Farrell Brumley
email: brumley@math.univ-montp2.fr

SECOND ORDER AVERAGE ESTIMATES ON LOCAL DATA OF CUSP FORMS 1

We specify sufficient conditions for the square modulus of the local parameters of a family of GL n cusp forms to be bounded on average. These conditions are global in nature and are satisfied for n ≤ 4. As an application, we show that Rankin-Selberg L-functions on GL n 1 × GL n 2 , for n i ≤ 4, satisfy the standard convexity bound.

Introduction

Let F be a number field and A its ring of adeles. Let π = ⊗ v π v be a cuspidal automorphic representation of GL n (A). At each finite place v = Ô of F there is associated with π Ô a semisimple conjugacy class A π (Ô) in GL n (C), the matrix of local (Langlands) parameters A π (Ô) = diag(α π (Ô, 1), . . . , α π (Ô, n)). The Ramanujan conjecture states that |α π (Ô, i)| ≤ 1 for all 1 ≤ i ≤ n, with strict equality when π Ô is unramified.

One may use the information supplied by the Ramanujan Conjecture to derive important analytic results for L-functions. In doing so one trades strong pointwise information for results that often have more to do with the average behavior of the local parameters, with α π (Ô, i) ranging over primes Ô and possibly over π is some family. One such consequence of the Ramanujan Conjecture is that any (appropriately normalized) L-function associated to π satisfies the optimal estimate O ǫ (C(π) ǫ ) on Re(s) > 1. The quantity C(π) is the analytic conductor of π (see Section 3 for the definition). When this property holds, we say that the L-function satisfies the standard convexity bound.

One technique used to demonstrate optimal bounds on sums of positive coefficients was introduced by Iwaniec [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF] for cusp forms π on GL 2 . Iwaniec uses a linearization process to show that if the coefficients λ(Ò, π) of L(s, π) don't begin to show O(1) behavior by the time NÒ is of size O ǫ (C(π) ǫ ) then this late excess will so propogate through the remaining coefficients via their multiplicative relations as to contradict the polynomial control granted by the Rankin-Selberg theory. Molteni [START_REF] Molteni | Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product[END_REF], working out the difficult combinatorics involved in implementing Iwaniec's idea in full generality, was then able to show that for π any cusp form on GL n the principal L-function L(s, π) satisfies the standard convexity bound.

To apply the same reasoning to the Rankin-Selberg L-function L(s, π × π) requires a more delicate analysis. It may come as a surprise to some that despite the recent breakthroughs in certain cases of subconvexity, it is still not known in complete generality and under no assumptions that L(s, π × π) satisfies the standard convexity bound. Molteni [START_REF] Molteni | Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product[END_REF] went some way toward this goal by showing that for π any cusp form on GL n as long as |α π (Ô, i)| ≪ NÔ 1/4 (1)

for all but finitely many primes Ô and 1 ≤ i ≤ n then L(s, π × π) satisfies the standard convexity bound (see his Hypothesis (R ′ )). At present, however, bounds of this quality are known only for cusp forms on GL 2 (A) where we have |α π (Ô, i)| ≪ NÔ 1/9 [START_REF] Kim | Cuspidality of symmetric powers with applications[END_REF].

In this paper we remove hypothesis (1) in certain cases, proving that L(s, π 1 × π 2 ) satisfies the standard convexity bound for pairs (π 1 , π 2 ) on GL n 1 × GL n 2 for n i ≤ 4. This improvement upon the range given by Molteni's work is due to a greater emphasis on global infomation and benefits from some recent advances in functoriality. Throughout the paper we take pains to describe what happens on higher rank in an effort to compare the strengths of our method with those of other approaches.

Main Theorem

We consider a Dirichlet series which acts as a majorizer of L(s, π × π). For π any cuspidal representation of GL n (A), define

L(s, π, |max| 2 ) := Ò λ(Ò, π, |max| 2 )NÒ -s := Ô r≥0 max i |α π (Ô, i)| 2r NÔ -rs .
We shall specify sufficient conditions under which this L(s, π,

|max| 2 ) is O ǫ (C(π) ǫ ) on Re(s) > 1.
We call this estimate the convexity bound at s = 1, detailing the specific point in this case since L(s, π, |max| 2 ), lacking a functional equation, does not allow for an interpolation to points to the left of 1. This function L(s, π, |max| 2 ) has the advantage over L(s, π × π) of being completely multiplicative in its coefficients. As we shall see in Proposition 10, Dirichlet series whose coefficients are positive and completely multiplicative can be subjected to Iwaniec's bootstrapping method with no additional assumption on the size of their coefficients.

The disadvantage of working with L(s, π, |max| 2 ) is that it is not an L-function coming from an automorphic form, making its analytic properties hard to unearth. To remedy this problem, we majorize λ(Ô, π, |max| 2 ) for Ô at which π Ô is unramified by a sum of the absolute values of certain more naturally arising coefficients (see Proposition 5). In doing so, we make use of the fact that for unramified Ô the unitarity of π Ô restricts the number of roots that can possibly violate the Ramanujan conjecture. The matrix of Satake parameters A π (Ô) in this case is forced to lie in the same semi-simple conjugacy class as A π (Ô) -1 , meaning that only ⌊n/2⌋ of the n roots can have size greater than 1. The function ⌊ • ⌋ is the "floor" function outputting the largest integer less than or equal to the imput value.

The following is our main theorem. For the definition of a strong isobaric lift consult Section 2. THEOREM 1. Let π a cuspidal representation of GL n (A). For an integer j ≥ 2 denote by ∧ j the exterior j-power representation of GL n (C). Assume that for every 2 ≤ j ≤ ⌊n/2⌋ there exists a strong ∧ j -isobaric lift. Then L(s, π, |max| 2 ) satifies the convexity bound at s = 1.

Applications

When n = 2 or 3, the conditions of Theorem 1 are empty, so the conclusion automatically holds. When n = 4 or 5 the sole condition in is that there exists a strong isobaric ∧ 2 lift. For n = 4, this condition was proven by Kim [9, Proposition 5.3.1], and so we can state unconditionally the following practical result. For cusp forms on GL 3 and GL 4 , this result is new. COROLLARY 2. Let π i be cuspidal representations of GL n i (A) where n i ≤ 4 for i = 1, 2. Then L(s, π 1 × π 2 ), as well as L(s, π i , ∧ 2 ) and L(s, π i , sym 2 ) for i = 1, 2, satisfy the standard convexity bound.

The proof of Corollary 2, essentially an application of the Cauchy-Schwartz inequality, is provided in sub-section 3.2.

We now given several examples where Theorem 1 can be used to replace the Ramanujan Conjecture or the hypothetical bounds [START_REF] Bushnell | An upper bound on conductors for pairs[END_REF]. The first is a large sieve inequality for long sums of Fourier coefficients of cusp forms on GL n . For cusp forms π on GL 2 /Q and their images sym 2 π under the Gelbart-Jacquet lift this large sieve inequality is a theorem, by Duke and Kowalski [START_REF] Duke | A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, With an appendix by Dinakar Ramakrishnan[END_REF] in the level aspect when π is holomorphic, by Luo [START_REF] Luo | Values of symmetric square L-functions at 1[END_REF] in the eigenvalue aspect when π is a Maass form.

For a parameter Q ≥ 1 let S n (≤ Q) be the set of all cusp forms on GL n /Q with analytic conductor bounded by Q. Under a remaining assumption giving polynomial growth on S n (≤ Q), the results of [START_REF] Duke | A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, With an appendix by Dinakar Ramakrishnan[END_REF] and [START_REF] Luo | Values of symmetric square L-functions at 1[END_REF] can be extended to n ≤ 4 using Theorem 1. COROLLARY 3. Let n = 3 or 4. Assume that there exists a number

d > 0 such that |S n (≤ Q)| = O(Q d ). Let α = 1 -(n 2 + 1) -1 and for π 1 , π 2 ∈ S n (≤ Q) define B = B(n) > 0 to be the exponent appearing in the convexity bound of L(s, π 1 × π 2 ) at s = α, so that L(α, π 1 × π 2 ) ≪ ǫ Q B+ǫ . Then for any ǫ > 0 the inequality π∈S(≤Q) n≤N a n λ(n, π) 2 ≪ ǫ (NQ) ǫ (N + Q B+d N α ) n≤N |a n | 2 holds for all complex numbers (a n ) 1≤n≤N .
The proof of Corollary 3 is by a well-known duality argument. We shall only sketch the details which pertain to the role of Theorem 1. The two terms on the right-hand side of the large sieve inequality come from a majorization of an integral involving the Rankin-Selberg L-function at the point s = 1 and along the line s = α+ǫ for small ǫ > 0. At s = 1 one uses Theorem 1 to show that the residue of L(s, π × π) is ≪ ǫ Q ǫ . On the line s = α + ǫ Theorem 1 is used to establish the convergence and negligibility of the correction factor that relates the bilinear Rankin-Selberg L-function to the true convolution. This correction factor, labeled H(s, π 1 × π 2 ) in [START_REF] Duke | A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, With an appendix by Dinakar Ramakrishnan[END_REF], is the product p H p (s, π 1 × π 2 ) of polynomials in p -s whose coefficients are symmetric polynomials in the local roots of π 1 and π 2 and whose linear term is zero. The product converges on Re(s) > α and satisfies H(s, π 1 ×π 2 ) ≪ ǫ Q ǫ in this region by the Luo-Rudnick-Sarnak bounds (see display [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF]) and Theorem 1.

As a final application we state a zero-density theorem for principal L-functions of cusp forms on GL n for n = 3 or 4. Kowalski and Michel [START_REF] Kowalski | Zeros of families of automorphic L-functions close to 1[END_REF] have proven a quite powerful zerodensity statement near the line 1 for L-functions of general GL n cusp forms over Q. They assume the bounds (1) on the local parameters, in this case to prove a mean-value theorem with pseudo-characters à la Selberg. As in Corollary 3 we may remove this assumption for n = 3 and 4 using our Theorem 1. See the introduction to [START_REF] Kowalski | Zeros of families of automorphic L-functions close to 1[END_REF] for applications of this type of zero-density statement to moments of L(1, π) for certain families of π.

