1 OPTIMIZATION ISSUES IN WEB

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1 1 OPTIMIZATIO ISSUES I WEB SEARCH EGIES Zhen Lu 1 and Phlppe an 2 1 IBM Research Hawthorne, Y 10532, USA zhenl@us.bm.com 2 IRIA B.P. 93, 06902, Sopha Antpols Cedex, France Phlppe.an@nra.fr Abstract: Crawlers are deployed by a Web search engne for collectng nformaton from dfferent Web servers n order to mantan the currency of ts data base of Web pages. We present studes on the optmzaton of Web search engnes from dfferent perspectves. We frst nvestgate the number of crawlers to be used by a search engne so as to maxmze the currency of the data base wthout puttng an unnecessary load on the network. Both the statc settng, where crawlers are always actve, and the dynamc settng where, crawlers may be actvated/deactvated as a functon of the state of the system, are addressed. We then consder the optmal schedulng of the vsts of these crawlers to the Web pages assumng these pages are modfed at dfferent rates. Fnally, we brefly dscuss some other optmzaton ssues of Web search engnes, ncludng page rankng and system optmzaton. Keywords: Web search engnes, web crawlers, schedulng, optmal control, queues; Markov decson process. 1.1 ITRODUCTIO The role of World Wde Web as a major nformaton publshng and retrevng mechansm on the Internet s now predomnant and contnues to grow extremely fast. The amount of nformaton on the Web has long snce become too large for manually browsng through any sgnfcant porton of ts hypertext structure. As a consequence, a number of Web search engnes have been developed n the last decade: startng from the poneerng search engnes such as Alta Vsta, Lycos, Infoseek, Magellan, Excte, to the most successful ones such as Yahoo and Google. 11

2 12 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS Search engnes have become an ndspensable utlty for Internet users. Accordng to a recent Pew Foundaton Internet and Project January 2005), Search engnes are hghly popular among Internet users. Searchng the Internet s one of the earlest actvtes people try when they frst start usng the Internet, and most users quckly feel comfortable wth the act of searchng. Users pant a very rosy pcture of ther onlne search experences., and as of January 2005, 84% of nternet users have used search engnes. On any gven day, 56% of those onlne use search engnes. Thus, technologes that enhance Web search engnes are of hgh practcal nterest. These search engnes consst of ndexng engnes for constructng a data base of Web pages, and n many cases crawlers for brngng nformaton to the ndexng engne. To mantan currency and completeness of the data base, crawlers perodcally make recursve traversals of the Web s hypertext structure by accessng pages, then the pages referenced by these pages, and so on. In the lterature one fnds other colorful terms for crawler, such as wanderer, robot or spder, and the noton of a crawler beng routed to or vstng a page. Ths chapter keeps wth the crawler and accessng termnology throughout. Tradtonally, crawlers vst and ndex the Web pages untl the data base reaches certan sze. Perodcally, ths process s repeated through the rebuldng of a brand new data collecton n replacement of the old one. Alternatvely, the data base can be refreshed or updated ncrementally. Such an operatonal mode s sometmes referred to as ncremental crawler, see e.g. Cho and Garca-Molna 2000b). Throughout ths chapter, we consder the latter mode,.e. the ncremental crawler, although most analyses apply to the former as well. Due to the crtcal role that these crawlers play n the Web search engnes, the optmzaton ssues are topcs of a number of research papers. In ths chapter we present some of these research problems. Rather than provdng comprehensve, but hghlevel, dscussons, we present detaled solutons to some of the techncal problems. More precsely, Secton 1.2 consders both the ssues of optmzng the number of the crawlers to be deployed when all crawlers are always actve statc settng Secton 1.2.1), and of fndng an optmal decson rule for the case where crawlers may be actvated/deactvated as a functon of the state of the system dynamc settng Secton 1.2.2). Performance of statc and dynamc polces are compared n Secton The optmal schedulng of the page vsts of these crawlers s studed n Secton 1.3. Fnally, we provde ponters to some other ssues such as page rankng and system optmzaton Secton 1.4). A word on the notaton n use: x respectvely x ) denotes the largest respectvely smallest) nteger less respectvely greater) than or equal to x. Also for any mappngs f and g, the relaton f x) x gx) s understood as lm x f x)/gx) = OPTIMIZIG THE UMBER OF CRAWLERS We frst address n Secton the stuaton where crawlers are always actve, regardless of the state of the system, and we determne the optmal number of crawlers to be deployed. Then, we move n Secton to the stuaton where crawlers may be actvated/deactvated as a functon of the state of the system, and we fnd an optmal

3 OPTIMIZATIO I WEB SEARCH EGIES 13 Ste 4 Ste 1 Ste 2 Ste 3 Robot 1 Robot 2 Indexng Engne Database where the nformaton collected by the robots s stored Source 1 Source 2 Server wth a fnte buffer Fgure 1.1 Model of search engne wth two crawlers decson rule for the number of actve crawlers at any tme. In both settngs the cost functon s a weghted sum of the starvaton probablty and loss rate. The results presented n ths secton are based on the work of Talm et al. 2001b) and Talm et al. 2001a). Practcal ssues of deployng parallel crawlers are dscussed n Cho and Garca-Molna 2002) The Statc Settng The search engne s modeled as a sngle server fnte capacty queue. The system capacty s K 2 ncludng the poston n the server), see Fgure 1.1. There are 1 crawlers: each crawler brngs new pages to the queue accordng to a Posson process wth rate λ > 0. These Posson processes are assumed to be mutually ndependent and ndependent of the ndexng servce) tmes. Hence, new pages are generated accordng to a Posson process wth ntensty λ. An ncomng page fndng a full queue s lost. Indexng tmes are assumed to be ndependent and dentcally random varables wth common dstrbuton Fx). Let 1/µ be the expected ndexng tme.

