Virtual memory of computers is usually implemented by demand paging. For some
page replacement algorithms the number of page faults may increase as the
number of page frames increases. Belady, Nelson and Shedler constructed
reference strings for which page replacement algorithm FIFO produces near twice
more page faults in a larger memory than in a smaller one. They formulated the
conjecture that 2 is a general bound. We prove that this ratio can be
arbitrarily large

Fornai, Peter

Ivanyi, Antal

English

arXiv

Fornai, Péter

Iványi, Antal

Repository of the Academy's Library

Acta Univ. Sapientiae, Informatica, 2, 1 (2010) 80–89FIFO anomaly is unboundedPe´ter FornaiEo¨tvo¨s Lora´nd UniversityDepartment of Computer AlgebraH-1117, Budapest, HungaryPa´zma´ny se´ta´ny 1/Cemail: peter.fornai@inf.elte.huAntal Iva´nyiEo¨tvo¨s Lora´nd UniversityDepartment of Computer AlgebraH-1117, Budapest, HungaryPa´zma´ny se´ta´ny 1/Cemail: tony@inf.elte.huAbstract. Virtual memory of computers is usually implemented by de-mand paging. For some page replacement algorithms the number of pagefaults may increase as the number of page frames increases. Belady, Nel-son and Shedler [5] constructed reference strings for which page replace-ment algorithm FIFO [10, 13, 36, 40] produces near twice more pagefaults in a larger memory than in a smaller one. They formulated theconjecture that 2 is a general bound. We prove that this ratio can bearbitrarily large.1 IntroductionLet us consider a computer with two-level virtual memory [12, 26, 38]. Physicaland secondary memory are divided into equal-sized blocks called frames. Theprocesses reside in secondary memory and are broken into fixed-sized blockscalled pages (the sizes of the frames and pages are the same). The run of theprocesses is described by reference strings.According to the pure demand paging the execution of a process startswith empty physical memory (in the following simply memory) and a page isbrought into the memory only when it is necessary. If a new page is demandedwhen the memory is full, the page replacement algorithm [38] chooses a pagebeing in memory to be replaced by the demanded one.Computing Classification System 1998: G.2.2Mathematics Subject Classification 2010: 68M20Key words and phrases: memory management, virtual memory, page replacement, FIFO,Belady anomaly80FIFO anomaly is unbounded 81We use the formal model proposed by P. Denning [12]. Let m, M, n, andp be positive integers (1 ≤ m ≤ M ≤ n < ∞), k a nonnegative integer,A = {a1, a2, . . . , an} a finite alphabet, Ak the set of k-length words and A∗be the set of finite words over A, where n is the number of pages, m is thenumber of frames in the small and M is the number of frames in the largememory, A is the set of pages.Page replacement algorithms [38] are handled as automata having a memoryof size m (or M), set of input signals X = A, set of output signals Y = A∪ {}and processing the sequence of input symbols R = (r1, r2, . . . , rp) or R =(r1, r2, . . . ). Memory state St (t = 1, 2, . . .) is defined as a set of symbolsstored in memory at the given moment t (after the processing of reference rt).Any page replacement algorithm starts with an empty memory that is S0 = {}.A concrete page replacement algorithm P is defined as a triple P = (QP, q0, gP),where QP is the set of control states, q0 ∈ QP is the initial control state and gPis the transition function determining the new memory state S ′, new controlstate q ′ and output symbol y using the old memory state S, old control stateq and input symbol x.We consider three page replacement algorithms: FIFO (First In First Out)[10, 13, 36, 40], LRU (Least Recently Used) [8, 10, 19, 26, 37] and MIN (Min-imal) [1, 2, 5, 28, 29, 32, 34].FIFO is defined by q0 = () andgFIFO(S, q, x) =(S, q, ), if x ∈ S,(S ∪ {x}, q ′, ), if x /∈ S and |S| < m,(S ∩ {y1} ∪ {x}, q", y1), if x /∈ S and |S| = m,where q = (y1, y2, . . . , yk), q ′ = (y1, y2, . . . , yk, x), and q ′′ = (y2, y3, . . . ,ym, x).LRU replaces the least recently used page of the memory, MIN replaces thepage having maximal distance up to its next occurrence in the reference string.The number of page faults (number of changes of the memory states) isdenoted by fP(R,m). If M > m and fP(R,M) > fP(R,m), then we havean anomaly and the ratio fP(R,M)/fP(R,m) is called anomaly ratio. Thefirst anomaly was observed in the practice [4, 5, 34] and is called Belady’sanomaly [28, 35]. Later other examples of unexpected events were described[6, 9, 14, 17, 18, 20, 21, 22, 23, 25, 27, 31, 33].Stack algorithms [28, 30, 39] do not suffer from anomaly.Mihnovskiy and Shor [24] and Gecsei et al. [15] proved independently thatMIN guarantees the minimal number of page faults at a fixed size of the82 P. Fornai, A. Iva´nyimemory.Arato´ [3] and Benczu´r et al. [7] proved that LRU is optimal in statisticalsense among the algorithms having no concrete information on the continua-tion of the reference string.A possible measure of the efficiency EP(R,m) of P is called paging rateand is defined by fP(R,m)/p for finite R = (r1, r2, . . . , rp) and byEP(R,m) = lim infk→∞ fP(Rk,m)kfor an infinite reference string R, where Rk = (r1, r2, . . . , rk).Let 1 ≤ m < n and C = (1, 2, . . . , n)∗ be an infinite cyclical reference string.Then the paging rates of FIFO and MIN are given in [16] EFIFO(C,m) = 1and EMIN(C,m) = (n−m)/(n− 1).2 Classical example and resultsIf we execute the reference string R = (1,2,3,4,1,2,5,1,2,3,4,5) [5] having 3frames in the memory (for the simplicity natural numbers are used to de-note the pages), then FIFO results the control state sequence q0 = (), q1 =(1), q2 = (1, 2), q3 = (1, 2, 3), q4 = (2, 3, 4), q5 = (3, 4, 1), q6 = (4, 1, 2), q7 =(1, 2, 5), q8 = (1, 2, 5), q9 = (1, 2, 5), q10 = (2, 5, 3), q11 = (5, 3, 4), q12 =(5, 3, 4) and 9 page faults occurred.The execution of R using 4 frames will end in the control state (2,3,4,5) and10 page faults occurred, so fFIFO(R,M)/fFIFO(R,m) = 10/9.Belady, Nelson and Shedler [5] gave a necessary and sufficient condition forthe existence of anomaly and constructed reference strings resulting anomalyratio which is close to 2.Theorem 1 [5] Let m and M be positive integers (1 ≤ m < M) and let Abe a finite alphabet of cardinality at least M+ 1. Then there exists a referencestring R ∈ A∗ which produces an anomaly if and only if M < 2m− 1.Theorem 2 [5] If m and M are positive integers satisfying the relations m <M < 2m − 1, then for sufficiently large p there exists a reference string R =(r1, r2, . . . , rp) resulting anomaly ratio fFIFO(R,M)/fFIFO(R,m) arbitrarilyclose to 2.FIFO anomaly is unbounded 83Belady, Nelson and Shedler [5] formulated the following conjecture.Conjecture [5] If m and M are positive integers and R is a reference string,then fFIFO(R,M)/fFIFO(R,m) ≤ 2.3 Disprove of the conjectureLet m = 5, M = 6, n = 7, k ≥ 1 and R = UVk, where U = (1,2,3,4,5,6,7,1,2,4,5,6,7,3,1,2,4,5,7,3,6,2,1,4,7,3,6,2,5,7,3,6,2,5) and V = (1, 2, 3, 4, 5, 6, 7)3.Now execution of U using m frames results the control state (7,3,6,2,5) and29 page faults. After this each execution of V results 7 new faults and thesame control state. Execution of U using M frames results the control state(2,3,4,5,6,7) and 14 page faults. After this each execution of V results 21 newpage fault and the same control state.Choosing k = 7 we get an anomaly ratio (14+7×21)/(29+7×7) = 161/78 >2. If the number of repetitions of V grows, then the anomaly ratio tends to 3.4 Anomaly is unboundedWe start with a well-known assertion of the number theory. Let [a]n be theequivalence class modulo n containing the