Aprendizaje individual y aprendizaje organizativo

Suppose that the average interconnection length is R and the total number of interconnections is S. Let ξ_lbe a random variable that denotes the length of the l-th interconnection, l = 1, 2, ..., S, then the average length of interconnections becomes R =P_S

l=1ξ_l/S. By taking the expectation, E[R] becomes

By connecting the average interconnections length to the physical mapping parameters, such as interconnection channels, two important random variables need to be defined:

(1) X_i,j = length of the i-th interconnection that is going through channel j.

(2) Y_j = total number of interconnections at channel j.

X_i,j and Y_j are assumed to be independent, as the length of the interconnections are the uti-lization of interconnects at each channel are usually considered separately in the routing algo-rithms. Suppose N channels are available for realizing these global communication links, each channel has W tracks and N W À S. Assume that the routing and placement are symmet-ric for the two sides of a link, as depicted in Fig. 3.3, the total interconnection length becomes P_S

E[R] becomes

By applying the identity¹E[P_Y

i X_i] = E[Y ]E[X], where Y is assumed to be a random variable that is independent of the X_i’s and X_i is independent of X_j for i 6= j. The expected interconnec-tions becomes:

In the following sections, the derivation for Y (j) and X(j) will be presented.

Derivation of Y (j)

Y (j) is the expected number of interconnections in channel j. Suppose that all interconnections for two modular blocks go through any one of the channels, Y (j) can be modeled as the number of interconnections that enter the channel j. This assumption simplifies the process of interconnect generation, which does not intend to model the exact routing mechanics. However, this provides a simple analytical model for the results of routing algorithms. Furthermore, this assumption yields an optimistic routing, as a shortest path is assumed for connecting two tiles through the routing channel. More sophisticate models can be extended based on this work to provide additional detail characterizations based on different routing assumptions.

1See proof in [Ros02].

3.3 A Stochastic Model for Interconnections Length Prediction 70

An example is depicted in Fig. 3.6. An interconnection is generated at tile (i, k) and traverses to join the channel j. The interconnection traverses through the shortest path to the nearest available channel, following the direction of the arrow in the figure. It is assumed that there are always switches and connection points available in the module for the interconnection to traverse locally.

However, when all tracks in the global interconnection channel are used up, the interconnection has to traverse further to find an available track to pass through the channel.

Let Q^j_(i,k)be the indicator variable for the event that an interconnection is generated at tile (i, k) and traverses to connect to one of the long wires at channel j. It is assumed that interconnections originate at the tile (i, k) with a probability κ, and that the interconnection originating at (i, k) moves toward channel j with probability θ. Following [ES81] to model the interconnection length, let D be the variable for the distance that the interconnection traverses before stopping at a long wire channel; D has a geometric distribution with parameter λ_d, as the geometric distribution is a discrete version of the exponential distribution. Further, this assumption has been verified from an experiment. A 256-bit link for connecting two modular blocks which are 50 tiles apart is placed and routed using Xilinx ISE design environment for a Virtex-4 (XC4VLX200) FPGA. The routing results can be obtained using the Xilinx FPGA Editor. With the help of the FPGA Editor, distribution of the distance parameter D can be obtained by counting the number of tiles that an interconnection traverses before joining the long range routing track.

Fig. 3.7 shows the distribution of distance that interconnections travel vertically before connecting to a long wire. The probability mass function (pmf) for D is denoted as h_λh,d(i,j), where d(i, j) =

|j + u/2 − i| is the distance between row i and channel j. Thus, the pmf for Q^j_(i,k) is z_ij = κθh_λh,d(i,j)= κθ(1 − λ_h)^d(i,j)λ_h.

The total number of interconnections at channel j becomes Y_j =P_u

i

P_v

kQ^j_(i,k), which is a joint probability of interconnections generated from all tiles that converges to channel j given that the total number of interconnections is S/2 and Y_j ≤ W . In the following, it is shown that Y_j can be approximated by a Poisson distribution with parameter λ_j = vP_u

i z_ij.

Lemma 1. For any interconnection that is generated at tile (i, k) and joins a long wire track at channel j with probability z_ij, the number of interconnections at channel j, Y_j⁰, can be ap-proximated by a Poisson distribution with parameter λ_j = vP_u

i z_ij where i = 1, 2, ..., u and j = 1, 2, ..., N .

0 5 10 15

Figure 3.7: Distribution of the parameter D, which is the distance an interconnection traversed before connecting to a long wire.

Proof. Let J_i^j be the total number of interconnections generated from row i and joining tracks at channel j. Thus, J_i^j =P_v

kQ^j_(i,k)can be obtained. Therefore P {J_i^j = n} =

µv n

¶

(z_ij)ⁿ(1 − z_ij)^v−n (3.6) It is obvious that J_i^jhas a binomial distribution with mean vz_ij(1 − z_ij). Therefore, this binomial distribution can be approximated by a Poisson distribution²with parameter λ_ij = vz_ij.

Since Y_j =P_u

i J_i^j and as J_i^j is a Poisson distribution variable with parameter λ_i, Y_j becomes a Poisson distribution variable with parameter λ_j = vP_u

i z_ij

There are N independent variables Y_j, j = 1, 2, ..., N , which have Poisson distribution with pa-rameters λ_j, j = 1, 2, ..., N . The probability mass function for Y_j is therefore

2To approximate binomial probabilities is one of the important applications of the Poisson probabilities. It can be easily see that when let λ = np be a constant, where n and p are the binomial parameters, the binomial pdf converges to a Poisson pdf when n → ∞.

