The independence of miss ratio on page size

The Independence of Miss Ratio on Page Size RONALD FAGIN AND MALCOLM C. EASTON I B M Thomas J . Watson Research Center, Yorktown Heiohts, New York AnsTRXCT A theoretical justification is given to the empirmal observation that in some computing systems with a paged, 2-level storage hierarchy, long-term miss ratio is roughly independent of page size Let M I S S be the expected working+set miss ratio in the independent reference model, with expected working set size C A P pages Now form blocks, by combining the B pages with the highest probabllitms of reference into one block, the B pages with the next-highest probabilities of reference into a second block, and so on Let M I S S * be the expected working-set miss ratio when all data are moved m blocks and when the expected working set size is again C A P pages, that is, C A P / B = C blocks. It ~s proved that I M I S S -- M I S S * I < (2/C) .-}- (33/C2). Thus, ff the expected working-set size (in blocks) is sufficiently large, then the miss ratios in the blocked and unblocked cases are approximately equal This result m used to argue the approximate independence of miss ratio on page size in more realistic models of page references. KEY WORDSAND PHRASES. page raze, working set, LRU, miss ratm, storage hierarchy, independent reference model cR c~'rEGo~.tms: 40, 4.3, 6.1, 6.20 1. Introductw~+ An important parameter in the design of a paged computing system is the page size, t h a t is, the number of bytes of information transferred from one level of a storage hierarchy to another in case of a page fault. Among the factors which influence the choice of page size are the page fault rate (or "miss ratio"), the fragmentation of memory, and the access and transfer times of secondary memory devices (see Gelenbe et al. [12] for a more detailed discussion). The research for this paper was sparked by an empirical observation of Bennett [3], who examined a page reference trace 1 from the I B M Advanced Administrative System (A.A.S.) [22], a large internal I B M data management system. Bennett found no consistent relationship between miss ratio and page size--for some main (first-level) memory sizes, the miss ratio was slightly laIger for the larger page size, and for other main memory sizes, slightly smaller. In all cases the size of main memory had a vastly greater effect on miss ratio than did page size, if the page size was sufficiently large (at least 1500 bytes). The cache multiprogramming trace of Kaplan and Winder [17] and the (main memory) program address traces of Lewis and Shedler [18] and of Anacker and Wang ([16], [2]) give similar results. In the examples cited, different but similar page replacement algorithms were employed, including the working-set memory management policy [5] and the closely-related L R U ("least recently used") memory management policy ([19], [1]). At first glance, some published data seem to contradict this insensitivity of miss ratio Copyright © 1976, Association for Computing Machinery, Inc General permission to republish, but not for profit, all or part of th2s material is granted provided that ACM's copyright notice is given and that reference is made to the pubhcation, to its date of issue, and to the fact that reprinting privileges were granted by permismon of the Association for Computing Machinery. Authors' present addresses R Fagin, IBM Research Laboratory, San Jose, CA 95193, M.C. Easton, IBM Thomas J. Watson Research Center, P O. Box 218, Yorktown Heights, NY 10598 1 All requests in the trace were to the data base--there were no index references or program references. Journal of the Assocmtlon for Computing Machinery, Vol 23, No 1, January 1976, pp 128-146 The Independence of Miss Ratio on Page Size 129 to page size. A dramatic example is found in Chu and Opderbeck's paper [4], in which LRU miss ratio seems to depend very heavily on page size. Their miss ratio curves asymptotically approach a value which is simply the number of initial loading misses (which is the same as the number of pages in the program, if main memory is large enough to hold the entire program) divided by the length of the observed page reference string. Of course, in general, there are approximately twice as many initial loading misses when the page size is half as large, and so the asymptotic value of their miss ratio curve is approximately twice as big when the page size is half as large. By contrast, in this paper we distinguish between initial loading misses on the one hand, and the "transientfree," or "long-term," miss ratio on the other hand (cf. [9]). In the case of LRU, "transient-free" means that the miss ratio is measured starting at a time after main memory has filled; in the working-set ease, "transient-free" means that the miss ratio is measured starting at a time greater than T, where T is the window size. In many cases, it is possible to anMyze the performance of a storage hierarchy more accurately by considering the effects of initial loading misses and of the transient-free miss ratio separately. From here on in this paper, "miss ratio" refers to the transient-free, or long-term, miss ratio The main result of this paper ~s a bound on the effect of page size on the expected working-set miss ratio, in the independent reference model (in whmh page ~ is referenced at time t with probability p,, independent of past history). Specifically, let M I S S be the expected working-set miss ratio in this model, where the window size is chosen so that the expected working-set size is CAP pages. Now form blocks, by combining the B pages with the highest probabilitms of reference into one block, the B pages with the next-highest probabilities of reference into a second block, and so on. Let M I S S * be the expected working-set miss ratio when all data are moved in blocks, and when the window size is chosen so that the expected working-set size is again CAP pages, that is, C A P / B = C blocks. It is proven that I M I S S -- M I S S * I < 2/C + 33/C ~. (1) Thus, if C is sufficiently large, then M I S S * ~ M I S S , that is, the expected miss ratios in the blocked and unblocked eases are approximately equal In Section 2 we will show that (1) implies the approximate independence of miss ratio on page size in certain more realistic models of page references than the independent reference model. This tells us that in some eases, transient-free miss ratio is not a key factor in the selection of page size. Of course, the effect of initial loading misses, which can be considered separately, is an important factor. 