number a and let Zn be the set ofall equivalence classes modulo n:[a]n = {a+ kn : k ∈ Z} and Zn = {[a]n : 0 ≤ a ≤ n− 1}.Lemma 3 [16] If the set c1, c2, . . . , cn is a complete residue system Zn (mod n),the greatest common divisor of a and n equals 1, then the set {ac1 + d, ac2 +d, . . . , acn+d} is also a complete residue system (mod n) for arbitrary integerd.Let a (mod b) denote the smallest positive representative of the residue classof numbers congruent with a (mod b).Lemma 4 Let n be an odd positive integer. Then the elements of the se-quence W = (w1, w2, . . . , wn) =(1 (mod n), 1+(n− 1)/2 (mod n), 1+ 2(n−1)/2 (mod n), . . . , 1+ (n− 1)(n− 1)/2 (mod n))have the following proper-ties.a) They form a complete system of residues (mod n).84 P. Fornai, A. Iva´nyib) If Wk = s1, s2, . . . , sk×n is the concatenation of k copies of W, then any nneighbouring elements of Wk form a complete residue system (mod n).c) If j ≥ 1, then sj+n+1 ≡ sj+1 (mod n).Proof. It is sufficient to use Lemma 3 and some elementary calculations. Lemma 5 Let n be an odd number (n ≥ 5), M = n − 1 and m = n − 2.If the small memory starts with control state (w3, w4, . . . , wn) and the largememory starts with control state (2, 3, . . . , n), then the reference string R =(1, 2, . . . , n)n−1)/2 results n page faults in the small memory, n(n−1)/2 pagefaults in the large memory and the initial control states in both memories.Proof. This is a consequence of Lemma 4. Lemma 6 Let 1 ≤ m < M < n and b1, b2, . . . , bm be arbitrary differentelements of A = {1, 2, . . . ,M + 1}. Then algorithm ANOMALY constructs areference string resulting q = (b1, b2, . . . , bm) in the small memory of size mwhile the pages are loaded into the large memory of size M in cyclical order1, 2, . . . , n.The next algorithm is written in PASCAL-like pseudocode [11].Construction algorithm ANOMALY(m,M,q,U).Input : m size of the small memory; M size of the large memory; q = (b1, b2,. . . , bm) the required control state of the small memory.Output : U = (u1, u2, . . .) reference string.Working variables: S actual state of the small memory; t index of the nextreference; A = {1, 2, . . . ,M,M+ 1} the set of pages.1. for k← 1 to M2. do uk ← k3. t←M+ 14. i← 05. while i < m do6. if bi+1 ∈ S7. then while bi+1 ∈ S do8. ut ← min{k|k ∈ A and k /∈ S}9. t← t+ 110. while b1 ∈ S(m) do11. uk ← min{k|k ∈ A and k /∈ S and k 6= bi+1}12. t← t+ 1FIFO anomaly is unbounded 8513. j← 114. while j < i+ 1 do15. ut ← bj16. t← t+ 117. j← j+ 118. ut ← bi+119. t← t+ 120. i← i+ 1Instead of a long proof of correctness of ANOMALY we explain how thisalgorithm generates U of the example used to disprove the conjecture. Theinput of the algorithm is m = 5, M = 6 and q = (7,3,6,2,5). Lines 1–2 resultthe references 1, 2, 3, 4, 5, 6 and control state q = (2,3,4,5,6) in the smallmemory. Now line 3 results t = 7, line 4 gives i = 0 and we start the whilecycle in line 5. Since b1 = 7 is missing from the small memory, we add u7 = 7to U (lines 18–20) changing the control state to q = (3, 4, 5, 6, 7), incrementt and i and return to line 5 where we observe that b2 = 3 is in the memory.Therefore we continue with the while cycle in line 7 and add the minimalmissing page to the reference string resulting control state (4,5,6,7,1). Sincepage 1 replaced b2, we go to the next cycle in lines 10–12 and add the minimalmissing pages differing from b2 = 3 (that is 2, 4, 5, and 6) to U changing thecontrol state to (1,2,4,5,6). Now this while cycle ends and using lines 13–17we add 7 then using lines 18–20 add 3 to U ending in control state (4,5,6,7,3).Now we return to line 5 and using lines 7-9 add 1 to U, then using lines 10–12add 2, 4 and 5 to U changing the control state to (3,1,2,4,5). Now using lines14–18 we add 7 and 3, then using lines 19–21 add 6 to U changing