Let

M(α, T ) = {z ∈ C| Re(z) ≥ α and |Im(z)| ≤ T } for α ∈ R, T ≥ 0. For any cuspidal automorphic representation π of GL n /Q, we let N(π; α, T ) = |{ρ ∈ M(α, T )|L(π, ρ) = 0}| (zeros counted with multiplicity). COROLLARY 4. Let n = 3 or 4. Let S n (≤ Q), Q ≥ 1, be as above. Assume that there exists a number d > 0 such that |S n (≤ Q)| = O(Q d ). Let α > 1 -(n 2 + 1) -1 and T ≥ 2.
Then there exist constants c, B > 0, depending only on n and d, such that

π∈Sn(≤Q) N(π; α, T ) ≪ T B Q c(1-α)
for all Q ≥ 1. The implied constant depends only on the choice of c.

Strength of method

We have tried in this paper to give the reader an idea of the strengths of our method relative to other approaches. It is for this reason that, despite the extremity of its hypotheses, Theorem 1 was stated for general n.

One weakness of the method we outline is that there is much information loss in passing from the conclusion of Theorem 1 to Corollary 2. This lost information is hard to quantify, and it is not at all clear that additional applications could be gleaned from the stronger result. To see the information loss, imagine trying to reverse the logic to deduce the convexity bound at s = 1 of L(s, π, |max| 2 ) from that of L(s, π, sym 2 ) and L(s, π, ∧ 2 ) alone. Note that convexity for these latter two implies the same for their product L(s, π×π), and indeed for any Rankin-Selberg pair L(s, π 1 × π 2 ) by the Cauchy-Schwartz inequality. But as n gets large (n greater that 5 will work), our Proposition 5 shows that many more representations are needed to control the modulus-squared of the roots.

Theorem 1 thus seems best suited for n ≤ 5 where the degree of the representations whose automorphy we assume is no larger than those to whose L-functions we apply the result. But even for n ≤ 5, where the standard and the exterior square representation suffice to control the square-modulus of the roots, there is information loss simply by the reduction to a completely multiplicative Dirichlet series. When applying Theorem 1 to L(s, π × π) for example, we are using the quantity (λ(Ô, π × π) + |λ(Ô, π, ∧ 2 )|) r to control λ(Ô r , π × π) for every r ≥ 0. When r = 1 the presence of the exterior square is clearly unnecessary. By treating all coefficients with essentially the same majorization, we neutralize the otherwise helpful effect of interior cancellation among the roots that might lead to some coefficients being small or zero.

Let us say more about other methods for proving the standard convexity bound for Rankin-Selberg L-functions. The most direct way to force the convexity bound for L(s, π× π) is by assuming the existence of both an exterior and symmetric square lift. For it is clear from the identity L(s, π × π) = L(s, π, sym 2 )L(s, π, ∧ 2 ) that the convexity bound for L(s, π × π) follows from that of both L(s, π, sym 2 ) and L(s, π, ∧ 2 ); by the results in [START_REF] Molteni | Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product[END_REF] again, this would follow from the (isobaric) automorphy of both sym 2 π and ∧ 2 π. Hence L(s, π × π), where π is a cusp form on GL 2 (A), satisfies the standard convexity bound for yet another reason: the Gelbart-Jacquet lift [START_REF] Gelbart | A relation between automorphic representations of GL(2) and GL(3)[END_REF].

In Section 2 we are able to shed some light on the relation between the hypothetical bounds (1) on the local roots and the direct assumption of functoriality of both ∧ 2 and sym 2 . In Corollary 6 we show that for n ≤ 5 the assumption of both functorial lifts is stronger than the condition (1). For n > 5 no such implication can be made by our method. In fact, working locally one unramified prime at a time, and using only unitarity as input, we show that many more functorial lifts are needed to break the 1/4 exponent that Molteni requires. Of course we lose lots of global information in setting up this implication, but it is interesting nonetheless to consider whether, for higher rank general linear groups, the existence of both functorial lifts, already such an extreme hypothesis, might actually be weaker than the bounds in [START_REF] Bushnell | An upper bound on conductors for pairs[END_REF].

Acknowledgements. This paper represents a chapter in my doctoral dissertation. I would like to thank my thesis advisor Peter Sarnak for suggesting this problem to me and Ramin Takloo-Bighash, Philippe Michel, and Akshay Venkatesh for their encouragement.

Consequences of unitarity

In the following proposition, we have chosen the exterior power lifts for simplicity. The proof uses only properties on the size of the eigenvalues α i . Alternative sets of representations of GL n (C) may be chosen, though one would then have to take into consideration the arguments of the α i . PROPOSITION 5. Let n ≥ 1 and m = ⌊n/2⌋. There exists a constant c n > 0 depending only on n such that for any matrix A = diag(α 1 , . . . , α n ) ∈ GL n (C) with A -1 and A lying in the same semi-simple conjugacy class

max i |α i | 2 ≤ c n 1 + m j=2 |Trace(∧ j A)| 2/j .
(

) 2
Proof:

The assumption on A means that there is some permutation σ of the indices such that α i α σ(i) = 1 for all i. The elements may be ordered by their size, say

|α 1 | ≥ • • • ≥ |α m | ≥ 1 ≥ • • • ≥ |α n |. (3)
We note that

Trace(∧ j A) = 1≤i 1 <•••<i j ≤n α i 1 • • • α i j . (4)
For the moment let R 1 , . . . , R m+1 be any array of positive real numbers satisfying

R 1 = 1 and 0 < R i < 1 for 2 ≤ i ≤ m + 1. It is clear that either (i) there exists some j ∈ {1, . . . m} such that |α i | ≥ R i |α 1 | for all 1 ≤ i ≤ j and |α j+1 | ≤ R j+1 |α 1 |; or else (ii) |α m+1 | ≥ R m+1 |α 1 |. In case (ii), we have |α 1 | 2 ≤ R -2 m+1 |α m | 2 ≤ R -2 m+1 by (3)
. Now let j be as in case (i). The leading term in (4) is α 1 • • • α j which has size

|α 1 • • • α j | ≥ j i=1 R i |α 1 | j .
From (3) all other terms are bounded in absolute value by

|α 1 • • • α j-1 α j+1 | ≤ R j+1 |α 1 | j .

Thus we have

Trace(∧

j A) = α 1 • • • α j + O((r j -1)R j+1 |α 1 | j ), (5)
where the implied constant is bounded by 1 and r j be the number of terms present in Trace(∧ j A), that is

r j = #{(i 1 , • • • , i j ) | 1 ≤ i 1 < • • • < i j ≤ n}.
The numbers R i should now be chosen to make the main term in (5) dominate the error term. For

1 ≤ j ≤ m + 1, set R j = j-1 i=1 R i r -1 j-1 .
Here we have put r

0 = 1. Thus R 1 = 1, R 2 = r -1 1 , R 3 = r -1 1 r -1 2
, and so on. By [START_REF] Henniart | Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique[END_REF] this implies

|α 1 | j ≤ j i=1 R i -1 1 - r j -1 r j -1 |Trace(∧ j A)| = R -1 j+1 |Trace(∧ j A)|.
Since R 1 > R 2 > . . . > R m+1 , we may encompass all cases by taking c n = R -2 m+1 . This completes the proof.

Let π v be any irreducible admissible representation of GL n (F v ). Let φ v : W ′ Fv → GL n (C) the parametrization of π v given by the local Langlands correspondence ([4], [START_REF] Henniart | Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique[END_REF], [START_REF] Langlands | On the classification of irreducible representations of real algebraic groups[END_REF]), where W ′ Fv is the Weil-Deligne group. The composition φ v •ρ is then the parametrization of an irreducible admissible representation ρ(π v ) of GL N (F v ). We may form the tensor product ρ(π) = ⊗ v ρ(π v ) over all places v of F . The result is an irreducible admissible representation of GL N (A). An automorphic representation

Π of GL N (A) is called isobaric if Π = Ind σ 1 ⊗• • •⊗σ k , for cuspidal representations σ i of GL n i (A), where n 1 +• • •+n k = N.
Langlands functoriality predicts that ρ(π) is automorphic.

We shall call an automorphic (respectively, isobaric) representation

Π ρ = ⊗ v Π ρ v of GL N (A) a weak ρ-automorphic (respectively, -isobaric) lift of a cuspidal representation π = ⊗ v π v of GL n (A)
if there exists a finite set S π of places, including the finite places v at which π v is ramified, such that Π ρ v ≃ ρ(π v ) for all v / ∈ S π . The lift is said to be strong if S π can be taken independent of π.

Using our Proposition 5 and additional assumptions on the existence of certain weak isobaric lifts, the Luo-Rudnick-Sarnak [START_REF] Luo | On the generalized Ramanujan conjecture for GL(n)[END_REF] bounds

|α π (Ô, i)| ≤ NÔ 1/2-(n 2 +1) -1 , (6)
valid for all primes Ô and 1 ≤ i ≤ n, can be dramatically improved. COROLLARY 6. Let π be a cuspidal automorphic representation of GL n (A). Put m = ⌊n/2⌋. Assume that there exists a weak sym 2 -isobaric lift and ∧ j -isobaric lifts of π for all 2 ≤ j ≤ m. There exists a δ n > 0 such that

NÔ -1/4+δn ≪ |α π (Ô, i)| ≪ NÔ 1/4-δn for 1 ≤ i ≤ n and almost all primes Ô. Proof: Using the trivial inequality |Trace(A)| 2 ≤ |Trace(sym 2 A)| + |Trace(∧ 2 A)|, Proposition 5 gives max i |α π (Ô, i)| 2 ≪ |Trace(sym 2 A π (Ô))| + m j=2 |Trace(∧ j A π (Ô))| (7)
for all Ô such that π Ô is unramified. Let ρ : GL n (C) → GL d (C) be a polynomial representation and Π = Π(ρ) a weak ρ-isobaric lift of π.