2. More Reahst~c Models In this section, we will show that our result about the insensitivity of miss ratio to page size in the independent reference model implies that this insensitivity holds in some more reahstic models of page reference patterns. For convenience in exposition, we will deal in this section with LRU miss ratio, rather than with working-set miss ratio. Indeed, we will show later that our main result (1) can be interpreted as saying that LRU miss ratio is insensitive to page size in the independent reference model, if the capacity (size of main memory) in, say, bytes, is held fixed (and if, as before, pages are blocked together in order of their probabilities). The independent reference model is, in general, inadequate. Various authors, including Lewis and Shedler [18], Denning, Savage, and Spirn [6], and Easton 17] have presented models of page reference patterns (or related quantities) in which page references are the result of two components, where the first is, roughly speaking, an "independent" or "random" component, and the second is a local component, such as a "locality of reference" or "sequential" component. We will first show that approximate independence of expected LRU miss ratio holds in one such model (Easton's), and then we will gem eralize the argument to cover other such models. 130 R. F A G I N A N D M.C. EASTON We brmfly describe Easton's model. If there are n pages, then there are n + 1 parameters, r, p t , • .. , p , , all between 0 and 1. Assume that page i was referenced at time t A t time t --t- 1, a coin is flipped, which comes up heads with probabihty r, and tails with probability 1 - r. If heads comes up, then page z is rereferenced, and we say t h a t the reference to page i (at time t ~ 1) was generated during "rereference" or "sequential" mode. If tails comes up, page3 is then referenced with probability p~, for 1 _< 3 <(including the case 3 = ~), and we say t h a t the reference to page j was generated during " r a n d o m " mode. Thus, if Q,j is the probability t h a t page 3 is referenced at time t ~ 1, given t h a t page ~ was referenced at time t, then j r + (1 - r ) p , , Q" = ( ( 1 - r)p~, J = z, j ~ Easton found [7] t h a t with appropriate choice of parameters, his model gives a good fit to the L R U miss ratio curve of A.A.S., which we referred to in the Introduction. Intuitively, this model "works" because if the page size is large enough, then "locality" can be approximately captured by rereferences to the same page. We will now show t h a t in this model, L R U miss ratio is insensitive to page size. Let S := R1 • - Rm be a finite-length page reference string generated b y this model. Thus, each R, is the name of a page. Assume that the string S is long enough that the effect of initial loading misses is neghgible. For each ~, let R~* be the name of the block which contains page R , . Then S* = R:* ... R,,* is the corresponding block reference string. We will show that the L R U miss ratio (where pages are the unit of transfer) over page reference string S and with capacity CAP pages is approximately the same as the L R U miss ratio (where blocks are now the unit of transfer) over block reference string S* and with capacity C A P / B blocks (i.e. CAP pages), when there are B pages per block. Let R~ , . . - , R,~ (where i~ < ~2 < " " ~ zk) be those page references which were generated after "tails" was flipped; thus, these are the page references which were generated during " r a n d o m " mode. So S~,d = R,~ .- R~ is the substring of S which conrains only the page references generated during random mode, and Srand * . .R~* . . R,k* is the corresponding substring of S*. Let z be the number of misses if we apply the L R U page replacement algorithm to string S~,.d, with capacity CAP pages. Each reference which was generated during sequential mode is automatically a " h i t " ; hence, the number of L R U misses (with capacity CAP pages) over page reference string S is also z. Now S~,,d can be looked at as a block reference string generated under the assumption of independent block references, where each block has reference probability equal to the sum of the reference probabilities of the pages which compose it. As we have said, our main result can be interpreted as saying that under the assumption of independent references, the L R U miss ratio is approximately the same in the blocked and unblocked cases. So z is also approximately the number of misses if we apply L R U block replacement to the block reference string S~,,d * with capacity C A P / B blocks. Again, there are exactly the same number of misses over the string S* as over S~,~d. So the miss ratios in the blocked and unblocked cases are approximately the same, as claimed. This result can be generalized to some other models with an "independent" component and a "locality of reference" component. The argument we will now outline will, of course, have to be tailored to fit each particular model. Assume t h a t the page size is large enough t h a t almost all "locality" references are hits. Then the miss ratio is essentially determined b y the independent component. So once again, the insensitivity of miss ratio to page size under the assumption of independent references implies this insensitivity under more realistic assumptions. We remark t h a t for models other t h a n the independent reference model, the difference in miss ratio between the blocked and unblocked cases will not necessarily be bounded b y the right-hand side of inequality (1). In our result about the independent reference model, we have assumed t h a t pages The Independence of Miss Ratio on Page Size 131 are blocked together in order of likelihood of reference. Of course, the real situation is much more complicated. However, we can justify this assumption on several grounds. 