Multi-step Quasi-Newton Optimization Algorithms Which Utilize Curvature Information J.A. FORD ? and I.A. MOGHRABIy ? Department of Computer Science , University of Essex , Wivenhoe Park , Colchester , Essex , United Kingdom yNatural Science Division , American Lebanese University , P.O. Box 13-5053 , Beirut , Lebanon An invited paper for the Fourth International Colloquium on Numerical Analysis , held in Plovdiv , Bulgaria -LRB- August , 1995 -RRB- ABSTRACT Multi-step quasi-Newton methods for unconstrained optimization were introduced by the authors -LRB- -LSB- 1 , 2 , 3 , 4 -RSB- -RRB- . At each iteration , these methods employ two polynomials , one to define a path interpolating recent iterates in the variable-space and the other to approximate the gradient as the path is followed . Numerical experiments described in -LSB- 1 , 3 -RSB- , for example , strongly indicated that several multi-step methods yield substantial computational gains over the standard -LRB- one-step -RRB- BFGS method . In this paper , we consider how to modify the structure of such methods to provide a more general model of the gradient , with the intention of improving the approximation . The results of numerical experiments on the new methods are reported and compared with those produced by existing methods . Utilizing curvature information in multi-step quasi-Newton methods 2 1 INTRODUCTION We address the problem of the numerical determination of an unconstrained minimum of the given objective function f -LRB- x -RRB- , where x fflRn and f is assumed to be twice continuously differentiable with gradient and Hessian denoted by g and G , respectively . The algorithms we consider here do not require G to be explicitly available and are iterative in character , generating from x0 -LRB- a given starting point -RRB- a sequence fxig of approximations to the desired minimum . In quasi-Newton methods , each Hessian G -LRB- xi -RRB- is approximated by a corresponding matrix Bi . A new estimate , xi +1 , is computed from the current one , xi , through use of a direction pi which is determined by analogy with Newton 's method : pi = ? B ? 1 i g -LRB- xi -RRB- ; -LRB- 1 -RRB- for example , we might execute a line-search along the ray fx -LRB- t -RRB- : x -LRB- t -RRB- ? xi + tpi ; t > = g. Before proceeding to the next iteration , it is necessary to obtain an estimate Bi +1 of the Hessian G -LRB- xi +1 -RRB- , so that the process described above may be repeated from xi +1 . Standard -LRB- one-step -RRB- quasi-Newton methods determine Bi +1 by using a formula of the type Bi +1 = Bi + U -LRB- Bi ; si ; yi -RRB- ; -LRB- 2 -RRB- where the correction term U -LRB- Bi ; si ; yi -RRB- is usually of rank one or two and where si def = xi +1 ? xi ; -LRB- 3 -RRB- yi def = g -LRB- xi +1 -RRB- ? g -LRB- xi -RRB- : -LRB- 4 -RRB- -LRB- In -LRB- 2 -RRB- , we draw special attention to the fact that the term U -LRB- Bi ; si ; yi -RRB- depends only on the most recent step-vectors , si and yi , and not on any previous ones . -RRB- The bestknown and most widely-used example of such a formula is the BFGS update -LRB- Broyden -LSB- 5 -RSB- , Fletcher -LSB- 6 -RSB- , Goldfarb -LSB- 7 -RSB- , Shanno -LSB- 8 -RSB- -RRB- -RRB- : Bi +1 = Bi ? BisisTi Bi sTi Bisi + yiyTi sTi yi : -LRB- 5 -RRB- This formula produces a matrix Bi +1 which satisfies the Secant Equation -LRB- as do most other such formulae -RRB- : Bi +1 si = yi : -LRB- 6 -RRB- The relevance of this equation is that it constitutes -LRB- Ford and Moghrabi -LSB- 1 , 2 -RSB- -RRB- an approximation to the Newton Equation G -LRB- xi +1 -RRB- x0 -LRB- o ? -RRB- = g0 -LRB- x -LRB- o ? -RRB- -RRB- ; -LRB- 7 -RRB- Utilizing curvature information in multi-step quasi-Newton methods 3 satisfied by G -LRB- xi +1 -RRB- itself . In equation -LRB- 7 -RRB- , x ? x -LRB- o -RRB- -LRB- where o fflR -RRB- is a continuously differentiable curve in Rn passing through xi +1 , while o ? is the value of o such that x -LRB- o ? -RRB- = xi +1 . The actual derivation , by this means , of the Secant Equation -LRB- 6 -RRB- from the Newton Equation is described in -LSB- 2 -RSB- . Ford and Moghrabi -LSB- 2 -RSB- proposed a \ multi-step '' approach to the issue of computing approximations to the Hessian matrix . In this new approach , x -LRB- o -RRB- is chosen to be a vector polynomial form in o of degree m which interpolates the m + 1 most recent iterates fxi?m + k +1 gmk = 0 . Correspondingly , an interpolating polynomial approximation ^ g -LRB- o -RRB- is constructed for g -LRB- x -LRB- o -RRB- -RRB- . This approximation has degree m and interpolates the known gradient values fg -LRB- xi?m + k +1 -RRB- gmk = 0 . By means of this technique , a new approximation Bi +1 ri = wi -LRB- 8 -RRB- to the Newton Equation is obtained -LRB- see -LSB- 2 -RSB- -RRB- . In equation -LRB- 8 -RRB- , the vectors ri and wi are defined by ri def = m ? 1 X j = 0 si?jf m X k = m?j L0k -LRB- om -RRB- g = x0 -LRB- om -RRB- ; -LRB- 9 -RRB- wi def = m ? 1 X j = 0 yi?jf m X k = m?j L0k -LRB- om -RRB- g ss g0 -LRB- x -LRB- om -RRB- -RRB- ; -LRB- 10 -RRB- where Lj -LRB- o -RRB- is the jth Lagrange polynomial of degree m corresponding to the real numbers fokgmk = 0 , so that Lj -LRB- oj -RRB- = 1 and Lj -LRB- oi -RRB- = for i < > j. -LRB- The abscissae fokgmk = 0 are the values of o associated with the iterates fxi?m + k +1 gmk = 0 on the path x -LRB- o -RRB- : x -LRB- ok -RRB- = xi?m + k +1 ; for k = ; 1 ; : : : ; m :-RRB- -LRB- 11 -RRB- It is now a simple matter to construct matrices satisfying the new condition -LRB- 8 -RRB- , in place of -LRB- 6 -RRB- . For example , we may use an updating formula such as -LRB- 5 -RRB- , where si and yi are now replaced by ri and wi , respectively . We note that such an approach employs not only the most recent step-vectors si and yi , but also previous ones -LRB- see equations -LRB- 9 -RRB- and -LRB- 10 -RRB- -RRB- . Ford and Moghrabi -LSB- 1 , 2 -RSB- reported that extensive numerical experimentation had shown that methods for which m = 2 generally performed better than those for which m = 3 . In the remainder of this paper , therefore , we will consider only two-step methods . Utilizing curvature information in multi-step quasi-Newton methods 4 2 A RATIONAL FORM FOR ^ g We wish to explore the possibility of obtaining a more accurate approximation to g0 -LRB- x -LRB- om -RRB- -RRB- to use in updating Bi . To this end , instead of using a polynomial approximation , we investigate the use of a rational form : ^ g -LRB- o ; ? -RRB- ? q -LRB- o -RRB- = -LRB- 1 + ? o -RRB- ; -LRB- 12 -RRB- where q -LRB- o -RRB- is a -LRB- vector -RRB- quadratic form in o . -LRB- Note , however , that we still continue to use a \ plain '' quadratic form for x -LRB- o -RRB- . -RRB- Because we are considering two-step methods , the curve ^ g -LRB- o ; ? -RRB- is still required to interpolate the three most recent values of g , for a set -LRB- to be specified in more detail below -RRB- of distinct values fojg2j = 0 : ^ g -LRB- oj ; ? -RRB- = g -LRB- xi?j +1 -RRB- def = gi?j +1 ; for j = ; 1 ; 2 : -LRB- 13 -RRB- We now stipulate -LRB- as for the methods A1 , A2 and A3 described in -LSB- 1 -RSB- -RRB- that the set fojg2j = 0 has been chosen such that o1 = ; -LRB- 14 -RRB- and we write o1 ? o0 = ? o0 def = aei ? 1 > 0 ; o2 ? o2 = o2 def = aei > ; -LRB- 15 -RRB- where , for example , the quantities aei ? 1 and aei could be defined -LRB- as they are in method A1 -LRB- -LSB- 1 -RSB- , where other ways of defining aei ? 1 and aei are also discussed -RRB- -RRB- by aei ? 1 = k si ? 