happen if the slow growth of the Bahama strait corals ends up by closing the passage of the Gulf Stream? […] the European and American civilization will be transformed, rich cities will turn into wastes, deserts will become populated, and so many events will be the consequence of a mere polypary." (Thoulet J. Ann.de Géol. t IV, p269, 1894- 1895).

 7 We should note that such a reference to the scientific community was also made by the contemporary pragmatist philosopher Richard Rorty as a model for a social organization (R. Rorty "Science as solidarity" in The Rhetoric of the human Science, J. S. Nelson, A. Megill and D. N. Mc Closkey eds, Univ. of Wisconsin Press 1987).

See N. Bouleau Philosophies des mathématiques et de la modélisation, L'Harmattan 1999, p.241 et seq.

By theory we mean deductively structured concepts linked to the real by correspondence rules or paradigmatic analogies.

Cf. M. Callon, P. lascoumes, Y. Barthes, Agir dans un monde incertain, essai sur la démocratie technique, Seuil 2001.

U. Beck, Risk Society, toward a new modernity (1986), Sage 1992.

For instance, A. Giddens, U. Beck, S. Lash, Reflexive Modernization, Politics, Tradition, and Aesthetics in the modern social order, Polity Press 1994.

Cf. A. Lebeau, L'engrenage de la technique, Gallimard 2005.

Cf. J.-Ch. Hourcade, V. Journé, "Monsieur Homais, les guides de montagne et le maître nageur, variations sur la négation des risques climatiques", Critique internationale, n°18, (2003) 65-79.


Ines De Courchelle
email: ines.de-courchelle@irit.fr

Thierry Monteil
email: monteil@laas.fr

Yann Labit
email: ylabit@laas.fr

Tom Guerout
email: tguerout@laas.fr

A data model for supplying a Data Center with several energy sources

This article presents a data modeling to handle the power flow in a smartgrid. The power flow is studied through RenewSim. It is a simulator dedicated to supply a Data Center which is simultaneously fed by renewable and non-renewable energy sources. Nowadays, the use and optimization of the energy consumption of Data Centers are very broad research areas. The development of energy called "green" and how to use them smartly, becomes an inevitable problematic because of the current ecological considerations. RenewSim is based on the use of several energy sources (electricity supplier, solar panels and batteries). This simulator has a center decision controller. It aims at make clever decisions for the use of heterogeneous sources. The contribution of this article is a data modeling which allows the power redirection by labeled and prioritized the different sources of energy. The priorities and the labels are presented and the results obtained are exposed.

I. INTRODUCTION

Areas such as home automation, smart buildings and cities want to operate largely with renewable energies. Smart grids are part of the implementation of such smart buildings throw Micro-grid. It aims at saving energy and reducing the CO2 foot print. In this article, the use of a smart-grid allows to feed in a green way a Data Center. Nowadays, the running cost of a Data Center is more and more expensive. The utilisation rate and users needs increase day after day (streaming, social network, and in clouds with the on-demand services). It becomes a real problematic because the "brown" energies (non-renewable) do not have an endless capacity. For example, the "Université de Paul Sabatier III" has consumed 148300MWh PE 1 in one year between 2008 and 2009 2 . The European Commision has estimated the Data Center consumption in the european area at 56 billion kW in 2008 and has evaluated 104 billion for 2020 3 . The use of different energy sources introduces some Fig. 1. A SmartGrid Energy sources management example new problematics related to the energy storage. Many works exist on the battery for load/unload models and their usage in a Data Center [1] [2]. This storage form may be considered as a safety for shortage case. Also, the use of solar panels is complex and unpredictable because it depends on the weather variations.

The Fig. 1 presents an architecture example of a Data Center supplied by heterogeneous energy sources and storage capacities. The control component aims at being able to take clever decision for the use of heterogeneous energy sources. Those observations are the article base dedicated to a data modeling for RenewSim.