1. In the case of certain large-data base systems, such as A.A.S., groups of sequentiMly stored records tend to have similar access properties. So in forming blocks in the usual way of grouping together sequentially stored records, those pages which are blocked together have approximately the same probabilities of being referenced; t h a t is, pages are blocked together approximately in order of probability of reference. 2. Of all possible ways of blocking pages together with B pages to a block, Yue and Wong [23] proved that in various storage applications and under various criteria of optimality, the blocking we have considered (in which pages are blocked together in order of probability of reference) is optimal. Hence, the use of this particular blocking is a natural assumption to make when one is discussing the performance of a storage hierarchy. We remark that the first author has found a counterexample to thc conjecture that expected LRU miss ratio is minimized in the independent reference model by this blocking [10]; however, this blocking seems to be near-optimal in the independent reference model. 3. Formal ~Stateme~t of Mai~t Result Let {pl, " " , p,} be a probability distribution (that is, ~ p , = 1 and each p, _> 0). Assume t h a t at each discrete time t, page z is referenced with probability p,, independent of past history. (This is the independent reference model.) The expected workzngset ntiss ratw (wzth window size T) [5] is the probability that the page referenced at time t was not one of the pages referenced over the course of the previous T (not necessarily distinct) references. Under the independent, time-invariant assumption of the independent reference model, it is clear t h a t this expected working-set miss ratio is independent of t, for t > T. Let CAP be the expected working-set size, t h a t is, the expected number of distinct pages to appear over the course of T references. Define M I S S ( C A P ) to be the corresponding expected working-set miss ratio. Thus, M I S S ( C A P ) is the expected working-set miss ratio with window size T, where the expected working-set size with window size T is CAP pages. Later on, we will discuss the close relationship between M I S S ( C A P ) and the expected L R U miss ratio with capacity CAP pages. We will now describe the blocked case. Let B (the "block size") be a positive integer which, for convenience, we assume divides n. Assume t h a t p~ > p2 >_ ' - ' _> p, ~ 0, and let B u, = ~ p(,-~)B+~, 1 _< i < n/B. Thus, ul = Pl + "'" + P n , u2 = PB*~ + "'" + P:B, etc. This corresponds to combining the B pages with the highest probabilities of reference into a block, the B pages with the next-highest probabflitms of reference into a second block, and so on. The blocked case corresponds to the independent reference model with block probabilities { u ~ , - . . , u~/BI. Define M I S S * ( C A P ) to be the expected working-set miss ratio (over the probability distribution [ u l , - • - , u~j,I ), when all d a t a are moved in blocks, and when the window size is chosen so t h a t the expected working-set size is C A P / B blocks ( C A P / B blocks contain the same number of bytes as CAP pages, and this is the quantity we hold fixed in comparing the blocked and unblocked cases.)2 Let C = CAP~B, and write M I S S and M I S S * for M I S S ( C A P ) and MISS*(CAP). The main result of this paper is IMISS- M I S S * I < 2/C + 33/C 2. (2) 2 [t may well happen that there is no integer T* such that the expected working-set stze wLth window size T* is CAP/B blocks. If so, then we interpolate, as we will see 132 a . F A G I N AND M.C, E A S T O N We can think of 2/C as the first-order error term, and 33/C ~ as the second-order error term. We will actually prove a slightly stronger statement than (2). Note that statement (2) is a distribution-free result: that is, the error terms do not depend on the values of the p, (or even on n, the number of pages). 4. Detads Abaut Main Result We. begin this section by giving an expression [5] for M I S S ( C A P ) , the expected workingset miss ratio when the expected working-set size is C A P pages. The expected workingset size, that is, the expected number of distinct pages which will be referenced over the course of T references, is S ( T ) = ~,~1 (1 - (1 - p,)r), because the probability that page z is referenced is 1 - (1 - p,)r. The expected working-set miss ratio with window p,)T is the probability that page i size T is M ( T ) = ~ p , (1 - p,)r, because p,(1 is the next page referenced and that page z did not appear in the last T references. Thus, if S ( T ) = CAP, then M I S S ( C A P ) is by definition M ( T ) = M ( S - ~ ( C A P ) ) . Note that M ( S - ~ ( x ) ) is well-defined for each real number x between 0 and ~, even if the intermediate parameter T = S-~(x) is not an integer. By this procedure, which amounts to an interpolation, we can define M I S S ( x ) = M ( S - I ( x ) ) for each x with 0 < x < , Similarly, in the blocked case, we define M I S S * ( x ) = M*(S*-~(x/B) ), for 0 < x < n, where - - n/B S*(T) = ~ (1 - (1 - u,)~), 0_< T < ~, n/B M*(T) = ~ u,(1 -- u,) ~, 0 _< T < ~. Thus, ff the expected