s J (n) = (n) + [ T ] 1 i=0 in .NET Creator 3 of 9 in .NET s J (n) = (n) + [ T ] 1 i=0

s J (n) = (n) + [ T ] 1 i=0 using barcode creator for .net framework control to generate, create code 39 full ascii image in .net framework applications. USPS POSTNET Barcode T 1 c(Q(i; x( ) .net framework Code39 ) . .

J (Q(t)),. n 1.. (11.53). The parameter i visual .net Code-39 s again a constant that may be chosen to attempt to minimize the variance of the uid estimator..

Control Techniques for Complex Networks Draft copy April 22, 2007. To implement the uid estimator we need to be able to compute J . This may be moderately time-consuming (computationally speaking) relative to the time taken to simply simulate the process Q. Be that as it may, the time taken to compute J is relatively insensitive to the congestion in the system.

Example 11.4.3.

Fluid estimator for the simple re-entrant line Shown at right in Table 11.1 is a summary of results obtained using the uid estimator. The best value of using the batch means method was found to be close to unity in each of the simulations, particularly at high loads where it was found to be within 5% of unity.

Observe that for low traf c intensities, the uid estimator yields reasonable variance reductions over the standard estimator. However, because it is more expensive to compute than the standard estimator, these results are not particularly encouraging. But as the system becomes more and more congested, the uid estimator yields signi cant variance reductions over the standard estimator, meaning that the extra computational effort per iteration is certainly worthwhile.

In summary, in the preceding example the quadratic estimator produces useful variance reductions in light to moderate traf c at very little additional computational cost. It is less effective in simulations of heavily loaded networks, but could potentially provide useful variance reductions in this regime if a better choice of D can be employed. The uid estimator provides modest variance reduction in light to moderate traf c, but appears to be very effective in heavy traf c.

In complex networks the additional computational overhead in computing the uid estimator can be substantial. In such cases we turn to a workload relaxation to construct an effective shadow function that is easily computed..

11.4.6 Shadow functions and workload We rst consider a special case for which we can obtain exponentially decaying error bounds in the smoothed estimator. Consider the two-dimensional CRW workload model, Y (t + 1) = Y (t) S(t + 1)1 + S(t + 1) (t) + L(t + 1), Y (0) Z2 , (11.54) +.

where S is a diag .net framework 3 of 9 barcode onal matrix sequence of the form {S(t) = diag (S1 (t), S2 (t)) : t 1} taking values in {0, 1}2 , and L is a two dimensional sequence taking values in Z2 . + The mean values i = E[Si (1)], i = E[Li (1)] satisfy i > i for i = 1, 2.

We restrict to the work-conserving policy, so that (11.54) becomes, Y (t + 1) = [Y (t) S(t + 1)1 + L(t + 1)]+ . The Markov chain Y is regular since the quadratic V (x) = 1 x 2 The uid model under the non-idling policy is expressed, y(t) = [y ( )t]+ ,.

solves (V3).. t 0, y R2 . + Control Techniques for Complex Networks If c : R2 R+ is piecewise linear, + Draft copy April 22, 2007. c(y) = max ci , y ,. 1 i c and if c( ) = visual .net Code 3/9 ci , for a unique i {1, . .

. , c }, then the assumptions of Proposition 5.3.

13 hold so that the uid value function is C 1 , with J(y) =. c(y(t)) dt,. y(0) = y. A smoothed estima tor for the steady state mean of c(Y (t)) is naturally obtained using = J, giving J (y) = E[J (Y (t + 1)) J(Y (t)) . Y (t) = y], y Z2 , and b + s J (n) = b [c(Y (t)) + J (Y (t))], b n 1.. (11.55). Proposition 11.4. 4.

Consider the two-dimensional CRW workload model (11.54) under the non-idling policy. It is assumed that i > i , i = 1, 2 and for some 0 > 0, E[e 0 .

L(1). ] < . Suppos e that c : R2 R+ is piecewise linear, and purely linear in a neighborhood + of the vector R2 . Then the smoothed estimator (11.

55) satis es the LDP + (11.10a,11.10b) with non-zero rate function: There exists > 0 such that for any r+ ( , + ), r ( , ), 1 s log P J (n) r+ b n n 1 s lim log P J (n) r b n n lim = IJ (r+ ) < 0 b = IJ (r ) < 0.

b. Proof. There are VS .NET barcode 3 of 9 two steps to the proof: First, we show that the function c + J b is uniformly bounded over all of R2 .

Next we establish geometric ergodicity: We + establish Condition (V3) with V = J, and then Condition (V4) with V = exp( for some > 0. Proposition 11.2.

4 then implies the desired result. The smoothed estimator is ordinary Monte-Carlo applied to the function, c(y) + J (y) = E[c(Y (t)) + J(Y (t + 1)) J(Y (t)) . Y (t) = y] . b J). (11.56). To show that this is bounded over R2 we rst apply Proposition 5.3.13 to conclude that + J is C 1 .

Moreover, the dynamic programming equation (5.50) holds in the form, D0 J = J, ( ) = c Consequently we can apply the Mean Value Theorem to obtain, J(Y (t + 1)) J(Y (t)) = J (Y ), Y (t + 1) Y (t) = J (Y (t)), Y (t + 1) Y (t) + J (Y ) J (Y (t)), Y (t + 1) Y (t) (11.57).

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