Ours challenges are to connect several energy sources and destinations together, how to redirect the energy produce by the solar panel in the smartgrid (Battery, Data Center, or supplier), and to take a decision to satisfy the energy cost at the weakest rate. The problematic is how to characterize the heterogeneous electricity sources to make a decision to reduce electricity bills ? In this article we will focus on a modeling for the heterogeneous electricity component. The paper is organized as follows: the Section 2 is the related work on the current simulators and the green Data Center supplied by batteries or solar panels. The section 3 describes smartgrid, green energies and the simulator RenewSim. The section 4 presents our modeling for RenewSim. The section 5 is the methodology used. The section 6 evaluates the model. Then, the section 7 is the balance sheet of this article and the further works.

II. RELATED WORK

Our approach with RenewSim is to feed a Data Center with renewable energy in a smart way, thanks to a control module. This one aims at optimizing the electric flow and the scheduler of a center of calculation in an electrical network powered by heterogeneous sources. Simgrid [START_REF] Casanova | Simgrid: A generic framework for large-scale distributed experiments[END_REF] is a simulator for distributed applications heterogeneous in distributed environments. It allows the execution of algorithms for scheduling a Data Center [START_REF] Liu | Energy-aware task clustering scheduling algorithm for heterogeneous clusters[END_REF], the simulation of storage in a Data Center [START_REF] Lebre | Adding storage simulation capacities to the simgrid toolkit: Concepts, models, and api[END_REF] and the use of the DVFS (Dynamic Voltage and Frequency Scaling) in a Data Center [START_REF] Charr | Energy consumption reduction with dvfs for message passing iterative applications on heterogeneous architectures[END_REF]. This simulator is in the improvement of the actual application in the field of grids. CloudSim [START_REF] Calheiros | Cloudsim: a toolkit for modeling and simulation of cloud computing environments and evaluation of resource provisioning algorithms[END_REF], historically based on GridSim [START_REF] Buyya | Gridsim: A toolkit for the modeling and simulation of distributed resource management and scheduling for grid computing[END_REF], is a simulator Cloud Computing of type infrastructure as a Service (IaaS). The architecture of the Data Center is finely modeled and can be easily modified by the user. It integrates the virtualization (migration), the addition of investment policies as well as the energy costs and operation induced. Another version [START_REF] Guerout | Energy-aware simulation with dvfs[END_REF] also integrates the DVFS. This simulator is purely dedicated to the internal management of the Data Center. GreenHadoop [START_REF] Goiri | Greenhadoop: leveraging green energy in data-processing frameworks[END_REF] is an evolution of GreenSolt [START_REF] Goiri | Greenslot: scheduling energy consumption in green datacenters[END_REF]. It allows the simulation of wind turbines, solar panels. GreenHadoop incorporates the principle of dynamic replication of data to ensure the availability of the latter. A module of the prediction of energy production is present with which the charging of the batteries is managed and incorporates five levels of priorities of jobs (Very High, High, Normal, Low, and very low). These two simulators [START_REF] Goiri | Greenslot: scheduling energy consumption in green datacenters[END_REF], [START_REF] Goiri | Greenhadoop: leveraging green energy in data-processing frameworks[END_REF] allow to optimize the consumption of the solar energy but do not propose to optimize simultaneously the energy consumption, and the financial cost. The Simulator RenewSim is a simulator dedicated to the use of solar panels. It uses states to qualify the servers in a Data Center, as in GreenHadoop. It is a simulator dedicated to the co-simulation. It will therefore be able to integrate the version [START_REF] Guerout | Energy-aware simulation with dvfs[END_REF] of CloudSim.

III. A MICRO-GRID TO REDUCE THE ENERGY COST

In this section, we will introduce RenewSim a simulator to supply a Data Center. It is a co-simulation tool based on a micro-grid architecture. RenewSim is a simulator for cosimulation based on the use of several energies (renewable or not). It uses batteries, solar panels and electricity suppliers. It aims at managing the power flow in a clever way between the several sources until the Data Center.

