Gestión del talento en la administración pública

We now wish to introduce long-range dependence into the source model and derive explicit estimates of boundsq…x†andQ…x†of Proposition 5.4.1 for such a case. As shown below, a natural way to do this is to assume a heavy tail for the distribution of either the silence period, the activity period, or both. In the rest of this section, we ®rst specify the LRD behavior of the rate process associated with heavy-tailed distributions for on=off durations, then provide the estimates for bounds q…x†and

Q…x† in such a case. We conclude this section by providing a weak convergence theorem for the limiting input process fOtg obtained in Section 5.3, which, once rescaled in space and time, converges toward the so-called fractional Brownian motion.

5.5.1 LRD Properties

Recall [6] that a stationary processfl_tg is said to exhibit short- (resp. long-) range dependence if its correlation functionrde®ned in Eq. (5.6) is such that the integral

„_‡1

0 r…t†dtis convergent (resp. divergent). If the distributions of on and off periods

A and B are Coxian, that is, their Laplace transforms a* and b* are rational functions, then standard results for Laplace transform inversion applied to expres- sion (5.7) entail that the correlation functiont7 !r…t†tends exponentially fast to 0 ast" ‡1, thus corresponding to short-term dependence for processfl_tg. Now, the situation appears totally different in the case when the distribution of either off durationAor on durationBis heavy tailed. A distribution functionFis here de®ned to be heavy tailed if

1 F…t† ˆh…t†_tr …5:28†

fort>0 and 1<r<2, where functionhhas a ®nite limith at in®nity. A typical example is the Pareto distribution, where

F…t† ˆ1 _t‡y_y

r

for some y>0. As a consequence of de®nition (5.28), the Laplace transform

f*:p>07 !„₀‡1e pt _{dF…t†of a heavy-tailed distributionFwith ®nite meanmand} powerr2 Š1;2‰veri®es

f*…p† ˆ1 mp‡hG_r…2₁ r†pr_‡o…pr_{† …5:29†} for small positivep, whereGis Euler's function (see Section 5.7.3). For 1<r<2, such an expression for the Laplace transform near 0 in fact characterizes the heavy- tailed behavior of the corresponding distribution F (in the case when rˆ2, Eq. (5.29) is identical to a classical Taylor expansion, corresponding to a regular distribution having a ®nite variance). We can then show the following.

Proposition 5.5.1. Assume that the Laplace transforms a*and b*of the off and on periods have expansions

a*…p† ˆ1 p

a‡arpr‡o…pr†;

respectively, withpositive coef®cients a_r and b_s and where powers r;s2Š1;2Šare suchthat r‡s6ˆ4. The covariance function of ratefl_tg then veri®es

r…t† r0

tq 1 …5:31†

for large t, withqˆmin…r;s†and

r0ˆ bn 3_a r G…2 r† resp:r0ˆ a…1 n†3bs G…2 s† ! if r<s(resp. s<r)and r₀ˆbn3ar‡a…1 n†3bs G…2 r† if rˆs.

In other terms, assuming such heavy-tailed distributions for A and B with

r;s2 Š1;2Š entails long-range dependence for process fl_tg since „₀‡1r…t†dt

diverges for 0<q 11. The proof of Proposition 5.5.1 can be performed simply by inserting expansions (5.30) for a* and b* into expression (5.7) and then using Tauberian Theorem 5.7.1 recalled in Section 5.7.3 for deriving the asymptotics ofr…t†for larget. Concerning input processfotgwithotˆ„0tludu, it is easy to express its varianceD2_{t†de®ned in Eq. (5.3) as}

D2_{t† ˆ2}…t

0…t u†r…u†du …5:32† in terms of the covariance functionrof processfltg. As a consequence, it readily follows that if

…_‡1

0 r…u†du<‡1;

then D2_{t† ˆO…t† as t" ‡1. On the other hand, if behavior (5.31) holds with} 1<q<2, we have instead

D2_{t† lt}3 q _5:33† ast" ‡1, where

lˆ 2r0

…3 q†…2 q† …5:34†

r₀ being expressed as in Proposition 5.5.1.

5.5.2 Impact on Queue Content Distribution

To make boundsq…x† and Q…x†of Proposition 5.4.1 explicit in the case of heavy- tailed distributions for on and off periods, we ®rst need some preliminary estimation of intermediate quantitiesm…t†andD…t†de®ned in expressions (5.21).

Lemma 5.5.2. Assume that expansions (5.30) hold for the Laplace transforms a*

and b*. Then quantitym…t†de®ned in Eq. (5.21) veri®es

m…t† 3 ₂ q

l

s2t2 q …5:35†

for large t, withqˆmin…r;s† and constants s2 _{and l de®ned in Eqs. (5.2) and}

(5.34), respectively.