Let Π = Ind σ 1 ⊗ • • • ⊗ σ k with σ i a cusp form on GL d i (A) and d 1 + • • • + d k = d. We have |Traceρ(A π (Ô))| = k i=1 Trace(A σ i (Ô)) ≤ k i=1 d i NÔ 1/2-(d 2 i +1) -1 ≤ dNÔ 1/2-(d 2 +1) -1 ,
for all Ô / ∈ S π by the Luo-Rudnick-Sarnak bounds [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF]. With d = max{deg(sym 2 ), deg(∧ m )} we apply this upper bound to each summand on the right hand side of (7) to get

max i |α π (Ô, i)| 2 ≪ NÔ 1/2-(d 2 +1) -1 .
Corollary 6 then follows from the unitarity of π Ô .

Global estimates

We construct a Dirichlet series which will be the focus of our attention for the rest of this paper. Let π be a cuspidal representation of GL n (A). Define

L(s, π, |max| 2 ) := Ò λ(Ò, π, |max| 2 )NÒ -s := Ô r≥0 max i |α π (Ô, i)| 2r NÔ -rs .
Let T be a finite set of primes such that Ô / ∈ T implies π Ô is unramified. Denote by Ø the (square-free) ideal which is the product of all primes in T . Write

L T (s, π, |max| 2 ) = Ô∤Ø r≥0 max i |α π (Ô, i)| 2 NÔ -rs = (Ò,Ø)=1 λ(Ò, π, |max| 2 )NÒ -s , (8)
and

L T (s, π, |max| 2 ) = Ô|Ø r≥0 max i |α π (Ô, i)| 2 NÔ -rs . ( 9
)
The following proposition is a consequence of Proposition 5 and the Luo-Rudnick-Sarnak bounds [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF]. The full strength of the bounds ( 6) is actually not used until the calculations involving ramified primes in Proposition 9.

PROPOSITION 7. Let π be a cuspidal representation of GL n (A). Then

L T (s, π, |max| 2 ) ≪ ♭ (Ò,Ø)=1 NÒ -σ ♭ (Ò,Ø)=1 λ(Ò, π × π)NÒ -σ m j=2 ♭ (Ò,Ø)=1 |λ(Ò, π, ∧ j )|NÒ -σ
uniformly on Re(s) = σ ≥ σ 0 > 1. The ♭ sign in the above sums indicates a restriction to square-free integral ideals.

Proof: From the Ô-th factor of L T (s, π, |max| 2 ) we may extract a linear term to ob-

tain r≥0 λ(Ô r , π, |max| 2 )NÔ -rs = (1 + λ(Ô, π, |max| 2 )NÔ -s ) r≥0 λ(Ô 2r , π, |max| 2 )NÔ -2rs .
The bounds λ(Ô, π, |max| 2 ) ≤ NÔ, guaranteed to hold by [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF], are strong enough to show convergence of the above geometric series to the right of 1. (We shall need the full strength of the bounds ( 6) to treat L T (s, π, |max| 2 ) in Proposition 9.) Thus

L T (s, π, |max| 2 ) ≪ Ô∤Ø (1 + λ(Ô, π, |max| 2 )NÔ -σ ) uniformly on Re(s) = σ ≥ σ 0 > 1. By Proposition 5, λ(Ô, π, |max| 2 ) ≪ 1 + λ(Ô, π × π) + m j=2 |λ(Ô, π, ∧ j )| for Ô ∤ Ø.
Applying this majorization gives

1 + λ(Ô, π, |max| 2 )NÔ -σ ≪ 1 + NÔ -σ + λ(Ô, π × π)NÔ -σ + m j=2 |λ(Ô, π, ∧ j )|NÔ -σ ≤ (1 + NÔ -σ )(1 + λ(Ô, π × π)NÔ -σ ) 2≤j≤m (1 + |λ(Ô, π, ∧ j )|NÔ -σ ).
Taking the product over all Ô ∤ Ø we obtain the proposition.

COROLLARY 8. Let π be a cuspidal representation of GL n (A). If L(s, π, ∧ 2 ) converges absolutely to the right of 1 for all 2 ≤ j ≤ ⌊n/2⌋, then L(s, π, |max| 2 ) converges to the right of 1.

When n = 2 or 3, the conditions of Corollary 8 are empty, so the conclusion automatically holds. When n = 4 or 5 the sole condition is that L(s, π, ∧ 2 ) be absolutely convergent to the right of 1. For n = 4, this property is proven by Kim in [9, Proposition 6.2]. Thus L(s, π, |max| 2 ) converges to the right of 1 for any π on GL n for n ≤ 4.

When applied to Dirichlet series which arise naturally in the theory of automorphic forms, Corollary 8 gives no new information. For to deduce the absolute convergence to the right of 1 of L(s, π, ∧ 2 ) from that of L(s, π, |max| 2 ) is just to repeat one of the hypotheses from which we derived the latter fact. The same can be said for L(s, π, sym 2 ). At this point, the loss of information in passing from L(s, π, |max| 2 ) to either L(s, π, ∧ 2 ) or L(s, π, sym 2 ) is just too great. The true strength of Proposition 7 will presently be seen to lie in questions regarding uniformity in the analytic conductor of π.

Gaining uniformity

Denote the local parameters of π at the infinite place v by µ π (v, i), 1 ≤ i ≤ n. Let q(π) be the conductor of π and define the analytic conductor to be C(π) = q(π)λ ∞ (π) where

λ ∞ (π) = v=∞ n i=1 (1 + |µ π (v, i)|).
For a pair of cusp forms π 1 , π 2 on GL n 1 (A) and GL n 2 (A) we define the analytic conductor using the parameters at infinity present in the gamma factors of the completed L-function. That is, for an infiinite place v,

L v (s, π 1,v × π 2,v ) = n 1 i=1 n 2 j=1 Γ Fv (s + µ π 1 ×π 2 (v, i, j))
for complex numbers µ π 1 ×π 2 (v, i, j). Above we have used the standard notation Γ R (s) = π -s/2 Γ(s/2) and Γ C (s) = 2(2π) -s Γ(s). When the infinite place v is unramified for either π and π ′ we have

{µ π×π ′ (v, i, j)} = {µ π (v, i) + µ π ′ (v, j)}. We define the analytic conductor of L(s, π 1 × π 2 ) to be C(π 1 × π 2 ) = q(π 1 × π 2 )λ ∞ (π 1 × π 2 ) where q(π 1 × π 2 ) is the conductor appearing in the functional equation for L(s, π 1 × π 2 ) and λ ∞ (π 1 × π 2 ) = v=∞ n 1 i=1 n 2 j=1 (1 + |µ π 1 ×π 2 (v, i, j)|).
The definitions of C(π) and C(π 1 × π 2 ) were first made by Iwaniec and Sarnak in [START_REF] Iwaniec | Perspectives on the analytic theory of L-functions, GAFA 2000[END_REF]. By the work of Bushnell and Henniart [START_REF] Bushnell | An upper bound on conductors for pairs[END_REF], q(π 1 × π 2 ) ≤ q(π 1 ) n 2 q(π 2 ) n 1 . It can also be shown that λ

∞ (π 1 × π 2 ) ≪ n 1 ,n 2 λ ∞ (π 1 ) n 2 λ ∞ (π 2 ) n 1 . Thus C(π 1 × π 2 ) ≪ n 1 ,n 2 C(π) n 2 C(π 2 ) n 1 . (10)
Definition. Let f (s, π) be a Dirichlet series associated with π which converges absolutely to the right of 1. We say that f (s, π) satisfies the convexity bound at s = 1 if f (s, π) = O ǫ (C(π) ǫ ) for every ǫ > 0 and all Re(s) > 1. When f (s, π) has a functional equation and nice analytic properties which allow for an interpolation to the left of 1, we drop the reference to any particular point and say simply that f (s, π) satisfies the standard convexity bound.

Our goal is to show that L(s, π, |max| 2 ) satisfies the convexity bound at s = 1. Uniform estimates in the conductor for a Dirichlet series are generally derived from a functional equation in which the conductor appears. Unfortunately L(s, π, |max| 2 ) satisfies no such functional equation. Proposition 7 will allow us to obtain uniform estimates for L(s, π, |max| 2 ) from those for L(s, π × π) and L(s, ∧ j π × ∧ j π) where 2 ≤ j ≤ ⌊n/2⌋. PROPOSITION 9. Let π be a cuspidal representation of GL n (A). If ∧ j π is strongly automorphic isobaric for 2 ≤ j ≤ ⌊n/2⌋, then L(s, π, |max| 2 ) = O(C(π) A ) on Re(s) > 1 for some A > 0.

Proof: For j ∈ {2, . . . , ⌊n/2⌋}, let ∧ j π = σ j,1 ⊞ • • • ⊞ σ j,ℓ j be the decomposition of ∧ j π into an isobaric sum of cusp forms σ j,i on GL n j,i . Then

L(s, ∧ j π × ∧ j π) = 1≤i 1 ,i 2 ≤ℓ j L(s, σ j,i 1 × σ j,i 2 ). (11)
The convergence of L(s, σ j,i 1 × σ j,i 2 ) to the right of 1 along with its functional equation [START_REF] Jacquet | Rankin-Selberg convolutions[END_REF] imply, through the Phragmen-Lindelof convexity principle, that

L(s, σ j,i 1 × σ j,i 2 ) = O(C(σ j,i 1 × σ j,i 2 ) B j,i 1 ,i 2 )
on Re(s) > 1 for some B j,i 1 ,i 2 > 0. By ( 10) and ( 11) we therefore have

L(s, ∧ j π × ∧ j π) = Ò λ(Ò, ∧ j π × ∧ j π) = O(C(π) B j ) (12)
on Re(s) > 1 where

B j = 1≤i 1 ,i 2 ≤ℓ j (n j,i 1 + n j,i 2 )B j,i 1 ,i 2 .
Similarly let B 1 > 0 be such that L(s, π × π) = O(C(π) B 1 ) on Re(s) > 1.