The Fig. 2 presents an example of a RenewSim architecture. It is possible to plug several sources and destinations. The main source is the solar panels production and the main destination is the Data Center. Each element plugged to the bus is connected to a converter. Those are modules for the conversion of DC/DC or DC/AC. This micro-grid is supervised Fig. 2. RenewSim Architecture by a control module. His role is to find the best possible configuration for both management of the energy flow (feeding the Data Center) and scheduling tasks in the Data Center. This involves to manage the power, the power sold from the solar panels to the suppliers, the power bought or sold to the supplier based on its cost and also the load/unload of the battery. The allocation of tasks depends on their characteristics (CPU and memory capacity, priority, preemptive or not) and the total amount of available energy. The Data Center has the ability to turn on and off the servers and allows tasks migration. RenewSim aims at reducing the energy cost by studying the power flow and the Data Center demand in a decision block.

IV. A MODEL TO IMPROVE RENEWSIM THE CONTROL

MODULE

To improve the RenewSim simulator, two possibilities are conceivable: manage the different degrees of freedom or specify the intrinsic behaviour of each block (batteries, solar panels, Data Center, Electric suppliers). Here, the model is based on the first possibility.

A. Freedom degrees

The model aims at being an input of RenewSim to decide which solution costs more, costs less or for example uses more renewable energy. The power asked by the Data Center does not let us a lot freedom degrees. Solar production does not depend on the system. As the data center load, it is a system input and does not have freedom degrees. Production of solar panels depends on the weather and time of day. The freedom degrees will be thanks to the batteries and the electricity suppliers. The supplier price depends of time. During the night the electricity is cheaper than during the day. The degree of freedom is the power of solar panel destination which is stored on the batteries or sold to the power supplier or supplied to the Data Center. Currently, RenewSim promotes renewable energy. What would happen if he favored monetary gain to renewable energy? For example, by promoting the electric suppliers instead of using renewable energy to feed the data center, we get two different functions: night and day.

1) Night: purchase energy required to operate the data center, fully charge the batteries from the electrical suppliers. 2) Day: Sell to the suppliers the solar overproduction, use the battery in case of underproduction.

For exploiting all the freedom degree, we need to characterize each element: the batteries, the solar panels, the Data Center and the suppliers. The Data Center and the solar panels can only be respectively destination and source. The other could be sources, destination, backup or overflow. The interest of the system is to optimize energy flow. For example, if the system takes a very small amount of energy from the battery to the Data center, the power flow is not efficient and involves too much losses in terms of energy. However, if the system takes a lot of amount of energy from the battery, for example it will be more interesting at time t to take half from the battery and the other half from the supplier, because it could be smarter to keep a amount of energy in the battery.

B. Element Characterization

To take a faster decision based on the freedom degrees, each element of the micro-grid (Fig. 2) except the decision module, is described by a model. An element can be a source of energy, a destination, a back-up and an overflow. The main source of RenewSim is solar panels. The destination is the main Data Center. The source and destination are inputs of RenewSim, they leave no degrees of freedom. One is the production Solar (Source) that depends on the weather and the day. The other is the Data Center Demand (Destination). The demand depends on the needs of users and servers at a time t. The batteries and the electric suppliers can be sources, destinations, overflow or backup. The backup allows to get power when the sources cannot answer the Data Center demand (under-production). The overflow allows to store energy when the sources provide more energy required to operate the Data Center (over-production). The table I below presents the modeling. Here, in Table I, the battery and solar panels supply the Data Center as sources. In case of under-production, the electric supplier (Backup = 1) feeds the Data Center. In case of over-production, we stock the excess energy in the battery (Overflow = 1). If the battery is full we sell energy to the electricity supplier (Overflow = 2). The modulation in the tables II and III favor the money than the renewable energy. During the day, the model is as follow (II): RenewSim use the solar panels to supply the Data Center. If there is an over-production of energy (Data Center demand ≤ solar panels production) the overfull is sold to the suppliers. If there is an under-production (Data Center demand ≥ solar panels production), the lack is taken from the backup 1. Here, it is the battery (unload). If the backup one is empty, the lack is taken from the backup 2 (purchase). Here, it is the supplier.