1 k2 ; aei = k si k2 : -LRB- 16 -RRB- Differentiating equation -LRB- 12 -RRB- with respect to o , we obtain ^ g0 -LRB- o ; ? -RRB- ? -LSB- q0 -LRB- o -RRB- ? ? ^ g -LRB- o ; ? -RRB- -RSB- = -LRB- 1 + ? o -RRB- : -LRB- 17 -RRB- Now , since q -LRB- o -RRB- ? -LRB- 1 + ? o -RRB- ^ g -LRB- o ; ? -RRB- is -LRB- by definition -RRB- a quadratic form in o , we can express it in its \ Lagrangian '' form : q -LRB- o -RRB- ? o -LRB- o ? o0 -RRB- o2 -LRB- o2 ? o0 -RRB- -LRB- 1 + ? o2 -RRB- gi +1 + -LRB- o ? o0 -RRB- -LRB- o ? o2 -RRB- o2o0 gi ? o -LRB- o ? o2 -RRB- o0 -LRB- o2 ? o0 -RRB- -LRB- 1 + ? o0 -RRB- gi ? 1 ; using the fact that o1 = 0 . Therefore , q0 -LRB- o -RRB- ? -LRB- 2o + aei ? 1 -RRB- aei -LRB- aei ? 1 + aei -RRB- -LRB- 1 + ? aei -RRB- gi +1 ? -LRB- 2o + aei ? 1 ? aei -RRB- aei ? 1aei gi + -LRB- 2o ? aei -RRB- aei ? 1 -LRB- aei ? 1 + aei -RRB- -LRB- 1 ? ? aei ? 1 -RRB- gi ? 1 -LRB- 18 -RRB- Utilizing curvature information in multi-step quasi-Newton methods 5 -LRB- using equations -LRB- 15 -RRB- -RRB- , from which it follows that q0 -LRB- 0 -RRB- = ? fffi ? 1 -LRB- 1 + ? aei -RRB- gi +1 ? -LRB- ffi ? 1 ? ffi -RRB- gi ? ffi -LRB- 1 ? ? aei ? 1 -RRB- gi ? 1g ; -LRB- 19 -RRB- where we define ffi def = aei = aei ? 1 ; ? def = -LRB- aei + aei ? 1 -RRB- ? 1 : -LRB- 20 -RRB- Therefore , using -LRB- 13 -RRB- and -LRB- 17 -RRB- , q0 -LRB- 0 -RRB- ? ? ^ g -LRB- 0 ; ? -RRB- = q0 -LRB- 0 -RRB- ? ? gi = ? fffi ? 1yi + ffiyi ? 1 + ? aei ? 1yi ? ? aeiyi ? 1g = ^ g0 -LRB- 0 ; ? -RRB- : -LRB- 21 -RRB- We now define a function OE , which is the restriction of the original objective function f to the path x -LRB- o -RRB- : OE -LRB- o -RRB- def ? f -LRB- x -LRB- o -RRB- -RRB- : -LRB- 22 -RRB- Then OE0 -LRB- o -RRB- ? x0 -LRB- o -RRB- T g -LRB- x -LRB- o -RRB- -RRB- ; -LRB- 23 -RRB- OE00 -LRB- o -RRB- ? x0 -LRB- o -RRB- T g0 -LRB- x -LRB- o -RRB- -RRB- + x00 -LRB- o -RRB- T g -LRB- x -LRB- o -RRB- -RRB- : -LRB- 24 -RRB- We focus attention on the curvature of OE at o = -LRB- corresponding to xi -RRB- : OE00 -LRB- 0 -RRB- = x0 -LRB- 0 -RRB- T g0 -LRB- x -LRB- 0 -RRB- -RRB- + x00 -LRB- 0 -RRB- T gi ss x0 -LRB- 0 -RRB- T ^ g0 -LRB- 0 ; ? -RRB- + x00 -LRB- 0 -RRB- T gi : -LRB- 25 -RRB- Now , since x -LRB- o -RRB- is a quadratic form in o , it is easy to show that x0 -LRB- 0 -RRB- = ? fffi ? 1si + ffisi ? 1g ; x00 -LRB- o -RRB- ? 2 ? 2f -LRB- 1 + ffi ? 1 -RRB- si ? -LRB- 1 + ffi -RRB- si ? 1g 8o : Therefore , using -LRB- 21 -RRB- and -LRB- 25 -RRB- , we obtain OE00 -LRB- 0 -RRB- ss ? 2fffi ? 1si + ffisi ? 1gT fffi ? 1yi + ffiyi ? 1 + ? aei ? 1yi ? ? aeiyi ? 1g +2 ? 2f -LRB- 1 + ffi ? 1 -RRB- si ? -LRB- 1 + ffi -RRB- si ? 1gT gi : -LRB- 26 -RRB- We define oeij def = sTi gj ; > = ij def = sTi yj = oei ; j +1 ? oeij : -LRB- 27 -RRB- Utilizing curvature information in multi-step quasi-Newton methods 6 Then -LRB- 25 -RRB- becomes OE00 -LRB- 0 -RRB- ss ? 2fffi ? 2 > = ii + > = i ; i ? 1 + > = i ? 1 ; i + ffi2 > = i ? 1 ; i ? 1 + 2 -LRB- 1 + ffi ? 1 -RRB- oeii ? 2 -LRB- 1 + ffi -RRB- oei ? 1 ; i + ^ ? -LRB- ffi ? 1si + ffisi ? 1 -RRB- T -LRB- ffi ? 1yi ? yi ? 1 -RRB- g ; -LRB- 28 -RRB- on defining ^ ? def = ? aei : -LRB- 29 -RRB- 3 ESTIMATION OF THE CURVATURE We now give brief descriptions of the two ways in which we obtain numerical estimates of OE00 -LRB- 0 -RRB- -LRB- using available values of OE and OE0 -RRB- and the consequent determination of ^ ? . The first approach uses the six values OE0 -LRB- ? aei ? 1 -RRB- , OE -LRB- ? aei ? 1 -RRB- = f -LRB- xi ? 1 -RRB- def = fi ? 1 , OE0 -LRB- 0 -RRB- , OE -LRB- 0 -RRB- = f -LRB- xi -RRB- def = fi , OE0 -LRB- aei -RRB- and OE -LRB- aei -RRB- = f -LRB- xi +1 -RRB- def = fi +1 . We therefore interpolate OE with a quintic polynomial -LRB- which we denote by ^ OE -RRB- in o and use the second derivative of ^ OE at o = 0 : ^ OE00 -LRB- 0 -RRB- = 4ae ? 1i -LRB- ffi ? 1 -RRB- OE0 -LRB- 0 -RRB- + 2 ? -LSB- OE0 -LRB- ? aei ? 1 -RRB- ffi2 ? OE0 -LRB- aei -RRB- ffi ? 1 -RSB- -LRB- 1 + ffi -RRB- +2 ? 2 -LSB- -LRB- fi ? 1 ? fi -RRB- ffi2 -LRB- 5 + 3ffi -RRB- + -LRB- fi +1 ? fi -RRB- ffi ? 2 -LRB- 5ffi + 3 -RRB- -RSB- -LRB- 1 + ffi -RRB- : -LRB- 30 -RRB- Equating the two approximations for the curvature given by -LRB- 28 -RRB- and -LRB- 30 -RRB- , we obtain a -LRB- linear -RRB- equation which is easily solved for ^ ? . We first define def = 2f -LRB- fi ? 1 ? fi -RRB- ffi2 -LRB- 5 + 3ffi -RRB- + -LRB- fi +1 ? fi -RRB- ffi ? 2 -LRB- 5ffi + 3 -RRB- g = -LRB- 1 + ffi -RRB- : -LRB- 31 -RRB- Then , we obtain ^ ? f -LRB- ffi ? 1si + ffisi ? 1 -RRB- T -LRB- ffi ? 1yi ? yi ? 1 -RRB- g = oeii -LRB- 1 ? 3ffi ? 2 ? 2ffi ? 1 -RRB- + oei ? 1 ; i -LRB- 3ffi2 + 2ffi ? 1 -RRB- + + foei ? 1 ; i ? 1 -LRB- 5ffi2 + 3ffi3 -RRB- ? oei ; i +1 -LRB- 5ffi ? 1 + 3ffi ? 2 -RRB- + oei ; i ? 1 -LRB- 1 ? ffi -RRB- + oei ? 1 ; i +1 -LRB- 1 ? ffi -RRB- g = -LRB- 1 + ffi -RRB- : If we define ? j def = 2ffj +1 ? fjg ? sTj fg -LRB- xj +1 -RRB- + g -LRB- xj -RRB- g ; then we have ^ ? f -LRB- ffi ? 1si + ffisi ? 1 -RRB- T -LRB- ffi ? 1yi ? yi ? 1 -RRB- g = f?iffi ? 2 -LRB- 5ffi + 3 -RRB- ? ? i ? 1 ffi2 -LRB- 5 + 3ffi -RRB- + -LRB- ffi ? 1 -RRB- -LRB- > = i ; i ? 1 ? > = i ? 1 ; i -RRB- g = -LRB- 1 + ffi -RRB- ; Utilizing curvature information in multi-step quasi-Newton methods 7 from which we find ^ ? . We denote the algorithm arising from this method of determining ^ ? by C1 . We note that if f is quadratic with Hessian A , then ? i = ? i ? 1 = and > = i ; i ? 1 ? > = i ? 1 ; i = sTi Asi ? 1 ? sTi ? 1Asi = ; which implies that ^ ? = 0 . Thus , in this case , the rational model for g would reduce to a quadratic model and the algorithm then becomes -LRB- depending on the precise choice of aei and aei ? 1 -RRB- the corresponding \ Accumulative-type '' method defined in -LSB- 1 -RSB- . Alternatively , we can obtain a simpler estimate of OE00 -LRB- 0 -RRB- by a divided difference : OE00 -LRB- 0 -RRB- ss fOE0 -LRB- aei -RRB- ? OE0 -LRB- ? aei ? 1 -RRB- g = -LRB- aei + aei ? 1 -RRB- = ? fx0 -LRB- aei -RRB- T gi +1 ? x0 -LRB- ? aei ? 1 -RRB- T gi ? 1g ; -LRB- 32 -RRB- from -LRB- 23 -RRB- . Now , it is easy to show , from the quadratic nature of x -LRB- o -RRB- , that x0 -LRB- aei -RRB- = ? f -LRB- 2 + ffi ? 1 -RRB- si ? ffisi ? 1g ; -LRB- 33 -RRB- x0 -LRB- ? aei ? 1 -RRB- = ? f -LRB- 2 + ffi -RRB- si ? 1 ? ffi ? 1sig : -LRB- 34 -RRB- Substituting these expressions into -LRB- 32 -RRB- , we obtain an estimate for OE00 -LRB- 0 -RRB- , which we then equate to the approximation -LRB- 28 -RRB- and thus derive a value for ^ ? . The algorithm defined by this value of ^ ? is denoted by C2 . Our final task is to obtain a suitable approximation to the Newton Equation -LRB- 7 -RRB- . To do so , we need expressions for x0 -LRB- o2 -RRB- = x0 -LRB- aei -RRB- and for ^ g0 -LRB- o2 ; ? -RRB- . Now , x0 -LRB- aei -RRB- is given in -LRB- 33 -RRB- , while we can determine ^ g0 -LRB- o ; ? -RRB- at o = aei by means of -LRB- 17 -RRB- and -LRB- 18 -RRB- : ^ g0 -LRB- aei ; ? -RRB- = ? -LRB- 1 + ^ ? -RRB- ? 1f -LRB- 2 + ffi ? 1 + ^ ? -RRB- yi + -LRB- ^ ? ? ffi -RRB- yi ? 1g : -LRB- 35 -RRB- Substituting these expressions into the Newton Equation -LRB- 7 -RRB- and cancelling common factors , we obtain the following condition for the Hessian approximation Bi +1 to satisfy : Bi +1 f -LRB- 2 + ffi ? 1 -RRB- si ? ffisi ? 1g = -LRB- 1 + ^ ? -RRB- ? 1f -LRB- 2 + ffi ? 1 + ^ ? -RRB- yi + -LRB- ^ ? ? ffi -RRB- yi ? 1g : -LRB- 36 -RRB- Utilizing curvature information in multi-step quasi-Newton methods 8 4 NUMERICAL EXPERIMENTS The algorithms C1 and C2 developed in Section 3 were compared -LRB- initially -RRB- with the standard -LRB- one-step -RRB- BFGS method . For both C1 and C2 , the distances aei and aei ? 