the controlstate to (4,5,7,3,6). Now lines 5–6 send us to lines 14–18 resulting referenceu22 = 2 and control state (5,7,3,6,2).We observe in lines 5–6 that b5 is in the memory therefore we add 1 to Uand so remove 2 from the memory. Now we continue U with 4 resulting thecontrol state (3,6,2,1,4). Finally lines 14–18 add 7, 3, 6 and 2 and lines 19–21add 5 to U implying the required control state.Lemma 7 Let n be an odd positive integer (n ≥ 5) and let M = n− 1, m =n−2. Then there exists a reference string R resulting anomaly ratio arbitrarilyclose to (n− 1)/2.Proof. This assertion is a consequence of Lemma 5 and Lemma 6. Main Theorem For any large number L there exist parameters m, M, andR such that the anomaly ratio fFIFO(R,M)/fFIFO(R,m) > L.86 P. Fornai, A. Iva´nyiProof. Let n be an odd integer satisfying n > 2L + 1 (and n ≥ 5). Then theparameters m,M and R in Lemma 5 result an anomaly ratio greater than L.5 SummaryIf the memory parameters m and M are fixed, then the anomaly is bounded.We suppose that the maximal anomaly occurs at cyclical reference strings,similar memory sizes, FIFO-like replacement in the large memory and MIN-like replacement in the small memory. If so then for fixed odd n our con-struction results the maximal possible anomaly ratio. We remark that there isa similar construction for even n resulting anomaly ratio arbitrarily close to(n− 2)/2.AcknowledgementsThe authors are indebted to Miha´ly Molna´r, former student of Eo¨tvo¨s Lora´ndUniversity who using genetic algorithms discovered the first reference stringresulting an anomaly ratio greater than 2. We thank the useful comments ofProfessor of Eo¨tvo¨s Lora´nd University Andra´s Benczu´r and of the unknownreferee.The research was supported by the project TA´MOP-4.2.1/B-09/1/KMR-2010-003 of Eo¨tvo¨s Lora´nd University.References[1] S. Albers, On generalized connection caching, in: ACM Symposium onParallel Algorithms and Architectures (Bar Harbor, 2000), Theory Com-put. Syst., 35, (3) (2002) 251–267. ⇒81[2] S. Albers, L. M. Favrholdt, O. Giel, On paging with locality of reference,J. Comput. System Sci., 70, 2 (2005) 145–175. ⇒81[3] M. Arato´, A note on optimal performance of page storage, Acta Cybernet.(Szeged), 3, 1 (1976/77) 25–30. ⇒82[4] L. A. Belady, A study of replacement algorithms for a virtual storagecomputer, IBM Syst. J., 5, 2 (1965) 78–101. ⇒81FIFO anomaly is unbounded 87[5] L. A. Belady, R. A. Nelson, G. S. Shedler, An anomaly in space-timecharacteristics of certain programs running in paging machine, Comm.ACM , 12, 1 (1969) 349–353. ⇒80, 81, 82, 83[6] L. A. Belady, F. P. Palermo, On-line measurement of paging behavior bythe multivalued MIN algorithm, IBM J. Res. Develop., 18 (1974) 2–19.⇒81[7] A. Benczu´r, A. Kra´mli, J. Pergel, On the Bayesian approach to optimalperformance of page hierarchies, Acta Cybernet. (Szeged), 3, 2 (1976/77)78–79. ⇒82[8] J. Boyar, L. M. Favrholdt, K. S. Larsen, The relative worst-order ratioapplied to paging, J. Comput. System Sci., 73, 5 (2007) 818–843. ⇒81[9] M. Caminada, L. Amgoud, On the evaluation of argumentation for-malisms, Artificial Intelligence, 171, 5–6 (2007) 286–310. ⇒81[10] M. Chrobak, J. Noga, LRU is better than FIFO, Algorithmica, 23, 2(1999) 180–185. ⇒80, 81[11] T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction toalgorithms (Second edition), The MIT Press/ McGraw-Hill Book Com-pany, Cambridge/Boston, 2009. ⇒84[12] P. Denning, Virtual memory. Comput. Surveys, 2, 3 (1980) 153–189. ⇒80, 81[13] L. Epstein, Y. Kleiman, J. Sgall, R. van Stee, Paging with connections:FIFO strikes again, Theoret. Comput. Sci., 377, 1–3 (2007) 55–64. ⇒80,81[14] M. A. Franklin, G. S. Graham, R. K. Gupta, Anomalies with variablepartition paging algorithms, Comm. ACM , 21, 3 (1978) 232–236. ⇒81[15] J. Gecsei, D. R. Slutz, I. L. Traiger, Evaluation