The proof is deferred to Section 5.7.4. Note that the average differencem…t† of input due to different initial conditionsl0ˆ0 orl0ˆ1 grows with increasing t rather than tending to a constant, an eloquent testimony to long-range dependence. It further follows from estimate (5.33) and the latter result that the variance D2…t†

introduced in Eq. (5.21) satis®es

D2…t† ˆD2_{t† m}2_t†s2_D2_{t† …5:36†} for largetsince 2…2 q†<3 q, whenq>1. These observations lead to the next proposition, the main result in this chapter.

Proposition 5.5.3. Assume that the distributions of either on or off periods have heavy tails with respective power r;s;2 Š1;2‰and that expansions (5.30) hold. Then limiting distributionx7 !h…x†of Proposition 5.4.1 has the logarithmic estimate

lim x"‡1 1 x2…1 H†logh…x† ˆ k2 2 …5:37† where H ˆ …3 q†=2; kˆ 1 HH_{1 H†}1 H gH  l p and withlspeci®ed in Eq. (5.34)

Proof. First, consider lower bound (5.22). As expansions (5.30) hold, estimate (5.33) applies to thatD2_{t† lt}2H _{for large t and 2Hˆ3 q. Use the variable} change tˆxu and write D…mx† ˆpl…ux†H_{‰1‡d…ux†Š, where d…t† !0 as}

t" ‡1. Lower bound (5.22) then reads

q…x† ˆsup u>0 F x1H l p r…u† 1‡d…ux† ! ; …5:38†

where

r…u† ˆ1‡_uHgu:

It is easily veri®ed that function u7 !r…u† has a unique minimum at point

u*ˆH=g…1 H† with r…u*† ˆgH_=HH_{1 H†}1 H_{. Supremum (5.38) entails, in} particular, thatq…x† q…~ x†with

~ q…x† ˆF x1H l p r…u*† 1‡d…u*x† ! :

Using the asymptotics F…z† exp… z2_=2†=zp_{2p for large z, we may write that} logF…z† ˆ z2_=2‡o…z2_{†. Since d…u*x† !0 as x" ‡1, lower bound q…~ x† thus} veri®es

logq…~ x† ˆ 1_{2 1‡}x1_d…uH_*x†r…u*†

l p !2 ‡o _1‡x1_d…uH_*x† !2 ˆ 1₂ x1 Hr…u*† l p 2 ‡o…x1 H†2: The latter consequently implies that

lim inf

x"‡1 1

x2…1 H†logq…x† lim infx"‡1 1

x2…1 H†logq…~ x†

ˆ k₂2 …5:39†

withkˆr…u*†=pl:

Second, consider upper bound (5.23). Performing again the variable change

tˆxuin the integral gives

Q…x† ˆ x1 H 2pl p …_‡1 0 exp x2…1 H† 2l Jx…u† ! Lx…u†du; …5:40† where Jx…u† ˆ r…u† 2 ‰1‡d…ux†Š2; Lx…u† ˆ R…ux;x† uH_{‰1‡d…ux†Š};

and with functionsu7 !r…u†and u7 !d…ux† de®ned as in Eq. (5.38). To evaluate integral (5.40), we apply the Laplace method [12] for the estimation of integrals with

exponential integrand. For ®xedx, the continuous functionu7 !J_x…u†tends to‡1

asu#0 andu" ‡1(in fact, it isO…u 2H_{†for small positiveuandO…u}2…1 H†_†for largeu). It therefore has a minimum at some pointu_x*. AsJ_x…u† !r2_{u†asx" ‡1} for boundedu, it is clear thatu_x*!u* as x" ‡1, where u* is the minimum of

u7 !r2_{u†. The contribution of the exponential factor to the value of the integral is} thus preponderant in the neighborhood of that minimum u_x*. Concerning the nonexponential factor L_x…u† in Eq. (5.40), we note from Lemma 5.5.2 with

Hˆ …3 q†=2 that

x…1‡gu†m…ux†s2 D…ux†2 !g

as x" ‡1 and u!u*. As D…ux† D…ux† for large x and ®xed u in view of estimate (5.36), we successively deduce from the above and expressions (5.21) that

S…ux;x† !0 and therefore R…ux;x† !s=p2p as x" ‡1 and u!u*. These limiting results entail thatLx…u† !LwithLˆs=…u*†H



2p

p

. FactorLx…u†tending to

a nonzero limit, integral (5.40) is therefore, up to powers ofx, of order exp x2…1 H†

2l Jx…ux†

!