V. METHODOLOGY

This section presents the environment for the different results (Section VI).

A. Algorithm

The destinations and the sources elements follow the equation 1, with S t is the sum of all the sources available at time t, and D t is the sum of all the destinations available at time t. To guarantee the equation 1 in case of over and under production, the equation 2 adds B t , the sum of all the backup resources available at time t, and O t the sum of all the overfull available. In the Algorithm 1, the sum of all the sources is greater than the destinations demand. This is an over-production. So, for each overfull element, in order of priority, the algorithm stores the over-production. There are three cases: the overfull element is over, the overfull can take all the energies or the overfull can store a part of the over-production. In the Algorithm 2, the sum of all the sources is lower than the destinations demand. This is an under-production. So, for each backup element, in order of priority, the algorithm takes the energy asked by the destination. There are three cases: the backup element is empty, the backup can offer all the energies or the backup can offer a part of the demand. For the experiments, the micro-grid used is made of one supplier, one battery, a Data Center and a solar panels park. In the (Fig. 3), CM1 to CM4 converters have the respective losses of 15%, 17%, 13% and 14%. The production of solar panels is derived from actual measurements. They come from ADREAM building 4 . For 48-hours, ADREAM solar produc-tion provides 1648 measures. The Fig. 4 shows two consecutive days of production with some production decline peaks. The decline peaks can be explained by cloud or bad weather at a time t. In Fig. 5, the lower bell is the solar production panel after the converter (15% loss). Another input of the system is the Data Center demand (dashed curve). This demand is not known at each time t by the simulator. In the following experiments (section V), the energy costs of the supplier are the following: 0.2366e per kWh from 8pm to 8am (night) and 0.6173e per kWh from 8am to 9pm (day). A(t) = P ee (t)/(1 -l)

S t = D t (1)
S t + B t = D t + O t (2

B. Study area

(3)

V (t) = V (t -1) + P ee (t) × (1 -l) (4)
The power purchase and sale to the supplier is respectively modeled by the equations 3 and 4 with the power purchased and the power sold with A the power purchased, V the power sold, P ee the power for the supplier before the converter module and l converter loss (CM4). The battery has a storage capacity of 10000 joules and a maximum power of 200 kW.

E b (t) = E b (t -1) + P bb × ∆ (5) E b (t) = E b (t -1) -P b × ∆ (6)
The battery load and unload is respectively modeled by the equations 5 and 6, with t the time, ∆ conversion time interval from kW to joules, t the time, P bb the power for the battery after the converter module (CM2), and E b the energy load into the battery (joules).

VI. RESULTS

In this section, we experiment two different states of our model. The initial state of the solar panel and the power load required by the Data Center is given by the Fig. 5 A. Experimentation 1: A renewable model For this experiment, we used the following modeling: In this case, RenewSim will purchase energy to the supplier only when the sum of all the destinations are upper than the sources (Equation 7) and also if the battery (backup element) is empty or cannot offer all the power (Algorithm 2). The battery load is represented by Fig. 6. The battery can supply the Data Center when the solar production slows down. This is the same noise around t= 15 to 17 hours we saw inFig.4). In Fig. 7, this is the sum of all the power purchase to the supplier.