1 were defined by equations -LRB- 15 -RRB- and the updating formula -LRB- 5 -RRB- was used , with the appropriate substitutions . We present below samples of the results obtained -LRB- more extensive tests are reported in Moghrabi -LSB- 9 -RSB- -RRB- . Eleven test functions -LRB- listed in the Appendix -RRB- were used , each with either one or two starting-points , giving a total of twenty problems . Several of the functions used were from the set described by Mor?e , Garbow and Hillstrom -LSB- 10 -RSB- and may be used with varying dimension - in such cases , the tests were carried out on a range of appropriate dimensions and the results were summed , since lack of space prevents a full tabulation . Further information on the functions and the starting-points used , together with details on the implementation of the algorithms , may be found in Ford and Moghrabi -LSB- 11 -RSB- . The results of this first set of experiments are presented in Table 1 . For each problem , the number of function/gradient evaluations required is given , followed by the number of iterations in brackets . The best performance for each problem -LRB- decided on the number of evaluations , with ties resolved by number of iterations -RRB- is indicated by y. Totals -LRB- including \ Scores '' , indicating the number of best performances -RRB- are given at the foot of the Table . As a further test , C1 and C2 were compared with the two-step method A1F introduced in -LSB- 11 -RSB- , which also uses function values to modify a two-step accumulative method . -LRB- In the case of A1F , this is accomplished by making the path x -LRB- o -RRB- a rational function and determining the additional parameter by the reduction in the function-value . -RRB- The results of this second comparison are given in Table 2 . 5 SUMMARY AND CONCLUSIONS Two new multi-step quasi-Newton methods have been constructed . These methods use a rational form to approximate the behaviour of the gradient as a quadratic path in the variable-space is traversed . They determine the remaining parameter in the rational form -LRB- after the interpolatory conditions have been satisfied -RRB- via numerical estimates of the curvature of the objective function -LRB- when restricted to the path in the variable-space -RRB- . Numerical tests show that , while the new methods exhibit substantial improvement when compared with the standard BFGS method , they are slightly inferior to the two-step Utilizing curvature information in multi-step quasi-Newton methods 9 Table 1 : Comparison of BFGS with C1 and C2 Problem BFGS C1 C2 1 -LRB- a -RRB- 542 -LRB- 530 -RRB- 441 -LRB- 429 -RRB- y 465 -LRB- 453 -RRB- 1 -LRB- b -RRB- 1653 -LRB- 1293 -RRB- 1512 -LRB- 1074 -RRB- 1487 -LRB- 1084 -RRB- y 2 -LRB- a -RRB- 631 -LRB- 612 -RRB- 569 -LRB- 526 -RRB- 523 -LRB- 493 -RRB- y 2 -LRB- b -RRB- 825 -LRB- 806 -RRB- 782 -LRB- 720 -RRB- 668 -LRB- 643 -RRB- y 3 -LRB- a -RRB- 159 -LRB- 122 -RRB- y 207 -LRB- 134 -RRB- 178 -LRB- 132 -RRB- 3 -LRB- b -RRB- 162 -LRB- 156 -RRB- 301 -LRB- 104 -RRB- 146 -LRB- 137 -RRB- y 4 -LRB- a -RRB- 807 -LRB- 386 -RRB- 998 -LRB- 446 -RRB- 788 -LRB- 364 -RRB- y 4 -LRB- b -RRB- 637 -LRB- 507 -RRB- 813 -LRB- 553 -RRB- 630 -LRB- 532 -RRB- y 5 -LRB- a -RRB- 634 -LRB- 596 -RRB- 542 -LRB- 480 -RRB- 541 -LRB- 493 -RRB- y 5 -LRB- b -RRB- 2390 -LRB- 2331 -RRB- 1843 -LRB- 1745 -RRB- y 1897 -LRB- 1834 -RRB- 6 -LRB- a -RRB- 2300 -LRB- 1786 -RRB- 1991 -LRB- 1098 -RRB- 1925 -LRB- 1232 -RRB- y 6 -LRB- b -RRB- 2136 -LRB- 2001 -RRB- 1719 -LRB- 1527 -RRB- 1716 -LRB- 1584 -RRB- y 7 -LRB- a -RRB- 280 -LRB- 226 -RRB- 265 -LRB- 222 -RRB- y 270 -LRB- 233 -RRB- 7 -LRB- b -RRB- 1967 -LRB- 965 -RRB- 1959 -LRB- 922 -RRB- 1879 -LRB- 893 -RRB- y 8 -LRB- a -RRB- 5099 -LRB- 2272 -RRB- 4870 -LRB- 1937 -RRB- 4746 -LRB- 1954 -RRB- y 8 -LRB- b -RRB- 2822 -LRB- 1969 -RRB- y 2880 -LRB- 1655 -RRB- 2829 -LRB- 1893 -RRB- 9 -LRB- a -RRB- 1034 -LRB- 987 -RRB- 780 -LRB- 718 -RRB- 769 -LRB- 722 -RRB- y 9 -LRB- b -RRB- 1779 -LRB- 1559 -RRB- 1673 -LRB- 1419 -RRB- 1633 -LRB- 1427 -RRB- y 10 132 -LRB- 126 -RRB- 130 -LRB- 124 -RRB- y 130 -LRB- 124 -RRB- y 11 1505 -LRB- 1262 -RRB- 1133 -LRB- 841 -RRB- y 1291 -LRB- 967 -RRB- Totals : 27494 -LRB- 20492 -RRB- 25408 -LRB- 16674 -RRB- 24511 -LRB- 17194 -RRB- Score : 2 5 14 Utilizing curvature information in multi-step quasi-Newton methods 10 Table 2 : Comparison of A1F with C1 and C2 Problem A1F C1 C2 1 -LRB- a -RRB- 443 -LRB- 429 -RRB- 441 -LRB- 429 -RRB- y 465 -LRB- 453 -RRB- 1 -LRB- b -RRB- 1487 -LRB- 1101 -RRB- 1512 -LRB- 1074 -RRB- 1487 -LRB- 1084 -RRB- y 2 -LRB- a -RRB- 485 -LRB- 465 -RRB- y 569 -LRB- 526 -RRB- 523 -LRB- 493 -RRB- 2 -LRB- b -RRB- 679 -LRB- 652 -RRB- 782 -LRB- 720 -RRB- 668 -LRB- 643 -RRB- y 3 -LRB- a -RRB- 174 -LRB- 129 -RRB- y 207 -LRB- 134 -RRB- 178 -LRB- 132 -RRB- 3 -LRB- b -RRB- 108 -LRB- 97 -RRB- y 301 -LRB- 104 -RRB- 146 -LRB- 137 -RRB- 4 -LRB- a -RRB- 765 -LRB- 344 -RRB- y 998 -LRB- 446 -RRB- 788 -LRB- 364 -RRB- 4 -LRB- b -RRB- 569 -LRB- 463 -RRB- y 813 -LRB- 553 -RRB- 630 -LRB- 532 -RRB- 5 -LRB- a -RRB- 549 -LRB- 485 -RRB- 542 -LRB- 480 -RRB- 541 -LRB- 493 -RRB- y 5 -LRB- b -RRB- 1973 -LRB- 1877 -RRB- 1843 -LRB- 1745 -RRB- y 1897 -LRB- 1834 -RRB- 6 -LRB- a -RRB- 1912 -LRB- 1092 -RRB- y 1991 -LRB- 1098 -RRB- 1925 -LRB- 1232 -RRB- 6 -LRB- b -RRB- 1772 -LRB- 1607 -RRB- 1719 -LRB- 1527 -RRB- 1716 -LRB- 1584 -RRB- y 7 -LRB- a -RRB- 266 -LRB- 225 -RRB- 265 -LRB- 222 -RRB- y 270 -LRB- 233 -RRB- 7 -LRB- b -RRB- 1936 -LRB- 917 -RRB- 1959 -LRB- 922 -RRB- 1879 -LRB- 893 -RRB- y 8 -LRB- a -RRB- 4696 -LRB- 1892 -RRB- y 4870 -LRB- 1937 -RRB- 4746 -LRB- 1954 -RRB- 8 -LRB- b -RRB- 2765 -LRB- 1819 -RRB- y 2880 -LRB- 1655 -RRB- 2829 -LRB- 1893 -RRB- 9 -LRB- a -RRB- 735 -LRB- 680 -RRB- y 780 -LRB- 718 -RRB- 769 -LRB- 722 -RRB- 9 -LRB- b -RRB- 1623 -LRB- 1379 -RRB- y 1673 -LRB- 1419 -RRB- 1633 -LRB- 1427 -RRB- 10 130 -LRB- 124 -RRB- y 130 -LRB- 124 -RRB- y 130 -LRB- 124 -RRB- y 11 1050 -LRB- 718 -RRB- y 1133 -LRB- 841 -RRB- 1291 -LRB- 967 -RRB- Totals : 24117 -LRB- 16495 -RRB- 25408 -LRB- 16674 -RRB- 24511 -LRB- 17194 -RRB- Score : 12 4 6 Utilizing curvature information in multi-step quasi-Newton methods 11 method A1F introduced in -LSB- 11 -RSB- . This suggests that methods based directly upon functionvalues -LRB- as A1F is -RRB- may be more successful than those utilizing function-values indirectly by means of curvature estimates . This conjecture is given further empirical support from the results obtained by Moghrabi -LSB- 9 -RSB- in testing a number of methods based on these two approaches . APPENDIX The test functions used in the numerical experiments were -LRB- the numbers correspond to the rows in Tables 1 and 2 -RRB- : -LCB- 1 -LCB- Discrete Boundary-Value Function -LCB- 2 -LCB- Extended Powell Singular Function -LCB- 3 -LCB- Variably-dimensioned Function -LCB- 4 -LCB- Penalty Function I -LCB- 5 -LCB- Modified Trigonometric Function -LCB- 6 -LCB- Penalty Function II -LCB- 7 -LCB- Extended Rosenbrock Function -LCB- 8 -LCB- Extended Wood Function -LCB- 9 -LCB- Watson Function -LCB- 10 -LCB- Broyden Tridiagonal Function -LCB- 11 -LCB- Oren and Spedicato Power Function ACKNOWLEDGEMENT One of us -LRB- I.A.M. -RRB- gratefully acknowledges the support of the Lebanese National Council for Scientific Research . Utilizing curvature information in multi-step quasi-Newton methods 12 References -LSB- 1 -RSB- J.A. Ford and I.A. Moghrabi , Opt . Methods and Software , 2 , 357-370 -LRB- 1993 -RRB- . -LSB- 2 -RSB- J.A. Ford and I.A. Moghrabi , J. Comput . Appl . Math. , 50 , 305-323 -LRB- 1994 -RRB- . -LSB- 3 -RSB- J.A. Ford and I.A. Moghrabi , Scientia Iranica , 1 , 327-334 -LRB- 1995 -RRB- . -LSB- 4 -RSB- J.A. Ford and I.A. Moghrabi , Comp . and Maths . with Appls . -LRB- in press -RRB- . -LSB- 5 -RSB- C.G. Broyden , J. Inst . Math . Appl. , 6 , 76-90 and 222-231 -LRB- 1970 -RRB- . -LSB- 6 -RSB- R. Fletcher , Computer J. , 13 , 317-322 -LRB- 1970 -RRB- . -LSB- 7 -RSB- D. Goldfarb , Maths . Comp. , 24 , 23-26 -LRB- 1970 -RRB- . -LSB- 8 -RSB- D.F. Shanno , Maths . Comp. , 24 , 647-656 -LRB- 1970 -RRB- . -LSB- 9 -RSB- I.A. Moghrabi , Ph.D. thesis , University of Essex -LRB- 1993 -RRB- . -LSB- 10 -RSB- J.J. Mor?e , B.S. Garbow and K.E. Hillstrom , TOMS , 7 , 17-41 -LRB- 1981 -RRB- . -LSB- 11 -RSB- J.A. Ford and I.A. Moghrabi , J. Comput . Appl . Math . -LRB- in press -RRB- .

