HEAPS: A Concept in Optimization

Transcription

1 J. Inst. Maths Applies (1974) 13, HEAPS: A Concept in Optimization P. J. BURVILLE University of Sussex f [Received 21 June 1973 and in revised form 19 August 1973] Information storage and retrieval systems have considerable importance in the world of telecommunications and computers. This paper describes a simple system which can be used to optimize such processes and offers guidance on its quantitative analysis. 1. Introduction THIS PAPER provides a basis on which some aspects of a well tried optimization technique can be quantified. The term HEAPS, which identifies one member of a family of information storage and retrieval configurations, is used as it has appropriate graphic connotations. It is also amenable to the acronym treatment e.g. Heuristically Evaluated Access Probability System. General In its most simple form the HEAPS concept, which has wide potential application, has (a) a primary source set of all the items of information to which many references are to be made; (b) a secondary sub-set of these items held in a heap; the heap is generally organized so that the last used item is placed on the top of the heap, and if the capacity of the heap is exceeded the item at the bottom of the heap is returned to the primary source (or destroyed if a copy was taken); (c) when an item is required a search is first made serially through the heap (starting at the top); only if the item is not in the heap is it necessary to obtain it from the primary source. A single item transfer process is assumed. The simple philosophy underlying the structure of the configuration is that the heap will tend to be made up of the items with the highest probability of being required. There are many situations where this philosophy would not be appropriate. Such a one is where once an item is called its probability of being called again within an interval of item demands is greatly reduced (perhaps to zero). Similarly one may be able to exploit the heap more efficiently in some situations by anticipating the items for which there will be a demand. Expressions for some of the heap parameters are given in the Appendix. For the calculations quoted in this paper no attempt has been made to correct for rounding errors. t Now at The London Graduate School of Business Studies

2 264 P. J. BURVILLE 2. Examples of Application The HEAPS optimization technique has the potential of wide and diverse deployment in many spheres of activity such as those of telecommunications and computers. It has been successfully deployed in a computer system where frequent access was required to a large file of item and equipment codes which could only be accommodated in a slow access-time store resulting in processing delay had there not been such a data access optimization technique available. In the context of computer usage of HEAPS due regard must be given in evaluating the expected accessing and processing time to any autonomous operation of the computer. Thus, if the accessing and processing time for the core-store is being evaluated it may be appropriate to regard retrieval from a backing store as having zero elapse time if such activity is autonomous and multi-programming is in operation. As the control system structure of the telephone network becomes more flexible, as it is with the introduction of electronic exchanges and eventually Stored Programme Control exchanges, information retrieval will be a task of significant size where optimization techniques may be used. An example of such a possible task is the translation process from the telephone number dialled by the customer to the actual routing data required to direct the call to the required destination. Wherever a call is made from in the United Kingdom the same Subscriber Trunk Dialling number is used where there are automatic facilities and the call is not a local one. Obviously the physical routing of the call through telephone exchanges and transmission links can be very different from one exchange to another and the probability of any given translation being required will vary with location and time. HEAPS is designed to accommodate such variations Terms and Definitions Let t n be the access time to (and including) cell n of the heap (e.g. t a is the time required to access cell H of the heap). tf be the access time to a given item or location in the primary source. T t be the (to be evaluated random) access time to item / wherever it is located. T a (k) be the calculation time of the algorithm (for the input parameter k). T p access time to the primary store (this does not include any time required to gain access to a particular item in the store). T s be the total travel time from one end of a store to the other. H be the number of cells in the heap (each item requires one cell). N be the number of (unique) items of information in the HEAPS. pr{e) be the probability of the event E; where the event is represented by a single character, e.g. B, p(b) may be used. Pi = pr {at any realization of the system item i is required}. Note that ^ipi = hpi> 0, for i = 1, 2,..., N. p(i, n) = pr {item / is resident in cell n of the heap}. Note that,/>(*,«) = 1, for n = 1,2,...,H /, n) = 1, for / = 1,2 N, and H = N.