4 14 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS The search engne s therefore modeled as the well known M/G/1/K queue see e.g. Cohen 1982,Chapter III.6)). In ths notaton we defne the cost functon as the weghted sum of two terms: the fracton of tme that the system s empty, hereafter referred to as the starvaton probablty; the expected number of tmes when an arrvng crawler fnds a full system per unt tme, hereafter referred to as the loss rate. Let X resp. X ) be the statonary queue-length at arbtrary epochs resp. statonary queue-length at arrval epochs) n a M/G/1/K queue wth arrval rate λ and servce rate µ. Wth ρ := λ/µ > 0 and for γ > 0 the cost functon s then defned as Cρ,γ,K) := γ ProbX = 0) + λ ProbX = K) 1.1) wth ProbX = 0) and λ ProbX = K) the starvaton probablty and the loss rate, respectvely. Snce ProbX = ) = ProbX = ) for = 0,1,...,K from the PASTA property Wolff 1982), 1.1) rewrtes as Cρ,γ,K) = γ ProbX = 0) + ρµ ProbX = K) 1.2) where λ n 1.1) has been replaced by ρµ. Throughout Secton we wll assume that ndexng tmes are exponentally dstrbuted. The general case where the ndexng tmes are arbtrarly dstrbuted s more nvolved, due to the lack of closed-form expressons for the M/G/1/K queue, and s dscussed n Talm et al. 2001b) The M/M/1/K Search Engne Model. We assume that the ndexng tmes are exponentally dstrbuted, namely, Fx) = 1 exp µx). In other words, we model the search engne as an M/M/1/K queue. In the M/M/1/K queue wth traffc ntensty ρ the statonary queue-length probabltes at arbtrary epochs are gven by Klenrock 1975): for = 0,1,...,K. Therefore, ProbX = ) = 1 ρ 1 ρ K+1 ρ 1.3) Cρ,γ,K) = 1 ρ)γ + µρk+1 ) 1 ρ K ) In partcular, Cρ,γ,K) = γ + µ)/k + 1) when ρ = 1. Lemma 1 shows the exstence of a unque mnmum for Cρ,γ,K) consdered as a functon of ρ. The proof s provded n Talm et al. 2001b). Lemma 1 For any γ > 0, K 2, the mappng ρ Cρ,γ,K) has a unque mnmum n [0, ), to be denoted ργ,k). Furthermore, 0 < ργ,k) < 1 f γ < γk), ργ,k) = 1 f γ = γk) and ργ,k) > 1 f γ > γk), wth γk) := µk + 2)/K.

5 OPTIMIZATIO I WEB SEARCH EGIES 15 We now return to the orgnal problem, namely the computaton of the number of crawlers that mnmzes the cost functon Cρ,γ,K) wth ρ = λ/µ. The answer s found n the next result whch s a drect corollary of Lemma 1. Proposton 1 For any γ > 0, K 2, let γ,k) be the optmal number of crawlers to use. Then, γ,k) = argmn n Cnλ/µ,γ,K) 1.5) wth n { ργ,k)µ/λ, ργ,k)µ/λ }. Furthermore, γ,k) µ/λ f γ < γk), γ,k) { µ/λ, µ/λ } f γ = γk), and γ,k) µ/λ f γ > γk). In the next secton we nvestgate the mpact of the parameter γ on the optmal number of crawlers Impact of γ on the Optmal umber of Crawlers. Recall that the parameter γ s a postve constant that allows us to stress ether the probablty of starvaton or the loss rate. Part of the mpact of γ on ργ,k), and therefore on γ,k), the optmal number of crawlers, s captured n the followng result. Proposton 2 For any K 2, the mappng γ ργ,k) s nondecreasng n 0, ), wth lm γ ργ,k) =. Proof. Pck two constants 0 < γ 1 < γ 2 and defne ρ,γ 1,γ 2,K) := Cρ,γ 2,K) Cρ,γ 1,K) = 1 ρ 1 ρ K+1 γ 2 γ 1 ). We assume that ργ 2,K) < ργ 1,K) and show that ths yelds a contradcton. Under the condton γ 1 < γ 2 the mappng ρ ρ,γ 1,γ 2,K) s decreasng n [0, ). Therefore, 0 < ργ 2,K),γ 1,γ 2,K) ργ 1,K),γ 1,γ 2,K) = [Cργ 2,K),γ 2,K) Cργ 1,K),γ 2,K)] +[Cργ 1,K),γ 1,K) Cργ 2,K),γ 1,K)] 0, 1.6) whch contradcts the fact that ρ ρ,γ 1,γ 2,K) s decreasng n [0, ). Therefore ργ 2,K) ργ 1,K) and the mappng γ ργ,k) s nondecreasng n [0, ). We may then defne L := lm γ ργ,k). From the dentty Cρ,γ,K)/ ρ = 0 for ρ = ργ,k) see Lemma 1) we obtan 0 = µργ,k) 2K+1) γk + µk + 2))ργ,K) K+1 +K + 1)µ + γ)ργ,k) K γ. 1.7)

6 16 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS K = 5 K= Fgure 1.2 γ γ,k) for 1/λ = 0.6 and µ = 1 Assume that L <. Lettng γ n 1.7) yelds KL K+1 K + 1)L K + 1 ) lm γ γ = µl K L K+2 K + 2))L + K + 1) ). 1.8) Snce L > 1 we have shown n Lemma 1 that ργ,k) > 1 for γ > µk + 2)/K) t s easly seen that KL K+1 K + 1)L K + 1 > 0, whch mples that the l.h.s. of 1.8) s nfnte whereas the r.h.s. s fnte. Therefore, 1.8) cannot hold f L < and lm γ ργ,k) =. Ths concludes the proof. Proposton 2 has a smple physcal nterpretaton. As the parameter γ ncreases the probablty of starvaton becomes the man quantty to mnmze. Hence, the mnmzaton s done by ncreasng the arrval rate or, equvalently, by ncreasng the number of crawlers, as shown n Proposton 2. Fgure 1.2 provdes two numercal examples llustratng the monotoncty of the optmal number of crawlers as a functon of γ Impact of K on the Optmal umber of Crawlers. In ths secton we examne the behavor of ργ,k) as a functon of K. The followng results hold see Talm et al. 2001b)): Proposton 3 a) If 0 < γ µ then the mappng K ργ,k) s nondecreasng n [2, ); b) If γ > µ then there exsts an nteger K 0 2u/γ λ) such that the mappng K ργ,k) s nondecreasng n [2,K 0 1] and non-ncreasng n [K 0, ). The next proposton examnes the lmtng behavor of ργ,k) as K ncreases to nfnty.