Denote by Π (j) the strong exterior j power lift of π. Denote by S j the set of finite primes outside of which Π (j) Ô ≃ ∧ j π Ô . Let T be the union of all the S j and the set of primes at which π is ramified. As before, denote by Ø the product of all primes in T . We note that NØ ≪ C(π).

To bound L T (s, π, |max| 2 ) (defined in [START_REF] Jacquet | Rankin-Selberg convolutions[END_REF]) polynomially in C(π) to the right of 1 we first note that for square-free ideals Ò, |λ(Ò, π,

∧ j )| ≤ 1 + |λ(Ò, π, ∧ j )| 2 = 1 + λ(Ò, ∧ j π × ∧ j π).
Secondly, Rudnick and Sarnak [START_REF] Rudnick | Zeros of principal L-functions and random matrix theory. A celebration of John F. Nash[END_REF]Appendix] have shown that the coefficients of L(s, Π × Π), where Π is any isobaric form on GL n , are non-negative. We may therefore remove the restriction of being square-free and relatively prime to Ø.

Thus ♭ (Ò,Ø)=1 |λ(Ò, π, ∧ j )|NÒ -σ ≤ Ò (1 + λ(Ò, π, ∧ j × ∧ j π))NÒ -σ .
An appeal to Proposition 7 and ( 12) gives L T (s, π, |max| 2 ) = (C(π) B ) where B = j B j and j runs through 1, . . . , ⌊n/2⌋. We now treat L T (s, π, |max| 2 ) (defined in ( 9)). Let δ = δ(n) = (n 2 + 1) -1 . Using the local bounds [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF] we have

L Ô (1, π, |max| 2 ) := r≥0 max i |α π (Ô, i)| 2r NÔ -r ≤ r≥0 NÔ -2rδ = 1 + cNÔ -2δ ,
for some constant c > 0 depending only on δ. Thus L Ô (1, π, |max| 2 ) ≤ 1+NÔ -δ for primes Ô such that NÔ ≥ c δ -1 . We have

L T (1, π, |max| 2 ) ≪ δ Ô|Ø NÔ≥c δ -1 (1 + NÔ -δ ) ≤ NÒ≤NØ NÒ -δ ≪ NØ 1-δ ≪ δ C(π) 1-δ .
With A = B + 1δ, we have proved the proposition.

Once we have polynomial control on L(s, π, |max| 2 ) to the right of 1 we can use a bootstrapping technique of Iwaniec [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF] to whittle down the exponent to be as small as we like. We now see the fruit of not having applied this technique straightaway to L(s, π × π), say, as in Molteni [START_REF] Molteni | Upper and lower bounds at s = 1 for certain Dirichlet series with Euler product[END_REF]: the complete multiplicativity of the coefficients λ(Ò, π, |max| 2 ) allows us to do without any further restriction on the size of the local roots. That is, no improvement on the Luo-Rudnick-Sarnak bounds [START_REF] Iwaniec | Small Eigenvalues of Laplacian for Γ 0 (N)[END_REF], already used in the proofs of Proposition 7 and Proposition 9, will be necessary. In fact, the following proposition could be applied to any Dirichlet series with completely multiplicative non-negative coefficients with polynomial control in the conductor to the right of 1.

We note that Propositions 9 and 10 combine to give Theorem 1.

PROPOSITION 10. Let π be a cuspidal representation of GL n (A). Assume that the function L(s, π, |max| 2 ) converges on Re(s) > 1. If there exists a constant A > 0 such that L(s, π, |max| 2 ) = O(C(π) A ) on Re(s) > 1 then L(s, π, |max| 2 ) satisfies the convexity bound at s = 1.

Proof: For convenience, put λ(Ò) = λ(Ò, π, |max| 2 ) and C = C(π). Set S(X) = NÒ≤X λ(Ò). Then the polynomial control and the positivity of the coefficients imply that for every σ > 1

S(X) ≤ X σ NÒ≤X λ(Ò)NÒ -σ ≤ X σ Ò λ(Ò)NÒ -σ ≪ C A X σ . (13)
By the complete multiplicativity of the λ(Ò) we have

S(X) 2 = NÑ,NÒ≤X λ(Ñ)λ(Ò) = NÑ,NÒ≤X λ(ÑÒ) = NÖ≤X 2 λ(Ö)τ (Ö),
where τ (Ö) is the number of divisors of Ö. Applying the bound τ (Ö) ≪ ǫ (NÖ) ǫ and ( 13)

we get S(X) 2 ≪ ǫ C A X 2+ǫ . Upon taking the square root we have S(X) ≪ ǫ C A/2 X 1+ǫ . Iterating this step M times, we obtain S(X) ≪ ǫ,M C A/2 M X 1+ǫ . For any ǫ > 0, we may take M > (log Alog ǫ)/ log 2 to obtain

S(X) ≪ A,ǫ C ǫ X 1+ǫ . (14)
Let NÒ ∼ M denote the diadic interval M ≤ NÒ < 2M. Using ( 14) along with the positivity of the coefficients we conclude that, for any ǫ > 0 and σ

≥ 1+2ǫ, L(σ, π, |max| k ) is M =2 k k≥0 NÒ∼M λ(Ò)NÒ -σ ≤ M =2 k k≥0 M -σ S(2M) ≪ A,ǫ C ǫ k≥0 2 -kǫ ≪ A,ǫ C ǫ .
This finishes the proof. Theorem 1 has now been proven. Note that even for large n the hypothesis that all ∧ j π, 2 ≤ j ≤ ⌊n/2⌋, be automorphic is not necessarily stronger than the condition α π (Ô, i) ≪ NÔ 1/4 on the local roots. Corollary 6 states that only when these exterior power lifts are combined with the symmetric square lift do the hypothetical bounds (1) follow. And yet the strength of the conclusion of Theorem 1 is much stronger than the convexity bound for only L(s, π × π).

Proof of Corollary 2

When Theorem 1 is combined with the (strong) automorphy of ∧ 2 π for π on GL 4 , a fact proved in [9, Theorem 5.3.1], we obtain the following corollary. COROLLARY 2. Let π i be cuspidal representations of GL n i (A) where n i ≤ 4 for i = 1, 2. Then L(s, π 1 × π 2 ), as well as L(s, π i , ∧ 2 ) and L(s, π i , sym 2 ) for i = 1, 2, satisfy the standard convexity bound.

Proof: For any integer r ≥ 0 and prime ideal Ô we have λ(Ô r , π × π) ≤ N (r, n) max i |α π (Ô, i)| 2r , where N (r, n) is the number of monomials in n variables of degree r. The same bound holds for λ(Ô r , π, ∧ 2 ) and λ(Ô r , π, sym 2 ). We can compute that N (r, n) ≪ r A(n) for some A(n) > 0. Theorem 1 therefore gives the convexity bound for each of L(s, π i × π i ), L(s, π i , ∧ 2 ), and L(s, π i , sym 2 ).

We deduce the convexity bound for L(s, π 1 ×π 2 ) from that of L(s, π i × π i ) for i = 1, 2. To do so, we avail ourselves of the notation and terminology of Macdonald's treatise [START_REF] Macdonald | Symmetric functions and Hall polynomials[END_REF]. For integers n ≥ 1 and r ≥ 0, let P n (r) be the set of partitions of r of length no greater than n. Let s λ denote the Schur function associated to a partition λ. Assume n 1 ≥ n 2 . If n 2 < n 1 then define α n 2 (Ô, i) = 0 for all n 2 < i ≤ n 1 . Put n = n 1 and set α π i (Ô) = (α π i (Ô, 1), . . . , α π i (Ô, n)). For primes Ô unramified for both π 1 and π 2 and integers r ≥ 0 the coefficients λ(Ô r , π 1 × π 2 ) are defined by (1α π 1 (Ô, i 1 )α π 2 (Ô, i 2 )NÔ -s ) -1 = r≥0 λ(Ô r , π 1 × π 2 )NÔ -rs .

It is then a standard identity in the theory of symmetric functions that λ(Ô r , π 1 × π 2 ) = λ∈Pn(r) s λ (α π 1 (Ô))s λ (α π 2 (Ô)).

Applying the Cauchy-Schwartz inequality to this, we obtain |λ(Ô r , π 1 × π 2 )| ≤ λ(Ô r , π 1 × π 1 ) 1/2 λ(Ô r , π 2 × π 2 ) 1/2 . A similiar inequality can be proven for ramified primes. When extended to all Ò this gives Ò λ(Ò, π 1 × π 2 )

Ò s ≤ Ò λ(Ò, π 1 × π 1 ) 1/2 Ò σ/2 λ(Ò, π 2 × π 2 ) 1/2 Ò σ/2
≤ L(s, π 1 × π 1 )L(s, π 2 × π 2 ), the last inequality again by Cauchy-Schwartz. The corollary immediately follows.

1≤i 1 ,i 2

 12 ≤n

2000 Mathematics Subject Classification. Primary: 11F66. Secondary: 11M41

The author was supported by a postdoctoral fellowship from the CNRS.


Zain A H Hammadeh
email: hammadeh@ida.ing.tu-bs.de

Rolf Ernst
email: ernst@ida.ing.tu-bs.de

Sophie Quinton
email: sophie.quinton@inria.fr

Rafik Henia
email: rafik.henia@thalesgroup.com

Laurent Rioux
email: laurent.rioux@thalesgroup.com

Bounding Deadline Misses in Weakly-Hard Real-Time Systems with Task Dependencies

Real-time systems with functional dependencies between tasks often require end-to-end (as opposed to task-level) guarantees. For many of these systems, it is even possible to accept the possibility of longer end-to-end delays if one can bound their frequency. Such systems are called weakly-hard.

In this paper we provide end-to-end deadline miss models for systems with task chains using Typical Worst-Case Analysis (TWCA). This bounds the number of potential deadline misses in a given sequence of activations of a task chain. To achieve this we exploit task chain properties which arise from the priority assignment of tasks in static-priority preemptive systems. This work is motivated by and validated on a realistic case study inspired by industrial practice and derived synthetic test cases.

