IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-1, NO. 4, OCTOBER 1971

Finally, sketches were made of the characters which wereincorrectly identified. A sample of 24 of these, from a totalof 56, is shown in Fig. 8. Two 5's contain full stops. Twosquashed patterns, a 6 and a 9, were thought to be zeros.An extension of the preprocessor to enable angularities tobe described may be the best way to deal with such mistakes.Many of the patterns are badly distorted and are usuallyeasy to reject. This recreation method has been found to bea useful tool to identify the deficiencies of the system andto plan its improvements.

VIII. CONCLUSIONS

A complete system for handwritten numeral recognitionhas been evaluated. The preprocessing part of the systemhas been designed as a hardware device. The classifyingsystem employs a simple statistical decision. The form ofthis is such that it might easily be turned into a set of lineardiscriminant functions and realized as a parallel network,giving a high speed for the total system. Methods used inthe study can be interpreted visually, and this facilitatesrapid improvements to the system.The scheme shows promise for extension to more dif-

ficult problems in character recognition. A useful step willbe to associate a set of edge segments which come from thesame region of constant density. In this way several charac-ters within the same picture field would be identifiable.Preprocessor extensions to the descriptions of angularitieswould be useful in that many nonnumeric pattern analysis

schemes (e.g., [10]) use angularities as their starting points.This would open up a new type of decision logic to comple-ment the existing statistical scheme.

ACKNOWLEDGMENT

The authors would like to thank the Post Office of GreatBritain for providing the raw data used in the experiment.

REFERENCES[1] W. H. Highleyman, "An analog method for character recognition,"

IRE Trans. Electron. Comput., vol. EC-10, Sept. 1961, pp. 502-512.[2] R. Bakis, N. Herbst, and G. Nagy, "An experimental study of

machine recognition of hand-printed numerals," IEEE Trans.Syst. Sci. Cybern., vol. SSC-4, July 1968, pp. 119-132.

[3] H. Genchi, K. Mori, S. Watanabe, and S. Katsuragai, "Recogni-tion of handwritten numerical characters for automatic lettersorting," Proc. IEEE, vol. 56, Aug. 1968, pp. 1292-1301.

[4] E. C. Greanias, P. F. Meagher, R. J. Norman, and P. Essinger,"The recognition of handwritten numerals by contour analysis,"IBM J. Res. Develop., vol. 7, Jan. 1963, pp. 14-21.

[5] T. Marill and D. M. Green, "Statistical recognition functions andthe design of pattern recognizers," IRE Trans. Electron. Comput.,vol. EC-9, Dec. 1960, pp. 472-477.

[6] R. G. Casey, "Moment normalisation of handprinted charac-ters," IBM J. Res. Develop., vol. 14, Sept. 1970, pp. 548-557.

[7] C. K. Chow, "An optimum character recognition system usingdecision functions," IRE Trans. Electron. Comput., vol. EC-6,Dec. 1957, pp. 247-254.

[8] B. J. Vieri and M. J. Gollifer, "An on-line television camera," inIndustrial Measurement Techniques for On-line Computers, Inst.Elec. Eng., Conf. Publ. 43, June 1968.

[9] G. Sebestyen and J. Edie, "An algorithm for non-parametricpattern recognition," IEEE Trans. Electron. Comput., vol. EC-1 5,Dec. 1966, pp. 908-915.

[10] W. F. Miller and A. C. Shaw, "Linguistic methods in pictureprocessing-a survey," in 1968 Fall Joint Computer Conf.,pp. 279-290.

A Stochastic Approximation MethodNARESH K. SINHA, SENIOR MEMBER, IEEE, AND MICHAEL P. GRISCIK, MEMBER, IEEE

Abstract-A new algorithm for stochastic approximation has beenproposed, along with the assumptions and conditions necessary forconvergence. It has been proved by two different methods that thealgorithm converges to the sought value in the mean-square sense andwith probability one. The rate of convergence of the new algorithm isshown to be better than two existing algorithms under certain conditions.Results of simulation have been given, making a realistic comparisonbetween the three algorithms.