: We therefore deduce from the above that

lim x"‡1 1 x2…1 H†logQ…x† ˆ _xlim_"‡1 Jx…ux*† 2l ; …5:41†

but the latter simply equalsr…u*†2_=2lˆk2_{=2 by continuity. The conjunction of} (5.39) and (5.41) together with the boundsq…x† h…x† Q…x†provide the desired

logarithmic estimate (5.37). j

The latter result consequently entails that the distribution of the content of a queue at heavy load, when fed by the superposition of an in®nite number of on=off sources with LRD characteristics, has a Weibullian tail at in®nity, namely,

h…x† ˆexp k₂2x2…1 H†‡o‰x2…1 H†Š

…5:42†

for large x. As shown in the proof of Proposition 5.5.3, only the leading term

D2_{t† lt}2H _{is used to derive the logarithmic estimate for h…x†. Re®ned} asymptotics for bounds q…x† and Q…x†, and thus estimates for the o‰x2…1 H†Š term in Eq. (5.42), can further be derived from theasymptotic expansionof the variance

D2_{t†for larget; that is,}

D2_{t† ˆlt}2H_‡l0_t2H0

‡o…t2H0

as t" ‡1, with H0_{<H. Using generalized Tauberian theorems [7, p. 142] and} identify (5.32), such an expansion can be derived from that of the Laplace transform

p7 !2r*…p†=p2 _{of t7 !D}2_{t† near pˆ0 and the use of expansions (5.30) in} formula (5.7) forr*…p†. Second-order powerH0_{proves, however, dif®cult to specify} for arbitrary powersr and s in expansions (5.30). By way of illustration, we just mention without proof that ifs<min‰r;…r‡1†=2Š, then a second-order expansion (5.43) forD2_{t†can be written with}

2Hˆ3 s;lˆ₃ 2_s†a…₂1 _s†n†_G3b₂s _s†; 2H0_{ˆ4 2s;l}0_ˆ a2…1 n†3b2s

…2 s†G…4 2s†:

Using Eq. (5.43) in formulas (5.22) and (5.23) then enables one to derive complete asymptotics forq…x†andQ…x†in the form

q…x† ek 0 kp2px1 Hexp k2 2 x2…1 H† ; Q…x† sek 0 gp2p  H 1 H r exp k₂2x2…1 H† ; …5:44† respectively, where k0_ˆ l0 2H2_l2g2:

We illustrate these approximations in the following numerical examples. Consider

Nˆ100 on=off sources, assuming that 100 sources is suf®cient for a useful application of our asymptotic results. We take the mean burst volume as unity

…1=bˆ1†. The mean activity probabilitynˆa=…a‡b†is set to 0.1…aˆ1

9†and the multiplexer load Nn=Cˆ …1‡g=npN† 1 _{to 0.9, implying C11:11 …gˆ}1

9†. Figure 5.1 shows a log plot of the complementary distribution function (interpreted as the over¯ow probability against buffer size) measured in units of mean burst volume, when sources have activity periods distributed as Pareto random variables. We also consider two values for H …Hˆ0:8 and H ˆ0:85). Curve A (B, resp.) corresponds to the limiting upper bound Q (lower bound q, resp.) given by Eq. (5.44). We observe that the buffer size required for a loss probability of 10 9 _is around 3104 _{times the mean burst size forH ˆ0:8 and more than 10}6_{times the} mean burst size forHˆ0:85. These values have to be compared to the case where both silent and activity periods are exponentially distributed. In the latter case, a buffer size of around 50 times the mean burst size guarantees a loss probability less than 10 9_.

5.5.3 Convergence to the FBM

The limiting Gaussian processfOtgde®ned in Proposition (5.3.6) of Section 5.3 has been derived as the integral of limiting rate processfY_tg, that is,

O_tˆ

…_t

0Yu du; t0:

Now, introduce the family of Gaussian centered processesfZ_t…y†gde®ned by

Z_t…y†ˆ 1

yHOyt; t0; …5:45† for all parameter values y>0 and with ®xed constant H. Process fZ_t…y†g is thus deduced fromfO_tg by the time and space scaling change …t;x† 7 ! …yt;xyH_{†. Each} processfYt…N†gde®ned by Eq. (5.15) being the superposition of i.i.d. rate processes distributed as fl_tg, its covariance function is independent of N and is therefore identical, along with that of its limiting processfY_tg, to the correlation functionrof the single rate processfl_tg. Similarly, in view of de®nition (5.18), the variance ofO_t equals that of anyO…tN†and is identical toD2…t† ˆVar…ot†de®ned in Eq. (5.3). The second-order properties of processesfZ…ty†g can then be readily derived as follows. Formula (5.32) ®rst entails that the variance ofZt…y† can be written as

Var…Zt…y†† ˆ2y2…1 H†

…_t

0dy

…_y

0r‰y…y x†Šdx: …5:46† Fig. 5.1 Over¯ow probability as a function of the buffer size with Pareto activity distribu- tion.