I. INTRODUCTION AND RELATED WORK

Timing performance analysis of real-time systems with concurrently executing task chains is notoriously difficult due to the complexity of timing interference between tasks. This is all the more true when task chains are derived from communicating threads [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF]. In this paper, we are interested in the analysis of end-to-end guarantees for weakly-hard systems with task dependencies, i.e., systems for which it is possible to accept the possibility of end-to-end deadline misses if one can bound their frequency [START_REF] Bernat | Weakly hard real-time systems[END_REF].

We present a method to compute end-to-end deadline miss models for static-priority preemptive systems with task chains. This bounds the number of potential deadline misses in a given sequence of executions of a task chain. Our approach is an extension of Typical Worst-Case Analysis (TWCA) [START_REF] Quinton | Formal analysis of sporadic overload in real-time systems[END_REF], [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF], for which we exploit task chain properties derived from the priority assignment of tasks in a way similar to [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF].

To the best of our knowledge, there is no state-of-the-art method for the computation of weakly-hard guarantees in realtime systems with task dependencies.

Extensive research has focused on the schedulability analysis of hard real-time systems with task dependencies. This includes approaches focusing on offset analysis [START_REF] Gutiérrez | Schedulability analysis for tasks with static and dynamic offsets[END_REF] but also more general precedence models [START_REF] Gutiérrez | Exploiting precedence relations in the schedulability analysis of distributed real-time systems[END_REF]. In [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF], an upper bound on the end-to-end latency of task chains in real-time systems is presented, on which we will base our work in this paper.

In contrast, there is little in the literature regarding the analysis of weakly-hard systems. Initial attempts [START_REF] Hamdaoui | A dynamic priority assignement technique for streams with (m, k)-firm deadlines[END_REF], [START_REF] Bernat | Weakly hard real-time systems[END_REF] can only handle periodic tasks (or sporadic tasks but using a coarse This work has been partially funded by the German Research Foundation (DFG) as part of the project "TypicalCPA" under the contract number TWCA ER168/30-1. interarrival time model) and no task dependencies. Recent work has focused on providing guarantees for systems with more complex activation patterns [START_REF] Quinton | Formal analysis of sporadic overload in real-time systems[END_REF], [START_REF] Hammadeh | Extending typical worst-case analysis using response-time dependencies to bound deadline misses[END_REF], [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] and [START_REF] Kumar | Quantifying the effect of rare timing events with settling-time and overshoot[END_REF], mostly relying on the so-called TWCA approach. None of these, however, can handle task dependencies.

The paper is organized as follows: Section II introduces our system model and formulates the problem that we address. Then, Section III explains the basic principles of TWCA. Section IV proposes an improved version of the worst-case latency analysis of [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF] which we use in V for the core contribution of our paper. Finally, Section VI shows our experimental results while Section VII proposes some conclusions.

II. SYSTEM MODEL

We consider uniprocessor systems consisting of a finite set of m disjoint task chains scheduled according to the Static Priority Preemptive (SPP) scheduling policy. A task chain is a sequence of distinct tasks which activate each other. Tasks in a system are required to belong to exactly one chain 1 . Formally, a task chain σ a , a ∈ [1, m], is defined by a finite sequence (τ 1 a , τ 2 a , . . . , τ n a ) of distinct tasks for some n ∈ N + , meaning that the output of τ i a is connected to the input of τ i+1 a for i ∈ [1, n -1]. Every task chain σ a is assigned an activation model (see definition below) defining the frequency of arrival at the input of τ 1 a ; and a relative deadline D a . The tasks in σ a are denoted τ i a , τ j a etc. Task τ i a denotes the i-th task in task chain σ a . The number of tasks in σ a is denoted n a . The first task in σ a is called the header task of σ a and the last one is called its tail task.

Figure 1 shows an example system with two task chains:

σ a = (τ 1 a , τ 2 a , τ 3 a , τ 4 a , τ 5 a , τ 6 a ), σ b = (τ 1 b , τ 2 b , τ 3 b
). We denote C the set of task chains. This set is partitioned into SC and AC, which contain respectively the synchronous and asynchronous chains. Synchronous and asynchronous chains are specified in the same way but behave differently at execution: In a synchronous chain σ a an incoming activation cannot be processed until the previous instances of σ a have finished [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF]. In an asynchronous chain σ b an incoming activation is processed independently from previous instances.

The activation models of task chains are defined using arrival curves as in e.g. [START_REF] Moy | Arrival curves for real-time calculus: The causality problem and its solutions[END_REF], i.e., functions η - a , η + a : N → N such that for any time window ∆T , η + a (∆T ) defines the ). We will also need the pseudo-inverse representation of arrival curves, namely δ - b , δ

+ b : N → N, such that δ - b (k) (respectively δ + b (k))
defines the minimum (respectively maximum) time that might pass between the first and the last activation in any sequence of k consecutive activations of σ b .

A task τ i a is defined by: (1) an arbitrary priority π i a and (2) an upper bound on its execution time C i a (we take 0 as a lower bound). The notation π i a > π j b is used to denote that τ i a has higher priority than τ j b . As a result, τ i a may preempt τ j b when it arrives. We also use the notation π i a π j b . The timing behavior of a task τ i a is an infinite sequence of instances defined by: an arrival time, possible preemption delays and a finish time. Preemption delays are due to the task being blocked by higher priority tasks from other task chains, but also by higher priority tasks from the same chain if it is asynchronous. In contrast, tasks in a synchronous chain cannot be preempted by other tasks of the same chain, even if they have higher priority. Task τ i a finishes latest after having been scheduled for C i a units of time. The timing behavior of a task chain σ a is an infinite sequence of task chain instances, where a task chain instance is made of one instance of each task in the chain such that the finish time of task τ i a corresponds to the arrival time of task τ i+1 a (assuming τ i a is not the last task in the chain). The arrival of task τ 1 a follows the activation model of σ a . The latency of an instance of a task chain σ a is the time interval between the activation of the header task of σ a and the finish time of the tail task of σ a . The worst-case latency of σ a is the maximum latency over all instances of σ a . An instance of σ b is said to miss its deadline if its latency exceeds the relative deadline of σ b . This can happen in weakly-hard real-time systems. We consider a simple, deadline-agnostic scheduler that does not anticipate, monitor or react to deadline misses but instead runs every instance to completion, independent of whether a deadline miss has occurred or not.

As usual in TWCA, we suppose that deadline misses are caused by rarely activated sporadic chains, e.g., interrupt service routines or recovery chains. These chains cause transient overload, increasing chain latencies which may cause deadline misses, hence their name: overload chains. We assume that the set of overload chains is identified and denoted C over . In this paper, we address the problem of computing DMMs of task chains in systems which contain overload task chains.

III. PRINCIPLE OF TYPICAL WORST-CASE ANALYSIS

Typical Worst-Case Analysis (TWCA) is a technique to compute deadline miss models which bound the number of deadline misses in a sequence of activations of a given task. TWCA applies to systems of independent tasks which may occasionally miss deadlines due to overload tasks. We recall here the principle of TWCA and refer to [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] for more detail.

Formally, a Deadline Miss Model (DMM) for a task τ i is a function dmm i : N + → N such that dmm i (k) bounds the maximum number of deadline misses that τ i may experience out of a sequence of k consecutive executions. The DMM computation is based on the analysis of unschedulable combinations, i.e., sets of overload tasks which, when activated together, may lead to a deadline miss. More formally, a combination, denoted c, is a set of overload tasks. c is schedulable (with respect to τ i ) if an instance of τ i is guaranteed to meet its deadline as long as only tasks in c experience overload activations in its level-i busy window, where a level-i busy window is a maximal time interval during which the processor has activations of τ i or higher priority tasks pending.

Let us consider a sequence of k activations of a given task τ i and focus on the computation of dmm i (k). Note that the sequence may span multiple busy windows. The activation model of the overload tasks bounds the number of activations of these tasks (also called overload activations) which may arrive during the considered sequence. Assuming we have all unschedulable combinations at hand, the problem is then to find how to assign overload activations to busy windows so as to pack as many unschedulable combinations as possible into the level-i busy windows under consideration. Therefore the problem becomes a multi-dimensional knapsack problem. Example. Figure 2 illustrates two possible packings of overload activations into 5 busy windows. Every row corresponds to one overload task while every column corresponds to one busy window of τ i . The number of activations per line is constrained by the activation models of the overload tasks. The number of deadline misses associated to a given packing depends on how many columns are unschedulable combinations. Here, any combination containing more than one task is unschedulable.

τ 1 τ 2 τ 3 X X X X X Busy windows case 1 case 2
Figure 2. Packing combinations into busy windows (X = deadline miss).

So far, TWCA can only handle independent tasks. In the rest of this paper we show how the state-of-the-art approach can be generalized to systems with task chains.

IV. LATENCY ANALYSIS REVISITED

Let us first revisit the worst-case latency analysis of systems with task chains [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF]. Consider two chains σ a and σ b . To quantify the interference of σ a on σ b we distinguish two cases:

1) some tasks in σ a have lower priority than all tasks in σ b ; in that case, σ a will be blocked by σ b every time it reaches one of those tasks. 2) In any other case, σ a is said to arbitrarily interfere with σ b . This means that every time σ a is triggered, we suppose that it may entirely execute before σ b can be scheduled again. As we will see later, there is no guarantee however that this will happen.

Definition 2. A chain σ a is said to be deferred by chain σ b if ∃i ∈ [1, n a ], π i a < min{π j b } n b j=1
Otherwise it is arbitrarily interfering with σ b .

The set of chains deferred by σ b is denoted DC(b) and the set of chains arbitrarily interfering with σ b is denoted IC(b).

For a chain σ a which is arbitrarily interfering with σ b , interference on σ b can be directly derived from the number of activations of σ a . If σ a is, however, deferred by σ b , then interference is defined based on the concept of segment of σ a w.r.t. σ b . Intuitively, a segment of σ a w.r.t. σ b represents a subchain of σ a that may interfere with σ b .