I. INTRODUCTIONS TOCHASTIC approximation may be defined as a

scheme for successive approximation of a sought quan-tity when the observations involve random errors due tothe stochastic nature of the problem. It can be applied to

Manuscript received December 21, 1971; revised May 9, 1971. Thiswork was supported by the National Research Council of Canada.N. K. Sinha is with the Department of Electrical Engineering,

McMaster University, Hamilton, Ont., Canada.M. P. Griscik is with Procter and Gamble of Canada, Ltd., Hamilton,

Ont., Canada.

any problem that can be formulated as some form ofregression in which repeated observations are made. Com-pared to conventional methods such as maximum likelihoodestimation it has the following advantages.

1) Only a small interval of data needs processing.2) Only simple computations are required, even when the

actual functional dependence of the regression function onthe parameters of interest is nonlinear.

3) A priori knowledge of the process statistics is notnecessary, nor is the detailed knowledge of the functionalrelationship between the desired parameters and the ob-served data. The only requirements are that the regressionfunction satisfy certain regularity conditions and that theregression problem have a unique solution. The methodcan be made asymptotically efficient when the a prioriknowledge concerning the statistics and the functionalrelationship is available.

Applications of stochastic approximation algorithms havebeen proposed in adaptive and learning control [1], system

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SINHA AND GRISCIK: STOCHASTIC APPROXIMATION METHOD

identification [2], adaptive communication, and patternrecognition [3].Major contributions to the area of stochastic approxima-

tion have been made by Robbins and Monro [4], Kieferand Wolfowitz [5], and Dvoretzky [6]. Essentially theRobbins-Monro procedure finds the root of a regressionfunction whose form is unknown but which can be observedand sampled. The Kiefer-Wolfowitz procedure is a methodfor finding the extremum, maximum or minimum, of aregression function, given only pertinent random observa-tions. Dvoretzky formulated and proved a general theoremto this effect, showing both convergence with probabilityone and in the mean-square sense for iterative solutionalgorithms convergent on regression functions obtainedfrom random observations.

Dvoretzky's work represents a major contribution to themathematical structure of stochastic approximation theory.Practical applications, however, are concerned with therate of convergence and their dependence on the parametersof the recursive solution algorithm. In the area of controltheory two stochastic approximation algorithms have beendeveloped and investigated by Fu et al. [7]. Further im-provement in the rate of convergence is desirable to reducethe number of iterations and the overall time to get a resultwith a given level of confidence.The objective of this paper is to show the development of

a new algorithm which has a considerably faster rate ofconvergence than the two proposed in [7]. It will be shownthat the new algorithm converges to the true value regardlessof the starting point. The conditions and limitations on theconvergence will be made explicit in the proof. Havingproven convergence, comparison will be made between thetwo existing algorithms and the new algorithm.

II. GENERAL STOCHASTIC APPROXIMATION PROCEDURE

A general algorithm of stochastic approximation of theform

Xn+1 = Xn + yn+ 1{f(rn+ 1) - ;n n = l,2, (1)

is said to be of the Dvoretzky type, where

i~cn nth estimate of x;s true value of the parameter being estimated;rn nth observation taken and used to calculate a

function;r^n+ 1 superscript T denoting transposition, = [r1,

r2'** ,rn+ I]T;f(rn + 1) scalar functional of rn + 1; andYn gain sequence.

It is required that the functionf(rn) be such that its expecta-tion

E[f(r,)] = x.

The first algorithm in [7] uses

f1(r.) = r.

where the observations

rn = x + Qn

(2)

(3)

Q, being a random component of zero-mean noise, and

Tn = 1/(n + cx) (5)where ac is constant. The selection of the constant cx isarbitrary, but if a priori statistics are known and the processis known to have a noise component of finite variance, then

a = (6)

where u2 is the variance of the distribution and 1o2 is theinitial value of the expected mean-square error. Selectingy,, and ac as in (5) and (6), respectively, gives the algorithm(3) the best convergence.The second algorithm in [7] uses

I n

f2(r.) =- r(7n i=l

where the samples rn are of the same form as (4), but thegain sequence is given by

Yn =n

n(n + 1)/2 + cx (8)

for the best convergence, where oc is again dependent upona priori statistics and defined by (6).The convergence of both the algorithms has been proved

[7] using Dvoretzky's theorem.