working-set size is C A P pages (i.e. C A P / B blocks), then M I S S * ( C A P ) is the expected working-set miss ratio in the blocked case. We will now briefly discuss the relationship between M I S S ( C A P ) on the one hand, and the expected L R U miss ratio M R ( C A P ) with capacity C A P on the other hand. Denning and Schwartz [5] make the intuitive observation that M I S S ( C A P ) ~ M R ( C A P ) . In w~rious simulations of the independent reference model, we found that M I S S differs from M R by around .01, when the number n of pages is several hundred. Indeed, the first author has recently proven [11] that in a certain precise sense, M I S S converges asymptotically to M R as the number n of pages gets large, m the independent reference model. So (2) imphes that in the independent reference model, the expected LRU miss ratio is approximately independent of page size, if the size of main memory is held fixed aad if pages are blocked together in order of their probabilities. (Of course, we are assuming that C is large enough that the right-hand side of inequality (2) is small, and that n is large enough that M I S S ( C A P ) ~ M R ( C A P ) . ) 5. Proof of Main Result We will prove the following theorem. THEOREM. Let M I S S be the expected working-set muss ratw in the independent reference model, with expected working-set szze C A P pages. Let M I S S * be the expected workzng-set ~zss ratw after blocking, where B old pages form each new block, where pages are blocked together ~n order of their reference probab~ht~es, and where the expected working-set size zs again C A P pages (z.e. C = C A P / B blocks). Assume that the original number of pages zs dwzsible by B. Then I M I S S - M I S S * I < 2/C + 33/C 2. PROOF. The theorem is trivial if C A P < 2B or C A P > n; hence, we will assume that 2B < C A P < n. We will actually prove - 1 . 9 2 ( B - 1)~CAP - 6 ( B / C A P ) 2 < M I S S - M I S S * < 1.01(B - 1)~CAP + 3 3 ( B / C A P ) 2 (3) The Independence of M~ss Ratw on Page Size 133 Of course, (3) lmphes { M I S S - M I S S * < 2 B / C A P + 3 3 ( B / C A P ) ~, which is the result of the theorem. There will be six main steps in the proof. Step0. S ( T ) < T, Stcpl. 0 < ~,~1(1- Step 2. 0 < B ~$!~(1 if T ~_ 28. Step 3. - . 7 3 6 ( B - 1 ) / T < ~ = 1 p,(1 - p,)r _ ~ = / ~ u,(1 - u , / 8 ) r < .736(B - 1)IT, if T k 0 . Step 4. 0 _< ~ = n[B 1 u,(1 - u , / 8 ) r - ~ / ~ if T > 2 8 . Step 5. H(T)/H(t) if T >_ 1 - u,/B) r < B- p,)7 _ B ~ = / ~ ( 1 ~n/B - u , / B ) r - B z.~=l (I < S(T)/S(t), if __ U z ) F / B 1, if T > 0. 1) + 6B2/T, < .184(B- u,(1 - u,) 7'/8 < .271(8 - 1 ) / T + 33B2/T 2, T > t > 1, where H(x) = 1 - M(x). Step 0 folh)ws immediately from the development m [5], provided T is an integer (which will not ahvays be the case for us--hence, we must prove It directly). Step 5 says that if the expected working-set size is increased, then the proportional increase in expected working-set hit ratio is bounded by the proportional increase in expected working-set size. Thus, if the expected working-set size is doubled, then the expected hit ratio is at most doubled. We will now show that these six steps imply statement (3). Instead of using the fuactions S* and 111" of Section 2, it will be convenient to define closely related functions S . and M A , as follows. n/B S~(T) = B ~(1 n/B MB(T) = ~u,(1 - (1 - us)T/B), - u J T/B. It is easy to see that M B ( S ~ ( C A P ) ) = M*(S*-~(CAP/B)) = MISS*(CAP), expected working-set miss ratio under blocking. T w o other functions we will find convenient to use are given by H ( T ) = 1 -- M ( T ) , the HB(T) = 1 - MB(T). I t is easy to see t h a t S and SB are each monotone, and each map onto the half-closed, half-open interval [0, n). So we can find Tt and TB such that S ( T O = C A P = S R ( T , ) . Now T1 >_ 1, since S(1) = 1 < C A P = S(T1), and since S is monotone increasing. So we can apply step 0, to obtain T~ > S(T1) = C A P > 2B. Hence T, > 2B. (4) TI B Now S ( T ) = n - ~ = ~ ( 1 - p,)r, and S , ( T ) = n - B ~ ! ~ ( 1 -- u~) . So if we add together the inequalities of steps 1 and 2, with T, substituted for T, we find t h a t 0 < SB(T,) -- S(T~) < Q, (5) where Q = 1.184 (B - 1) + 6 (B~/T,). (6) Since S(T1) = CAP, statement (5) says 0 < SB(T,) - C A P < Q. (7) So So(T1) > C A P = S s ( T B ) . Since S~ is monotone increasing, Tx > T~. (8) The functions H s and S B / B have the same form as H and S, with u, substituted for p , , with n / B substituted for n, and with T / B substituted for T. So it follows from step 5 t h a t if ( T / B ) > ( t / B ) > 1, that is, if T > t > B, then H~(T)/HB(t) < (S~(T)/B)/(Sa(t)/B) = SB(T)/SB(t). (9) 134 R. F A G I N A N D M.C. EASTON Now T~ > T , ~ B : f i r s t , Ti ~" T B b y (8), and S , ( T , ) = C A P > B = S , ( B ) , s o since S , is monotone increasing, this implies that T , > B. Therefore, we can substitute Ti for T and T , for I in (9) ; then H,(T1)/H,(T~) < S~(Ti)/S,(T,) = S,(T~)/CAP. So HB(T~) _~ = _< < H~(TB)(S,(T~)/CAP) H ~ ( T , ) ( 1 --[- ((SB(T~) -- C A P ) / C A P ) ) g s ( T ~ ) ( 1 + ( Q / C A P ) ) b y (7) H~(T~) -k ( Q / C A P ) , since H ~ ( T ~ ) _~ 1 We have lust shown that H~(T~) < H , ( T ~ ) + ( Q / C A P ) . (lO) It is easy to see t h a t HB is monotone increasing. So, from (8), II~(T~) _> H ~ ( T , ) . (11) From (10) and (11), it ~mmednttely follows t h a t 0 < H , ( T i ) -- H , ( T ~ ) ~ Q/CAP, and hence -Q/CAP ~ Ms(T1) -- M~(TB) ~ O. (12) Adding together the inequalities of steps 3 and 4, with T~ substituted for T (which is all right b y (4)), we find t h a t --.736 (B - 1)/T1 < M(T1) - MB(T1) < 1.01 (B - 1)/T~ W 33 (B~/Ti'). (13) Since Ti > C A P b y step 0, statement (13) implies - . 