7 OPTIMIZATIO I WEB SEARCH EGIES 17 Proposton 4 For any γ > 0, lm ργ,k) = ) K Proof. Let M := lm K ργ,k), where the exstence of the lmt follows from Proposton 3. Lettng now K n 1.7) we see that the r.h.s. converges to γ f M < 1 and converges to nfnty f M > 1, thereby showng that necessarly M = 1, whch concludes the proof. Proposton 4 shows that the optmal arrval rate converges to the servce capacty when the buffer sze ncreases to nfnty. The lmtng result 1.9) can be used to derve an approxmaton for the optmal number of crawlers to be deployed when K s large. Indeed, the relaton lm γ,k) = lm K arg mn K n { µ/λ, µ/λ } Cλn/µ,γ,K), 1.10) whch follows from 1.5), suggests the followng approxmaton, for large K µ/λ f Cρ +,γ, ) Cρ,γ, ) γ,k) K µ/λ f Cρ +,γ, ) > Cρ,γ, ) 1.11) wth the notaton Cρ,γ, ) := lm K Cρ,γ,K), ρ + := λ/µ) µ/λ and ρ := λ/µ) µ/λ. Snce Cρ,γ, ) = γ1 ρ) for ρ 1 and Cρ,γ, ) = µ1 ρ) for ρ 1 from 1.4), we may rewrte 1.11) as µ/λ f µ1 ρ + ) γ1 ρ ) γ,k) K µ/λ f µ1 ρ + ) > γ1 ρ ). 1.12) The mappng K ργ,k) s dsplayed n Fgure 1.3 for γ < µ and n Fgure 1.4 for γ > µ. Table 1.1 gves γ,k) for dfferent values of K and compare these values wth the approxmaton 1.12) last column n Table 1.1). The approxmaton 1.12) appears to be farly senstve to model parameters; however, n all but one case 1.12) les wthn 10% of the exact value as soon as K 10. We also observe that the qualty of the approxmaton ncreases when γ ncreases wthn 10% of the exact value for γ = 2 for all K 2).

8 18 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS Fgure 1.3 K ργ,k) for γ = 0.5 and µ = Fgure 1.4 K ργ,k) for γ = 2 and µ = 1

9 OPTIMIZATIO I WEB SEARCH EGIES 19 Table 1.1 K γ,k) for λ = 0.01 and µ = 1 K : γ = γ = γ = γ = γ = The Dynamc Settng In ths secton we assume that the number of actve crawlers may vary n tme accordng to the backlog n the queue and to the number of crawlers already actve. To address ths stuaton we wll cast our model nto the Markov Decson Process MDP) framework Bertsekas, 1987; Puterman, 1994; Ross, 1983). The ndexng engne s agan modeled as a fnte-capacty sngle-server queue. Servce tmes stll consttute ndependent random varables wth common negatve exponental dstrbuton wth mean 1/µ) and the buffer may accommodate at most K 2 customers, ncludng the one n servce, f any. There are avalable crawlers and each of these crawlers, when actvated, brngs pages to the server accordng to a Posson process wth rate λ. We assume that these Posson processes are mutually ndependent and further ndependent of the servce tme process. The new feature n ths secton s that the number of actve crawlers may be modfed at any arrval and at any departure epoch. When an arrval occurs, the ncomng crawler s deactvated at once; the controller may then decde to keep t dle or to reactvate t. When a departure occurs the controller may ether decde to actvate one addtonal crawler, f any avalable, or to do nothng.e. the number of actve crawlers s not modfed). The objectve s to fnd a polcy to be defned) that mnmzes a weghted sum of the statonary starvaton probablty and the loss rate. We now ntroduce the MDP settng n whch we wll solve ths optmzaton problem. Snce the tme between transtons s varable we wll use the unformzaton method Bertsekas, 1987,Sec. 6.7). At the n-th decson epoch t n the state of the MDP s represented by the trple x n = q n,r n,s n ) {0,1,...,K} {0,1,...,} {0,1,2}, wth q n and r n the queue-length and the number of actve crawlers just before the n-th decson epoch, respectvely, and s n the type arrval, departure, fcttous see below) of the n-th decson epoch.

10 20 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS The successve decson epochs {t n,n 1} are the jump tmes of a Posson process wth ntensty ν := λ +µ, ndependent of the servce tme process. In ths settng, the n-th decson epoch t n corresponds to an arrval n the orgnal system wth probablty λr n /ν n whch case s n = 1), to a departure wth probablty µ/ν provded that q n > 0 s n = 0) and to a fcttous event wth the complementary probablty r n )λ+µ)/ν s n = 2). Let a n {0,1} be the acton chosen at tme t n. We assume that a n = 1 f the decson s made to actvate one addtonal crawler, f any avalable, and a n = 0 f the decson s made to keep unchanged the number of actve crawlers. By conventon we assume that a n = 0 f the n-th decson epoch corresponds to a fcttous event s n = 2). From the above defntons we see that states of the form,0,1) and 0,,0) are not feasble, as an arrval cannot occur f all crawlers are nactve and a departure cannot occur f the queue s empty, respectvely. Therefore, the state-space for ths MDP s {q,r,s),0 q K,0 r,s = 0,1,2} {0,r,0),q,0,1),0 q K,0 r }. However, ths set contans one absorbng state, the fcttous state 0, 0, 2). To remove ths undesrable state we wll only consder polces see formal defnton below) that always choose acton a = 1 when the system s n state 1,0,0) so that 0,0,2) can never be reached. Ths s not a severe restrcton snce a polcy that never actvates crawlers when the system s empty s of no nterest. In concluson, the state space for ths MDP s X := {q,r,s),0 q K,0 r,s = 0,1,2} {0,0,2),0,r,0),q,0,1),0 q K,0 r } and the set A x of allowed actons when the system s n state x = q,r,s) X s gven by {0} f s = 2 A x = {1} f q,r,s) = 1,0,0) {0, 1} otherwse. To complete the defnton of the MDP we need to ntroduce the one-step cost c and the one-step transton probabltes p. Gven that the process s n state x = q,r,s) and that acton a s made, the one-step cost s defned as cx) = γ1q = 0) + ν1q = K,s = 1), 1.13) ndependent of a. We wll show later on n ths secton that ths choce for the one-step cost wll allow us to address, and subsequently to solve, the optmzaton problem at hand.