techniques for storagehierarchies, IBM Syst. J., 9, 2 (1970) 78–117. ⇒81[16] A. M. Iva´nyi and L. N. Korolev, Classification of paging algorithms, in:Problems of Interpretation of Experiments (eds. A. N. Tihonov, P. N.Zaikin and V. Y. Galkin), Moscow State University, Moscow, 1975, pp.105–125. ⇒82, 8388 P. Fornai, A. Iva´nyi[17] A. M. Iva´nyi, On dumpling-eating giants, in: Finite and Infinite Sets (eds.A. Hajnal, L. Lova´sz and V. T. So´s), North-Holland, Amsterdam, 1984,pp. 379–390. ⇒81[18] A. M. Iva´nyi, R. L. Smelyanskiy, Elements of theoretical programming (inRussian), Moscow State University, Moscow, 1980. ⇒81[19] Ya. A. Kogan, A class of hierarchical paging algorithms, MTASza´mita´stechn. Automat. Kutato´ Int. Tanulma´nyok (3rd Visegra´d WinterSchool Theory of Operating Systems, Visegra´d, 1977), No. 69 (1977) 7–13.⇒81[20] S. Kolahi, Dependency-preserving normalization of relational and XMLdata, J. Comput. System Sci., 73, 4 (2007) 636–647. ⇒81[21] R. Korf, Linear-time disk-based implicit graph search, J. ACM , 55, 6(2008) Art. 26, pp. 40. ⇒81[22] K. V. Malkov, D. V. Tunitskii, On an extremal problem of adaptive ma-chine learning that is connected with the detection of anomalies (in Rus-sian), Avtomat. i Telemekh., 2008, no. 6, 41–52; translation in Autom.Remote Control , 69, 6 (2008) 942–952. ⇒81[23] B. Mans, C. Roucairol, Performances of parallel branch and bound algo-rithms with best-first search, Discrete Appl. Math., 66, 1 (1996), 57–74.⇒81[24] S. D. Mihnovskiy, N. Z. Shor, Estimation of the number of page faults inpaged virtual memory (in Russian), Kibernetika, 1, 5 (1965) 18–20. ⇒81[25] D. A. Naumann, Observational purity and encapsulation, Theoret. Com-put. Sci., 376, 3 (2007) 205–224. ⇒81[26] H. Raquibul, R. Sohel, DesynchLRU: An efficient page replacement algo-rithm with desynchronized cache and RAM, in: Abstracts of Int. Conf.on Applied Informatics (Eger, January 27–30, 2010). ⇒80, 81[27] S. Roosta, Parallel processing and parallel algorithms, Springer-Verlag,New York, 1999. ⇒81[28] A. Silberschatz, P. Galvin and G. Gagne, Applied operating system Con-cepts, John Wiley and Sons, New York, 2000. ⇒81FIFO anomaly is unbounded 89[29] D. Sleator, R. E. Tarjan, Amortized efficiency of list update and pagingrules, Comm. ACM , 28, 2 (1985) 202–208. ⇒81[30] A. J. Smith, Analysis of the optimal, look-ahead demand paging algo-rithms, SIAM J. Comput., 5, (4) (1976) 743–757. ⇒81[31] A. M. Sokolov, Vector representations for efficient comparison and searchfor similar strings (in Russian), Kibernet. Sistem. Anal., 43, 4 (2007), 18–38, 189; translation in Cybernet. Systems Anal., 43, 4 (2007) 484–498. ⇒81[32] H.-G. Stork, On the paging-complexity of periodic arrangements, Theoret.Comput. Sci., 4, 2 (1977) 171–197. ⇒81[33] N. B. Waite, A real-time system-adapted anomaly detector, Inform. Sci.,115, 1–4 (1999) 221–259. ⇒81[34] Wikipedia, Belady’s algorithm,http://en.wikipedia.org/wiki/Belady%27s Min#Belady.27s Algorithm,2010. ⇒81[35] Wikipedia, Belady’s anomaly,http://en.wikipedia.org/wiki/Belady%27s anomaly, 2010. ⇒81[36] Wikipedia, FIFO page replacement algorithm,http://en.wikipedia.org/wiki/FIFO, 2010. ⇒80, 81[37] Wikipedia, LRU page replacement algorithm,http://en.wikipedia.org/wiki/LRU, 2010. ⇒81[38] Wikipedia, Page replacement algorithm,http://en.wikipedia.org/wiki/Page replacement algorithm, 2010. ⇒ 80,81[39] C. Wood, Christopher, E. B. Fernandez, T. Lang, Minimization of demandpaging for the LRU stack model of program behavior. Inform. Process.Lett., 16, 3 (1983) 99–104. ⇒81[40] N. E. Young, On-line paging against adversarially biased random in-puts, (Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SanFrancisco, 1998). J. Algorithms, 37, 1 (2000) 218–235. ⇒80, 81Received: October 17, 2009 • Revised: March 8, 2010