3 HEAPS 265 E{X) be the expectation of the random variable Z(Kingman & Taylor, 1966). ~ u and v depend on a parameter which tends to a limit; then "u ~ v" if u/v approaches 1 as the parameter tends to the limit. Unless stated otherwise natural logarithms (to base e) will be used Evaluation of Examples In dimensioning the HEAPS and its performance there are two variables of primary interest (a) the expected processing time, which is normally in terms of the expected item access time; (b) the storage capacity required which can be desk accommodation, computer core store etc. For the three examples only the timing aspect of the dimensioning will be considered. For random item requirement probability distributions the completely random case (Pi = l/n, for all i) offers the least prospect of benefit arising from the use of HEAPS, hence evaluations of the completely random case are given as examples. Algorithm Example. The expected access time (T t ) to item i is given by: T t = i ^O,n)+(f fl +r fl (i))ri-s p(i,n)\. n=l L»=1 J If the algorithm is to determine the factorial of the number /(/!), the time for the evaluation T a (i) will not be constant with i and this may well influence the heap strategy; e.g. there may be an integer k such that ifi<k then the heap is not deployed and the algorithm is used. In this situation the access time is given by n=l _ n=l i = k, fc+1 N. The expected access time (for the situation where all the items are liable to be in the heap) is given by 1} = I pit {PV> n)(t n -t H -T a (i))}]+t B + E p t TJS). 1=1 Ln=l J «=1 Assuming that the demand for items is completely random and letting T it = 2" + 2N' then /»(/, n) = p, l/n (see the Appendix, part (f)), for i = 1,2,..., N, and n = 1, 2,..., #, and lfh(l-h) Hr(3N+l)- r(3n+l) E{Tl] " NI^ ' 4NJ +H ' +

4 266 P. J. BURVILLE Serial Primary Source. The primary source is a serially structured store (such as a normal magnetic tape) in which the items of information are stored with the location of each item known. If the item retrieval device stays at the last required item and moves linearly directly to the next required item, then for a completely random requirement probability the average travel time to the required item is TJ3 (where T s is the time to traverse the whole store end to end). The expected access time to item I is given by = t tj(i,n)ht B +T p +TJ3) J3)\l- n=l Then the expected access time for the completely random item requirement probability case, with t tt = nt, and letting T = T p +TJ3, is given by L n=l +Ht+T. Composite Heap and Primary Source. The primary source is serially structured such that (a) it is effectively an extension of the heap and hence has N H cells; (b) the reordering philosophy of the heap is extended so that an item being discarded from the bottom cell of the heap is accommodated in the top cell of the primary source, with the appropriate adjustments to the other cells to facilitate the change. The expected access time to item / is given by With / = t f, for n > H Ti= t np (i,n)+ n=l n=l The expected access time is given by Let t n = nt, then But /=ff+l E{T,} = * E Pi np(i, n). i=l n=l t,p(i,f). E "I**.») =!+ -X-. n = l (see the Appendix, part (d)). Hence j*ipi + Pj * = 1, 2 AT,

5 Letp t = (l-0)0'"-\ 0 < 0 < 1, N = oo, then HEAPS 267 fc. 4 Let 0 = e~", a > 0. Then which becomes Since e* -* 1, as a -> 0, as 0 -> 1. Thus thus log 2 " 0 * *** ^ 1 i aft a e-* x dx 0" Iog2 Iog2 -log0 1-0 ' From the Appendix, part (e), it will be seen that for this example E{T,} = tfi, Suppose that for a given 0 an approximation for E{T t } is required, which has an accuracy of say d, that is the approximation E{T t } must lie in the interval E{Tf}+d. Now, Let GO Qk oo nk y J. y (_n»-i_l_ i^i+0* jt 1 ;! 1-0** and l tr k=1 S n = E (-I)*' 1 ",, forn = l,2,...