III. THE NEW ALGORITHM

It is proposed that the functionf be chosen as

f3(rn+1) = IRr +i(l)11/2 (9)

where krn+, (1) is an estimate of the sample autocorrelationfunction of the samples r1,r2, ,r +, with 1 - 1 < n,and where IRrn+ (l)I is the absolute value of r,+ i(l).The convergence of this algorithm will now be proved.

An alternative proof based on Dvoretzky's theorem isgiven in the Appendix. From the definition the estimate ofthe sample autocorrelation.

R,n(l) = E[rnr.+1]and recalling (4), one may write

RrA(l) = E[(x + c.)(X + 'n+ I)]

= X2 + IRAO(l)I

(10)

(1 1)

since x is a constant and n,, and (n+ are elements of azero-mean random process. Hence (1) may be written as

n+l n + yn+l{[X + tIRk+1(l)I]1/2 -_ Xn (12)

Subtracting x from both sides of (12) and rearranging terms,

(Xn±i - X) = (1 - + 1)(n - X) + Yn+l,n+1 (13)where

frn+l = x 1 + R +1(1)1]'/2 - . (14)

If (13) is iterated, then an expansion can be developed in(4) terms of the initial mean-square error. After n + 1 itera-

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tions, the following equation is obtained:

(Otn+l - x) = ( - Y.+ 1)(l - Y (l - Y2)( - Yi)

(Co x) + (1- Y,+,1)(, Y.)..

( - Y2)YIOI' + (I - Yn+ 1)(l -Yn

( - Y3)Y2A2 + + (1 - Yn+l)Ynl//

+ Yn+ lI0n+ 1 (15)Equation (15) may be written into the following closed

form:

(in+ I - x) =(n+ I nn+ I(fl+l (1 - yi)} (_o - X) +

X+1

i= I i= I

n+ 1H (I-yj) (16)

j=i+ I

where all void products are taken as unity. Squaring (16)and substituting for t i from (14) gives, after some re-arrangement,

(tn+1 X){+12

= (.o X) (1 -i = I

n + 1 n + I 2+x2 E Yi nl O yi=1X =i+,

Note that (21) is valid not only for white noise, but also forcolored noise if / is taken sufficiently large.

Therefore, for the mean-square error to become zeroregardless of the starting value and for the algorithm toconverge, it is necessary and sufficient that

n+ Ilim H (1 - yi) -. 0.n-'coo i= I

(22)

The selection of a y-sequence that satisfies (22) is arbitrarywithin the restrictions of the limit given in the preceding.One may therefore try to select the sequence in such amanner as to maximize the rate of convergence of thealgorithm. An attempt to obtain the optimum y-sequenceusing the discrete maximum principle did not prove fruitful,however, due mainly to the fact that the resulting equationswere quite formidable, and also a knowledge of the initialerror as well as some statistical properties of the noise wererequired.

It was therefore decided arbitrarily to select two y-se-quences which appeared good and satisfied (22). These are

Yn = I/n (23)

and

Yn = I - (I/n). (24)

[n+ I n+ 1+ E Yi{X2 + IRA4(I)I}1/2 H (1i= 1 j=i+ 1

)]It will now be shown that the new algorithm (9) provides

an unbiased estimate. By definition, an estimator 5in of xis unbiased if

{n+ I

+ 2(Go - x) H ( - Yi)~n+ I n+ 1

yiy{x2 + IA4(j)I}1/2 H (1 yj)= 1 nj=i+ 1

n+Ii1n n+l

2x(O _x) H (l- Y); E Yi H yi)i= 1 i=1 j=i+ 1

{n+ I n+ I-2X Y,Yi Hn lyi)Jn+1 n+ 1

yi{x2 + (l)I}1/2 H (iU- ) * (17)j=i+ I

In general the autocorrelation function of the noise may

be written as

k<k(l) = E[Xkik+X] E o12e (18)

where III represents the magnitude of I. As the spectrum ofthe noise becomes wider, i.e., as it approaches white noise,the autocorrelation function becomes an impulse,