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Almeidayx

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virgilio@bu.edu

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Bestavrosy

-LRB-

best@bu.edu

-RRB-

Mark

Crovellay

-LRB-

crovella@bu.edu

-RRB-

Adriana

de

Oliveiraz

-LRB-

dri@dcc.ufmg.br

-RRB-

Department

of

Computer

Science

Boston

University

Boston

MA

Abstract

As

the

World

Wide

Web

-LRB-

Web

-RRB-

is

increasingly

adopted

as

the

infrastructure

for

large-scale

distributed

information

systems

issues

of

performance

modeling

become

ever

more

critical

In

particular

locality

of

reference

is

an

important

property

in

the

performance

modeling

of

distributed

information

systems

In

the

case

of

the

Web

understanding

the

nature

of

reference

locality

will

help

improve

the

design

of

middleware

such

as

caching

prefetching

and

document

dissemination

systems

For

example

good

measurements

of

reference

locality

would

allow

us

to

generate

synthetic

reference

streams

with

accurate

performance

characteristics

would

allow

us

to

compare

empirically

measured

streams

to

explain

differences

and

would

allow

us

to

predict

expected

performance

for

system

design

and

capacity

planning

In

this

paper

we

propose

models

for

both

temporal

and

spatial

locality

of

reference

in

streams

of

requests

arriving

at

Web

servers

We

show

that

simple

models

based

only

on

document

popularity

-LRB-

likelihood

of

reference

-RRB-

are

insufficient

for

capturing

either

temporal

or

spatial

locality

Instead

we

rely

on

an

equivalent

but

numerical

representation

of

a

reference

stream

a

stack

distance

trace

We

show

that

temporal

locality

can

be

characterized

by

the

marginal

distribution

of

the

stack

distance

trace

and

we

propose

models

for

typical

distributions

and

compare

their

cache

performance

to

our

traces

We

also

show

that

spatial

locality

in

a

reference

stream

can

be

characterized

using

the

notion

of

self-similarity

Self-similarity

describes

long-range

correlations

in

the

dataset

which

is

a

property

that

previous

researchers

have

found

hard

to

incorporate

into

synthetic

reference

strings

We

show

that

stack

distance

strings

appear

to

be

stongly

self-similar

and

we

provide

measurements

of

the

degree

of

self-similarity

in

our

traces

Finally

we

discuss

methods

for

generating

synthetic

Web

traces

that

exhibit

the

properties

of

temporal

and

spatial

locality

that

we

measured

in

our

data

This

work

has

been

partially

supported

by

NSF

-LRB-

grants

and

-RRB-

and

by

CNPq-Brazil

yComputer

Science

Department

Boston

University

Boston

MA

USA

zDepto

de

Ciencia

da

Computac

ao

Universidade

Federal

de

Minas

Gerais

Belo

Horizonte

MG

Brazil

xOn

sabbatical

at

Boston

University

from

Universidade

Federal

de

Minas

Gerais

Introduction

The

principle

of

locality

of

reference

has

very

important

consequences

for

computer

systems

design

Reference

streams

exhibiting

temporal

locality

can

benefit

from

caching

and

reference

streams

exhibiting

spatial

locality

can

benefit

from

prefetching

The

application

of

these

principles

in

e.g.

memory

systems

is

well

understood

-LSB-

-RSB-

In

order

to

apply

these

principles

to

the

design

of

World

Wide

Web

-LRB-

Web

-RRB-

caching

and

prefetching

systems

it

's

important

to

characterize

the

degree

of

temporal

and

spatial

locality

present

in

typical

Web

reference

streams

Good

measurements

of

these

properties

would

allow

us

to

generate

synthetic

reference

streams

with

accurate

performance

characteristics

would

allow

us

to

compare

empirically

measured

streams

to

explain

differences

and

would

allow

us

to

predict

expected

performance

for

system

design

and

capacity

planning

Previous

research

in

the

locality

properties

of

symbolic

reference

streams

have

shown

the

benefits

of

transforming

the

reference

stream

into

an

equivalent

but

numerical

representation

a

stack

distance

stream

-LSB-

-RSB-

This

transformation

preserves

all

of

the

information

of

the

original

trace

except

for

the

specific

reference

names

and

so

it

is

invertible

allowing

reconstruction

of

the

original

trace

-LRB-

albeit

with

synthetic

names

-RRB-

Two

properties

of

the

stack

distance

stream

can

be

used

to

measure

the

notions

of

temporal

and

spatial

locality

marginal

distribution

and

correlation

structure

-LSB-

-RSB-

The

marginal

distribution

measures

the

likelihood

that

two

references

to

the

same

object

are

separated

by

some

number

of

intervening

references

in

the

stream

The

correlation

structure

of

the

stream

measures

the

likelihood

that

we

can

predict

future

references

based

on

the

stream

of

past

references

In

this

paper

we

use

quantitative

methods

based

on

stack

distance

to

measure

both

the

temporal

and

spatial

locality

properties

for

Web

reference

streams

We

characterize

reference

streams

collected

at

four

Web

servers

propose

a

model

for

marginal

distribution

of

stack

distance

-LRB-

and

thus

temporal

locality

-RRB-

and

show

typical

ranges

of

the

relevant

parameters

We

also

propose

a

model

for

spatial

locality

of

our

reference

streams

based

on

the

correlation

structure

of

stack

distances

Using

these

measurements

we

show

how

to

generate

synthetic

reference

streams

of

arbitrary

length

whose

performance

properties

mimic

those

of

our

empirically

measured

data

A

novel

feature

of

our

approach

is

the

use

of

statistical

self-similarity

to

capture

the

correlation

structure

of

the

reference

stream

That

is

we

show

evidence

that

Web

stack

distance

streams

are

statistically

self-similar

and

that

this

property

can

be

used

to

explain

aspects

of

the

reference

stream

that

are

the

result

of

spatial

locality

The

presence

of

self-similarity

in

the

reference

stream

means

that

correlations

between

object

references

can

occur

at

widely

varying

timescales

-LRB-

long-range

dependence

-RRB-

A

timeseries

with

long-range

dependence

appears

much

more

bursty

than

series

with

only

short-range

or

no

dependence

which

agrees

with

generally

observed

characterizations

of

reference

locality

Long-range

dependence

in

the

stack

distance

series

is

also

related

to

previously

noted

fractal

properties

of

cache

miss

patterns

-LSB-

-RSB-

and

we

interpret

prior

work

in

fractal

reference

patterns

in

terms

of

our

observations

This

method

of

measuring

spatial

locality

also

provides

an

interpretation

for

the

notion

of

spatial

locality

when

reference

streams

are

composed

of

symbols

rather

than

numbers

We

consider

object

A

to

be

close

to

object

B

if

there

is

a

high

likelihood

that

a

reference

to

A

will

immediately

follow

a

reference

to

B

That

is

the

distance

between

objects

is

the

inverse

of

the

transition

probability

of

object

references

considered

as

a

first-order

Markov

chain

While

this

definition

does

not

correspond

to

our

usual

notions

of

a

norm

-LRB-

it

is

not

symmetric

-RRB-

it

does

capture

the

important

property

of

spatial

locality

for

the

purposes

of

prefetching

the

likelihood

that

prefetching

will

be

successful

Our

methods

focus

on

the

locality

properties

of

reference

streams

without

regard

to

document

size

Thus

in

our

simulations

we

consider

caches

to

be

sized

to

hold

a

fixed

number

of

documents

rather

than

a

fixed

amount

of

data

This

assumption

stems

from

the

fact

that

we

are

interested

in

locality

which

is

a

property

of

reference

streams

independent

of

any

cache

size

In

the

remainder

of

this

section

we

describe

the

data

on

which

our

measurements

and

trace

simulations

are

based

In

section

we

characterize

the

popularity

of

Web

documents

and

show

that

a

cache

reference

model

based

on

popularity

is

not

expressive

enough

to

capture

the

temporal

and

spatial

locality

of

reference

properties

exhibited

in

actual

Web

traces

Evidence

of

these

locality

of

reference

properties

is

presented

in

section

In

sections

and

we

characterize

these

properties

and

present

statistical

models

that

capture

these

characteristics

In

section

we

examine

related

work

We

conclude

in

section

with

a

summary

and

directions

for

future

work

Data

Collection

The

Web

is

a

large-scale

distributed

information

system

based

on

a

client-server

architecture

Thus

the

workload

viewed

from

a

server

standpoint

consists

of

a

number

of

requests

originated

at

many

different

clients

In

order

to

analyze

reference

locality

we

examined

the

access

logs

for

different

Web

servers

namely

the

NCSA

Web

server

located

at

the

National

Center

for

Supercomputing

Applications

-LRB-

NCSA

-RRB-

the

SDSC

Web

server

at

the

San

Diego

Supercomputer

Center

-LRB-

SDSC

-RRB-

the

EPA

Web

server

located

at

Research

Triangle

Park

NC

and

the

Web

server

at

the

Computer

Science

Department

at

Boston

University

-LRB-

BU

-RRB-

In

the

case

of

the

NCSA

the

data

refers

only

to

one

server

-LRB-

Costello

-RRB-

The

SDSC

and

EPA

logs

are

available

at

the

Internet

Traffic

Archives

-LSB-

-RSB-

the

NCSA

log

was

obtained

after

contacting

the

staff

at

the

site

and

the

BU

logs

were

collected

at

the

departmental

Web

server

The

logs

have

one

line

of

information

per

request

processed

by

the

server

Each

line

contains

the

name

of

the

host

making

the

request

the

timestamp

the

request

was

made

the

filename

of

the

requested

object

and

size

in

bytes

of

the

reply

Table

summarizes

the

statistics

about

the

logs

of

the

four

Web

servers

Item

NCSA

SDSC

EPA

BU

Access

Log

Duration

day

day

day

weeks

Access

Log

Start

Date

-LRB-

in

-RRB-

Dec

Aug

Aug

Oct

Total

Requests

Unique

Requests

Table

Summary

of

Access

Log

Data

Characterizing

Document

Popularity

The

highly

uneven

popularity

of

various

Web

documents

is

a

well-documented

phenomenon

that

has

been

exploited

in

previous

caching

replication

and

dissemination

studies

In

these

studies

it

was

shown

that

Popular

files

are

very

popular

-LSB-

-RSB-

In

-LSB-

-RSB-

Cunha

Bestavros

and

Crovella

characterized

the

popularity

of

Web

documents

requested

by

clients

and

confirmed

the

applicability

of

Zipf

's

law

-LSB-

discussed

in

-LSB-

-RSB-

-RSB-

to

Web

documents

Zipf

's

law

was

originally

applied

to

the

we

have

shown

in

previous

work

-LSB-

-RSB-

that

document

size

is

correlated

with

expected

reference

frequency

this

assumption

is

somewhat

limiting

for

direct

use

in

cache

performance

analysis

For

that

reason

we

are

currently

extending

this

work

to

include

document

sizes

in

our

analysis

Nonetheless

we

believe

that

the

current

methods

provide

a

simple

and

effective

way

to

capture

both

the

spatial

and

temporal

locality

of

Web

reference

streams

relationship

between

a

word

's

popularity

in

terms

of

rank

and

its

frequency

of

use

It

states

that

if

one

ranks

the

popularity

of

words

used

in

a

given

-LRB-

denoted

by

ae

-RRB-

by

their

frequency

of

use

-LRB-

denoted

by

P

-RRB-

then

P

ae

Note

that

this

distribution

is

a

parameterless

hyperbolic

distribution

i.e.