7 3 6 (B - 1 ) / C A P < M(T1) - MB(T1) < 1.01 (B - 1 ) l e A P + 33 ( B / C A P ) ~. (14) Also, T1 ~_ C A P imphes (from statement (6)) t h a t Q _~ 1.184 (B - 1) + 6 (B~/CAP). (15) If we add together the inequalities in statements (12) and (14), and substitute for Q the right-hand side of (15), we get --1.92 (B - I ) / C A P - 6 ( B / C A P ) 2 < M ( T i ) - MB(TB) < 1.Ol ( B - 1 ) ~ C A P + 33 ( B / C A P ) ~. Since as we observed, M(T1) = M I S S ( C A P ) and MB(TB) = M I S S * ( C A P ) , this gives us statement (3), as desired. I t remains to prove steps 0-5. 6. Prehm~narzes One of our basic techniques will be the use of Schur functions ([21]; see also [20]). Assume for convenience throughout t h a t all functions considered are infinitely differentiable Definitwn. Assume t h a t u = ( a l , . - . , a,~), where al >_ a~ ~_ . . - ~_ am, and t h a t = (31, ' " , 3~), where fll >_ g2 ~ "-" ~_ tim- We say ~ > ~ if ~ - 1 c ~ . >_ ~ - l f l , , 1 < k < re, and ~ - 1 ~ , = ~ - 1 3 ~ . Definztion. A real-valued function f of m real variables x l , • • • , x~ is a Schur ]u~ctzon if for each pair i ~ 3, (x, - x,) (O/Ox,)f - (O/Ox,)f ~ O. THEOaEM (Schur). Let f be defined for x, > . . . > x,~. Then f ( a , , . . . , am) _> f(31, "'" , 3m) whenever ¢l > ~, ~ff f is a Schur functzon. Speczal Case 1 [13, p. 89]. Let ~ be a real convex function of one real variable, t h a t is. ~b((x "-b y ) / 2 ) ~ ½(4~(x) --t-~(y)). (16) The Independence of M i s s R a t w on Page Szze T h e n the function (x~, .. 135 , Xm) ---~ ~ 1 ¢ ( X , ) is a Schur function. Hence, if a )- ~, S p e c m l Case 2. Let ~ be continuous a n d concave, t h a t is, the inequality sign in (16) is reversed. T h e n - ~ is convex. Hence, ff a > ~, t h e n ~ - ~ ¢ ( a , ) < ~,2.~ ¢(fl,). An infinite sum ~ : ~ a~ of real n u m b e r s a, is alternating if 1. a, ~ 0 ~ f f a , + l ~ 0, 1 _~ ~ < ~¢. ~. la, I _> la~÷ll, 1 < ~ < ~. 3. a , - - ~ 0 as t--~. ~ . A n alternating s u m is always convergent, and its value lies between a n y two successive partial sums ( L e i b n i t z ' s test [15, p. 68]). 7. P r o o f of Step 0 We will prove t h a t S ( T ) ~ T, if T > 1. Let ~b be the functmn x --~ 1 - (1 - x) v, with d o m a i n the closed interval [0, 1]. I t is easy to see t h a t ¢ is concave. Clearly (pl , " " • , p~) ~ ( 1 / n , . • , l / n ) . So b y specml case 2 of Section 6, ± S(T) o = ¢(p,) < ~ O ( 1 / n ) Hence, we need only show t h a t n ¢ ( 1 / n ) (1 - = ,~¢(1/n). ~ T. This is equivalent to l/n)7 ~_ 1 - T/,. (17) If T ~ ~, then the right-hand side of (17) is negative, a n d (17) follows lmmedmtely. If T ~ n, then the binomial expansion of (1 - 1 / n ) r is an alternating sum, and (17) again follows. 8. P r o o f of Step 1 We will show t h a t n/B 0 < (l/B) ~ (1 - p , ) r _ ~ (1 - u J B ) r < ( B - 1)/B, (18) i f T ~ 0. Let ¢ he the function x ~ (I - x) ~, with domain the closed interval [0, I]. It is easy to see that ¢ is convex and monotone decreasing with range [0, 1]. Since 4~ is convex, it follows [13, p. 72] that ¢ ( (xl zc • • q- x B ) / B ) If we substitute xl = p(z-1)B+l , J'2 < (q~(xl) + • • -k ¢ ( x , ) ) / B . = p(~--I)B+2 , "'" , X. = (19) PuB into (19), for 1 < ~ < n / B , we find t h a t (1 u,/B) r ~ -- (l/B)((1 -- p(,_~)B+,) r + . . . + (1 -- p , , ) r ) . (20) Adding together the inequalities (20), for 1 < ~ < n / B , we obtain n/~ (1 -- u , / B ) r ~ ( l / B ) ~I (1 -- p,)V, ~I which proves the first inequality in (18). If 1 ~ x~ > . . . ~ xB > 0, t h e n since 4~ is m o n o t o n e decreasing, 4~(xl) -~ " . . -b -- ~ ( x~ zc " " " + < (B - 1)~b(x,) + 4~(x,) _ 4~(x,) - B B B 1 (4~(x~) - 4~(x,)). (21) 136 a , FAGIN AND M.C. EASTON Substituting x~ = p<,-x)~+~, x~ = p(,-~)B+2 , • "" , x~ = p,s into (21), for 1 _< i _< n / B , we obtain (1/B)(ch(p<,-~)s+~) + . . + O(P,B)) -4~(u,/B) _~ ( ( B - 1 ) / B ) ( o ( p ~ a ) -4~(P(~-a)B+0). (22) Now ¢'(p(,--1)B+I) _> ¢(P<,--1)B), SO we obtain from (22) (1/B)(~k(p(,-~)s+1) + " " + ¢#(p,.) ) --¢(u,/B) _< ( ( B - l)/B)(¢(p,,) --¢(p(,_~),)). (23) Adding together inequality (22) for ~ = 1 to inequalities (23) for 1 < z < 7~/B, the right-hand side telescopes to give (lIB) ~+(p,) - ~+(u,IB) < ( ( B -- 1)/B)(q~(p.) - ¢ ~ ( p x ) ) _< ( B - I)/B. (24) 1) + 6(Be~T), (25) This is the right-hand inequality of (18). 9. Proof of Step 2 We will show that n/B 0 < B ~(1 n/IB - u,/B) r- B ~(1 - u,) r/~ < . 1 8 4 ( B - if T > 2B. We will first demonstrate the tirst inequality. We need only show t h a t for each ~, (I - u , / B ) ~ > (1 - u,) T:~. (26) This is equivalent to showing t h a t (1 - u , / B ) B > (1 - u,). (27) I t is straightforward to check that the binomial expansion of the left-hand side of (27) is an alternating sum. So (27) follows We will now prove the second inequMity of (25). For each u, 0 _< u < 1, and each nonnegative number 0, define ao(u) = (1 - u / B ) zs°, bo(u) = (1 - u ) °, co(u) = ae - bo. (28) If ½ < u < 1, then e g u ) _< ao(u) _< (1 - 1/2B) Bo. In particular, if 0 = T / B , then er/n(u) _< (1 - 1/2B) r. (29) Assume from here on t h a t 0 < u < ½. We will write e0 for to(u), etc Let x and y be nonnegative real numbers. Then inequality (26), with B x substituted for T and u substituted fol u , , gives a~ > b,. Hence a~ + b~ < 2a~. (30) Multiplying together inequality (30) and the equality a~ -- b~ = ~, we obtain a~ay + %b~ - a~bu -- b~b~ < 2a=%. (31) Clearly, a=a~ = a=+~, b=b, = b=+~. (32) Substituting into (31) using (32), and replacing a,+~ - b=+u b y ~,+u and rearranging terms, we obtain ~=+u --< 2 a ~ + a~bu -- aub~. (33) T h e I n d e p e n d e n c e of M , s s R a t i o on P a g e S , z e 137 S u b s t i t u t i n g a. -- ~. for b. a n d a~ -- e~ for b~ in (33), we find ~+~ _< ~.a~ + ~a~. (34) We now claim t h a t if 3 is a positive integer, then % < j~,a~ -~ (35) This is obvious if :/ = 1. Assume inductively t h a t it is true for.7 = N T h e n from (34), eN+~ < e~a~ -4- ,~a~ < Ne~a~ ~ -4- ~ a ~ b y inductive assumption = N ~ a ~ ~v -4- e l a l N = (N -4- 1)e,a~~'. Hence (35) holds, b y induction, for every positive integer 3. Write T / B = j -4- r, j a positive integer and 0 < r < 1. Then ~T/B _< eja, + e~aj by _< .l~la~- lat -4- ~,aj b y (35) = 3*~a~+~-~ -4- ~a~ J since ar = a~~ and since al < 1 and 3e~a~ +r-I + era~+r-t (34) a~ = a~~ r < 1 (36) < ( T / B ) ~ l a ~ T / ' ) - I - - t- era~ T/B)-I = (T/B)e~(1 How big is E~? As sum. Hence we -- u / B ) T-B -4-4-e.