6 {"YlilllYITI K*t\J total TABLE 1 Exact evaluation ofthepq, n), with the approximation shown in parenthesis (H = N = 10). System A (0-1) (0-1) Oil ( ) (010738) (010579) (010394) ( ) (009894) (009523) (008968) (007857) (009123) (009262) (009421) (009606) (009828) ( ) (010477) ( ) (012143) (010438) (010369) (010290) (010197) ( ) (009947) (009762) (009484) (008928) (009562) (009631) (009711) ( ) (009914) (010053) ( ) (010516) ( ) (010088) ( ) (010058) (010039) (010027) (009989) (009952) (009897) (009786) (009912) (009926) (009942) (009961) (009983) (010011) ( ) (010103) (010214) ( ) (010040) (010029) (010020) (010009) (009995) (009976) (009948) (009893) ( (009963) (009971) (009980) (009991) ( ) ( ) (010052) (010107) Row total

7 HEAPS 269 Then it will be seen that since the (finite) u k decrease and tend to zero as n approaches infinity, the series converges and the sum S will lie between S tt and S a+u for n = 1,2,... Thus the series can be summed to a given accuracy, say ± d/t, by summing the terms until 5 n -5 n+1 < d/t. Let * = S n +S tt+1 then 5 will lie in the interval S±d/2t. Let E{T,} = t(l+2s), * then E{T,} will lie in the interval E{T t }±d, as required. I o 12 on 009 -/=3 -/ Cell n of heap FIG. 1. A plot of the figures from Table 1 for items 3, 4, 5, 6, 7, and 8 for the exact evaluation. System A.

8 n\' Column total 1 TABLE 2 Exact evaluation ofthep{i, n), with the approximation shown in parenthesis (H = N= 10). System B ( ) (018765) (014383) ( ) (017377) (013688) ( ) ( ) ( ) (019844) (013937) (011969) (014288) (011715) (010858) (007343) (008937) (009469) ( ) (005234) (007617) ( )( ) (004839) (-O-43583X-O-11433X-O-O0716) (0-1) (o-d (008247) (008525) (008842) (009213) (009657) (010213) (010953) (012064) ( ) (005617) (006312) (007105) (008031) (009142) (010531) ( ) (015161) ( ) (004653) (005500) (006469) (007598) (008954) (010648) (012908) (016296) ( ) (002988) (004099) (005369) (006850) (008628) (010850) (013813) (018258) ( ) (002111) (003361) (004790) (006456) (008456) (010956) (014290) (019290) ( ) (001322) (002697) (004269) (006102) (008302) (011052) (014719) ( ) ( ) Row total

9 HEAPS Heap Probabilities Implicit in most evaluations of a HEAPS configuration is a quantification of the probabihty that item i is in cell n of the heap [p(i, n)] for all the items and cells. In the Appendix exact and approximate methods of determining these probabilities are obtained. As an indication of the form of the results which can be expected the probabilities are determined for two ten item systems with System A the probability of item requirement independent and approximately equal (see Table 1 and Fig. 1). System B the probability of item requirement independent and disparate (see Table 2 and Fig. 2). The probability that item i is required (p t ) is given by p(i, 1) in Tables 1 and r O I 0 5 S O Cell nof heap FIG. 2. A plot of the figures from Table 2 for items 1, 2, 3, 4, 6, 8, and 10 for the exact evaluation. System B.