11 OPTIMIZATIO I WEB SEARCH EGIES 21 For x X, the one-step transton probabltes p x,x a) are gven by µ 1q > 1) ν f x = q 1,mn{r + a,},0) λr p x,x a) = f x = q 1,mn{r + a,},1) ν 1.14) 1 f s = 0, a = 0,1; µ ν λr + a 1) p x,x a) = ν µ1q > 1) λr ν f x = q 1,mn{r + a,},2) f x = mn{q + 1,K},r + a 1,0) f x = mn{q + 1,K},r + a 1,1) 1.15) f s = 1, a = 0,1; 1 µ + λr + a 1) ν p x,x 0) = µ 1q > 0) ν f λr ν f x = mn{q + 1,},r + a 1,2) x = q,r,0) f x = q,r,1) 1.16) 1 µ1q > 0) + λr ν f x = q,r,2) f s = 2. All other transton probabltes are equal to 0. Wthout loss of generalty we wll only consder pure statonary polces snce t s known that nothng can be ganed by consderng more general polces Puterman, 1994,Ch. 8-9). Recall that n the MDP settng a polcy π s pure statonary f, at any decson epoch, the acton chosen s a non-randomzed and tme-homogeneous mappng of the current state Bertsekas, 1987; Puterman, 1994; Ross, 1983). We defne an admssble statonary polcy as any mappng π : X {0,1} such that πx) A x. For later use ntroduce Pπ) := [p x,x πx))] x,x ) X X, the transton probablty matrx under the statonary polcy π. Let P be the class of all admssble statonary polces. For any polcy π P ntroduce the long-run expected average cost per unt tme [ 1 n W π x) = lm n n E π cx ) x 1 = x =1 ], x X. 1.17) The exstence of the lmt n 1.17) s a consequence of the fact that π s statonary and X s countable Puterman, 1994,Proposton 8.1.1).

12 22 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS We shall say that a polcy π P s average cost optmal f W π x) = nf π P W πx) x X. 1.18) In order to use results from MDP theory for average cost models we frst need to determne to whch class recurrent, unchan, multchan, communcatng, etc.) the current MDP belongs to. Consder the followng example: = 2 and let π be any statonary polcy that selects acton 1 n states,r,1) for r {1,2} and n state 1,0,0), and acton 0 otherwse. It s easly seen that ths polcy nduces a MDP wth two recurrent classes X {,1, )} and X {,2, )} and a set of transent states X {,0, }). We therefore conclude from ths example that the MDP {x n,n 1} s multchan Puterman, 1994,p. 348). An MDP s communcatng Puterman, 1994,p.348) f, for every par of states x,x ) X X, there exsts a statonary polcy π such that x s accessble from x, that s, f there exsts n 1 such that P n x,x π) > 0, where P n x,x π) s the x,x )-entry of the matrx P n π). Lemma 2 The MDP x n,n 1) s communcatng. The proof of Lemma 2 s gven n Talm et al., 2001a). The next result follows from Lemma 2 and Proposton 4 n Bertsekas, 1987,Sec. 7.1): Proposton 5 There exsts a scalar θ and a mappng h : X IR such that, for all x X, θ + hx) = cx) + mn a A x p x,x a)hx ) 1.19) x X wth θ = nf π P W π x) for all x X, whle f π x) attans the mnmum n 1.19) for each x X, then the statonary polcy π s optmal. The optmal average cost θ and the optmal polcy π n Proposton 1.17 can be computed by usng the followng recursve scheme, known as the relatve value teraton algorthm. Proposton 6 Let ˆx be a fxed state n X and 0 < τ < 1 be a fxed number. For k 0, x X, defne the mappngs h k, k 0) as wth h k+1 x) = 1 τ)h k x) + τt h k )x) T h k ) ˆx)) T h k )x) := cx) + mn a A x p x,x a)h k x ), x X where h 0 ˆx) = 0 but otherwse h 0 s arbtrary. Then, the lmt hx) = lm k h k x) exsts for each x X, θ = τt h) ˆx), and the optmal acton π x) n state x s gven by π x) argmn a Ax x X p x,x a)hx ). Proof. Snce the MDP s communcatng cf. Lemma 2) the proof follows from Puterman 1994,Sec. 8.5,9.5.3) see also Bertsekas 1987,Prop. 4, p. 313 )).

13 OPTIMIZATIO I WEB SEARCH EGIES 23 We now return to our ntal objectve, namely, mnmzng a weghted sum of the statonary starvaton probablty and the loss rate. To see why the soluton to ths problem s gven by the soluton to the MDP problem formulated n ths secton, t suffces to show that the average cost 1.17) s a weghted sum of the statonary starvaton probablty and the loss rate. It should be clear, however, that ths result cannot hold for polces that nduce an average cost 1.17) that depends on the ntal state x as, by defnton, the statonary starvaton probablty and the loss rate are ndependent of the ntal state. We wll therefore restrct ourselves to the class P 0 P of polces that generate a constant average cost, namely, P 0 = {π P : W π x) = W π x ), x X}. The set P 0 s non-empty as t s well-known that t contans, among others, all unchan polces Puterman, 1994,Proposton 8.2.1). Among such polces s the statc polcy π that always mantan crawlers actve, namely, π x) = 1 for all x =,,s) X wth s = 0,1 and π x) = 0 for all x =,,2) X. We may also note that reducng the search for an optmal polcy to polces n P 0 does not yeld any loss of generalty as t s also known that there always exts an optmal polcy wth constant average cost n the case of communcatng MDP s Puterman, 1994,Proposton 8.3.2). Fx π P 0. Introducng 1.13) nto 1.17) yelds W π x) = γs π x) + L π x) wth [ 1 n ] S π x) = lm n n E π 1q = 0) x 1 = x =1 [ 1 n L π x) = ν lm n n E π 1q = K,s = 1) x 1 = x =1 In the followng we wll drop the argument x n S π x) and L π x) snce these quanttes do not depend on x from the defnton of P 0. Let us now nterpret S π and L π. S π s the statonary probablty that the system s empty at decson epochs. Snce the decson epochs form a Posson process, we may conclude from the PASTA property Wolff, 1982) that S π s also equal to the statonary probablty that the system s empty at arbtrary epoch wth s nothng but the statonary starvaton probablty. Let us now consder L π. Recall that {t n,n 1}, the successve decson nstants, s a Posson process wth ntensty ν and assume wthout loss of generalty that t 1 = 0 a.s. Defne At) as the total number of customers that have arrved to the queue up to tme t, ncludng customers whch have been lost, and let Qt) be the queue length at tme t. We assume that the sample paths of the processes {At), t 0} and {Qt), t 0} are rght-contnuous wth left lmt. Wth these defntons and the dentty E[t n ] = n/ν we may rewrte L π as [R E tn π 0 L π = lm n 1Qt ) = K)dAt)]. E[t n ] In other words, we have shown that L π s the rato, as n tends to nfnty, of the expected number of losses durng the frst n decson epochs over the expected occurrence tme of the n-th decson epoch. ].