R4kk/= a2 3(l) (19)

and for I > I,

R4k(l) -. 0. (20)

Hence, for > 1, (17) reduces to

n+- 2

(Rn+1 _ x)2 = (X X0)2 f]i (I- Yi) .(21)

E [Snj = x. (25)

Again using the assumption that the noise has a fairly widespectrum, for large I, one may approximate (11) as

rn+ I(l) = x2. (26)

Using this approximation in (16) and taking the expecta-tions of both sides, it is seen that

E[n + I- x] = 0. (27)

Thus it can be said that with a choice of the y-sequencesatisfying (22) and sufficiently large 1, the algorithm con-verges and gives an unbiased estimate of x. In practice (iis often a white-noise sequence, and it may be permissibleto make I as small as 2 or 3.

IV. COMPARISON OF THE THREE ALGORITHMS

The rate of convergence of the various algorithms woulddepend considerably upon the initial guess, as well as they-sequence selected for each case. One way to make a faircomparison would be to plot the normalized expectedmean-square error as a function of the number of iterationsn. This is particularly convenient as an analytical expressioncan easily be obtained for this quantity for each of thethree algorithms. It has been shown [7] that for the firstalgorithm of Fu et al., the expected mean-square errorafter n + I iterations V, , is given by

vn+ I = (1 - Yn+1)2 + Yn+202 (28)

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SINHA AND GRISCIK: STOCHASTIC APPROXIMATION METHOD

2[A2n-Y

4.curve, f(i) Vfn

____- I/n

_._ 2 I/n

3 1- I/n

.......... 3 I/n

A.R2

I"-13

S.S.E.

2 3 4 5 6

Fig. 1. Normalized expected mean-square error as function of n.(a2 = 1)

where a2 is the variance of the noise. Hence a curve of VJ,may be plotted against n, assuming that V0 and a are

equal to one. This would be the plot of the normalizedexpected mean-square error against the number of iterations.The expected mean-square error for the second algorithm

of [7] is given by2 (1 - 1+)21K2 + y ia2/n)2 (29)

1. _

100

2-

10 --

Io -~

0

105100

20'0

200

103 -

T.S.S.E. _ _ _102 =

10

0- 260

Fig. 2. AP, SSE, and TSSE as function of n for A = 1.50. - - - -firstalgorithm of [7]; --second algorithm of [7]; new algorithm.

The expected mean-square error after n + 1 iterationsas a function of the initial error for the new algorithm isobtained from (21) and may be written as (subject to theapproximation involved):

{n+ 1 2

?1+1 vo Il (1 - i)i= 1

A plot of these three equations against n for the y-sequence

Vn-=1/n is shown in Fig. 1. In addition a plot for thesequence yn= - (1/n) is also shown for the new al-gorithm. It will be seen that the performance of the new

algorithm is better than of the two older ones, particularlyif the sequence Yn = 1 - (1/n) is used. It may be notedthat for the first two algorithms the sequence Vn= 1/n may

be regarded as the optimal in the absence of a prioristatistical information.

V. SIMULTANEOUS RESULTSTo test the relative effectiveness of the various algorithms

a simple stochastic approximation problem was simulatedon the CDC 6400 digital computer at the McMasterUniversity Computation Center. A zero-mean pseudo-random noise sequence Xn with a given value a of thestandard deviation was generated and added to a constantx in order to obtain the set of observations rn as definedin (4). Each of the three algorithms based its sample in-formation on these rn at every n and calculated an estimate

the first estimate being taken as the first sample rl.The calculations were made for different values of the rms

4-

A. P

2-

(30)

0

3103

102

S.S.E.

10 -.