ae

is

raised

to

exactly

so

that

the

nth

most

popular

document

is

exactly

twice

as

likely

to

be

accessed

as

the

most

popular

document

Our

data

shows

that

Zipf

's

law

applies

quite

strongly

to

Web

documents

serviced

by

Web

servers

This

is

demonstrated

in

Figure

for

all

documents

accessed

in

the

BU

trace

The

figure

shows

a

log-log

plot

of

the

total

number

of

references

-LRB-

Y

axis

-RRB-

to

each

document

as

a

function

of

the

document

's

rank

in

overall

popularity

-LRB-

X

axis

-RRB-

The

tightness

of

the

fit

to

a

straight

line

is

strong

-LRB-

-RRB-

as

is

the

slope

of

the

line

-LRB-

shown

in

the

figure

-RRB-

Thus

the

exponent

relating

popularity

to

rank

for

Web

documents

is

very

nearly

as

predicted

by

Zipf

's

law

-LRB-

Number

of

References

-RRB-

-LRB-

Document

Rank

-RRB-

Figure

Zipf

's

Law

Applied

to

Web

Documents

Our

measurements

suggest

that

Zipf

's

law

applies

not

only

to

single

requests

for

documents

but

also

to

request

strides

A

request

stride

is

a

sequence

of

requests

from

the

same

client

where

the

time

between

successive

requests

in

that

stride

is

not

larger

than

a

given

stride

threshold

shows

a

log-log

plot

of

the

total

number

of

references

-LRB-

Y

axis

-RRB-

to

strides

of

length

k

-LRB-

for

k

-RRB-

as

a

function

of

the

stride

's

rank

in

overall

popularity

-LRB-

X

axis

-RRB-

The

trace

used

for

this

plot

is

the

BU

trace

and

the

stride

threshold

is

set

at

seconds

Again

the

tightness

of

the

fit

for

the

family

of

curves

in

to

straight

lines

slope

close

to

suggest

that

Zipf

's

law

does

apply

to

sequences

of

requests

Given

the

above

observations

one

possible

access

model

-LSB-

-RSB-

is

to

generate

document

requests

according

to

the

popularity

profile

of

documents

We

call

this

model

the

Zipf-based

model

To

test

the

validity

of

this

model

we

measured

the

miss

rate

at

the

server

when

requests

are

generated

's

law

has

subsequently

been

applied

to

other

examples

of

popularity

in

the

social

sciences

References

Rank

Length

Length

Length

Figure

Zipf

's

Law

Applied

to

Sequences

of

Web

Documents

synthetically

to

satisfy

the

observed

popularity

profile

of

documents

at

BU

and

compared

these

measurements

with

the

measurements

obtained

when

the

actual

BU

trace

is

used

to

drive

the

simulation

The

synthetic

trace

was

generated

by

applying

a

random

permutation

to

the

actual

trace

for

BU

The

result

of

this

comparison

is

shown

in

Figure

Obviously

using

popularity

as

the

basis

of

a

cache

model

for

Web

documents

does

not

yield

acceptable

results

Figure

shows

that

for

a

cache

size

of

items

the

miss

rate

that

results

from

a

Zipf-based

cache

model

is

nearly

less

than

the

actual

miss

rate

The

main

weakness

of

a

Zipf-based

model

is

that

it

does

not

capture

the

potential

locality

of

reference

properties

that

may

be

present

in

actual

traces

In

particular

while

the

Zipf-based

synthetic

trace

preserves

the

general

popularity

profile

of

documents

it

fails

to

capture

the

temporal

and

spatial

locality

of

reference

properties

In

the

next

section

we

show

evidence

that

these

properties

indeed

exist

in

Web

traces

and

in

sections

and

we

provide

simple

yet

expressive

statistical

models

thereof

Evidence

of

Reference

Locality

Temporal

Locality

Temporal

locality

implies

that

recently

accessed

documents

are

more

likely

to

be

referenced

in

the

near

future

Based

on

the

concepts

proposed

in

-LSB-

-RSB-

we

define

a

stack

distance

model

that

captures

the

temporal

locality

relationships

present

in

a

request

stream

Let

us

consider

a

reference

stream

Rt

rt

where

rt

denotes

the

name

of

the

object

requested

at

virtual

time

t

Our

unit

of

time

will

be

one

request

so

that

virtual

time

t

indicates

that

t

requests

have

already

arrived

at

a

server

Let

us

also

define

the

LRU

stack

St

which

is

an

ordering

of

all

objects

of

a

server

by

recency

of

usage

Thus

at

virtual

time

t

the

LRU

stack

is

given

by

St

oNg

-LRB-

-RRB-

500 100015002

Miss

Rate

Cache

Size

Zipf-Synthetic

Workload

BU

Trace

Figure

Miss

rate

for

a

Zipf-based

synthetic

workload

and

for

an

actual

trace

where

oN

are

objects

of

the

server

and

is

the

most

recently

accessed

object

the

next

most

recently

referenced

etc.

If

is

the

most

recently

accessed

document

then

rt

Whenever

a

reference

is

made

to

an

object

the

stack

must

be

updated

Considering

that

rt

oi

then

the

stack

becomes

St

foi

oi

oi

oNg

Suppose

now

that

St

oNg

and

rt

oi

Then

we

could

say

that

request

rt

is

at

distance

i

in

stack

St

Let

dt

denote

the

stack

distance

of

the

document

referenced

at

time

t

We

then

have

the

following

relation

if

rt

oi

then

dt

i

-LRB-

-RRB-

Thus

for

any

request

string

Rt

rt

there

is

a

corresponding

distance

string

ffi

dt

We

can

then

consider

that

the

distance

string

and

the

request

string

are

equivalent

in

terms

of

reference

information

Since

the

distance

string

reflects

the

pattern

in

which

Web

users

request

documents

from

a

server

rather

than

the

actual

identity

of

the

documents

it

will

be

used

as

our

model

for

object

reference

pattern

The

marginal

distribution

of

distance

probabilities

is

an

indication

of

temporal

locality

because

it

measures

the

number

of

intervening

references

between

two

references

to

the

same

object

Small

stack

distances

result

from

frequent

references

to

a

document

In

other

words

the

documents

referenced

most

frequently

tend

to

be

those

close

to

the

top

of

the

stack

As

a

measure

of

the

temporal

locality

present

in

our

traces

we

compare

the

average

stack

distance

found

in

the

BU

trace

with

that

of

the

scrambled

BU

trace

The

results

shown

in

table

indicate

that

the

average

stack

distance

is

apparently

greater

for

the

scrambled

trace

meaning

that

temporal

locality

has

been

decreased

Spatial

Locality

The

existence

of

spatial

locality

of

reference

could

be

established

by

comparing

the

total

number

of

unique

sequences

observed

in

a

trace

and

the

total

number

of

unique

sequences

that

would

be

found

in

a

random

permutation

of

such

a

trace

Notice

that

a

random

permutation

of

a

trace

-LRB-

henceforth

Item

BU-original

BU-scrambled

Mean

Stack

Distance

Standard

Deviation

Table

Evidence

of

Temporal

Locality

called

a

scrambled

trace

-RRB-

preserves

the

popularity

profile

of

the

various

documents

in

that

trace

but

that

it

destroys

the

spatial

locality

of

reference

that

may

exist

in

such

a

trace

-LRB-

by

uncorrelating

the

sequence

of

references

-RRB-

If

references

are

indeed

correlated

then

one

would

expect

the

total

number

of

unique

sequences

observed

in

a

trace

to

be

much

less

than

the

total

number

of

unique

sequences

observed

in

a

scrambled

trace

Figure

shows

the

total

number

of

unique

sequences

-LRB-

Y

axis

-RRB-

of

length

k

-LRB-

X

axis

-RRB-

observed

in

the

first

week

-LRB-

the

first

references

-RRB-

of

the

BU

trace

which

we

call

the

server

trace

Three

cases

are

shown

The

top

curve

shows

the

total

number

of

unique

k-long

sequences

observed

in

a

scrambled

trace

The

middle

curve

shows

the

total

number

of

unique

k-long

sequences

observed

in

the

trace

Since

traces

arriving

at

the

server

reflect

an

interleaving

of

a

number

of

individual

client

traces

we

plot

in

the

bottom

curve

the

total

number

of

unique

k-long

sequences

observed

when

separately

considering

individual

clients

in

the

trace

LengthNumber

of

Unique

Sequences

Server

Server

Trace

-LRB-

Random

Permutation

-RRB-

Client

Trace

Figure

Unique

sequences

observed

in

the

BU

trace

All

three

curves

show

an

increase

in

the

total

number

of

unique

sequences

as

k

increases

with

the

randomized

trace

showing

the

steepest

increase

and

the

client

trace

showing

the

slowest

increase

These

results

confirm

the

existence

of

considerable

spatial

locality

in

client

access

patterns

For

example

while

up

to

different

sequences

of

length

are

observed

in

the

scrambled

reason

that

all

three

curves

level-off

is

related

to

the

limited

length

of

the

trace

In

particular

it

is

impossible

to

observe

any

more

than

l

k

unique

k-long

sequences

in

a

trace

of

length

l.