Definition 3. A segment of σ a w.r.t σ b is a maximal subchain (τ i a , τ i+1 a , . . . , τ i+k a ) of σ a , i ∈ [1, n a ] and k ∈ [0, n a -1]
, with the convention 2 that task identifiers should be read modulo n a and such that

∀l ∈ [0, k], π i+l a min{π j b } n b j=1
Note that we (conservatively) assume that a segment may span over two instances of σ b . S b a denotes all such segments. Example. Chain σ a in Figure 1 We now revisit the worst-case latency analysis introduced in [START_REF] Schlatow | Response-time analysis for task chains in communicating threads[END_REF] and propose a description that is similar to worst-case response-time analysis as explained in [START_REF] Quinton | Formal analysis of sporadic overload in real-time systems[END_REF]. 2 That is, if i + l > na then it should be read (i + l) mod na. Definition 6. A σ b -busy-window is a maximal time interval during which (at least) one instance of σ b is pending, i.e., it has been activated but has not finished yet.

Definition 7. The q-event busy time of a chain σ b is the maximum time it may take to process q activations of σ b within a σ b -busy-window starting with the first of these q activations. Theorem 1. The q-event busy time of σ b is bounded by

B b (q) = q × C b + max(0, η + b (B b (q)) -q) × C s header b if σ b ∈ AC + σa∈IC(b) η + a (B b (q)) × C a + σa∈AC∩DC(b) η + a (B b (q)) × C s header a,b + s∈S b a C s + σa∈SC∩DC(b) C s crit a,b (1)
where C x denotes the sum of the execution time bounds of the tasks in segment or chain x.

Proof. The above equation is made of five components:

1) The first line corresponds to the time needed to actually perform the q computations; 2) The second component accounts for the interference of additional activations of σ b which may arrive while the q activations under consideration are being processed. Note that these instances will at most interfere until they have to execute the lowest priority task in σ b . This component only applies to asynchronous chains; 3) The third element represents the interference from arbitrarily interfering chains, synchronous or asynchronous; 4) The fourth line deals with interference from deferred, asynchronous chains. Instances can arbitrarily queue up which allows the header segment to interfere arbitrarily. For all other segments at most one instance can be backlogged because tasks between segments have lower priority than tasks within segments. Each such instance can interfere for at most one segment (see below). 5) The fifth component in the equation accounts for the interference from deferred, synchronous chains. Here only one instance per chain may interfere for at most one segment (see below).

The correctness of the last two components in Equation (1) relies on the following property. Lemma 1. Tasks of a chain σ a that are in different segments cannot execute instances corresponding to the same chain instance in the same σ b -busy-window.

Proof. Segments are maximal sequences of tasks with a priority higher than or equal to the lowest priority task, say τ i b , in σ b . This means that between two segments of σ a there is at least one task, say τ j a , that has lower priority than τ i b . In order to execute these two segments for the same instance of σ a , one has to execute τ j a . Since τ j a has lower priority than all the tasks in σ b , this can only happen after σ b closes its current σ b -busy-window.

Theorem 2. The maximum number of activations of σ b in a σ b -busy-window is

K b = min{q 1 | B b (q) δ - b (q + 1)}
The latency of a task chain σ b is bounded by

WCL b = max q∈[1,K b ] {B b (q) -δ - b (q)}
Proof. This proof proceeds exactly as the proofs in [START_REF] Quinton | Formal analysis of sporadic overload in real-time systems[END_REF].

The main objective of TWCA is to bound the number of deadlines misses of a task chain σ b which may be caused by an activation at the input of an overload task chain σ a . For that, we need to know over how many σ b -busy-windows a instance of σ a may span.

We already know that, in a chain σ a , the execution of tasks corresponding to the same instance of σ a cannot take place in the same σ b -busy-window if those tasks are in different segments. This implies that an instance of σ a spans over at least as many σ b -busy-windows as there are segments of σ a w.r.t. σ b .

Note that there is no guarantee that a segment of σ a will be executed within one σ b -busy-window. As an example, in Figure 3 the execution of segment (τ 1 a , τ 2 a , τ 3 a ) spans over two σ b -busy-windows. We therefore introduce the notion of active segment, which applies to subsegments which are guaranteed to be executed in the same σ b -busy-window. Example. In Figure 1, chain σ a has three active segments: (τ 1 a , τ 2 a ), (τ 3 a ), (τ 5 a ). Lemma 2. The execution of an active segment of σ a w.r.t. σ b cannot span over more than one σ b -busy-window. 3 Here, i + l is always smaller than or equal to na.

Definition 8. An active segment of σ a w.r.t σ b is a subchain 3 of a segment (τ i a , τ i+1 a , . . . , τ i+k a ) of σ a where i ∈ [1, n a ] and k ∈ [0, n a -i] such that ∀l ∈ [1, k], π i+l
Proof. Once the execution of an active segment of σ a w.r.t. σ b has started, τ tail b will not be able to execute because the active segment is blocking it or a task preceding it, and therefore the current σ b -busy-window cannot be closed, until the whole segment has finished executing. This lemma is illustrated in Figure 3, where every active segment of chain σ a executes within one σ b -busy-window.

Note that an active segment is part of a segment in the sense of Definition 3. As a result, we easily conclude from Lemma 1 and 2 that two active segments of chain σ a may be executed within one σ b -busy-window if and only if they are part of the same segment of σ a .

V. TWCA FOR TASK CHAINS

We now have all the ingredients needed to show how we extend TWCA to handle task chains. We follow here the same approach as the one for systems with independent tasks explained in Section III. For the rest of the section we suppose given a chain σ b and k 1 and focus on the computation of dmm b (k), that is, a bound on the number of deadlines that σ b can miss out of a k-sequence, i.e., k consecutive activations. Similar to [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF], we assume that there is at most one activation of an overload chain σ a in a σ b -busy-window. As a result, we can without loss of generality consider our overload task chains as synchronous.

A. Combinations for TWCA of task chains

For the case where tasks are independent, a combination is defined as a set of overload tasks. The DMM computation based on this definition heavily relies on the fact that one overload activation impacts exactly one busy window. In the context of task chains, we have seen in the previous sections that one instance of a task chain σ a may span over several σ b -busy-windows. As a result, the impact of one overload activation is not here limited to one σ b -busy-window. We have however also shown that the execution an active segment of σ a is restricted to a single σ b -busy-window. Hence our choice to define combinations based on active segments rather than tasks or task chains. Note that our definition excludes combinations which cannot execute within one σ b -busy-window based on our definition of segment.

Example.

There are four possible combinations of the active segments of chain σ a in Figure 1:

{(τ 1 a , τ 2 a )}, {(τ 3 a )}, {(τ 5 a )}, {(τ 1 a , τ 2 a ), (τ 3 a )}. Definition 10. A combination c is schedulable (w.r.t. σ b ) if σ b
is guaranteed not to miss any deadline in a σ b -busy-window in which only the active segments in c execute (in addition to non-overload chains). Otherwise c is said to be unschedulable.

B. An ILP formulation for the DMM

Having clarified the notion of combination that we use, we can now state our main theorem, similar to [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF].

Theorem 3. Let us define dmm b (k) as max N b c∈U x c | ∀σ a ∈ C over , ∀s ∈ S a , {c∈U | s∈c} x c Ω a b (2)
where • N b is the maximum number of deadlines that σ b can miss in one busy window; • U is the set of unschedulable combinations; • x c is the variable constraining the number of busy windows that could contain one activation of the k-sequence and suffer from an overload corresponding to c ∈ U; • S a denotes the set of active segments of σ a ;

• Ω a b is the maximum number of activations of σ a which could impact the considered k activations of σ a . Then dmm b (k) is a DMM for σ b .

The formal definition of N b and Ω a b is given below. Because U can be too large to be statically constructed, Section V-C discusses an efficient criterion to determine whether a combination is in U. The x c are the variables of our ILP problem. Proof. Assume that we have Ω a b for all chains σ a , i.e. the maximum number of activations of σ a which could impact the k-sequence. In the worst case, each active segment of σ a also impacts σ b Ω a b times. As in Section III, we here also face a multi-dimensional knapsack problem where items correspond to unschedulable combinations and capacities to Ω a b for every line s associated with an active segment of overload chain σ a . So considering that x c stands for the number of times that a combination c is used in the packing under consideration, we want to find the packing that maximizes the number of deadline misses of σ b -which is equal to the number of unschedulable combinations used multiplied by the maximum number of deadline misses due to each combination. This packing is constrained by the fact that active segments cannot be used in more combinations than is allowed by their corresponding Ω a b . Let us now formally define N b and Ω a b . Lemma 3.

N b = #{q ∈ [1, K b ] | B b (q) -δ - b (q) > D b } Proof.
The proof proceeds exactly like that of Theorem 2.

Lemma 4. The maximum number Ω a b of activations of σ a which could impact the considered k activations of σ a is

Ω a b = η + a (δ + b (k) + WCL b ) + 1 Proof.
Clearly, activations of chain σ a which occur after the first instance of chain σ b in the k-sequence is activated and before the last activation in the k-sequence finishes may have an impact on the latencies in the k-sequence. There are at most η + a (δ + b (k) + WCL b ) such activations. In contrast, an instance of σ a which arrives after the last instance of chain σ b in the ksequence has finished does not impact the k-sequence. Finally, we have assumed that there is at most one activation of σ a in a σ b -busy-window so that at most one activation of σ a before the k-sequence can impact it.

C. Criterion of schedulability

As already mentioned, U can be too large to be statically constructed. We present here an efficient criterion to determine whether a combination c is in U or not. Let us reorganize Equation 1 for the multiple busy-time computation to show explicitly the contribution of the overload chains of a combination in the multiple busy time (and the latency) of σ b .

B c b (q) = q × C b + max(0, η + b (B c b (q)) -q) × C s header b if σ b ∈ AC + σa∈IC(b)\Cover η + a (B c b (q)) × C a + σa∈AC∩DC(b) η + a (B b (q)) × C s header a,b + s∈S b a C s + σa∈SC∩DC(b)\Cover C s a b + σa∈Cover s∈Sa C s × r c s (3)
where r c s is a Boolean which holds exactly when s ∈ c.