A

iK

100 200

(/./

100 200

104n10

T.S.E.

10

0 100 200

Fig. 3. AP, SSE, and TSSE as function of n for A = 3.0. first

algorithm of [7]; -second algorithm of [7]; new algorithm.

.2

.1

I li

I/

*i -

A Y -1 -_- _

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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, OCTOBER 1971

1000-

4 -

A. P

2 -

0

05

104

i3S.S.E.

102

10

0

10 -1

0410-

103TSS.E.

102

10

-

100o-

100 200

.S.E.

10-

II

00 2601 -

/r

100 200

Fig. 4. AP, SSE, and TSSE as function of n for A = 2.0 ---- firstalgorithm of [7]; - -second algorithm of [7]; new algorithm.

noise-to-signal ratio A, defined as

A = alx (31)

and from these the approximation trajectory (AP), i.e., thepath of the successive approximation xflh, normalized withrespect to x, was calculated for each algorithm. To obtainsuitable measures for comparison, along with these werecomputed the sample-square error (SSE) and the timesample-square error (TSSE) defined as

1nSSE = E(i-x)2 (32)n i=i

and

TSSE = - i(.k- x)2. (33)n i=i

For I = 20, y,, = l/n, and A = 1.5, the AP, SSE, andTSSE for the three algorithms are shown in Fig. 2, andthese have been repeated in Fig. 3 with A = 3. Fig. 4shows a comparison with the y-sequence Yn = 1 - (I/n)using A = 2, and I = 10. It will be seen that the resultsobtained from the new algorithm are better in each case.

It should be noted, however, that for both y-sequences,there is a noise level, that is, a value of A at which the newalgorithm has the same sum of square errors as the secondalgorithm in [7]. This value of A is called the value ofequi-utility for the two algorithms being compared. Ev-idently, this depends on the y-sequence chosen, as well asthe values of I and n. Fig. 5 shows the plot of the SSEagainst A for 1 = 10, n = 100, and the two y-sequencesmentioned earlier. It will be seen that the new algorithmis better for values of A less than or equal to 3.

0

0

0

0

11

+. + + + . 0

iV.l I I

0 1 2 3 4

AtxFig. 5. Sum of square errors (SSE) as function of A . + ++

second algorithm of [7]; . . . .new algorithm with r-=I /n; o o o onew algorithm with r, = 1- I/n.

VI. CONCLUSIONA new algorithm has been proposed for stochastic

approximation, and its convergence has been established.It has been shown that the normalized mean-square erroras a function of the number of iterations decreases at amuch faster rate with the new algorithm than with theexisting algorithms. Results of simulation indicate that thenew algorithm gives a smaller sum of square errors thanthe second algorithm in [7] as long as the rms noise-to-signal ratio is less than 3.

APPENDIXALTERNATIVE PROOF BASED ON DVORETZKY'S THEOREM

Since the new algorithm is of the Dvoretzky type, theproperty of convergence can be proved using his theorem.First a statement of Dvoretzky's theorem will be given,outlining the conditions and the results. Then it will beshown that the algorithm satisfies all the requirements ofthe theorem.

TheoremLet an, fin, and yn, n = 1,2,, be nonnegative real

numbers satisfying the following conditions:

lim an = 0

noo

1: fin < (x)7n= 0

;, Yn = 00.n=l1

(34)

(35)

(36)

I'l

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Let 0 be a real number and T7 be measurable transforma-tions satisfying

IT,P(p1,p2,. ,p,,) - 01 < max [aj,(l + f3n)Ip0 - 01 - y.](37)

for all real P1,P2,. * ,p,,. Let X1 and Y, be random variablesand define

X"+ 1 = Tn[X1,X2,.* .,Xn] + Yn (38)

for n > 1, satisfying the conditions

E[X12] < xc00

, E[fYy2] < xc

E[Yn X15X2, * Xn,] = 0

with probability one. Then as n -+ oc, X,, approaches 0 withprobability one, and also in the mean-square sense, that is,

P[lim Xn = o] = 1n-oo

lim E[(Xn- 0)2] = 0.n- oo

(39)

(42)

(43)

An extension, six generalizations, and two corollarieswere also proved by Dvoretzky, but they will not be neededhere and hence have not been stated.