14 24 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS The nterpretaton of L π as a loss rate now follows from the dentty [R E tn π 0 1Qt ) = K)dAt)] L π = lm n E[t n ] [ Z 1 T ] = lm T T E π 1Qt ) = K)dAt), π P 0, 1.20) 0 upon notcng that the latter quantty represents the mean number of losses per unt tme or the loss rate. The second dentty n 1.20) s a drect consequence of the theory of renewal reward processes Ross, 1983,Theorem 7.5) and of the defnton of the set P 0. In summary, we have shown that for any polcy π n P 0 the average cost s W π = γs π + L π, wth S π the starvaton probablty and L π the loss rate. The optmal polcy has been computed for dfferent values of the model parameters. Fgures dsplay the optmal polcy for = 16, K = 5, λ = 0.1, µ = 1 and for dfferent values of γ γ < γk) = 1.4, γ = γk) and γ > γk)). The results were obtaned by runnng the value teraton algorthm gven n Proposton 6 wth the stoppng crteron max x X h k+1 x) h k x))/h k x) < , 255 and 256 teratons were needed to compute the optmal polcy dsplayed n Fgures 1.5, 1.6 and 1.7, respectvely). We see from these fgures that the optmal polcy s a monotone swtchng curve, namely, there exst two monotone decreasng here) nteger mappngs f s : {0,1,...,} {0,1,2,...}, s {0,1}, such that π x) = 1 f s r) q) for all x = q,r,s) X wth s = 0,1 we must also have f 0 0) 1 so that π 1,0,0) = 1 as requred). We conjecture that the optmal polcy always exhbts such a structure but we have not able been to prove t.

15 OPTIMIZATIO I WEB SEARCH EGIES 25 r q r q s = 0 s = 1 Fgure 1.5 Optmal polcy γ = 1, Cost = ) r q r q s = 0 s = 1 Fgure 1.6 Optmal polcy γ = 1.4, Cost = )

16 26 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS r q r q s = 0 s = 1 Fgure 1.7 Optmal polcy γ = 2, Cost = ) Statc Versus Dynamc Polces In ths secton we compare statc and dynamc polces n the case where the ndexng tmes are exponentally dstrbuted. The results are reported n Tables 1.2 and 1.3. Throughout the experments µ = 1. For dfferent sets of parameters λ,k,γ, we frst computed the optmal number of crawlers s gven by Proposton 1) and the average cost C s gven n 1.4)) n the statc settng. Then, for each set of parameters λ,k,γ, we set the value of the number of avalable crawlers to s and determned, va the relatve value teraton algorthm gven n Proposton 6 wth τ = the closer τ s from 1 the faster the algorthm converges), the optmal average cost C d gven n 1.18)) as well as the mnmum mn ) and the expected ) number of crawlers actvated by the optmal dynamc polcy. These results can be found n Table 1.2. We stopped the numercal procedure when the relatve error between two consecutve terates was unformly) less than The number of teratons ter ) and the relatve mprovement 100% C s C d )/C d ) are also reported n Table 1.2. Last, we computed the overall optmal dynamc polcy by removng the restrcton on the number of avalable crawlers. The optmal average cost C d as well as the mnmum mn ), expected ) and maxmum max ) number crawlers used by the overall optmal dynamc polcy are gven n Table 1.3. We observe that substantal gans may be acheved by dynamcally controllng the actvty of the crawlers. When the number of avalable crawlers s set to s Table 1.2) the relatve mprovement w.r.t. to the optmal statc polcy ranges from 4% to 103% for the consdered model parameters; when the restrcton on the number of avalable crawlers s removed then the mprovement ranges from 6% to 3226%! The gan appears to be an ncreasng functon of the queue sze K and of the arrval rate λ.

17 OPTIMIZATIO I WEB SEARCH EGIES 27 Table 1.2 Statc vs. dynamc polces wth µ = 1 and τ = ) Statc Approach Dynamc Approach λ K γ C s s C d mn ter Rel. Impr % % % % % % % % % % % % % % % % % % % % % % % % % % %

18 28 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS Table 1.3 Statc vs. dynamc polces wth µ = 1 and τ = ) Statc Approach Dynamc Approach λ K γ C s s C d mn max Rel. Impr % % % % % % % % % % % % % % % % % % % % % % % % % % % 1.3 OPTIMAL SCHEDULIG OF THE CRAWLERS We now turn to the problems of schedulng a crawler that mantans the currency of exstng pages n search-engne data bases. For sake of arguments, we assume that the set of Web pages s fxed. However, as we shall see, our results can be promoted as

19 OPTIMIZATIO I WEB SEARCH EGIES 29 heurstcs that acqure new pages and drop old pages over tme. A specfc objectve wll be to fnd crawler schedules that mnmze the obsolescence of the data base n some useful sense. For example, assume there are Web pages, labeled 1,2,...,, whch are to be accessed repeatedly by a crawler, the duraton of each access beng an ndependent sample from a gven dstrbuton. Assume also that the contents of page are modfed at tmes that follow a Posson process wth parameter µ. A page s consdered up-to-date by the ndexng engne from the tme t s accessed by the crawler untl the next tme t s modfed, at whch pont t becomes out-of-date untl the crawler s next access. Let r be the fracton of tme page spends out-of-date. The problem s to fnd relatve page-access frequences and a sequencng polcy that realzes these frequences such that the objectve functon C = 1 c r, s mnmzed, where the c are gven weghts. Under smplfyng but plausble assumptons on the weghts, page access tmes, and the class of allowed polces, we obtan explct solutons to ths problem. From a theoretcal pont of vew, our problem s closely related to those multplequeue sngle-server systems usually called pollng systems n the queueng lterature. Indeed, the crawler can be consdered as the server and the pages as the statons n the pollng system. The duratons of consecutve page accesses correspond to swtchover tmes and the page modfcatons correspond to customer arrvals. The servce tmes n ths pollng system are zero. Our two-stage approach of optmzng crawler schedules determnng access frequences and then fndng a schedule that realzes them) s smlar to the approach n Borst 1994), Borst et al. 1994) and Boxma et al. 1993) of optmzng vst sequences n pollng systems. An extensve lterature exsts on the analyss and control of pollng systems. The nterested reader s referred to the book of Takag 1986) for general references; the specal ssue of the journal Queueng Systems, Vol ) on pollng models and the recent thess of Borst 1994) can be consulted for more recent developments. In partcular, the pollng systems wth zero servce tmes were motvated by communcaton networks such as teletext and vdeotex where pages of nformaton are to be broadcast to termnals connected to a computer network Ammar and Wong, 1987; Dykeman et al., 1986; Lu and an, 1992). However, the problem here has not been analyzed. Indeed, n the usual analyss of pollng systems wth unbounded buffers, nterest centers on mean watng tmes and mean queue lengths, whereas n our problem, the performance measure of nterest, vz., the obsolescence tme, corresponds to the maxmum watng tme of a customer durng a vst cycle of the server. An alternatve vew of our model dentfes t wth a pollng loss system havng unt buffers, n whch our obsolescence tme becomes the watng tme. Wth ths pont of vew, our model has potental use n mantenance applcatons. The next subsecton s devoted to a precse formulaton of our model, and a revew of some useful concepts n stochastc orderng theory. Secton begns by provng two propertes of crawler schedulng polces: ) expected obsolescence tmes ncrease as the page-access tme ncreases n the ncreasng-convex-orderng sense, and ), by Schur-convexty results, accesses to any gven page should be as evenly spaced as possble. We then derve a tght lower bound on the cost functon C assumng that the weghts c are proportonal to the µ. These results yeld a formula for