(1 - u / B ) T-È observed, the binomial expansion of (1 - u / B ) B is a n alternating (1 - u / B ) A < 1 - u -4- ( B -- 1 ) u ~ / 2 B . (37) So er = a t - - b, = (1 -- u / B ) Br -- (1 -- u ) ~ _< (1 -- u + ( B - - 1 ) u 2 / 2 B ) r - (1 - u) r b y (37) = (1 - u ) ' [ ( 1 + ( B -- 1 ) u ~ / 2 B ( 1 - u ) ) ~ -- 1]. (38) Let z = ( B -- 1 ) u ~ / 2 B ( 1 -- u ) . Since 0 < u < ½, it is easy to see t h a t 0 < z < 1. Now the binomial expansion of (1 -4- z) r - 1 is an alternating s u m i f 0 < z < 1 a n d 0 < r < 1 ( t h a t is, there is an alternating s u m if we consider only the second, third, . . . terms of the binomiM expansion of (1 -4- z)~). P u t t i n g this together with (38), we o b t a i n u ) ' [ r ( B -- 1 ) u 2 / 2 B ( 1 - u)] = r ( B 1 ) u ~ / 2 " B since (1 - u ) > ½ < .54 ( ( B -- 1 ) u 2 / B ) , er _< (1 - 1)u~/2B(1 - u ) ~-~ _< r ( B - since we find b y elementary calculus t h a t the m a x i m u m of r / 2 ~, 0 < r ~ log 2) < .54. So er _< .54(B - 1 ) u 2 / B < .54 u ~. 1, is 1/(e (39) How big is ~1 ? F r o m (37), we find immediately t h a t ~ _< ( B - 1)u2/2B. (40) If we substitute into the last line of (36) using (39) a n d (40), we o b t a i n ~r/B _< u2(1 - u / B ) T - n ( ( T ( B -- 1 ) / 2 B ~) -4- .54). We are interested in obtaining an upper b o u n d for B ~ n a m e l y u~, can be greater t h a n ½. So from (29) a n d (41), (41) ~T/B(U,). A t most one u , , n/B B ~ ~r/a(u,) < B(1 - 1/2B) r n/B + ((T(B -- 1 ) / 2 B ) -4- .54B) ~ u , 2 ( 1 -- u , / B ) r-~. (42) 138 R. F A G I N A N D M.C. EASTON 0 PW Pmox I~'e B Fro. 1 How big can z..,~-I u~ (1 - us~B) ~-" be? Let ~ be the function x --, J:2(1 - x / B ) r-~, with domain [0, B]. By using elementary calculus, we find that ~ has its maximum at Pm~, = 2 B / ( T + 2 - B ) , and two points of inflection; the first such point of inflection, P*, lies strictly between 0 and P . . . . and the second point of inflection, P**, is bigger than P . . . . The second derivative of ¢ is positive between 0 and P*; negative, between P* and P**; and positive, between P** and B. So the shape of ¢ is roughly as in Figure I. Let VEC1 = {(xl, - . . , xm) :m a positive integer, ~,"21 x~ < 1, and 1 ~ xz ~_ x2 >_ • • > x ~ > 0}. Note t h a t the length of a tuple in VEC1 is arbitrary (but finite). I t is not important t h a t the x~'s are in descending order, b u t it will make the exposition simpler. Say we can find an upper bound M for { ~ ~#(x~) : x E VECz}, where x is an abbreviation ,K.-,~/s for ( x l , . ' . , x~). Then z_,,-1 u~2(1 - u , / B ) r-~ _~ M. Let VEC2 = { x : ~ x ~ < I, and Pm~ >__ xl ~_ x2 ~_ . . - > x , ~ 0}. Then s u p ~ 4 ~ ( x ~ ) :x E VEC~} = s u p l ~ ( x ~ ) : x E VEC2}, since if x E VEC~ and y, = r a i n ( x , , Pm~) for each i, then y E V E C 2 . Let v be a tuple in VEC2; then v can be written as the concatenation ~ of a tuple y (with all entries between P* and P ~ ) and a tuple z (with all entries between 0 and P*); if any entry of x is exactly P*, we place that entry in z. Assume that ~ z , = a. Let 1 be the unique nonnegative integer and 6 the unique real number, 0 < 6 < P*, such t h a t a = lP* + ~. Let a be the vector (P*, • • , P*, 6, 0 , . - . , 0) with length the same as that of z, and with 1 occurrences of P*, oDe occurrence of ~, and the rest O's. Clearly, a > z. Now ~b"(x) > 0 for each x in [0, P*]; hence [13, p. 76] ~ is convex in [0, P*]. Applying special case 1 of Section 5, we find t h a t ~ - ~ ~b(a~) > ~ _ ~ ~b(z,). We have shown that if v = y~"~is an arbitrary tuple in VEC~, then there is a tuple w = y (P*, , P*, ~) such t h a t ~ b ( v ~ ) < ~ b ( w , ) . VECa = {x : ~ In other words: let x, < 1, Pm,~ ~ zz >_ x2 ~_ "'" >_ x,,-1 >_ P*, P ~ x >_ xm >_ 0, where m is the length of x}. Then sup l ~ b ( x , ) : x ~ VEC3} = sup{ ~ ~b(x,) : x 5 VEC~}. Now let v be an arbitrary tuple in V E C s . Write v = y~'~, where each entry of y lies between P* and Pm~x, and where z is a tuple of length 1, whose entry lies between 0 and P ~ . Assume t h a t y is of length k, and t h a t ~ y~ = a. Let a be the tuple ( a / k , • • • , a / k ) of length k. Then y >- ~. Now ~ (x) ~ 0 for each x in [P*, Pmax[; hence, 4~ is concave in [P*, P ~ ] . Applying special case 2 of Section 5, we find t h a t ~ 4~(Y,) _~ ~b(a,). In other words: let VEC4 = Ix : ~ x, < 1, xz . . . . . xm-~ , where nm is the length of x}. Then s u p { ~ q , ( x , ) : x ~ VEC~} = s u p { ~ ¢ ( x , ) : x ~ VEC~}. Let v = {¢, ¢, • • • , ~, ~} be an arbitrary tuple in V E C 4 . Then ~(v,) _< ( 1 / ~ ) l ~ ( ~ ) l + ~(~) -< ( ~ ( ~ ) / ~ ) + _< ~(~) (max ~(x)/x) + (m~ ~(~)). (~3) O~x~l O~z~l The Independence of M i s s R a t w on Page Size 130 Now ¢ ( x ) / x = x(1 - x / B ) r-" Let k be arbitrary for the moment, and let ~ be the function x --, xk(1 - x / B ) T-B. We are interested in finding max0~<l ~b for k = 1 and k = 2 (and, in proving step 4, we will be interested in the case k = 3). By elementary calculus, we find that ~b has its maximum at k B / ( T + k - B), with maximum value (kB/(T + k - B ) ) k (1 - k/(T + k -- B ) ) r-B. (44) We will now estimate (1 - k / ( T + k - B ) ) 7-B. I t i s well known that (1 - l / x ) ~ ~ 1/e as x -~ o~ Also, the function x --> (1 - 1 / x ) ' is a monotone, strictly increasing function of x for x > 1, fol, the derivative of this function, evaluated at z > 1, is easily found to be = 1-- 1 . . . . . . . . . . 