Identifying T7n(p, ,p,,) with (1 - yn+ 1)WWn and pn withW1, condition (37) would be satisfied if

(1 - Yn+ 1)l WnI < max [PXn,( + f3n - yJ)IWn;]. (51)(Although (51) is slightly different from (37), it has beenshown as a stronger condition in one of Dvoretzky'sgeneralizations.)

Since 1 - Yn+1 is positive for all n for both the y-se-quences specified in (50), the inequality (51) is equivalent to

(P.n + Yn+I - vn)IW.I .t 0

and, it is always satisfied by the inequality since(40) K2 + K + 2nK + n

(41) nn(n+K +1)(n +K) >

(52)

(53)

when

1=n=

n + K

and

fln+ Y+ n-" ~ 1

I1 > 0 (54)

n2 n + K n + K + 1

when

Application to the New AlgorithmFrom (13) in Section III the new algorithm may be written

as

(;*n+ 1- x) = (1 - On+1l)(Gn - x) + y,n+1in+1 (44)

where, as in (14),

= {X2 + R 1(l)1}1/2 - x.

Consider the transformation

;2n - x = Wn.Applying this transformation to (45) gives

Wn+1= (1 - Yn+l)Wn + Yn+/lIn+ (

Equation (47) is a zero-seeking algorithm and is analogousto the case when 0 = 0 in (42) and (43). Thus, if the con-ditions of the theorem are satisfied by (47), the convergenceof Wn + to zero with probability one as n -+ oo impliesthat in will converge to x in the same manner as well as inthe mean-square sense.

Let

in = 1/n (48)3= I1/n2 (49)

1

n+= or 1- 1n + K

(50)

where K is a nonnegative real constant. It can easily beshown that these sequences satisfy the conditions (34), (35),and (36), respectively.

Ynn + K

Now, identifying (38) with (47) yields

Yn = (n+4 In+l

where kn+ 1 was defined in (45). Hence

(55)

(45)=y2.= . [{x2 + A+1(1)I}1/2 - X]2

= yn2[2x2 + IRkn+,(l)I - 2x1x + Ik,4n+i(1)1]. (56)(46) Therefore,

00 %0

E[Y,E2]=E[Yn2n=l n=l

= E E Yn2{2X2 + IR Cn+l,(0

- 2x1x2 + f4n+1( )1}] (57)

As x is a constant, and as RXn+4 (l) is finite (except when1 = 0) and decreasing with 1, and because of the nature ofY,,, it can be shown that the right-hand side of (57) is finite.Thus condition (40) is satisfied.

Also, from (55) and (45)

E[Yn X1,X2, ... Xn] = yn, E [{x2 + IR4n+,(l)1}"2 - x].(58)

If 1 is taken sufficiently large, R",,4+,(l)l -+ 0. Hence theright-hand side of (58) approaches zero, satisfying condition(41) with probability one. Thus all the conditions of the

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theorem are satisfied proving that the algorithm convergesto the correct value in the mean-square sense and withprobability one.

REFERENCES[1] J. Sklansky, "Learning systems for automatic control," IEEE

Trans. Automat. Contr., vol. AC-Il, Jan. 1966, pp. 6-19.[2] G N. Saridis, Z. Z. Nikolic, and K. S. Fu, "Stochastic approxima-

tion algorithms for system identification, estimation and decom-position of mixtures," IEEE Trans. Syst. Sci. Cybern., vol. SSC-5,Jan 1969, pp. 8-15.

[3] D. B. Cooper, "Adaptive pattern recognition and signal detectionusing stochastic approximation," IEEE Trans. Electron. Comput.