20 30 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS optmal access frequences. Our technques can be extended to general c, but explct formulas are not attanable n general. To motvate the assumpton on weghts, note that a useful choce for the c s the customer page-access frequency, for n ths case the total cost can be regarded as a customer total error rate. The specal case where the customer access frequency c s proportonal to the page-change rate µ s reasonable under ths nterpretaton - the greater the nterest access frequency), the greater the frequency of page modfcaton. Sectons and deal wth the problem of sequencng page accesses optmally, or near optmally, so as to realze a gven set of access frequences. Ths materal s prefaced by a dscusson at the end of Secton whch relates our schedulng problem to those that come under the headng of generalzed round-robn or templatedrven schedulng. In Secton 1.3.3, we ntroduce randomzed page accessng, where each access s determned by an ndependent and dentcally dstrbuted..d.) sample from a dstrbuton { f }. We show how to fnd that choce for ths dstrbuton whch mnmzes C. In Secton 1.3.4, we develop a polcy that performs well when s large. It s based on work of Ita and Rosberg 1984) n an entrely dfferent settng) and yelds a cost wthn 5% of optmal. Results presented n ths secton are based on the work of Coffman Jr. et al. 1998). A related study was conducted n Cho and Garca-Molna 2000a) Prelmnares Let {X k } be the sequence of duratons of consecutve page accesses by the crawler, each X k beng dstrbuted ndependently as a random varable X. For schedulng polcy π, let π n {1,2,...,} be the schedulng decson for the n-th access,.e., the ndex of the n-th page to be accessed by the crawler under π. Defne the nter-access dstance d j π) = n j π) n j 1 π), where n j π) s the ndex of the j-th access of page,.e., n j π) = nf{n > n j 1 π) π n = }, and where n 0 π) 0. Let X j = X j π) be the j- th nter-access tme of page,.e., the tme between the j 1)-st and j-th page- access completon tmes. We have X j = n j k=n j 1 +1 X k, so the random varables X j are mutually ndependent. ote that, f page access tmes X k are exponentally dstrbuted, then Xj has an Erlang dstrbuton of d j stages. Hereafter, except n defntons, the polcy π wll normally be omtted from our notaton; n such cases, the polcy wll always be clear n context. Let Z j = Z j π) be the tme that page s out-of-date durng the j-th nter-access tme of page. Let m n = m nπ) be the number of accesses of page among the frst n accesses: m n = n k=1 1{π k = }, where 1{ } s the ndcator functon. Hereafter, we consder only statonary schedulng polces n the sense that, for each such polcy, the lmt m n f = f π) = lm n n 1.21) exsts and s strctly postve for all, 1. We call f the access frequency of page. We also requre that the lmts lm n n j=1 Z j /n and lm n n j=1 E[Z j ]/n

21 OPTIMIZATIO I WEB SEARCH EGIES 31 exst and be equal. These last assumptons hold under farly mld condtons, e.g., when the sequence {d j π)} j s statonary and ergodc cf. Kngman 1968)). The obsolescence rate r = r π) of page s the lmtng fracton of tme that page s out of date; precsely, t s defned as r = lm m n n j=1 Z j m n j=1 X j = m n j=1 lm Z j n n m n j=1 lm X j n n = 1 E[X] lm m n j=1 E[Z j ] n n. 1.22) In partcular, when polcy π s cyclc wth cycle length K,.e., when π nk+k = π n 1)K+k for all 1 k K and all n = 1,2,..., then r = 1 m K KE[X] j=1 E[Z j], 1.23) where m K s the number of page- accesses durng a cycle. The cost functon to be mnmzed s the weghted sum of the obsolescence rates: C = Cπ) = =1 c r, 1.24) where c are gven postve real numbers and the mnmzaton s to be over all statonary schedulng polces. A few bascs n stochastc orderng conclude ths secton. For two m-dmensonal real vectors x and y, x majorzes y, wrtten x y, f k =1 x [] k =1 y [], for k = 1,...,m 1 and m =1 x [] = m =1 y [], where x [] s the th largest component of x. Intutvely, y s better balanced than x. A functon h s sad to be Schur-convex f hx) hy) whenever x y. See Marshall and Olkn 1979) for more detals about ths and related propertes. A random varable Y 1 s sad to be no greater than a random varable Y 2 n the convex orderng sense, denoted Y 1 cx Y 2, f E[hY 1 )] E[hY 2 )] for all convex functons h, provded the expectatons exst. If n ths defnton convex s replaced everywhere by ncreasng and convex, then we wrte Y 1 cx Y 2. As s easly verfed, Y 1 cx Y 2 mples that Y 1 has the same mean but smaller varance than Y 2. It s also easy to see that Y 1 cx Y 2 mples Y 1 cx Y 2. See Stoyan 1933) for equvalent defntons and further propertes Schur Convexty and a Lower Bound Recall that a page s consdered out-of-date from the tme t s modfed untl the next tme t s accessed by the crawler. Thus, f page s not modfed durng ts j-th nteraccess nterval, then the obsolescence tme s Z j = 0. Otherwse, Z j s the tme that elapses from the frst moment page s modfed durng ts j-th nter-access nterval untl the end of that nterval. Recall also that the modfcaton or mutaton) epochs of page follow a Posson process wth parameter µ. By the memoryless property of the