For

our

experiments

l

hence

the

observed

ceiling

trace

only

-LRB-

more

than

less

-RRB-

different

sequences

of

length

are

observed

in

the

client

traces

This

spatial

locality

is

diluted

to

some

extent

-LRB-

but

still

very

much

evident

-RRB-

in

the

server

trace

-LRB-

about

less

-RRB-

different

sequences

of

length

are

observed

in

the

server

trace

Characterizing

Temporal

Locality

Since

it

is

clear

that

real

traces

and

scrambled

traces

differ

significantly

in

the

aggregate

properties

of

their

stack

distance

traces

we

examine

the

marginal

distributions

of

stack

distances

As

discussed

in

Section

the

marginal

distribution

of

stack

distance

captures

the

temporal

locality

present

in

the

trace

That

is

if

D

is

a

random

variable

corresponding

to

stack

distance

-LRB-

with

distribution

function

FD

-RRB-

and

M

-LRB-

C

-RRB-

is

the

miss

rate

of

a

cache

that

can

hold

C

files

-LRB-

using

the

same

replacement

algorithm

that

was

used

to

generate

the

stack

distance

trace

-RRB-

then

P

-LSB-

D

C

-RSB-

FD

-LRB-

C

-RRB-

M

-LRB-

C

-RRB-

Thus

knowledge

of

the

complementary

distribution

function

FD

provides

enough

information

to

predict

the

performance

of

a

cache

of

any

size

for

the

given

trace

To

estimate

distributions

for

the

four

traces

we

consider

histograms

are

shown

in

Figure

This

figure

shows

that

most

of

the

stack

distances

are

close

to

-LRB-

the

traces

show

good

temporal

locality

-RRB-

and

yet

there

are

long

tails

to

the

distributions

We

considered

a

number

of

long-tailed

distributions

to

model

this

data

In

particular

we

rejected

the

Pareto

distribution

-LRB-

-LSB-

-RSB-

-RRB-

because

although

the

tails

are

long

they

do

not

seem

to

follow

a

power-law

The

data

were

found

to

be

best

fit

by

a

lognormal

distribution

Data

that

follows

a

lognormal

distribution

has

the

property

that

its

logarithm

is

normally

distributed

The

resulting

distribution

has

a

very

long

tail

but

is

not

heavy-tailed

in

the

strict

sense

of

following

a

power-law

in

the

tails

To

illustrate

the

use

of

lognormal

distributions

to

model

stack

distances

Figure

shows

the

empirically

measured

distributions

and

the

fitted

distributions

Each

plot

shows

a

histogram

of

the

base

logs

of

the

stack

distance

data

in

addition

the

plot

shows

the

shape

of

the

corresponding

fit

of

a

lognormal

distribution

to

the

data

While

the

data

sets

may

appear

to

vary

from

normal

to

some

degree

in

reality

these

differences

have

been

magnified

by

the

log-transformation

process

In

fact

the

fit

to

the

data

is

quite

close

as

we

will

show

below

The

parameters

used

to

generate

the

fitted

distributions

shown

in

Figure

are

given

in

Table

The

and

oe

columns

show

the

mean

and

standard

deviation

of

the

base

log

of

the

datasets

The

table

shows

that

for

the

traces

we

studied

the

oe

varies

only

slightly

Most

of

the

variation

between

the

datasets

occurs

in

the

parameter

Trace

oe

BU

NCSA

SDSC

EPA

Table

Parameters

of

the

Lognormal

Distribution

for

Traces

The

ability

of

lognormal

distributions

to

predict

actual

miss

rates

in

finite

caches

is

shown

in

Figure

As

discussed

above

we

plot

P

-LSB-

D

C

-RSB-

since

it

is

equivalent

to

miss

rate

In

each

plot

DistanceNumber

of

DistanceNumber

of

DistanceNumber

of

DistanceNumber

of

Occurrences

Figure

Empirical

Stack

Distance

Distributions

-LRB-

Upper

BU

NCSA

Lower

SDSC

EPA

-RRB-

-LRB-

Stack

Distance

-RRB-

-LRB-

Stack

Distance

-RRB-

-LRB-

Stack

Distance

-RRB-

-LRB-

Stack

Distance

-RRB-

Figure

Lognormal

Distributions

of

Stack

Distance

Traces

-LRB-

Upper

BU

NCSA

Lower

SDSC

EPA

-RRB-

we

show

the

predicted

miss

rate

using

the

lognormal

stack

distance

model

and

the

actual

miss

rate

obtained

from

the

data

The

agreement

between

model

and

data

in

each

case

is

good

except

in

the

case

of

the

EPA

trace

which

has

very

poor

locality

A

comparison

of

the

traces

in

terms

of

their

fitted

parameters

is

shown

in

Figure

This

figure

shows

how

temporal

locality

-LRB-

measured

by

miss

rate

-RRB-

varies

across

the

traces

and

the

corresponding

parameter

values

that

capture

that

locality

The

figure

shows

that

as

the

temporal

locality

of

a

trace

decreases

the

mean

of

lognormal

distribution

increases

-LRB-

from

about

up

to

-RRB-

and

the

standard

deviation

of

the

lognormal

distribution

also

tends

to

increase

although

to

a

lesser

degree

-LRB-

from

up

to

-RRB-

500 100015002

Miss

Rate

Cache

Size

Actual

Data

Model

500 1000 1500 2000

2500 3000 3500 4000

Miss

Rate

Cache

Size

Actual

Data

Model

200 400 600 800

Miss

Rate

Cache

Size

Actual

Data

Model

Miss

Rate

Cache

Size

Actual

Data

Model

Figure

Actual

and

Predicted

Miss

Rates

using

Lognormal

Models

-LRB-

Upper

BU

NCSA

Lower

SDSC

EPA

-RRB-

Characterizing

Spatial

Locality

Having

characterized

the

temporal

locality

present

in

our

traces

we

now

present

models

for

spatial

locality

Previous

work

has

shown

that

the

correlation

structure

of

the

stack

distance

stream

is

useful

for

modeling

spatial

locality

-LSB-

-RSB-

however

previous

attempts

to

model

the

correlation

structure

of

reference

strings

have

used

models

with

only

short-range

dependence

such

as

finite

first-order

Markov

chains

As

pointed

out

in

-LSB-

-RSB-

such

short-range

dependent

models

are

not

capable

of

capturing

the

long-term

structure

of

reference

strings

which

includes

phase

change

behavior

Miss

Rate

Cache

Size

SDSC

-LRB-

m

s

-RRB-

NCSA

-LRB-

m

s

-RRB-

BU

-LRB-

m

s

-RRB-

EPA

-LRB-

m

s

-RRB-

Figure

Comparison

of

Temporal

Locality

and

Parameters

Measuring

Stack

Distance

Self-Similarity

To

capture

the

long-range

dependence

in

stack

distance

strings

we

employ

the

statistics

of

selfsimilarity

We

say

that

a

series

Yt

is

H-self-similar

if

for

any

positive

stretching

factor

c

the

rescaled

process

with

time

scale

ct

which

is

c?HYct

is

equal

in

distribution

to

the

original

process

Yt

That

is

Yt

=d

c?HYct

for

all

c

t

-LRB-

-RRB-

This

means

that

the

series

Yt

has

fractal-like

properties

detail

is

present

at

all

scales

As

a

result

using

only

the

single

parameter

H

-LRB-

related

to

the

fractal

dimension

-RRB-

we

can

capture

variation

at

a

wide

range

of

scales

Equation

applies

to

a

motion

process

such

as

Brownian

motion

-LRB-

for

which

H

-RRB-

For

stationary

processes

like

our

stack

distance

traces

an

equivalent

definition

is

used

which

relates

the

stationary

process

to

its

corresponding

motion

process

-LRB-

formed

by

taking

successive

sums

of

the

stationary

process

-RRB-

Definition

Let

fXtg

be

a

stationary

process

with

continuous

time

parameter

t

Then

we

say

that

Xt

is

H-self-similar

if

for

any

positive

aggregation

factor

m

the

series

obtained

by

m-aggregating

Xt

is

equal

in

distribution

to

Xt

That

is

Xt

=d

m?H

tm

X

i

-LRB-

t

-RRB-

m

Xi

for

all

m

N

t

-LRB-

-RRB-

Because

of

the

correlations

present

on

all

scales

that

is

implied

by

Equation

stationary

self-similar

processes

also

have

the

property

of

long-range

dependence

A

long-range

dependent

series

typically

has

bursts

-LRB-

values

above

or

below

the

mean

-RRB-

that

occur

on

all

time

scales

That

is

regardless

of

the

scale

at

which

the

series

is

viewed

bursty

patterns

can

be

observed

This

property

is

in

strong

contrast

to

short-range

dependent

models

-LRB-

such

as

Markovian

processes

or

Poisson

processes

-RRB-

which

appear

smooth

at

long

timescales

The

property

of

burstiness

at

all

timescales

is

noticeably

present

in

stack

distance

series

it

is

this

property

that

has

been

difficult

to

capture

in

the

past

using

only

short-range

dependent

models

An

example

of

this

effect

is

shown

in

Figure

The

figure

compares

the

BU

stack

distance

trace

with

a

trace

formed

by

first

randomly

scrambling

the

BU

dataset

before

calculating

stack

distances

Note

that

both

traces

consist

of

exactly

the

same

set

of

file

references

however

in

the

case

of

the

scrambled

dataset

the

spatial

locality

has

been

destroyed

The

figure

shows

the

two

stack

distance