3.3 A Stochastic Model for Interconnections Length Prediction 72 truncated Poisson, as Y_j has a Poisson distribution with parameter λ_j. Therefore,

P {Z_j = y} = P {Z_j = y, Z_j ≤ W }

It is needed to compute the joint probability mass function (pmf) for variables. Let q⁰be the pmf forP_N

j Z_j=1and q_j be the pmf ofP_N

k6=j,n=1Z_k. The variables q⁰and q_j can be readily computed by convolutions as follows,

q_j(y) = P

where (∗) denotes convolution³of probability mass functions. By substituting Eqs. 3.10, 3.11 and 3.8 into Eq. 3.9, it yields,

P

Given the total number of bits in the link (S), the equation expresses the number of interconnec-tions travelling via channel j limited by the number of available wires (W ) in the channel.

Approximation of Y (j)

Consider a case where W , channel width, is arbitrarily large or is adjustable. Then, the expression for Y can be further simplified. Y_j has a Poisson distribution with mean λ_j and by using the property of Poisson random variable,P_N

j Y_j has a Poisson distribution with parameter P_N

j λ_j, then

3The convolution of f and g, where f , g are independent (not necessary identical) pmf, is defined as f ∗ g = P_n

kf (k)g(n − k)

3.3 A Stochastic Model for Interconnections Length Prediction 74

In other words, the conditional distribution for Y_j given that P_N

j Y_j = S/2, has a binomial

It is interesting to interpret the distribution of interconnections in routing channels as a binomial distribution, assuming that the channel width is arbitrarily large. It gives the probability as shown in Eq. 3.14 of getting exactly y successes in S/2 trials in generating interconnection across the two channels.

Derivation of X(j)

The length of an interconnection at channel j in the communication link X_j can be considered as a sum of three segments. It is best illustrated using Fig. 3.8. For net (a), the length would be the sum of the length from the tile in the left area to M₁ (Λ), (L) and the length from M₂to the tile in the right area (Λ⁰). Similarly, for net (b), the lengths are the sum of three segments. Thus, X_j = L + Λ_j+ Λ⁰_jcan be obtained and

X(j) = E[X_j] = L + 2E[Λ_j] (3.16)

where E[Λ_j] = E[Λ⁰_j].

Figure 3.8: Illustration of parameters for computing the X(j). (a) Direction interconnection. (b) Fringed interconnection.

The pmf z_ij defined earlier for the probability of an interconnection connects a track at channel j from any tile at row i. By assuming an uniform distribution of the placement, a pmf ζ_i,j for channel j that connected from any of the tile at row i can be obtained as follows,

ζ_i,j = z_ij vP_u

i z_ij (3.17)

Thus, all possible interconnections length can be readily explored by analyzing Manhattan distance shown in Fig. 3.9. Basically, E[Λ_j] can be clearly defined based on two different cases, which are j ≤ u/2 and j > u/2, for direct and fringed interconnections, denoted by E[Λ^j_j≤u/2] and E[Λ^j_j>u/2] respectively. To compute E[Λ^j_j≤u/2], the constraint area is clearly defined (in Fig. 3.9).

The expected value would be the sum of enumerating all possibilities in these three panels.

3.3 A Stochastic Model for Interconnections Length Prediction 76

Figure 3.9: An example of computing the Manhattan distance within a modular block. Each number in the block represents the Manhattan distance to connect the particular tile to the point at M₁.

Hence,

E[Λ^j_j≤u/2] =

u/2+j−1X

l=0

Xv k=1

(k + l)ζ_u/2+j−l,j + Xv k=1

kζ_u/2+j,j +

u/2−jX

l=1

Xv k=1

(k + l)ζ_u/2+j+l,j (3.18)

For the case of E[Λ^j_j>u/2],

E[Λ^j_j>u/2] = Xv

i=1

Xu l=1

(k + l + j − u/2 − 2)ζ_u−l+1,j (3.19)

0 5 10 15 20 25 30

Figure 3.10: Plots of number of interconnections, Y (j), and average interconnection length, X(j).

3.3.2 Summary

The expected number of interconnections that go through channel j is Y (j) and the expected length of interconnection at channel j is X(j). Fig. 3.10 shows the plot of Y (j) (upper graph) and X(j) (lower graph) as a function of j. The channel is indexed as the distance from the center of the placement area. Three cases of Y (j) are shown. The approximation case assumes that the number of long wires per channel W is a very large number, so that the number of interconnections at each channels can be approximated by a simple express in Eq. 3.15. For different channel capacity, W = 24 and W = 18, the number of interconnections in the channel can be approximated by Eq. 3.13. Clearly, when the number of available tracks in a channel is smaller (W = 18), more channels are required to fit all interconnections, as depicted in the figure. Interconnections lengths can be modeled by Eq. 3.16. Interconnections in a channel that is further away from the placement area have longer length. Specifically, for the direct interconnection, the length is modelled by Eq. 3.18 and for the fringed interconnection, it is modelled by Eq. 3.19. Based on X(j) and Y (j), the overall average and variance of interconnection lengths can be derived and thus compute the delay.

3.3 A Stochastic Model for Interconnections Length Prediction 78