22 32 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS Posson process, the tme that elapses from the begnnng of page s j-th nter-access nterval to the frst subsequent mutaton has an exponental dstrbuton wth parameter µ. Let R 1,R 2,... be an..d. sequence of such random varables, so that Z j d = X j R j) +, 1.25) where x + denotes maxx,0) and d = denotes equalty n dstrbuton. As an mmedate consequence, we obtan Proposton 7 If the page access tme s decreased n the ncreasng convex orderng sense, then the obsolescence rate s decreased for all pages under any schedulng polcy. Proof. Let {X k } be a sequence of access tmes dstrbuted ndependently as X, and defne {X j } j and {Z j } j as for X k. Assume that X cx X. Then, d + = X j R j) cx X j R + d= j) Z j, Z j and so E[Z j ] E[Z j ]. Thus, as desred. r = 1 E[X ] lm m n j=1 E[Z j ] 1 n n E[X] lm m n j=1 E[Z j ] = r, n n Returnng to our man problem, where the dstrbuton of page access tmes s assumed gven, we now show that the obsolescence rate s a Schur convex functon of the vector of nter-access dstances. For ths, we need the followng calculaton whch wll also be useful for later results. Defne h = E[e µ X ], the Laplace transform of X evaluated at µ. Lemma 3 For any page, E[Z j] = d je[x] 1 ) 1 h d j. µ Proof. Let G j be the probablty dstrbuton of X j. We have from 1.25) that E[Z j] = = = = = Z 0 Z 0 Z Z 0 z Z Z x 0 Z 0 PZ j > z)dz PX j R j > z)dz 1 e µ x z) ) G jdx)dz ) 1 e µ x z) dzg jdx) 0 x 1 ) e µ x G jdx) µ

23 OPTIMIZATIO I WEB SEARCH EGIES 33 whch yelds the lemma. We can conclude from the above proof that the result of Proposton 7 stll holds when the ncreasng convex orderng s replaced by the weaker Laplace-transform orderng see Stoyan 1933)). It follows from a result of Schur cf. Marshall and Olkn 1979,Proposton 3.C.1, page 64)) and Lemma 3 that Proposton 8 For any fxed number n of page- accesses, the expected total obsolescence tme of page, n =1 E[Z j ] s a Schur convex functon of the dstances d j, j = 1,...,n. Thus, n order to mnmze the expected obsolescence tme, the accesses to any partcular page should be as evenly spaced as possble. An algorthm that computes a schedule of the crawler that mplements a gven set of access frequences n the sense of 1.21) s called an accessng polcy. In these terms, the schedulng polces proposed n ths paper consst of two stages; the frst computes a set of access frequences { f } and the second s an accessng polcy that mplements { f }. The even-spacng objectve of accessng polces yelds a lower bound, as follows. Proposton 9 The obsolescence rate under any accessng polcy mplementng the access frequences { f } satsfes for each, r 1 E[X] f + f ) h 1/ f. E[X] µ µ Proof. r = = = = = 1 E[X] lm m n j=1 E[Z j ] n n m n d je[x] 1µ 1 h d j 1 E[X] lm 1 n n j=1 1 E[X] lm 1 ne[x] m n + 1 n n µ µ 1 E[X] f + 1 m 1 n lm E[X] µ µ n n j=1 1 E[X] f + 1 m n lm E[X] µ µ n 1 E[X] f + f ) h 1/ f, E[X] µ µ m n j=1 h d j n hn/m n h d j ) )) where the nequalty comes from the Schur convexty of m n Proposton 8). j=1 hd j n the d j s cf.

24 34 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS The above lower bound can be acheved only n specal cases. For nstance, f the frequences are all equal, the polcy that accesses pages 1,2,..., cyclcally yelds ths optmal obsolescence rate. Another example where we can fnd a feasble accessng polcy achevng the lower bound s when the frequences are of the form f = 1/2 k, where k s an nteger for every. We return to the general case after consderng the cost-mnmzaton theorem. The proof of the followng theorem gves a soluton technque applcable to general weghts c and shows that the technque leads to explct results n an nterestng specal case. Proposton 10 Assume that the weghts n the cost functon are proportonal to the mutaton rates of the pages,.e., c = c 0 µ for all = 1,2,...,. Then for any schedulng polcy, ) C = c 0 µ r c 0 µ 1 E[X] + 1 E[X] h > 0, 1.26) where µ = =1 µ. =1 Proof. For the moment, let the c be general. Followng Proposton 9, we have C C, where C s the soluton to the followng optmzaton problem: subject to x 0 and C = mn =1 =1 c 1 1 x + 1 ) x h 1/x E[X]µ E[X]µ =1 x = 1. To solve the above problem, we use Lagrange multplers and defne Lx 1,...,x,λ) = =1 +λ c 1 1 x 1 =1 x + 1 ) x h 1/x E[X]µ E[X]µ ). 1.27) By the convexty of the functon =1 c 1 1 µ E[X] x + 1 ) µ E[X] x h 1/x n the vector x 1,...,x ), the soluton satsfes the necessary and suffcent condtons: L = c x µ E[X] L λ = =1 1 h 1/x + lnh ) h 1/x + λ = ) x x 1 = )

25 OPTIMIZATIO I WEB SEARCH EGIES 35 Observe that h < 1, so that h 1/x < 1. One can easly check that the functon 1 y + ylny s strctly decreasng n y for y < 1. Thus, under the assumpton that c s proportonal to µ, we conclude from 1.28) that all h 1/x are dentcal and that the mnmum s acheved, by 1.29), when x = lnh =1 lnh = lnh ) 1 =1 lnh. 1.30) 1 ) Ths soluton s postve so t s also the soluton to the mnmzaton problem n 1.27). Hence, ote that so =1 =1 C = c 0 µ c 0 E[X] = c 0 h = E [ = c 0 exp and the proof s complete. { x + c 0 x h 1/x =1 E[X] =1 }) µ 1 E[X] + 1 E[X] exp{ lnh =1 µ 1 E[X] + 1 E[X] =1 µ X }] > 1 E µ 1 E[X] + 1 E[X] =1 =1 h ). [ =1µ X ] h > 0, = 1 µe[x], When the weghts n the cost functon are not proportonal to the mutaton rates of pages, one can stll use Lagrange multplers to solve the optmzaton problem. As noted earler, however, we do not have closed-form solutons n general. It s also worthwhle notcng that the optmal access frequences cf. 1.30)) n the above lower bound are not necessarly proportonal to the page mutaton rates µ, a fact that has emerged n the context of other pollng systems see, e.g., Borst et al. 1994) and Boxma et al. 1993)). Rather, they are proportonal to lnh ) 1 = 1. ln E[e ]) µ X Proportonalty to the µ occurs only when X s a constant. ote also 1 that the magntude of the dfference between µ E[X] and ln E[e ]) µ X s large f VarX) s large or X s large n the convex orderng sense). To summarze, the results of ths secton show that, f the weghts n the cost functon are proportonal to the mutaton rates of the pages, then an accessng polcy that comes close to the lower bound n Proposton 9 wth the f nearly proportonal to the lnh ) 1 wll come close to mnmzng C.