FIFO anomaly is unbounded

Abstract. Virtual memory of computers is usually implemented by demand paging. For some page replacement algorithms the number of page faults may increase as the number of page frames increases. Belady, Nelson and Shedler [5] constructed reference strings for which page replacement algorithm FIFO [10

Péter Fornai

Antal Iványi

CiteSeerX

Acta Univ. Sapientiae, Informatica, 2, 1 (2010) 80–89
FIFO anomaly is unbounded
Péter Fornai
Eötvös Loránd University
Department of Computer Algebra
H-1117, Budapest, Hungary
Pázmány sétány 1/C
email: peter.fornai@inf.elte.hu
Antal Iványi
Eötvös Loránd University
Department of Computer Algebra
H-1117, Budapest, Hungary
Pázmány sétány 1/C
email: tony@inf.elte.hu
Abstract. Virtual memory of computers is usually implemented by de-
mand paging. For some page replacement algorithms the number of page
faults may increase as the number of page frames increases. Belady, Nel-
son and Shedler [5] constructed reference strings for which page replace-
ment algorithm FIFO [10, 13, 36, 40] produces near twice more page
faults in a larger memory than in a smaller one. They formulated the
conjecture that 2 is a general bound. We prove that this ratio can be
arbitrarily large.
1 Introduction
Let us consider a computer with two-level virtual memory [12, 26, 38]. Physical
and secondary memory are divided into equal-sized blocks called frames. The
processes reside in secondary memory and are broken into fixed-sized blocks
called pages (the sizes of the frames and pages are the same). The run of the
processes is described by reference strings.
According to the pure demand paging the execution of a process starts
with empty physical memory (in the following simply memory) and a page is
brought into the memory only when it is necessary. If a new page is demanded
when the memory is full, the page replacement algorithm [38] chooses a page
being in memory to be replaced by the demanded one.
Computing Classification System 1998: G.2.2
Mathematics Subject Classification 2010: 68M20
Key words and phrases: memory management, virtual memory, page replacement, FIFO,
Belady anomaly
80
FIFO anomaly is unbounded 81
We use the formal model proposed by P. Denning [12]. Let m, M, n, and
p be positive integers (1 ≤ m ≤ M ≤ n < ∞), k a nonnegative integer,
A = {a1, a2, . . . , an} a finite alphabet, Ak the set of k-length words and A∗
be the set of finite words over A, where n is the number of pages, m is the
number of frames in the small and M is the number of frames in the large
memory, A is the set of pages.
Page replacement algorithms [38] are handled as automata having a memory
of size m (or M), set of input signals X = A, set of output signals Y = A∪ {ε}
and processing the sequence of input symbols R = (r1, r2, . . . , rp) or R =
(r1, r2, . . . ). Memory state St (t = 1, 2, . . .) is defined as a set of symbols
stored in memory at the given moment t (after the processing of reference rt).
Any page replacement algorithm starts with an empty memory that is S0 = {}.
A concrete page replacement algorithm P is defined as a triple P = (QP, q0, gP),
where QP is the set of control states, q0 ∈ QP is the initial control state and gP
is the transition function determining the new memory state S ′, new control
state q ′ and output symbol y using the old memory state S, old control state
q and input symbol x.
We consider three page replacement algorithms: FIFO (First In First Out)
[10, 13, 36, 40], LRU (Least Recently Used) [8, 10, 19, 26, 37] and MIN (Min-
imal) [1, 2, 5, 28, 29, 32, 34].
FIFO is defined by q0 = () and
gFIFO(S, q, x) =