2X2 3x~ 4x ~ by using the Taylor expansion of log(1 - l/x) >0 So 1>(1 e- x+l = 1 1) 1 x+l = x~-I 1 -- 1 x+l )2 x x+l' (45) If we multiply (45) through by (x + 1)/x, weflnd that (1 1 x+l )" lx+l <---e x (46) So 1 T-t-k-- B = < 1 --x+l " ' 7 wherex = ( T - B)/k by (46) So the maximum value of ~b, which we found to be expression (44), is bounded by (from(47) ) (T+k B - So from (43), < B 4B ~ + e ( T -- B) e2(T -- B ) 2" Tracing back what we have demonstrated, this means that ~/B (1-- u~_}\r-" _< B + 4B ~ (49) We will now estimate the subexpression (1 - 1/2B) T of (42). It will be convenient for work later on to find a more general estimate than we need now. We will show that i f s > l a n d t > 0, then (1 -- l / s ) t < m!s'~/t '~ for each integer m ~ 0. (50) First, (1 - - l / s ) t = (1 - - l / s ) "('/') < e-'/', (51) 140 R. FAGIN AND M.C. EASTON since as we showed, (I -- I / x ) ~ P I/e. Now if z > 0, then e" > z'~/m! for each integer nn > 0, since z'~/m ! is one term of the Taylor expansion of e~. Hence e-~ < m !/z m. Applying this to the right-hand term of (51), (1 -- 1/s) t < e-~/" < m! s~/t '~, as desired. Hence (1 - 1/2B) r < 2 B / T , (52) where we let n~ = 1 in (50). Substituting into (42) using (49) and (52), we obtain ~/" 2B 2 B~,_a,TjB(u,) < - T - /T(B-+ \ 2B 1) ~( B .... 4 i f , -t- . 5 4 B ] \ e ( T B) + e ~ ( T B) 2J .54ff 2.16ff - 2B ~ + (B - 1)T + + 2 B T ( B - 1) + T 2e(T -- B) e(T -- B) e2(T - B) ~ e~(T - B) 2 2B 2 B - 1 T .54B B -< -'T- + 2--'~-- T - B + e T - B 2B T B 2.16B B B + e-'TT-BT-B -b e~ T - B T - B " (53) We will find simple upper bounds for T / ( T - B) and B / ( T - B), given that T > 2B. T- B- 'r 1+ T -1+ T -- B -< T < 1, sinceB/(T-B) B <l+2B T----~--~ _ T by (54). (54) (55) Substituting into (53) using (54) and (55), we obtain "/" B ' ~ ~T/~(u,) < 2B ~ + B-l( < B - ] 2--~--+-~ 1+ 2+~ _~_) .54B 2B + --- ] + e + ~ + ~ ~ + e~ ] , (56) where we have expanded out and replaced the term B__y__ ~ 1. ~B -by B 2 If we replace BaIT 2 on the right-hand side of (56) by B~/T, and numerically evaluate, we find n]B B ~ ~T/B(U,) < .lS4 (B -- 1) + 5.56 ( B 2 / T ) , which implies the second inequality of (25). 10. Proof of Step 3 We will show that i p,(1 - p , ) r _ ~,,l~ u,(1 - u , / B ) T < .736(B - 1 ) / T , (57) if T > 0. Let ¢ be the function x -+ x(l - x) r, with domain [0, 1]. We find by elementary calculus that ~khas its maximum at p = 1 / ( T + 1), with maximum value 1(, M A X = +------i T 1 < ~ by (46). T + 1 (58) The Independence of Miss Ratzo on Page Size 141 Further, ~ is monotone increasing between 0 and t~, and monotone decreasing between and 1. If 1 > Xl >_ . . . > x~ ~ ~, then since ~ is monotone decreasing between /, and 1, we find as in (21) of the proof of step l that ((¢(xl) + ... + ~(~,) )/B) - ¢~( ( ~ + . . . + x , ) / B ) _< ( ( B -- 1 ) / B ) (~(xB) -- ~b(xl)). (59) And, (~b(xl "-k "" -Jr x , ) / B ) -- (~k(xl) + " " -b ¢ z ( x , ) ) / B _< ¢~(~.~) - ( ( B - 1)¢~(zi) + ~ ( x , ) ) / B = ((B - 1)/B)(~b(x~) -- ~b(xl)). (60) Putting together (59) and (60), we obtain I (~(xl) + "'" + ~(x~))/B - ~((xl + .. + x ~ ) / B ) I < ((B - 1)/B)(¢(xB) -- ~ ( x , ) ) . (61) Let k be the maximal integer such that pks ~ u. T h e n l >_ px > p: >_ • • > pk, >_ u. By using (61) in an analogous way to our use of (21) in the proof of step 1, we obtain, as in (24) of step 1, that i kB (l/B) k ~b(p,) - B ,~l~(u'/B) <_ ( ( B - 1)/B)(~b(p~8) - ~b(p,)) _< ( ( B - 1)/B)~b(pkB). (62) We know t h a t p(~+~)e < u. There are now two cases to consider. Case 1. PkB+I__< tz- Assume t h a t ~ > xl _> x2 _> . . - _~ xs ~ 0. Since ~/, is monotone increasing between 0 and u, we find, by a similar argument to that used to prove (61) and (62), t h a t ] (~b(xl) + . . . + ~ , ( x B ) ) / B -- ~k((xx + .. + XB)/B) I _< ( ( B 1)/B)(¢(x~) (1/B) ~ ~,(p,) - B ~2 ~ ( u , / B ) < ((B - - ~b(x,)). 1)/B)~,(pk~+~). (63) (64) Hence, from (62) and (64), (l/B) ~,(p~) -- B ~ ~,(u,/B) _< ( ( B -- 1)/B)(g,(p~B) + ~(P~.+i)) (2(B -- 1)/B)MA_X < 2(B - 1 ) / e B T b y (58). Since 2/e < .736, this gives us (57). Case 2. p,~+l > u. We ~411 show t h a t [ (&(pEn+,) + . . . + ~b(p(~+,)~))/B -- ~b (u~+ffB)[ _< ( ( B -- 1 ) / B ) ( M A X -- ~ (p,8+~)) + ((B- 1)/B)(MAX - ~k(p(~+~)a)). (65) We will first show t h a t this is sufficient to prove (57). As in the proof of (64) of case 1, n (l/B) ~ n/B ~b(p,) - B ~ ~b(u,/B) ~_ ( ( B -- 1)/B)~b(p(~+~)n+~). (66) Since p~,+~ > ~, and since ~b is monotone decreasing between u and 1, it follows t h a t ¢,(pk~) _~ ~h(pka+l). So from (62), (l/B) ~ ~(u,/B) ,-, ~k(p,) - B ~ ,-1 < ( ( B -- 1)/B)~k(p~+~). (67) 142 lt. F A G I N A N D M.C. EASTON Since p(k+~)B < V, and since 6 is monotone increasing between 0 and v, it follows that ~b(p,k+l)B) >_ .¢(p(k+l)B+l). So (65) gives (1/B) I ¢dp~+l) + ... + ¢(p~+~) - B ¢,(u~+,/B) I < ((B -- 1 ) / B ) ( M A X - q~(pkB+~)) + ((B -- 1 ) / B ) ( M A X - ¢,(p(~+~)B+i). (68) If we add together (66), (67), and (68), and use the triangle inequality, we obtain (I/B) ~(p,) - B ~¢(u,/B) _< (2(B - 1 ) / B ) MAX, and as in the conclusion of ease 1, this gives us (57). It remains to prove (65). Assume that 1 > xl >_ . - . ~_ xB > 0. To prove (65), we must show ] ((iV(x,) + • • + ~k(x,))/BI - ~k((xl + < ((B - ... + xB)/B) I 1)/B)(MAX -~b(x,)) + ( ( B -- 1 ) / B ) ( M A X - ¢ ( x . ) ) . (69) Assume that ~ ( x , ) ~_ .~(x~). The proof ~s sinular if ~b(x~) > ~k(x~). Since ~ ( x . ) > ~b(x~), clearly ~b(x,) > ~(Xl), ~ = 1, . . . , B (~(x~) + .. + ~ ( z ~ ) ) / B - ~((Xl + ... + xD/B) < ((B - 1) M A X + ~ ( x , ) ) / B - ¢ (x,) = ((B - 1 ) / B ) ( M A X - ~(x,)) _< ((B - 1 ) / B ) (MAX -- ~ (x,)) + ( ( B -- I ) / B ) ( M A X - ¢(xB)). (70) And, ~ ( ( x l + . . . + x ~ ) / B ) - (~(xl) + . . . + ~b(x~))/B _< MAX - ((B -- 1)~(xl) + qJ(xs))/B < M A X -- ( ( B - 1)~(x~) + ~b(xB))/B + ((B - 2 ) / B ) ( M A X - ~ ( x s ) ) = ( ( B -- 1 ) / B ) ( M A X -- ¢(x~)) + ( ( B -- 1 ) / B ) ( M A X - ¢dxa)). (71) Putting together (70) and (71), we obtain (69). Remark The result of step 3 can be improved (reducing the right-hand side of (57) by less than a factor of 2), by taking advantage of more properties of ~ than that it is monotone increasing and then monotone decreasing. 