traces

at

varying

levels

of

detail

In

the

bottom

graphs

each

point

in

the

series

is

formed

by

summing

consecutive

points

-LRB-

in

non-overlapping

blocks

-RRB-

from

the

original

traces

Moving

up

the

next

series

is

formed

by

summing

points

from

the

lower

series

and

so

on

until

at

the

top

level

each

data

point

consists

of

the

sum

of

data

points

from

the

original

stack

distance

series

In

each

plot

we

have

only

plotted

points

so

as

to

show

equivalent

levels

of

detail

at

the

top

this

corresponds

to

the

entire

series

at

lower

levels

the

plots

are

over

ranges

that

are

selected

arbitrarily

When

summing

data

in

this

way

short

range

dependences

are

averaged

out

if

the

series

is

long-range

dependent

bursts

will

still

remain

This

can

be

seen

by

comparing

the

scrambled

trace

-LRB-

on

the

right

-RRB-

with

the

original

trace

-LRB-

on

the

left

-RRB-

The

bursts

that

are

present

in

the

original

trace

are

evidence

of

very

long-range

correlations

in

the

original

dataset

which

corresponds

to

long

periods

of

very

large

stack

distances

-LCB-

as

would

be

caused

by

phase

changes

in

file

referencing

behavior

The

degree

of

self-similarity

in

a

series

is

captured

by

the

parameter

H

which

takes

values

between

and

If

H

the

series

has

no

long-range

dependence

and

acts

like

Gaussian

noise

-LRB-

whose

motion

process

is

Brownian

motion

-RRB-

As

H

the

burstiness

of

the

series

becomes

more

pronounced

at

high

levels

of

aggregation

To

estimate

the

H

parameter

for

our

series

we

use

four

methods

Three

methods

the

variance-time

plot

the

R/S

plot

and

the

periodogram

are

graphical

These

methods

are

useful

for

assessing

whether

assumptions

about

the

data

are

correct

-LRB-

such

as

stationarity

-RRB-

and

for

providing

a

single

estimate

of

H

The

fourth

method

is

the

maximum

likelihood

estimator

for

H

called

the

Whittle

estimator

which

provide

confidence

intervals

as

well

Interested

readers

are

referred

to

-LSB-

-RSB-

for

a

discussion

of

these

methods

in

more

detail

Plots

of

the

graphical

estimators

for

the

BU

trace

are

shown

in

Figure

On

the

left

is

the

variance-time

estimator

linearity

on

this

plot

is

evidence

of

long-range

dependence

and

the

slope

of

the

line

yields

an

estimate

of

H

which

in

this

case

is

-LRB-

the

line

corresponding

to

H

is

also

plotted

for

reference

In

the

center

is

a

plot

of

the

R/S

statistic

again

linearity

is

evidence

of

long-range

dependence

and

the

slope

yields

an

estimate

of

H

In

this

plot

the

lines

corresponding

to

H

and

H

are

plotted

for

reference

On

the

right

is

the

periodogram

estimator

significant

scatter

is

typical

in

this

plot

but

a

least-squares

fit

to

the

points

also

yields

and

estimate

of

H

in

this

case

Each

of

these

plots

shows

evidence

of

self-similarity

and

the

graphical

results

for

the

other

three

stack

distance

traces

show

similar

linearity

In

addition

to

the

graphical

estimators

we

also

used

the

Whittle

MLE

estimator

to

estimate

H

for

our

datasets

In

all

cases

we

found

reasonable

agreement

between

the

estimators

and

values

of

H

significantly

above

The

results

for

our

four

datasets

are

shown

in

Table

The

values

of

H

range

from

about

for

the

BU

dataset

to

about

for

the

EPA

dataset

We

considered

the

possibility

that

the

process

of

transforming

the

original

traces

into

stack

distance

strings

may

have

been

responsible

for

introducing

dependence

into

the

data

To

answer

this

question

we

randomly

rearranged

each

dataset

calculated

the

corresponding

new

stack

distance

series

and

estimated

the

H

values

of

the

resulting

series

The

results

are

shown

in

Table

These

estimates

are

all

very

close

to

indicative

of

no

long-range

dependence

in

the

series

As

a

result

we

conclude

that

the

presence

of

long-range

dependence

in

these

timeseries

reflects

actual

correlations

among

the

symbol

pattern

in

the

original

reference

traces

Figure

Stack

Distance

Scaling

Behavior

of

Original

-LRB-

left

-RRB-

and

Scrambled

-LRB-

right

-RRB-

BU

Traces

-LRB-

Normalized

Variance

-RRB-

-LRB-

m

-RRB-

-LRB-

r/s

-RRB-

-LRB-

d

-RRB-

-LRB-

periodogram

-RRB-

-LRB-

frequency

-RRB-

Figure

Graphical

Estimators

of

H

for

BU

Trace

Whittle

V-T

R/S

Period

MLE

-LRB-

conf

-RRB-

BU

-LRB-

-RRB-

NCSA

-LRB-

-RRB-

SDSC

-LRB-

-RRB-

EPA

-LRB-

-RRB-

Table

Estimates

of

H

for

Datasets

Used

in

This

Study

Whittle

V-T

R/S

Period

MLE

BU

NCSA

EPA

SDSC

Table

Estimates

of

H

for

Scrambled

Datasets

Relationship

to

Fractal

Cache

Miss

Patterns

Another

way

to

look

at

Figure

is

that

progressing

from

top

to

bottom

we

are

zooming

in

on

the

data

The

fact

that

the

plots

on

the

left

of

the

figure

show

roughly

similar

degrees

of

burstiness

at

all

scales

is

evidence

of

fractal-like

behavior

of

the

stack

distance

trace

In

fact

a

statistically

self-similar

series

Xt

like

the

one

shown

on

the

left

of

Figure

has

an

almost

sure

fractal

dimension

-LRB-

in

the

sense

of

either

the

Hausdorff

dimH

or

box-counting

dimension

dimB

-RRB-

of

-LSB-

-RSB-

dim

-LRB-

Xt

-RRB-

H

Thus

the

fractal

dimension

of

such

a

series

is

for

Gaussian

noise

-LRB-

H

-RRB-

and

decreases

to

as

the

correlation

present

in

the

series

increases

-LRB-

H

-RRB-

The

corresponds

with

intuition

a

series

with

more

dependence

is

more

predictable

and

fills

space

less

completely

The

fractal

properties

of

stack-distance

series

serve

to

place

earlier

work

by

Voldman

et

al.

-LSB-

-RSB-

in

a

slightly

broader

context

Those

authors

found

that

patterns

of

cache

misses

often

form

random

fractals

One

of

their

results

is

that

if

cache

miss

events

are

plotted

on

the

number

line

using

the

sequence

number

of

the

reference

as

the

coordinate

the

result

is

a

fractal

pattern

of

dimension

between

and

Our

results

are

consistent

with

that

work

as

follows

Cache

miss

patterns

can

be

seen

to

be

the

points

of

intersection

between

the

stack

distance

trace

and

a

line

parallel

to

the

x

axis

corresponding

to

some

fixed

cache

size

The

intersection

of

fractal

X

having

dimH

-LRB-

X

-RRB-

and

a

line

L

-LRB-

dimH

-LRB-

L

-RRB-

-RRB-

is

almost

surely

a

new

fractal

X

with

dimH

-LRB-

X

-RRB-

dimH

-LRB-

X

-RRB-

-LSB-

-RSB-

This

means

that

a

cache

miss

fractal

derived

from

a

stack

distance

fractal

such

as

our

traces

should

have

a

dimension

between

and

In

fact

we

have

observed

this

effect

and

have

found

that

cache

misses

in

our

traces

show

fractal

characteristics

that

are

consistent

with

the

patterns

previously

observed

by

Voldman

et

al.

Generating

Synthetic

Web

Reference

Traces

To

generate

synthetic

Web

reference

traces

it

is

important

to

mimic

as

closely

as

possible

both

the

temporal

and

spatial

locality

present

in

real

traces

To

do

so

using

the

results

in

this

paper

suggests

the

following

process

Select

parameters

and

oe

reflecting

temporal

locality

and

H

reflecting

spatial

locality

These

may

be

selected

based

on

empirical

measurement

of

traces

that

are

to

be

imitated

or

from

the

range

of

values

measured

in

the

traces

reported

here

Generate

a

stack

distance

trace

with

marginal

distribution

determined

by

and

oe

and

longrange

dependence

determined

by

H.

Invert

the

stack

distance

trace

to

form

a

sequence

of

file

names

On

first

glance

step

seems

difficult

Fortunately

recent

results

have

shown

that

marginal

distribution

and

long-range

dependence

can

be

considered

as

separate

problems

for

trace

generation

-LSB-

-RSB-

That

work

shows

that

given

a

long-range

dependent

series

Xt

with

H

HX

and

marginal

cumulative

probability

function

FX

-LRB-

x

-RRB-

we

can

generate

another

long-range

dependent

series

Yt

with

H

HX

and

arbitrary

marginal

cumulative

probability

function

FY

-LRB-

x

-RRB-

This

is

achieved

by

individual

transformation

of

the

elements

of

Xt

as

follows

Yt

F

Y

-LRB-

FX

-LRB-

Xt

-RRB-

-RRB-

-LRB-

-RRB-

The

authors

in

-LSB-

-RSB-

show

that

this

transformation

preserves

long-range

dependence

with

the

same

value

of

H.