(S, q, ε), if x ∈ S,
(S ∪ {x}, q ′, ε), if x /∈ S and |S| < m,
(S ∩ {y1} ∪ {x}, q", y1), if x /∈ S and |S| = m,
where q = (y1, y2, . . . , yk), q ′ = (y1, y2, . . . , yk, x), and q ′′ = (y2, y3, . . . ,
ym, x).
LRU replaces the least recently used page of the memory, MIN replaces the
page having maximal distance up to its next occurrence in the reference string.
The number of page faults (number of changes of the memory states) is
denoted by fP(R,m). If M > m and fP(R,M) > fP(R,m), then we have
an anomaly and the ratio fP(R,M)/fP(R,m) is called anomaly ratio. The
first anomaly was observed in the practice [4, 5, 34] and is called Belady’s
anomaly [28, 35]. Later other examples of unexpected events were described
[6, 9, 14, 17, 18, 20, 21, 22, 23, 25, 27, 31, 33].
Stack algorithms [28, 30, 39] do not suffer from anomaly.
Mihnovskiy and Shor [24] and Gecsei et al. [15] proved independently that
MIN guarantees the minimal number of page faults at a fixed size of the
82 P. Fornai, A. Iványi
memory.
Arató [3] and Benczúr et al. [7] proved that LRU is optimal in statistical
sense among the algorithms having no concrete information on the continua-
tion of the reference string.
A possible measure of the efficiency EP(R,m) of P is called paging rate
and is defined by fP(R,m)/p for finite R = (r1, r2, . . . , rp) and by
EP(R,m) = lim inf
k→∞ fP(Rk,m)k
for an infinite reference string R, where Rk = (r1, r2, . . . , rk).
Let 1 ≤ m < n and C = (1, 2, . . . , n)∗ be an infinite cyclical reference string.
Then the paging rates of FIFO and MIN are given in [16] EFIFO(C,m) = 1
and EMIN(C,m) = (n−m)/(n− 1).
2 Classical example and results
If we execute the reference string R = (1,2,3,4,1,2,5,1,2,3,4,5) [5] having 3
frames in the memory (for the simplicity natural numbers are used to de-
note the pages), then FIFO results the control state sequence q0 = (), q1 =
(1), q2 = (1, 2), q3 = (1, 2, 3), q4 = (2, 3, 4), q5 = (3, 4, 1), q6 = (4, 1, 2), q7 =
(1, 2, 5), q8 = (1, 2, 5), q9 = (1, 2, 5), q10 = (2, 5, 3), q11 = (5, 3, 4), q12 =
(5, 3, 4) and 9 page faults occurred.
The execution of R using 4 frames will end in the control state (2,3,4,5) and
10 page faults occurred, so fFIFO(R,M)/fFIFO(R,m) = 10/9.
Belady, Nelson and Shedler [5] gave a necessary and sufficient condition for
the existence of anomaly and constructed reference strings resulting anomaly
ratio which is close to 2.
Theorem 1 [5] Let m and M be positive integers (1 ≤ m < M) and let A
be a finite alphabet of cardinality at least M+ 1. Then there exists a reference
string R ∈ A∗ which produces an anomaly if and only if M < 2m− 1.
Theorem 2 [5] If m and M are positive integers satisfying the relations m <
M < 2m − 1, then for sufficiently large p there exists a reference string R =
(r1, r2, . . . , rp) resulting anomaly ratio fFIFO(R,M)/fFIFO(R,m) arbitrarily
close to 2.
FIFO anomaly is unbounded 83
Belady, Nelson and Shedler [5] formulated the following conjecture.
Conjecture [5] If m and M are positive integers and R is a reference string,
then fFIFO(R,M)/fFIFO(R,m) ≤ 2.
3 Disprove of the conjecture
Let m = 5, M = 6, n = 7, k ≥ 1 and R = UVk, where U = (1,2,3,4,5,6,7,1,2,
4,5,6,7,3,1,2,4,5,7,3,6,2,1,4,7,3,6,2,5,7,3,6,2,5) and V = (1, 2, 3, 4, 5, 6, 7)3.
Now execution of U using m frames results the control state (7,3,6,2,5) and
29 page faults. After this each execution of V results 7 new faults and the
same control state. Execution of U using M frames results the control state
(2,3,4,5,6,7) and 14 page faults. After this each execution of V results 21 new
page fault and the same control state.
Choosing k = 7 we get an anomaly ratio (14+7×21)/(29+7×7) = 161/78 >
2. If the number of repetitions of V grows, then the anomaly ratio tends to 3.
4 Anomaly is unbounded
We start with a well-known assertion of the number theory. Let [a]n be the
equivalence class modulo n containing the number a and let Zn be the set of
all equivalence classes modulo n:
[a]n = {a+ kn : k ∈ Z} and Zn = {[a]n : 0 ≤ a ≤ n− 1}.
Lemma 3 [16] If the set c1, c2, . . . , cn is a complete residue system Zn (mod n),
the greatest common divisor of a and n equals 1, then the set {ac1 + d, ac2 +
d, . . . , acn +d} is also a complete residue system (mod n) for arbitrary integer
d.
Let a (mod b) denote the smallest positive representative of the residue class
of numbers congruent with a (mod b).
Lemma 4 Let n be an odd positive integer. Then the elements of the se-
quence W = (w1, w2, . . . , wn) =
(
1 (mod n), 1+(n− 1)/2 (mod n), 1+ 2(n−
1)/2 (mod n), . . . , 1+ (n− 1)(n− 1)/2 (mod n)
)
have the following proper-
ties.
a) They form a complete system of residues (mod n).
84 P. Fornai, A. Iványi
b) If Wk = s1, s2, . . . , sk×n is the concatenation of k copies of W, then any n
neighbouring elements of Wk form a complete residue system (mod n).
c) If j ≥ 1, then sj+n+1 ≡ sj+1 (mod n).