10 272 P. J. BURVILLE For both evaluations the exact and approximate probabilities are given in the tables. In the figures only exact evaluations are used, and these for only some items, in the interests of clarity. The results in these tables andfiguresillustrate that (a) the approximation for p(i,ri)is quite accurate for System A (Table 1) in which the item requirement probabilities are near to the completely random value (l/n). For System B (Table 2), where the item requirement probabilities are disparate the approximation can be very inaccurate (see items 1, 2 and 3 where negative probabilities are offered!); (b) the approximation for p(i,ri)is a function of the probability p t only and takes no account of the distribution of item requirement probabilities (compare the accuracy of item 1 in Table 1 with item 4 in Table 2); (c) the monotonic character of p(i, ri) in Fig. 1 (the approximation for p(i, ri) is a monotonic function); (d) the p(i,ri) in Fig. 1 are nearly equal (to I IN) in cells 6 and 7 of the heap as is expected from the approximation determined in part (b) of the Appendix. For the disparate probabilities plotted on Fig. 2 no such tendency is detectable. The expected position of the items in the heap for the two systems are given in Table 3. TABLE 3 Expected position of the items in the heap System A Item System B The expected position of the general item in the heap is for System A and for System B. It is left as an exercise for the interested reader to compare these values with the boundary values obtained in the Appendix. The evaluation of the expected item access time for the Serial Primary Source example given above is recorded in Table 4 for the item systems A and B. This evaluation is obtained from = Pi\t i=l Ln=l

11 HEAPS 273 with t n = nt. Plots of the expected access times for systems A and B for various values of T (with t set to unity) with the full range of heap sizes are given in Figs 3 and 4 respectively. lot r=io I s o 7 o c a 5 8 4'f 0 I Number of cells In the heap (H) FIG. 3. The expected item access time for System A. The results in Table 4, and Figs 3 and 4 illustrate (a) the greater potential advantage to be obtained from the deployment of a heap with system B than with system A; (b) that for the completely random case if Nt < T, then the expected access time is a monotonically decreasing function with increasing heap size; (c) that if advantage is gained from the use of a heap, then greater advantage can be realized using a heap up to size N.

12 274 P. J. BURVILLE Number of cells In the heap IH) FIG. 4. The expected item access time for System B TABLE 4 Serial Primary Source, expected item access time evaluation System A 10/ l-899/ r 2-699/ r 3-399f r 3-998/+0-499ir 4-497f T 4-896f T 5-195H T 5-394f r 5-494/+00r Heap size (cells) System B l-0f T l-794/ r 2-405f r 2-856r r 3-173/ T 3-380/ r 3-503/+00615r 3-565f T 3-587/ r 3-590/+00r

13 HEAPS Conclusion In this article some simple HEAPS examples have been evaluated and some results of a general nature derived which may at least offer a guide in assessing many further configurations constructed to meet particular needs. A configuration of particular interest is that with a dynamic heap size. The decision function determining the heap size could produce benefit, in terms of item access time and storage capacity, both in the initial setting-up phase from an empty heap and in the longer term steady state. Such a dynamic configuration is (potentially) more accommodating where the storage capacity available varies, and to the situation frequently met where the item access probabilities, at least in the short term, are very variable perhaps with a large percentage of the items having zero access probability. W. J. Hendricks (1972) obtains the stationary probability distribution for a Markov chain based on a model having similarities to HEAPS. In his article Hendricks poses certain problems whose solutions are offered in this article and in a paper by Professor J. F. C. Kingman and the author (1973). The author wishes to thank Professor J. F. C. Kingman for his help and encouragement. The author also wishes to acknowledge the British Post Office with whose sponsorship he was able to attend the University of Sussex. REFERENCES BURVILLE, P. J. & KINGMAN, J. F. C /. appl. Prob. 10, Cox, D. R. & MILLER, H. D The theory of stochastic processes. London: Methuen. HENDRICKS, W. J /. appl. Prob. 9, KINGMAN, J. F. C. & TAYLOR, S. J Introduction to measure and probability. Cambridge University Press. Appendix The properties of the system determined in this paper are based on the existence of the limiting occupation probability that item i is in cell n of the heap [p(i,«)]. This probability exists for an irreducible ergodic Markov chain (Cox & Miller, 1968), which requires that the following conditions exist (i) the chain is irreducible that is all states intercommunicate; this is so since all permutations of item orders in the heap can be transformed into any other by the normal action of the system; and (ii) the chain is aperiodic. Both the above conditions are met and hence there are a unique set of limiting occupation probabilities. It is readily seen that the sequence of states of the heap is a Markov chain with N(N 1)(...)(N H+1), possible states, each state transforming in one realization of the system into one of N states (including no change). For the results given in the Appendix where no derivation is given the reader is referred to the Kingman & Burville paper. (a) Probability that item i is in cell n of the heap [p{i,«)] t P(i,n) = pt (-)'( N 7 + / ) (p,+p h + +p Jm _ m J = 0 \ * / t where the pj, are distinct; j x # /; N is the number of items.