A combination c is schedulable if B c b (q) -δ - b (q) D b for all q ∈ [1, K b ]. Now, let us define L b (q) as follows. L b (q) = q × C b + max(0, η + b (δ - b (q) + D b ) -q) × C s header b if σ b ∈ AC + σa∈IC(b)\Cover η + a (δ - b (q) + D b ) × C a + σa∈AC∩DC(b) η + a (δ - b (q) + D b ) × C s header a,b + s∈S b a C s + σa∈SC∩DC(b)\Cover C s a b (4
) Then we now have a much simpler sufficient condition for schedulability:

c is schedulable if ∀q ∈ [1, K b ], L b (q)+ σa∈Cover s∈Sa C s ×r c s δ - b (q)+D b (5)
We have now shown how we can reuse the ILP solution of [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] for systems with task chains with limited changes.

VI. EXPERIMENTAL RESULTS

We have experimented with a case study directly derived from industrial practice at Thales Research & Technology. The system is a single-core processor scheduled according to SPP. Figure 4 shows the specified task set and the real-time attributes of each task. In the following experiments we focus on providing DMMs for σ c and σ d . Experiment 1. We first compute the worst-case latency WCL of task chains σ c and σ d as described in Section IV. The analysis results show that the system is not schedulable as σ c can in the worst-case miss its deadline, see Table I Let us provide additional details resulting from this DMM computation. Both chains σ a and σ b arbitrarily interfere with σ c because neither has a task with a priority lower than 1 which is the lowest priority in σ c . As a result σ a and σ b have only one segment, respectively (τ 1 a , τ 2 a ) and (τ 1 b , τ 2 b , τ 2 b ). These two segments are also active segments because the priority of the tail task of chain σ c is lower than all priorities in these segments (see figure 4). Therefore no constraints on combining active segments are needed. Our set of combinations thus has three elements: c1 = {(τ 1 a , τ 2 a )}, c2 = {(τ 1 b , τ 2 b , τ 3 b )}, and c3 = {(τ 1 a , τ 2 a ), (τ 1 b , τ 2 b , τ 3 b )}. Based on the schedulability criterion we introduced in the previous section we conclude that c3 is the only unschedulable combination, so in this case the TWCA is fairly simple.

We now want to generalize the results obtained on our industrial case study, while preserving practical relevance. For that purpose, we arbitrarily modify the priority assignment so as to generate random systems with different scenarios. Experiment 2. We arbitrarily assign priorities to show the impact of priority assignments on the schedulability and the deadline miss models. In this experiment we randomly choose 1000 assignments to test our analysis intensively. Figure 5 shows dmm c [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] and dmm d [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF]. Notice first that out of . dmmc [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] and dmm d [START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] the 1000 assignments generated, chain σ c is schedulable (misses no deadline) 633 times. More interestingly, chain σ d is schedulable only 307 times out of 1000. TWCA in that case is very useful as for more than 500 of the remaining systems it can guarantee that no more than 3 out 10 deadlines can be missed. Note that we have repeated our experiment 30 times and observed similar results.

VII. CONCLUSION

In this paper we present the first method for computing end-to-end deadline miss models for systems with task dependencies, using Typical Worst-Case Analysis (TWCA). This bounds the number of potential deadline misses in a given sequence of activations of a task chain. Our approach addresses uniprocessor systems with Static Priority Preemptive scheduling. We show how state-of-the-art TWCA can be extended using recent results in the analysis of hard realtime systems with task dependencies. Specifically, we show how we can formulate our problem as a knapsack problem. Our approach is validated on a realistic case study inspired by industrial practice and synthetic variants of it.

This paper is an important step towards using TWCA for the practical design of distributed embedded systems.

Definition 1 .

 1 A deadline miss model for a task chain σ b is a function dmm b : N + → N such that dmm b (k) bounds the maximum number of deadline misses in a window of k consecutive executions of σ b .

  of σ a w.r.t. σ b defined as the subchain (τ 1 a , τ 2 a , . . . , τ i a ) where i ∈ [1, n a -1] is the smallest integer such that τ i+1 a has lower priority than all tasks in σ b .

2 bFigure 3 .

 23 Figure 3. Number of impacted busy windows of chain b.

  task of σ b .

Definition 9 .

 9 A combination c is a set of active segments w.r.t. σ b such that if two active segments of the same chain σ a are in c then they are part of the same segment of σ a w.r.t. σ b .

Figure 4 .

 4 Figure 4. Model of our case study. We use the following notations: task chains are specified as σ[δ -(2) : D] and tasks with τ [π : C]. Chains σc and σ d are periodically activated while σa and σ b are sporadic, overload chains.

Figure 5

 5 Figure 5. dmmc[START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF] and dmm d[START_REF] Xu | Improved deadline miss models for real-time systems using typical worst-case analysis[END_REF]

  has 2 segments w.r.t. chain σ b : The critical segment of a chain σ a deferred by σ b , denoted s crit a,b , is the segment (τ i a , τ i+1 a , . . . , τ i+k a ) of σ a w.r.t. σ b that maximizes computation time, i.e., 0 l k C i+l a . Definition 5. Consider an asynchronous chain σ a . We denote: if σ a is deferred by σ b then we denote s header

	(τ 1 a , τ 2 a , τ 3 a ) and (τ 5 a ). Note that τ 4 a and τ 6 a have lower priority than τ 2 b and are therefore not part of any segment.
	Definition 4. • s header a the subchain (τ 1 a , τ 2 a , . . . , τ i a ) where i ∈ [1, n a -1] is the smallest integer such that τ i+1 has the lowest priority a in σ a . If τ 1 a has the lowest priority then s header a is empty.

•

Table I WCL

 I . OF TASK CHAINS σc AND σ d A second analysis, in which all overload chains are abstracted away, reveals that the system is schedulable and σ c meets its deadline if neither σ a nor σ b are activated. We thus perform TWCA as presented in this paper. The computed DMM of σ c is shown in TableIIσ d is schedulable and therefore does not need a DMM.

		task chain	WCL	D
		σc	331	200
		σ d	175	200
	task chain		DMM
	σc	dmmc(3) = 3, dmmc(76) = 4, dmmc(250) = 5
		Table II
		dmm(k) FOR TASK CHAIN σc

To analyze systems that are not only made of disjoint task chains but also contain forks and joins (but no cycle), one can additionally define paths, i.e. sequences of distinct task chains. This is out of the scope of this paper.


E Mermoz

J M Linares

A Bernard

Benefits and limitations of parametric design implementation in helicopter gearbox design phase

Introduction

One main constraint for any company is to increase its performances as compared to its competitors. It requires the improvement of the products quality and the reduction of the time to market. These generic constraints are clearly applicable to helicopter market. The lead time to develop a new helicopter is representing a several years engineering effort. Its duration is mainly the conjunction of 2 parameters which are the complexity of the system to be developed and the level of safety to ensure. Gearboxes development is one of the major contributors to the The purpose of this paper is to present a parametric design methodology has been successfully implemented in Eurocopter to gearboxes.

Recall of helicopter gearbox design characteristics 2.1. The gearbox product

The role of the main gearbox is to transform the mechanical power generated by the engines (from 0.3 to 4 MW depending on the size of the helicopter). The engine rotational speed (from 6000 RPM to 23,000 RPM) is reduced according the rotor needs to lower values (from 400 RPM to 160 RPM).

A helicopter gearbox is a complex system based on several combinations of power gear trains with specific power bearings, installed into complex housing (Fig. 1). The configuration of the gear train is the result of geometrical constrain given by the helicopter architecture together with power gears sizing. The role of the gearboxes on helicopter is not limited to mechanical power transformation and distribution. Indeed, gearboxes also provide a geometrical and mechanical links between many subsystems of the helicopter dynamic system such as engines, servo actuators, tail drive shafts and rotors systems.

The gearbox design process

During the design process of a new helicopter, all the dynamic system components have to be developed simultaneously. As helicopter development lead time. described in Ref. [START_REF] Meier | Industrial Product-Service Systems-IPS 2[END_REF], the helicopter requirements will be progressively cascaded and completed down to subsystem levels. Like any specification for aeronautic systems, a specific attention reduce the duration of the design phase of the helicopter will be paid to weight, safety or life cycle cost. However, at the beginning of a development program, nobody is in a position to specify from a chief engineer point of view some geometrical dimensions for interfaces between helicopter subsystems. Significant pre-design efforts have to be done at subsystem levels in order to first define ''what is the best architecture'' and then ''what are the technological solutions'' to be used at interfaces between subsystems. This activity, that can be called concept generation [START_REF] Ullman | The Mechanical Design Process[END_REF], generates several iteration loops on the specification of subsystems.

These iterations are vitals to reach the best tradeoff between all subsystems. In the same time, these loops bring a lot of uncertainties to the geometrical requirements of a subsystem like the main gearbox (MGB).

As the MGB is composed of hundreds of components, the variability at the interfaces can generate many long redesign loops. The duration of these iterations will in fact directly limit the creativity of the design team, who is already under pressure by lead-time and non recurring cost (NRC) targets.

Keywords:

Computer aided design (CAD), Design method, Gearbox This paper describes a parametric design methodology implemented to reduce gearbox design phase. A helicopter gearbox is a complex mechanical system, in which parts are heavily loaded in order to save weight but that have to run for thousands of hours without failure. Such a system is strongly optimized therefore the smallest change in specifications or in geometrical environments can drastically jeopardize the scheduling. In order to speed up the gearbox design process and to strengthen its robustness regarding specifications variability, a parametric design methodology has been developed thanks to a close collaboration between the laboratories and Eurocopter.

Why change the gearbox design process?

On today programs, the development targets on both lead-time and NRC are the result of the global competitions on the worldwide market. The later has forced the introduction of project management ''best practices'' everywhere. But at the end, the customer is buying the added value brought by the products and this added value is highly linked with the quality of design activity.