11. Proof of Step 4 We will prove n/B 0 < ~ tml u , ( 1 -- u , / B ) r - ~B (1 -- u,) T'B < . 2 7 1 ( B - - 1 ) / T + 33 ( B / T ) 2 (72) tml if T > 2B. Statement (26) of the proof of step 2 says (1 -- u , / B ) r > (1 - u,) r/B. Hence u , (1 - u J B ) ~ > u , (1 - u,) T/, , and the first mequMlty • of (72)-- follows. If we adopt the notation of the proof of step 2, then we are concerned with finding .,,-1 u,e~/B(u,). As before, if ½ < u~ < 1, then i = 1 and u,er/s(u,) _< (1 -- 1/2B) T < 8 ( B / T ) 2, by (50) with m = 2. (73) If 0 < u, < ½, then from (41) of step 2, u,~r/~(u,) < u, s (1 -- u , / B ) r-B ( ( T ( B -- 1)/2B 2) "4- .54). (74) The Independence of Miss Ratio on Page Size 143 So from (73) and 74), n/B n]B u:r/~(u,) < 8 ( R / T ) 2 -q- ~ ~--1 u,3(1 -- u~/B) r-B ( ( T ( B -- 1)/2B 2) + .54). (75) z--I By exactly the same method as in step 2, where we now let ~ be the function x 13(1 - x / B ) T-', we find that ~ u , 3 ( 1 - u , / B ) r-" _< (max ( ¢ ( x ) / x ) ) + (max ~(x)) 0_~<i _< 4B2/e2(T - (76) o_o_<I B) 2 -F 27B2/e3(T - B) 3 , since (48) gives the maxima. If we substitute into (75) using (76), we obtain "/'u,~,/,(u,)< 8 (~)~ "F (--4B2 o-, \e2(T -- B)2 "F 27B3 "~(T(B - I) -'I-.54) e3(T - B) 3] \ 2B 2 B 2.16( +-~~ B )' 2 T 14.58( +--~~ B )' (77) where we expanded out and made one substitution of B for B - 1. If we now use (54) and (55) to substitute 2 B / T for B / ( T -- B) and (1 + 2 B / T ) for T / ( T - B) in (77), we obtain ./~ 2 2(B- 1) 1-t- -I-~ 1-t- + 2.16 e~T _ + + 8+~+~+8.64 + ~+~+ d J (78) where we expanded and occasionally substituted B for B -- 1. If we replace ( B / T ) 3 on the right-hand side of (71) by ( B / T ) ~ and numerically evaluate, we obtain n/B u:r/B(u,) < .271 (B -- 1 ) / T + 32.68 ( B / T ) 2, which implies the second inequality of (72). 12. Proof of Step 5 We will prove that H(T)/H(t) ifT>t> 1. We will show that H ( T ) / S ( T ) ( 1 -- ~- p,(1 -- p , ) r < S(T)/S(t) (79) _< H ( t ) / S ( t ) , that is, (1 -- (1 -- p , ) r ) _< ( 1 - - ~ p , ( 1 Assume for convenience that each p, is nonzero. - - p , ) t ) / ~ ( 1 ,-, -- (1 (80) 144 R. Let a, = I - (1 - p , ) ' B, = 1 - (1 - p~ f~, < FAGIN AND M.C. EASTON p,)T, 1 < i < n. T h e n (80) is e q u i v a l e n t to p, oq L e t a , ' = a, / 5 : °~ - l a j , = B, / Z ~ - 1 ~ , S t a t e m e n t (81) is e q u i v a l e n t to ~,. (81) 1 _< i _< ~. T h e n 5: a, ' = = 1. 5: p,~/ < 5: p,~,'. (82) N o w the function f, given by x --~ 5:~-1 p , x , , with domain ~-tuples ( x l , . . . , x,) with x~ > x2 _> . - . >_ x~, is a Schur function, since (O/Ox,)f = p , , 1 < i < n, and hence (d/Ox,)f > (O/Ox,)f iff p, > p~ iff ~ > .~ iff x, > xj . So, to prove (82), we need only show a' > ~'. (83) To p r o v e (83), we need the following lemma. I,EM~A. I f 0 < r~ < ~'~ < 1, the, (1 -- r,t)/(1 -- r~ t) > (1 -- r,~')/(1 -- r ~ ) . (84) PROOF. I t is easy to see t h a t (84) holds if r~ = 0; hence, assume rl > 0. Let f~ be the function x --~ (1 - rl~)/(1 - r2~), with domain [1, ~ ). We need only show t h a t f~ is m o n o t o n e decreasing, t h a t is, t h a t f~' < 0. A simple calculation shows t h a t f ' _< 0 iff rl ~ l o g r ~ / ( 1 - rl ~) > r2~ log r2/ (1 - r~x) for e a c h x > 1. (85) If we m u l t i p l y b o t h sides of (85) b y x, and let gl = r~", g2 = r ( , we obtain g~ log g~/(1 - g~) ~ g2 log g2/(1 - g~). (86) So we need only show t h a t (86) holds w h e n e v e r 0 g g~ _~ g2 < 1. Let sl = 1 s~ = 1 - g~. T h e n (86) becomes (1 - Sl)log(1 -- s~)/s~ ~_ (1 -- s2)log(1 -- s~)/s2. g~, (87) We need only show t h a t (87) holds w h e n e v e r 0 < s~ .~ Sl ( 1. Let f2 be the function x ~ (1 - x)log(1 - x ) / x , with domain (0, 1). W e m u s t show t h a t f~ is m o n o t o n e increasing. If we replace log(1 - x) by its T a y l o r expansion, t h e n ( f(z)=(1-~) = --(l--x)( -z-y--5-¥ .... l + x ~ + ~ x2 + ¥ + x' )/ ... ) which ~s clearly m o n o t o n e increasing. T h e l e m m a is now proved. To prove (83), we m u s t show k 5:~,' k >_ ~ , ' , i < ~ < ~. ( W e h a v e already n o t e d t h a t ~ , " - ~ a,' = ~ - ~ / ~ ' = 1.) f W e will first p r o v e a ~ ~_ /~1'. I n t h e l e m m a , i f w e l e t r ~ = 1 t h e n we get ~,/a,+~ _> ~ J ~ , + , , 1 _< ~ < n. (88) p , , r~ = 1 -- p,+~, (89) W e will now show t h a t (89) implies ~,' > ~,'. (9o) The Independence of Miss Ratio on Page S~ze For, find o-i, - • • , 0"n--1and "rl, T h e n (89) implies 145 • .,T,-lsuch t h a t a~+l = a , a , , B,+l = v,O,, 1 ~ i < n. 1 < ~ < ~. ow # Ogl = = = ,~U(~I + '" + (91) '~,,) a l / a l ( 1 -k- aa 4- ala2 "4- o'ia:aa q'- . ' . ) 1 / ( 1 + ol + o'1o2 + o'1o2o3 --P .. ) . (92) Similarly 8a' = 1/(1 + T1 "4- TiT2 • T1T2T3 -~ " ' " ). (93) T h e n (91), (92), and (93) imply (90). We will close b y showing t h a t al' -4- c~2' >_ ¢~1' "4- ~ ' . T h e other inequalities of (88) are proved very similarly. Let ~ = (al q- a~, ~3, a4, . - . , a , ) and ~ = (/31-1- t3~,/~,134, . . . , B , ) . W e w i l l first prove t h a t a,/&,+l 7> D,/D,+i (94) for e a c h / . This follows for ~ _> 2 b y (89). If i = l, then (94) says (al --k a 2 ) / a 3 _> (~1 + ~ ) / ~ 3 . (95) B u t (95) holds, since al/a~ _> Ol/f~ a n d a~/a~ ~_ t8~/~33, each b y an application of the lemma, just as in the proof of (89). Let ~ be the normalization of ~, t h a t is, ~ , = a , / ~ - i ~, • Similarly, define ~'. (94) implies ~'1 _> ~'1, (96) just as (89) implies (90). B u t (96) is equivalent to al' -4- a~' _> /~l' + B~', which was to be shown. 13. Summary We have shown t h a t the working-set miss ratio is insensitive to page size, in the indep e n d e n t (page) reference model. We argue t h a t this implies t h a t the insensitivity also holds in more realistic models. ACKNOWLEDGMENT. 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