(Corresp.), vol. EC-13, June 1964, pp. 306-307.[4] H. Robbins and S. Monro, "A stochastic approximation method,"

Ann. Math. Statist., vol. 22, 1951, pp. 400-407.[5] J. Kiefer and J. Wolfowitz: "Stochastic estimation of the maximum

of a regression function," Ann. Math. Statist., vol. 23, 1952,pp. 462-466.

[6] A. Dvoretzky, "On stochastic approximation," in Proc. 3rdBerkeley Symp. on Mathematical Statistics and Probability, vol 1.Los Angeles: Univ. of California Press, 1956, pp. 39-56.

[7] K. S. Fu, Z. Z. Nicolic, T. Y. Chien, and W. G. Wee, "On thestochastic approximation and related learning techniques,"Purdue Univ., Lafayette, Ind., Rep. TR-EE66, Apr. 1966.

[8] Y. T. Chien and K. S. Fu, "On Bayesian learning and stochasticapproximation," IEEE Trans. Syst. Sci. Cybern., vol. SSC-3,June 1967, pp. 28-38.

A User-Oriented Evaluation of a Time-SharedComputer System

SAKARI T. JUTILA, MEMBER, IEEE, AND GIORA BARAM, STUDENT MEMBER, IEEE

Abstract-Criteria for the evaluation of computer systems aretraditionally computer science oriented. This exploratory investigation isconcerned with evaluations of users' satisfaction with a time-sharedcomputer system. The first part of the investigation is an experimentindicating that the type of programming language, i.e., Basic versusFortran, is the relevant significant factor in determining the learningrates for the use of the system. The second part of the investigation isdevoted to the study of an additive users' utility function. The variablesof this function are the waiting time in the queue for a time-sharedcomputer terminal and the users' problem solving time (users' turn-around time at the terminal). It was found that users became rapidlydissatisfied if these times exceeded 10-15 min. A probability densityfunction for the users' turnaround time at the terminal was obtained in thesetting of an industrial laboratory. It turned out to be nearly exponentialwith a mean of about 12 min.

INTRODUCTIONTHERE HAS BEEN extensive research to develop

techniques for the evaluation of computer and informa-tion systems and to set up criteria for the selection of suchsystems. The main emphasis, however, has been in thedirection of computer sciences and not in the direction ofusers' satisfaction with various computer systems [I]-[24].Relatively few recent articles [22]-[24] start divertingfrom a pure computer science approach. Sackman andGold [22] emphasized the lack of experimental methods formeasuring human behavior in man-computer communica-tions: "experimental skills, statistical know-how, the

Manuscript received July 6, 1970; revised May 10, 1971. This workwas supported in part by the General Electric Co.The authors are with the Department of Operations Analysis,

College of Business Administration, and the Department of IndustrialEngineering, College of Engineering, University of Toledo, Toledo,Ohio.

mysteries of human learning and motivation, and thevagaries of human individual differences are not part of thelegacy of the computer sciences."

Lesser and Ralston [20] explain the reason for developinga regional computation center, including a time-sharedsystem, on a brute force trial and error process, rather thanon the basis of a comprehensive experimental study:"there is little knowledge about the operation and manage-ment of such a service, with regard in particular to eco-nomics, reliability, customer acceptance, and variety andnature of user access." One might add that this situation isstill a typical one.

Recent studies by Nickerson et al. [25] and Carbonel et al.[26] treat the man-computer interaction from the users'point of view. However, no specific empirical results aregiven. Independently from the aforementioned investiga-tions, this exploratory study presents actual empiricalresults from such a point of view.

EXPERIMENTAL DETERMINATION OF LEARNING RATESIN USE OF TIME-SHARED COMPUTER SYSTEM

An important aspect of users' satisfaction with anycomputer system is the rate at which they learn to use thesystem effectively. If the process of learning is slow andtedious, users tend to feel dissatisfied. If the learning processis rapid and relatively uncomplicated, the users tend to feelsatisfied. An experimental design was developed to detectthe predominant factors affecting the users' learning ratesfor a particular type of application utilizing a time-sharedcomputer system. Thereafter, learning curves of users wereestimated in terms of the predominant factors affecting thedesirability of a time-shared computer system.

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