Thus

our

approach

to

step

is

as

follows

First

generate

a

self-similar

series

with

given

H

This

can

be

done

in

a

number

of

ways

a

fast

approach

is

described

in

-LSB-

-RSB-

Then

apply

the

transformation

of

Equation

to

obtain

a

series

with

the

proper

lognormal

marginal

distribution

We

have

begun

to

use

the

method

for

generating

web

reference

traces

and

are

currently

evaluating

the

properties

of

the

resulting

traces

Initial

experiments

involving

models

of

only

temporal

locality

properties

are

encouraging

Related

Work

In

this

section

we

review

previous

research

related

to

our

work

namely

work

in

reference

locality

modeling

use

of

fractals

for

modeling

program

behavior

and

work

in

exploiting

Web

reference

locality

Reference

Locality

Modeling

Denning

and

Schwartz

-LSB-

-RSB-

established

the

fundamental

properties

that

characterize

the

phenomenon

of

locality

in

hierarchical

structures

of

memory

In

-LSB-

-RSB-

the

authors

studied

stack

algorithms

and

introduced

stack

distance

as

a

means

for

analyzing

behavior

of

demand-paged

memory

systems

They

also

discussed

the

significance

of

stack

distance

strings

for

evaluating

performance

of

memory

management

schemes

Then

Spirn

-LSB-

-RSB-

proposes

the

use

of

distance

string

models

to

represent

program

behavior

Distance

strings

are

equivalent

in

information

content

to

the

reference

string

but

are

more

easily

handled

by

mathematical

models

The

distance

string

model

and

the

LRU

stack

model

are

equivalent

since

it

is

possible

to

determine

the

LRU

stack

at

each

time

from

the

knowledge

of

the

distance

string

However

the

proposed

distance

string

model

does

not

capture

the

long-range

dependencies

among

references

As

pointed

out

by

the

author

the

model

does

not

provide

an

accurate

characterization

of

page

faults

which

tend

in

real

programs

to

occur

in

clusters

When

changes

in

locality

occur

gradually

page

by

page

it

is

relatively

simple

to

model

the

reference

locality

However

in

a

real

program

the

locality

set

of

execution

is

completely

disrupted

by

the

execution

of

a

new

phase

of

the

program

Usually

the

abrupt

changes

in

locality

generate

large

distances

in

distance

string

model

Markov

distance

string

models

are

capable

of

predicting

bursts

based

on

the

most

recent

generated

distances

Thus

the

work

in

-LSB-

-RSB-

argues

that

Markov

models

are

not

able

to

capture

the

behavior

of

real

programs

which

exhibit

some

form

of

long-range

dependencies

The

author

also

points

out

that

those

models

based

on

the

distance

probabilities

are

not

suitable

for

simulation

studies

of

program

behavior

because

they

lack

the

ability

to

mimic

long-range

effects

Application

of

Fractals

to

Reference

Modeling

The

use

of

fractal

geometry

and

self-similarity

to

analyze

behavior

of

computer

systems

was

pioneered

by

the

work

of

Voldman

et

al

-LSB-

-RSB-

The

authors

looked

at

memory

reference

traces

of

three

software

environments

and

found

that

the

distributions

of

intermiss

distances

are

hyperbolic

The

cache

misses

are

grouped

over

time

in

clusters

which

are

statistically

self-similar

As

a

result

of

the

analysis

they

conjecture

that

the

fractal

dimension

of

a

given

cache

miss

distribution

is

a

measure

of

the

complexity

of

the

underlying

software

is

a

vast

literature

on

characterizing

and

exploiting

the

locality

of

reference

properties

exhibited

in

computer

systems

Covering

this

literature

is

beyond

the

scope

of

this

paper

Based

on

the

work

in

-LSB-

-RSB-

paper

-LSB-

-RSB-

models

the

patterns

generated

by

a

processor

accessing

memory

as

random

walk

with

fractal

dimension

The

author

proposes

that

in

an

asymptotic

behavior

the

number

of

misses

that

a

program

encounters

in

an

infinite

cache

is

given

by

a

hyperbolic

function

of

the

fractal

dimension

The

paper

also

presents

an

approximate

analytical

model

for

cache

misses

The

model

has

two

distinct

phases

one

called

forced

mode

which

represents

the

cold

start

process

The

fractal

mode

phase

models

the

stable

phase

of

a

cache

However

the

model

is

not

easily

used

because

of

the

empirical

process

of

obtaining

and

adjusting

its

parameters

from

plots

of

the

traces

Also

the

proposed

model

does

not

represent

multiprogramming

environments

where

context

switches

occur

frequently

Our

paper

extends

the

work

referenced

in

-LSB-

-RSB-

to

the

context

of

large

distributed

systems

and

also

introduces

new

parameters

that

fully

describe

locality

aspects

of

a

reference

stream

that

arrives

at

a

Web

server

Exploiting

Web

Reference

Locality

Previous

studies

in

caching

replication

dissemination

and

prefetching

protocols

for

large

distributed

information

systems

have

recognized

and

exploited

the

various

locality

of

reference

properties

examined

in

this

paper

In

the

remainder

of

this

section

we

present

a

brief

review

of

these

protocols

Most

of

the

caching

studies

for

large

distributed

information

systems

concentrate

on

client-based

caching

whereby

recently/frequently

accessed

information

is

cached

at

the

client

-LRB-

or

at

a

proxy

thereof

-RRB-

in

anticipation

of

future

accesses

to

the

same

information

Traditionally

this

was

done

in

the

realms

of

distributed

file

systems

-LSB-

-RSB-

but

more

recently

there

have

been

some

attempts

at

extending

these

techniques

to

distributed

information

systems

such

as

FTP

and

HTTP

Caching

to

reduce

the

bandwidth

requirements

for

the

FTP

protocol

on

the

NSFNET

has

been

studied

in

-LSB-

-RSB-

In

this

study

a

hierarchical

caching

system

that

caches

files

at

Core

Nodal

Switching

Subsystems

is

shown

to

reduce

the

NSFNET

backbone

traffic

by

The

effect

of

data

placement

and

replication

on

network

traffic

was

also

studied

in

-LSB-

-RSB-

where

file

access

patterns

are

used

to

suggest

a

distributed

dynamic

replication

scheme

A

more

static

solution

based

on

fixed

network

and

storage

costs

for

the

delivery

of

multimedia

home

entertainment

was

suggested

in

-LSB-

-RSB-

Multi-level

caching

was

studied

in

-LSB-

-RSB-

where

simulations

of

a

two-level

caching

system

is

shown

to

reduce

both

network

and

server

loads

In

-LSB-

-RSB-

a

dynamic

hierarchical

file

system

that

supports

demand-driven

replication

is

proposed

whereby

clients

are

allowed

to

service

requests

issued

by

other

clients

from

the

local

disk

cache

A

similar

cooperative

caching

idea

was

suggested

in

-LSB-

-RSB-

Glassman

-LSB-

-RSB-

presents

one

of

the

earliest

attempts

for

caching

on

the

Web

whereby

satellite

relays

-LRB-

proxy

caches

-RRB-

are

organized

into

a

tree-structured

hierarchy

with

cache

misses

in

lower

relays

percolating

up

through

higher

relays

until

the

requested

object

is

found

The

performance

of

this

caching

system

for

a

single

relay

with

a

rather

small

cache

size

indicated

that

it

is

possible

to

maintain

a

fairly

stable

hit

rate

The

performance

of

such

a

system

for

large

cache

sizes

was

extrapolated

using

a

Zipf-based

model

In

particular

it

was

estimated

that

with

an

infinitesize

cache

the

maximum

achievable

hit

rate

is

Another

early

attempt

at

characterizing

Web

access

patterns

for

the

purpose

of

engineering

Web

caching

systems

is

the

work

of

Recker

and

Pitkow

-LSB-

-RSB-

in

which

a

model

for

Web

information

access

is

proposed

based

on

two

metrics

borrowed

from

psychological

research

on

human

memory

namely

the

frequency

and

recency

rates

of

past

accesses

The

first

comprehensive

study

of

client-based

caching

for

the

Web

was

conducted

by

our

Oceans

group

In

that

study

-LSB-

-RSB-

the

effectiveness

of

session

caching

host

caching

and

LAN

proxy

caching

cs-www

were

established

using

a

unique

set

of

client

traces

-LRB-

almost

URL

requests

-RRB-

which

were

obtained

by

intrumenting

Mosaic

as

detailed

in

-LSB-

-RSB-

This

study

concluded

that

LAN

proxy

caching

while

effective

in

reducing

response

time

by

as

much

as

a

half

is

ultimately

limited

by

the

low

level

of

sharing

of

remote

documents

amongst

clients

of

the

same

site

This

finding

agrees

with

Glassman

's

predictions

-LSB-

-RSB-

and

was

further

confirmed

for

general

proxy

caching

by

Abrams

et

al

-LSB-

-RSB-

Williams

et

al

-LSB-

-RSB-

present

a

taxonomy

-LRB-

and

compare

the

performance

-RRB-

of

a

number

of

proxy

cache

replacement

policies

Their

work

suggests

that

proxy

cache

management

should

consider

document

sizes

when

making

replacement

decisions

This

was

also

confirmed

by

Bolot

and

Hoschka

-LSB-

-RSB-

who

showed

the

usefulness

of

using

information

about

document

size

and

network

load

in

the

replacement

algorithms

of

Web

caches

based

on

time

series

analysis

techniques

of

Web

traffic

Similar

results

about

the

inadequacy

of

classic

LRU

cache

replacement

are

presented

in

-LSB-

-RSB-

In

-LSB-

-RSB-

Bestavros

proposed

an

alternative

to

client-based

caching

which

attempts

to

capitalize

on

the

global

knowledge

amassed

at

servers

about

geographical

locality

of

reference

the

property

that

an

object

accessed

by

a

client

at

a

site

is

likely

to

be

accessed

again

in

the

future

by

a

client

in

a

nearby

site

To

that

end

he

proposed

a

popularity-based

dissemination

protocol

which

allows

servers

to

cache

documents

at

proxies

that

are

closer

to

clients

in

such

a

way

that

the

distance

between

a

client

and

a

document

server

-LRB-

or

proxy

thereof

-RRB-

is

inversely

proportional

to

the

popularity

of

that

document

A

similar

philosophy

was

sketched

in

-LSB-

-RSB-

whereby

the

placement

of

popular

documents

was

based

on

geographical

information

-LRB-

such

as

the

distance

in

actual

miles

between

servers

and

clients

-RRB-

In

-LSB-

-RSB-

Markatos

examines

the

potential

performance

gains

from

using

Main

Memory

Web

Caches

He

shows

that

a

small

amount

of

main

memory

could

yield

substantial

improvement

in

performance

if

cache

management

methods

that

are

sensitive

to

document

popularity

and

to

user

preferences

for

small

documents

-LSB-

-RSB-

are

employed

Exploiting

the

spatial

locality

of

reference

properties

exhibited

in

Web

access

patterns

was

investigated

in

two

recent

prefetching

studies

of

our

Oceans

group

In

-LSB-

-RSB-

Bestavros

and

Cunha

propose

a

protocol

that

allows

a

client

to

prefetch

Web

documents

based

on

Markov

models

built

from

the

client

's

previous

access

patterns

Preliminary

results

indicate

that

client-initiated

prefetching

is

extremely

effective

for

access

patterns

that

involve

frequently-traversed

documents

but

not

effective

at

all

for

access

patterns

that

involve

newly-traversed

documents

For

such

access

patterns

Bestavros

proposed

an

alternative

protocol

-LSB-

-RSB-

that

allows

prefetching

to

be

initiated

by

servers

-LRB-

or

by

clients

as

a

result

of

hints

generated

by

servers

-RRB-

based

on

a

Markov

model

built

by

analyzing

the

server

's

access

logs

Conclusion

In

this

paper

we

have

measured

the

locality

present

in

reference

traces

arriving

at

four

servers

in

the

World

Wide

Web

and

shown

statistical

models

that

capture

the

locality

properties

of

our

datasets

These

models

can

be

used

to

compare

different

reference

streams

predict

cache

performance

and

generate

synthetic

reference

streams

with

realistic

properties

We

first

showed

that

simple

methods

for

generating

reference

streams

based

only

on

document

popularity

capture

neither

the

temporal

nor

the

spatial

locality

present

in

a

trace

We

showed

that

some

method

that

measures

locality

explicitly

is

needed

in

order

to

generate

traces

with

the

necessary

characteristics

As

a

result

to

measure

locality

directly

we

have

relied

on

the

technique

of

transforming

reference

strings

into

distance

strings

and

studying

the

statistical

properties

of

those

distance

strings

We

then

measured

the

temporal

locality

properties

of

the

reference

traces

by

characterizing

the

distribution

of

stack

distances

in

each

trace

We

've

shown

that

stack

distances

have

long

tails

and

that

a

reasonable

model

for

stack

distance

distribution

is

the

lognormal

distribution

Typical

parameters

for

the

lognormal

distributions

fitted

to

our

data

were

given

and

the

resulting

models

were

shown

to

exhibit

cache

miss

rates

that

are

close

to

the

original

traces

over

the

entire

range

of

cache

sizes

To

model

spatial

locality

we

argued

that

models

incorporating

long-range

dependence

were

necessary

in

order

to

capture

the

long-term

behavior

of

reference

patterns

We

showed

that

stack

distance

traces

seem

to

show

properties

of

self-similarity

and

we

measured

the

relevant

parameters

for

each

of

our

traces

The

resulting

insight

into

the

nature

of

correlation

in

stack

distance

series

allowed

us

to

interpret

previous

work

in

fractal

cache

miss

patterns

in

a

broader

context

Ongoing

work

is

extending

these

models

of

locality

to

incorporate

explicit

modeling

of

document

size

In

addition

although

our

work

is

based

on

traces

obtained

from

servers

we

are

also

interested

in

the

locality

properties

of

client

reference

traces

and

how

those

properties

change

when

streams

are

captured

close

to

the

client

as

compared

to

close

to

the

server

Acknowledgements

The

authors

thank

John

Dilley

for

comments

that

improved

the

content

and

presentation

of

this

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