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class PaymentWorker attr_accessor :user_id def initialize(user_id) selfuser_id = user_id end def perform user = Userfind(user_id) payment_info = userpayment_info if payment_infocharge(1999)

0.20 0.15 0.10 0.05 0.00 0 2 4

Figure 10.4.3 Convergence of extinction rate versus number of realizations and number of iterations for fractional volume f = 15% and N = 2000 and 4000. The extinction rate is normalized to the independent scattering case. Other parameters are Es = 3.2Eo and ka = 0.2.

( j)

0.25..,.---------------..., 0.20 0.15

Figure 10.4.4 Convergence of extinction rate versus number of realizations and number of iterations for fractional volume f = 25% and N = 2000 and 4000. The extinction rate is normalized to the independent scattering case. Other parameters are Es = 3.2E o and ka = 0.2.

We solve (10.4.37) by iterations. In this section we present results for dielectric spheres of permittivity Es = 3.2Eo . In Fig. 10.4.3 we show the results of Ke/(Ke)ind at a fractional volume of 15% as a function of realization by using 4 and 6 iterations. We note that for 2000 spheres there is practically no difference between 4 and 6 iterations. For 4000 spheres, the results of 4 and 6 iterations are practically identical. This demonstrates numerically the convergence with iterations for the simulated extinction rate. The results in

8.ooe-4 7.00e-4

~.,.,

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Under the de nitions and predictive ow control algorithm de ned above, if Vmax < and [( p V )/( V )] H1 (1) 1, we have [81] q(n) q0 (n) 2C1 , where C1 is a constant that does not depend on n. For l = n n 0 we de ne the accumulated error as

'" c:

6.00e-4 5.ooe-4 4.00e 4 3.ooe-4 2.00e-4

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l.vvg-....

----:~ """7'",,..

...../

( j) =

. ,

......

.,....

a-c:~;Y-:;::';;------------

10 15 20 25 fractional volume (In "10)

( j)

Figure 10.4.5 Extinction rate normalized to the free-space wave number as a function of the fractional volume of scatterers. The plots show calculations based on independent scattering, Foldy's formula, QCA-PY, QCACP-PY, and Monte Carlo simulations. Other parameters are Os = 3.200 and ka = 0.2. Fractional Volume 5% 15% 25% 0.748 0.381 0.224 (4th)* (6th) (9th) 0.732 0.398 0.213 (4th) (6th) (9th) 1 1 1 0.969 0.915 0.870 0.318 0.150 0.674 0.402 0.215 0.732

When the user signs up for the premium service, we would add a task to the system that charges her credit card immediately, with a high priority, such as this:

Method Monte Carlo 4000 spheres Monte Carlo 2000 spheres Independent scattering Foldy QCA PY QCA-CP-PY

Table 10.4.1 Numerical values of the ratio h;e/(h;e)ind. (*The numbers in parentheses below the Monte Carlo values denote the numbers of iterations used.)

Fig. 10.4.3 also demonstrate convergence with the number of realizations. However, there is a small difference between the results with 2000 and 4000 spheres. The small difference can be attributed to the difference in the number of spheres that lie close to the edge of the cubic box in the cases of 2000 and 4000 spheres. The case of 25% is shown in Fig. 10.4.4 with 6 and 9 iterations. In Fig. 10.4.5 we compare the simulated results of 5% (30 realizations), 15% (20 realizations), and 25% (20 realizations) with analytic approximations of independent scattering, Foldy's approximation, QCA-PY, and QCA-CP-PY. We plot Kelk as a function of fractional volume f. We note that independent scattering predicts a linear increase with fractional volume, while Foldy's approximation predicts a monotonic increase with fractional

( j)

volume at a slower rate than that of independent scattering. The QCA~PY and the QCA-CP~PY predict saturation and a decrease with further increase of fractional volume and are in good agreement with the simulations. Note that these simulations are for hard spheres without interparticle force. The results are tabulated in Table 10.4.1. 4.4 Simulation of Absorption Coefficient

For particles with absorption, the absorption coefficient can also be simulated. Upon solving the Foldy-Lax multiple scattering equations of (10.4.37), we obtain the coefficients w~~)(l) and wc:J(I) which gives the final exciting field of particle l. The power absorbed by particle l is then, from (2.8.84) in 2 of Volume I,

W(l)

= __1_ '"' {lw(M)(1)1 2 (ReT(M)(I) + IT(M)(l)12) 2

+ Iw~J(l) 12(ReT~N)(I) + IT~N)(I)

C ( j1 j2) =

(10.4.73)

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L W~l)

(10.4.74)

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