Wientahl et al. compare the time to market and the costs induced by product specification changes in the western countries, USA and Japan in 1998. The authors show that the western companies are less competitive, due to late changes of the product [START_REF] Wientahl | Logistic Orientated Product Design[END_REF].

Consequently, Eurocopter's methodology for designing MGB has been modified according 2 main objectives:

-realizing the conceptual design of a MGB in 2 weeks of work, in order to quickly and efficiently propose several concepts to chief engineers, -performing parametric CAD modeling during the conceptual design phase, in order to be robust to some changes of development phase.

3. The parametric approach

Background

The parametric capacities of CAD tools have been developed for many years, and several examples of parametric models directly managed by the CAD construction tree are presented in Refs. [START_REF] Andrews | Design Reuse in a CAD Environment[END_REF][START_REF] Chih-Hsing | Computer Aided Parametric Design for 3D Tire Mold Production[END_REF][START_REF] Sergio | The Design of a Washing Machine Prototype[END_REF].

´ functional aspect from the design one and apply this idea to the design of a washing machine. The HOMEBOX project developed in [START_REF] Sergio | The Design of a Washing Machine Prototype[END_REF] shows a remarkable work around the multiple and specialized knowledge integration on a unique product. Jiang et al. [START_REF] Jiang | Research and Development of a Constraint-Based Product Family Design and Assembly[END_REF] apply the concept of product family to mass customization. They propose to integrate in the definition of the part customizable parameters and a set of rules. They introduce also the Gen-Part concept. According to this concept, the classical geometrical definition of the part is replaced by a prototype of its functional description and the product becomes a set of Gen-Part. Ma et al. [START_REF] Ma | Associative Feature Modeling for Concurrent Engineering Integration[END_REF] and Ranta et al. [START_REF] Ranta | Integration of functional and feature-daseb product modeling -the IMS/GNOSIS Experience[END_REF] have studied the integration of structured knowledge in products, using a feature concept. In particular, Ma et al. have introduced and applied an interesting concept called associative features.

Even with such a rich background, Tomiyama et al. underlined 3.2. Why a specific parametric approach is needed? possibilities of parameter propagations.

Some Eurocopter specificities

It was chosen to work with CATIA V5 to take advantages of powerful 3D representations together with its native parametric modeling capabilities. Another choice was to manage equations that drive the design parameters out of CATIA from Excel files. The target was to simplify the work of designers during the building of the model and also to ensure the protections of the Eurocopter knowledge that are linked with the choice of the parameters. During the development phase, the CAD model are indeed used for data exchanges between Eurocopter, engineering subcontractors, partners or manufacturing suppliers, so the model must not integrate specific know how.

Organizing the functional parameters

The wish to reach quickly a MGB conceptual design puts in light the need to make extensive design re-use of CAD models (Fig. 2). As the power transmitted by MGB are very different from one aircraft requirements that always happens in the early stage of to another, the first step was the creation of different classes of parametric CAD models for kinematic components like roller and ball bearings, spur and spiral bevel gears. These models are stored in a database in which designers come to take the parametric models that fit best with the topology of their mechanical assemblies under construction. Each parametric CAD model is linked with an Excel file that contained a fixed list of specifications. It gives the general sizing of the bearings or gears.

As the MGB is an organized assembly of power gear trains, the kinematic links and geometrical positioning of all axes have to be Sergio et al. [START_REF] Sergio | The Design of a Washing Machine Prototype[END_REF] present another way that consists in separating the managed at the assembly level. In order to manage these constraints the best, a new file called ''MGB functional skeleton'' has been built to be used as reference for all CATParts or CATProducts. This CATPart file contains some high level geometrical constraints which are linked to subsystem interfaces or aircraft geometrical configurations. For example, the following elements are taken into account:

-geometrical positioning of engines axes, -geometrical positioning of main and tail rotor axes,

The challenge for Eurocopter was to develop a robust parametric approach for a product with high ''inborn complication'', as defined in Ref. [START_REF] Lu | Complexity in Design of Technical Systems[END_REF], without increasing the ''acquired complexity'' during the design phase. The parametric design experiences recalled in the previous paragraph were all limited to isolated parts or simple assemblies. A MGB is composed of hundreds of components, so the methodology has to be robust to the size of the CAD model used.

In order to limit the duration of the design phase, the design of a MGB is done by a team of designers. The methodology has to ensure the consistency of the design parameters across several CAD models under construction at the same time. Some design parameters are piloted by the maturity of other helicopter subsystems like engines or servo actuators. The approach has to allow some interactions between the MGB and other subsystems design parameters. As helicopter subsystems digital models are managed in context thanks to a PDM, the parametric approach has to comply with the PDM environment, which offers few

) T ( ] [ G $ I D F

in [START_REF] Tomiyama | Design Methodologies: Industrial and Educational Applications[END_REF], a significant gap between the number the design methodologies proposed by academics and the number that find industrial applications [START_REF] Roy | Recent Advances in Engineering Design Optimisation: Challenges and Future Trends[END_REF][START_REF] Ge | An Axiomatic Approach for ''Target Cascading'' of Parametric Design of Engineering Systems[END_REF][START_REF] Vaneker | What-if as Synthesizing Working Method in Product Design[END_REF]. -inputs and output speeds, -input power, etc.

The design parameters of this ''functional skeleton'' are also piloted by an Excel file (Fig. 3).

CATProduct is the key to ensure the robust propagation of design parameters towards CAD models inside a PDM Software.

Organizing the design files

As the MGB conceptual design can easily be made of more than one hundred parts, the CATIA and excel files have to be organized. The MGB is split in several modules that will be under the responsibility of a given designer. Each module is positioned thanks to interface files that are stored separately. If a module is complex with many parts or housings, the designer can built his own ''local skeleton'' that will contain all information to ease the positioning of all parts. The designers use design parameters via CATIA publications from the ''functional skeleton'' or from interface drawings. Fig. 4 gives a synthesis of this new file organization that allows a simultaneous work of several designers on an MGB parametric design. By this way, the consistency of the design parameters is ensured for all users. The propagations of changes made on high level requirements can then be done in a controlled manner whatever the numbers of CATParts.

3.6. Organizing the design team creation of a new position that has been called the ''Design Architect''.

The ''Design Architect'' is responsible for the integrity of the ''functional skeleton'', and for the propagations of design parameters inside the CAD model. He has also to detect conflicts between modules and to help designers to solve them. Thanks to the architecture of data that has been built, the designers do not modify any data at the assembly level because these are managed by the ''Design Architect''. It also avoids having several workers on the same file, which will limit the risks of data corruption or involuntary data deleting.

The benefits of the approach

The hard points mentioned in Section 3. This parametric approach has proven its robustness on several This new way to use a CATPart to manage the geometrical Eurocopter MGB conceptual designs in the last years. This positioning instead of using the assembly features of the approach gives really a breakthrough in the lead-time to produce helicopter MGB conceptual design. The lead-time to reach a 3D model mature enough to discuss efficiently with other subsystems design teams has been reduced to only 2 or 3 weeks depending the size of the MGB. This improvement of design speed allows being more ambitious in the preliminary concept and so gives more room to designers creativity. Moreover this is a true benefit for the development of innovative design that has generally a very high mortality rate in the first month of a development program. The externalisation of design parameters calculation in Excel files allows protecting Eurocopter design knowledge when the CAD models are exchanged outside Eurocopter (Fig. 5).

Some limitations

Such an approach, on which the CAD Part design is guided by some external high level publications, seems to be intuitive because it follows a top down cascading of the design specifications. In fact, it is not obvious seen from the designer position working on the CAD tool.

During the building of the CAD parts, designers have to be extremely rigorous as they import the publications from the ''functional skeleton'' in order to drive design features. The help of the ''Design Architect'' is often mandatory.

As the CAD tools have become incredibly rich, some simple series of holes on a given diameter can be done in several ways. The experience has shown that depending on the designer background, the propagations of parameters will work perfectly even with some significant modifications in the complex CATParts like housing. The The usage of a parametric approach requires the proper organization of the design team. Traditionally, the design team is built around a work package manager who is in charge of the project, and of 1-15 designers. The need to guaranty a proper propagation of parameters into all CAD models has led to the experience has also shown that using specific CAD features can easily jeopardize some parametric links when the CAD part under modification. The recovery of the links with design parameters will be later a difficult task that could lead to the complete redesign of the CATPart.

It's still difficult to fix a complete methodology down to the lowest level of CAD design because the topology of the MGB housing is always different.

The interfaces and the gear trains axes are driven by external publications, but all other features which need to define bearing locations, fixation holes or wall thickness are managed locally by the designers. The parametric modeling of complex housings is still a field on which the design methodology has to be improved in order to easily generate a full 3D solid model.

Conclusions

The development of this parametric approach has brought many benefits in the lead-time reduction and in the quality of Eurocopter gearboxes conceptual design. Even if there is a huge benefit for MGB concept generations, a design choice is also highly linked with concept evaluations that will be mainly based on deflection and stress analyses. Therefore one of the next challenges for Eurocopter is the propagation of the design parameters not only in the CAD model, but also in finite elements models (FEM) (Fig. 6).

Mastering a parametric approach that links the CAD model, the meshing, the loads and boundaries conditions are new challenges that can significantly contribute to mechanical design lead time reduction.

T I D ) ( F [ $ ] G

Fig. 1 .

 1 Fig. 1. Example Eurocopter main gearbox CAD Model.

Fig. 2 .

 2 Fig. 2. Example of parametric roller bearing CAD model.

  2 have been solved thanks to a new operational design methodology based on the usage of CATPart as ''functional skeleton'', a new organization of the design files and a new organization of the design team.

Fig. 3 .

 3 Fig. 3. Example of main gearbox ''functional skeleton''.

FFig. 4 .

 4 Fig. 4. Cartography of data files used to implement parametric design methodology.

Fig. 5 .

 5 Fig. 5. MGB housing with axes driven by external publications.

Fig. 6 .

 6 Fig. 6. Example of MGB FEM analysis.