S t < D t (7)

B. Experimentation 2: An economic model

For this experiment, we used the following modelings seen previously: the Table II and III (section IV. B). In this case, RenewSim will always use the battery as a source during the day. It will store energy on it only during the night. RenewSim will never use the battery during the night (Fig. 8), because the electricity is cheaper. It will buy the necessary power to supplier when it is cheaper by night (Fig. 9). All the over-full of energy is sold to the supplier to earn money. During the second day, RenewSim never bought energy. The battery and the production is enough to feed the Data Center. In the Fig. 8, the battery aims at supply the Data Center when the solar panel production is lower (15 hours).

VII. CONCLUSION

The goal of this paper was to present a model to represent each element of a micro-grid and improve the RenewSim control module. The model allows a modular architecture which aims at make clever decision for the use of heterogeneous sources. The contribution of this article is a data model which allows the power redirection by labeled and prioritized the different sources of energy. In the future, this model will allow us to include predictions model and learning machine into the control module of RenewSim. Through two distinct scenarios, the control module can take decision in a purpose: use more renewable energy, or purchased as a low price. This model generalize all the process where we can move the power load, for example, water heater in apartment. There is a sociological study to do in the future to see what is the acceptability of the load shift and how to increase it. The more we come to accept people to increase it, the more we will have savings. Other scenario are possible as use more brown energy in order to keep the battery load for next hour. The RenewSim expansion prospects consist include: multiobjective optimization studies via heuristics, algorithms of learning and predictions (production / use) as well as most successful models. Finally, the purpose of RenewSim is also to be a co-simulation tool allowing communication simulators dedicated to each component.

Fig. 3 .

 3 Fig. 3. RenewSim Architecture for the experimentation

Fig. 4 .

 4 Fig. 4. Solar panel production for the experiments

Fig. 6 .

 6 Fig. 6. Exp 1: Battery load at time t during two days

Fig. 7 .

 7 Fig. 7. Exp 1: Sum of all the power purchase to supplier during two days

Fig. 8 .

 8 Fig. 8. Exp 2: Battery load at time t during two days

TABLE I A

 I MODELING EXAMPLE: HOW TO USE LESS BROWN ENERGY

		Source	Destination	Backup	Overflow
	Solar Panels	1	0	0	0
	Data Center	0	1	0	0
	Battery	1	0	1	1
	Supplier	0	0	2	2

  )

	Algorithm 1 Over-Production: P > 0
	Require: P > 0
	tab overf ull ⇐ sort table overf ull(tab overf ull)
	for i=0;i;len(tab_overfull);i++ do
	if tab overfull[i] over==true then
	continue;
	end if
	if tab overfull[i] enough==true then
	update tab overf ull(overf ull)
	end if
	if tab overfull[i] not enough==true then
	update tab overf ull
	(tab overf ull[i](storage available))
	end if
	end for

  Algorithm 2 Under-Production: P < 0 Require: P < 0 tab backup ⇐ sort table backup(tab backup)

	for i=0;i;len(tab_backup);i++ do
	if tab backup[i] empty==true then
	continue;
	end if
	if tab backup[i] enough==true then
	update tab backup(demand)
	end if
	if tab backup[i] not enough==true then
	update tab backup
	(tab overf ull[i](quantity available))
	end if
	end for

TABLE IV A

 IV MODELIZATION EXAMPLE: HOW TO USE LESS BROWN ENERGY

		Source	Destination	Backup	Overflow
	Solar Panels	1	0	0	0
	Data Center	0	1	0	0
	Battery	0	0	1	1
	Supplier	0	0	2	2

PE, Primary Energy is used in reference value to represent the energy footprint of a building. The energy vector weighting is set by countries for each sector. Ep = f inalenergies × energiesvector

http://developpementdurable.cpu.fr/img/documents/universites/Fiche de synthese Universite Paul Sabatier Toulouse III.pdf

"Code of Conduct on Data Center Energy Efficiency" by the European Commission in October 2008

https://www.laas.fr/public/en/adream

ACKNOWLEDGMENT

The authors would like to thank the Université Paul Sabatier for the funding of a PhD grant within the framework of neOCampus project.