26 36 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS Fndng good accessng polces that realze a gven set of access frequences s the subject of the next two sectons. In Secton 5, we develop an optmal randomzed accessng polcy, and n Secton 6, we adapt the well-studed golden-rato polcy to our problem, prmarly as a canddate for good asymptotc performance; we wll see that ths polcy gves an obsolescence rate wthn 5% of the lower bound, n the lmt of large. We remark that ths problem s closely related to the desgn and analyss of pollng/ splttng sequences n the context of queueng and n partcular, communcaton) systems Andrews et al., 1997; Aran and Levy, 1992; Borst et al., 1994; Boxma et al., 1994; 1993), where algorthms are descrbed as template drven or generalzed round robn. Wth future research n mnd, we note that these studes suggest other approaches worth nvestgatng, e.g., extensons of the mathematcal programmng technques n Borst et al. 1994) and the algorthms Aran and Levy, 1992) derved from Hajek s results on regular bnary sequences Hajek, 1985). Although the latter lack the establshed performance bounds of the golden-rato polcy, smulatons n the earler queueng models show they are superor algorthms. Thus, they make promsng canddates for our page-accessng model Randomzed Accessng and Its Optmal Soluton Let f 1, f 2,..., f be gven access frequences. Accordng to the randomzed schedulng polcy, at each decson pont, the crawler chooses to access page wth probablty f ; the decson s made ndependently of all prevous decsons. One can easly see that {d j } j, {X j } j and {Z j } j are three sequences of..d. random varables for all. Moreover, d j has a geometrc dstrbuton: Pd j = n) = f 1 f ) n 1. Thus, we have Lemma 4 For gven frequences f 1, f 2,..., f, r = 1 E[X] f + f E[X] µ µ Proof. As d j has a geometrc dstrbuton, we obtan E[X j] = n=1 f h 1 h + f h f 1 f ) n 1 ne[x] = E[X] f, and by Lemma 3 we have, E[Z j] = f 1 f ) ne[x] n ) h n n=1 µ µ = E[X] f h, f µ µ 1 h + f h so elementary renewal theory and 1.22) mply r ρ) = E[Z j ρ)] E[Xj ρ)] = 1 E[X] f + f E[X] µ µ ). f h 1 h + f h ).

27 OPTIMIZATIO I WEB SEARCH EGIES 37 It s nterestng to compare the lower bound of Proposton 9 wth the obsolescence rate of the randomzed polcy. One can see that when f s small close to 0) or large close to 1), the dfference between r and the lower bound tends to 0. More precsely, ths dfference s h µ E[X] f 2 + o f ) 2 ) when f goes to 0, and s h µ E[X] 1 + lnh ) f 1) + o1 f ) when f goes to 1. We now consder the problem of fndng the optmal access frequences under the randomzed polcy. Frst, we have the followng lower bound over all frequences. Proposton 11 Assume that the weghts n the cost functon are proportonal to the mutaton rates of the pages,.e., c = c 0 µ for all = 1,2,...,. Then ) C = c 0 µ r c 0 µ 1 =1 E[X] =1 h 1 1) ) =1 h 1 1) Moreover, ths lower bound s acheved when the access frequences are proportonal to h 1 1. The proof uses agan Lagrange multplers and can be found n Coffman Jr. et al. 1998). ote that f the weghts n the cost functon are not proportonal to the mutaton rates of the pages, the bound s stll vald, see dscussons n Coffman Jr. et al. 1998), provded c mn 1 µ c =1 µ h 1 1 ) 1 + =1 h 1 1 ) Asymptotc Optmalty and the Golden Rato Polcy In ths secton we consder the asymptotc large- behavor of schedulng polces. A smlar study was carred out by Ita and Rosberg 1984) n the context of the control of a multple-access channel. Some of the results here are analogous to thers. We defne asymptotcally optmal polces wth respect to the lower bound n Proposton 10. Hence, we assume throughout ths secton that the weghts n the cost functon are proportonal to the mutaton rates of the pages,.e., c = c 0 µ for all = 1,2,...,. We say that a polcy π s asymptotcally optmal f lm Cπ) C = 0. ote frst that f the total mutaton rate µ tends to zero, then all cyclc polces are asymptotcally optmal. Indeed, consder an arbtrary cyclc polcy wth cycle length

28 38 HADBOOK OF OPTIMIZATIO I TELECOMMUICATIOS K. It follows from 1.23) and Lemma 3 that C = c 0 = =1 c 0 KE[X] µ r m K µ =1 j=1 = c 0 µ c 0 KE[X] = c 0 µ c 0 KE[X] = Oµ 2 ), {d je[x] )} 1µ 1 h d j m K =1 j=1 m K =1 j=1 1 1 d j µ E[X] + Oµ 2 )) ) d j µ E[X] + Oµ 2 ) ) so f µ 0, then C 0. Thus, we assume that when, the total mutaton rate µ, as well as the expected access tme E[X], s fxed. However, for any, 1, we have µ 0 when. Under such assumptons, the lower bound C n Proposton 10 becomes lm C = c 0 µ c 0 E[X] 1 lm =1 h ) = c 0 µ 1 E[X] + 1 E[X] e µe[x] ), 1.32) where we used the facts that E[e µ X ] = e µ E[X] + oµ α ) and that =1 µα 0 for all 1 < α < 2. ow consder the followng cyclc schedulng polcy, called the Golden Rato polcy, and studed n Ita and Rosberg 1984) for the control of a multple-access channel. The polcy s defned n terms of the Fbonacc numbers F k = φk 1 φ) k 5, k = 0,1,..., where φ = 5 + 1)/2, and where φ 1 = 5 1)/ s the golden rato. For any fxed k, let γk,) denote the golden rato polcy wth F k the cycle length, and the total number of pages. It s assumed that F k. Let Mk, be the number of page- accesses n each cycle of γk,); these numbers satsfy f F k M k, f F k and =1 M k, = F k, where f are the optmal access frequences gven by 1.30) f = lnh =1 lnh.