On the other hand, it is known that for any centered process fO_tg with stationary increments, the covariance E…O_uO_t† can be expressed in terms of the variance functiont7 !D…t†2_ˆVar…O

t†only as

E…O_uO_t† ˆ1

2‰D…t†2‡D…u†2 D…t u†2Š …5:47† withut. The covariance structure and, therefore, the distribution of each Gaussian processfZ_t…y†gare thus entirely de®ned through formulas (5.46) and (5.47).

From expression (5.46), in particular, we deduce that the convergence of the family of processesfZ_t…y†gasy" ‡1toward some limiting process is governed by the behavior of the covariance functionrin the neighborhood of in®nity. In view of factory2…1 H† _{in Eq. (5.46), it is then natural to expect that the variance Var…Z}…y†

t †

and, therefore, the sequence of processesfZt…y†ghave a nontrivial limit asy" ‡1in the case wherer…t†behaves as a power of tfor larget. Such a situation has been exempli®ed in Proposition 5.5.1 of this section, where it is shown that the covariance functionrof rate process fltgis preciselyO…t1 q†for larget and someq2 Š1;2‰, provided the distribution of either on or off periods is heavy tailed. A weak convergence result for the family of processesfZt…y†g,y>0, is therefore plausible in this framework.

Before stating the formal result, it is useful to recall here that [14, p. 318] there exists a unique centered Gaussian processfZ_tghaving

continuous paths withZ₀ˆ0,

stationary increments,

and with covariance function de®ned by

E…Z_uZ_t† ˆ1

2‰jtj2H‡ juj2H jt uj2HŠ …5:48† for allu;t2Rand for some parameterH 2 ‰1

2;1‰.

Note that, inviewof the stationarity of increments offZtgand of general property (5.47), this de®nition is equivalent to stating thatE…Z2

t† ˆ jtj2Hfor allt2R. This processfZtg is known as the fractional Brownian motion (FBM) with Hurst parameterH.

Proposition 5.5.4. Assume that the covariance function t7 !r…t†of rate process fl_tgveri®es asymptotics (5.31) for large t and some power q2 Š1;2‰. The processes fZ_t…y†g de®ned in Eq. (5.45) with

Hˆ3 q

2

then converge weakly asy" ‡1toward processpl fZtg, wherefZtgis an FBM

withHurst parameter H andlis de®ned in Eq. (5.34).

Proof. To ensure the convergence of the sequence of centered Gaussian processesfZ_t…y†g, one could, for instance, rely on Theorem 5.3.2. Here we content

ourselves with the veri®cation of the ®rst half of this theorem's hypotheses, that is, convergence of the ®nite-dimensional distributions. The latter holds if the covariance functions converge toward the covariance function of some identi®ed centered Gaussian process. In the present case, using estimate (5.31) in (5.46), we readily obtain Var…Zt…y†† 2y2…1 H† …_t 0dy …_y 0 r₀ ‰y…y x†Šq 1dx; …5:49†

where the conditionq2 Š1;2‰ensures that the integral is ®nite. As

…_t 0dy …_y 0 dx ‰y xŠq 1ˆ …_t 0 y2 q 2 qdyˆ t3 q …3 q†…2 q†;

the right-hand side of Eq. (5.49), withHˆ …3 q†=2, is consequently asymptotic to yq 1_y2r_q 0_{1…3 q†}t3 ₂q _q†ˆlt2H

asy" ‡1. Using general property (5.47) for processes with stationary increments, we then deduce that the covariance function offZt…y†gconverges to

1

2‰ljtj2H‡ljsj2H ljt sj2HŠ

as y" ‡1. Up to multiplicative constant l, the latter is equal to the covariance function of the FBM introduced in Eq. (5.48). We therefore conclude that the sequence of Gaussian centered processesfZt…y†gconverges weakly toward the latter

Gaussian process. j

5.6 CONCLUSION

As noted by Boxma and Dumas [4] in the case of a ®nite number N of on=off sources, no matter how heavy the tail of the off duration distribution, the distribution ofV₀…N† is light tailed (i.e., has ®nite moments) whenever the on duration distribution is light tailed. On the other hand, for an on duration distribution with Pareto tail, the distribution ofV₀…N† also has a Pareto tail whatever the off duration distribution. In contrast, the Weibullian distribution derived in this chapter forV₀…N† asN" ‡1in

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