14 276 P. J. BURVILLE Note that the summation is being carried out over the set of distinct items i,j\,j 2,..,,j N - n+l, in which change of item order does not alter the set. It will be seen that in all cases p{i, 1) = p h i= 1,2,..., N. The general result for p(i,ri)is not very amenable to rapid evaluation since N can be very large (even infinitely so) and it would be of interest to have the result in a form in which N does not appear explicitly. The approximation given below is easier to evaluate and by its structure can be readily built into an iterative procedure. Let then to the first order in S t, Note that Pi = jj+s t, N "~ 1 1 I N r n " 1 (b) Cell location for equal p(i, n),for all the items Consider the approximation then the condition for/>0', n) = l/n, is that either S t, or its coefficient, is equal to zero. The latter situation arises when n-l 1 V 1 Now or n-2 1 n-l I t>n-l Letting n = N(l e ), and taking limits iv -* oo, for the bounds N and Thus as the number of items in the system approaches infinity the probability of any item occupying cell A^l-e" 1 ) tends to be equal (to l/n). This characteristic is

15 HEAPS 277 illustrated in Fig. 1 where the monotonic character of the p(i, n) is also shown for the situation where the <5 f are not large. [N(\-e~ l ) = 6-32, in Fig. 1]. (c) The probability that item i is in the first m cells of the heap An approximation for this is obtained below using the approximation for p(i, n) given in part (a) of this Appendix.,., m S,mN 8 ( N " "- 1 1 and n=l " ^ l J Y - i n=lk=o m n 1 -i m 1,*»^ f'^-m, "Z^j-m C m ~ 2 x-m, Thus the lower bound for the approximation is ^+^ZTL 1 " (JV " m)l and the upper bound for the approximation is The smaller 5, is the more accurate in this approximation, (d) Mean position of item i in the heap \ji t ] Let Now and This gives an upper limit for n,. = 2-p,.

16 278 P. J. BURVILLE Thus p ' ^ **' "" Pi' It will be noted that the upper bound is only of interest if/?,- ^ l/n. (e) Expected position of the general item in the heap [ {/ij] Let \i = Efai}, then 1 AJ = 2 J i=l j = Let /7j ^ /> 2 ^ > p N. Then ^; > ^, and It will be seen that in addition to the completely random case n is equal to the lower bound when the items are allocated to the heap cells in an order based on the ordering of the respective p t (that is the item with the fcth highest probability is allocated to cell k of the heap). Thus the maximum expected position (or travel down the heap to the required item) is only twice the least possible which to be realized requires the ordering mechanism described. (f) Results for the equal probability completely random case For this case 7^ = 1/Af, i = 1, 2,..., N, and the item requirement probabilities are independent. As one might expect p(i, ri) = p h In the almost all equal case with PJPJ i = 1,2,..., N n=l,2,...,h. Pi = PI Pi = P3 = ' " " = PN = JV-1 ' n, (JV-i)(N-2X...)(N->... / P(1 ' H) = (N-2+pXJV-3+2pX..)(N-«+(n-r- P(1 ~ P) Thus, as iv -> CXD p(l,n)->xl- J P) B " 1, which is a geometric distribution. (ii) ft JV n=l (iii) H As for /ij, JV+1