Proof. It is sufficient to use Lemma 3 and some elementary calculations. 
Lemma 5 Let n be an odd number (n ≥ 5), M = n − 1 and m = n − 2.
If the small memory starts with control state (w3, w4, . . . , wn) and the large
memory starts with control state (2, 3, . . . , n), then the reference string R =
(1, 2, . . . , n)n−1)/2 results n page faults in the small memory, n(n−1)/2 page
faults in the large memory and the initial control states in both memories.
Proof. This is a consequence of Lemma 4. 
Lemma 6 Let 1 ≤ m < M < n and b1, b2, . . . , bm be arbitrary different
elements of A = {1, 2, . . . ,M + 1}. Then algorithm ANOMALY constructs a
reference string resulting q = (b1, b2, . . . , bm) in the small memory of size m
while the pages are loaded into the large memory of size M in cyclical order
1, 2, . . . , n.
The next algorithm is written in PASCAL-like pseudocode [11].
Construction algorithm ANOMALY(m,M,q,U).
Input : m size of the small memory; M size of the large memory; q = (b1, b2,
. . . , bm) the required control state of the small memory.
Output : U = (u1, u2, . . .) reference string.
Working variables: S actual state of the small memory; t index of the next
reference; A = {1, 2, . . . ,M,M+ 1} the set of pages.
1. for k← 1 to M
2. do uk ← k
3. t←M+ 1
4. i← 0
5. while i < m do
6. if bi+1 ∈ S
7. then while bi+1 ∈ S do
8. ut ← min{k|k ∈ A and k /∈ S}
9. t← t+ 1
10. while b1 ∈ S(m) do
11. uk ← min{k|k ∈ A and k /∈ S and k 6= bi+1}
12. t← t+ 1
FIFO anomaly is unbounded 85
13. j← 1
14. while j < i+ 1 do
15. ut ← bj
16. t← t+ 1
17. j← j+ 1
18. ut ← bi+1
19. t← t+ 1
20. i← i+ 1
Instead of a long proof of correctness of ANOMALY we explain how this
algorithm generates U of the example used to disprove the conjecture. The
input of the algorithm is m = 5, M = 6 and q = (7,3,6,2,5). Lines 1–2 result
the references 1, 2, 3, 4, 5, 6 and control state q = (2,3,4,5,6) in the small
memory. Now line 3 results t = 7, line 4 gives i = 0 and we start the while
cycle in line 5. Since b1 = 7 is missing from the small memory, we add u7 = 7
to U (lines 18–20) changing the control state to q = (3, 4, 5, 6, 7), increment
t and i and return to line 5 where we observe that b2 = 3 is in the memory.
Therefore we continue with the while cycle in line 7 and add the minimal
missing page to the reference string resulting control state (4,5,6,7,1). Since
page 1 replaced b2, we go to the next cycle in lines 10–12 and add the minimal
missing pages differing from b2 = 3 (that is 2, 4, 5, and 6) to U changing the
control state to (1,2,4,5,6). Now this while cycle ends and using lines 13–17
we add 7 then using lines 18–20 add 3 to U ending in control state (4,5,6,7,3).
Now we return to line 5 and using lines 7-9 add 1 to U, then using lines 10–12
add 2, 4 and 5 to U changing the control state to (3,1,2,4,5). Now using lines
14–18 we add 7 and 3, then using lines 19–21 add 6 to U changing the control
state to (4,5,7,3,6). Now lines 5–6 send us to lines 14–18 resulting reference
u22 = 2 and control state (5,7,3,6,2).
We observe in lines 5–6 that b5 is in the memory therefore we add 1 to U
and so remove 2 from the memory. Now we continue U with 4 resulting the
control state (3,6,2,1,4). Finally lines 14–18 add 7, 3, 6 and 2 and lines 19–21
add 5 to U implying the required control state.
Lemma 7 Let n be an odd positive integer (n ≥ 5) and let M = n− 1, m =
n−2. Then there exists a reference string R resulting anomaly ratio arbitrarily
close to (n− 1)/2.
Proof. This assertion is a consequence of Lemma 5 and Lemma 6. 
Main Theorem For any large number L there exist parameters m, M, and
R such that the anomaly ratio fFIFO(R,M)/fFIFO(R,m) > L.
86 P. Fornai, A. Iványi
Proof. Let n be an odd integer satisfying n > 2L + 1 (and n ≥ 5). Then the
parameters m,M and R in Lemma 5 result an anomaly ratio greater than L.

5 Summary
If the memory parameters m and M are fixed, then the anomaly is bounded.
We suppose that the maximal anomaly occurs at cyclical reference strings,
similar memory sizes, FIFO-like replacement in the large memory and MIN-
like replacement in the small memory. If so then for fixed odd n our con-
struction results the maximal possible anomaly ratio. We remark that there is
a similar construction for even n resulting anomaly ratio arbitrarily close to
(n− 2)/2.
Acknowledgements
The authors are indebted to Mihály Molnár, former student of Eötvös Loránd
University who using genetic algorithms discovered the first reference string
resulting an anomaly ratio greater than 2. We thank the useful comments of
Professor of Eötvös Loránd University András Benczúr and of the unknown
referee.
The research was supported by the project TÁMOP-4.2.1/B-09/1/KMR-
2010-003 of Eötvös Loránd University.
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Received: October 17, 2009 • Revised: March 8, 2010


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