Serial Correlation in a Simple Dam Process

Serial Correlation in a Simple Dam ProcessAuthor(s): Nils BlomqvistSource: Operations Research, Vol. 21, No. 4 (Jul. - Aug., 1973), pp. 966-973Published by: INFORMSStable URL: http://www.jstor.org/stable/169147 .

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Serial Correlation in a Simple Dam Process

Nils Blomqvist

University of Gothenburg, Gothenburg, Sweden

(Received May 2, 1972)

This paper deals with a dam model similar to one due to P. A. P. MORAN. It assumes that both the inflow and release of water are exponentially distributed random variables and studies the dam during its steady state. It derives the serial correlation in the sequence of water levels and compares it with previous results for a corresponding queuing system. Since knowledge of the serial correlation is essential in designing simulation experiments, the paper suggests that similar investigations be performed on other systems of standard type.

IN STUDYING systems commonly appearing in operations research, such as queuing, inventory, and water-storage systems, it is often of interest to know the

correlation structure of the stochastic process defined by the main variable of the system. This is the case particularly in simulation experiments, since the variance of the observed (steady-state) average under general conditions is approximately equal, for large n, to

var (x)H[var (x) /n] p (h() 1 )

where p (h) } are the serial correlation coefficients. In the extensive literature on systems of this kind, very few papers deal with the

correlation aspects. In fact, to my knowledge, only the single-channel queue has been studied thoroughly from this point of view (See BLOMQVIST [2,31 and DALEY [51),

although some work has been done for multichannel queues. [1 10 11

This paper carries this matter a little further. It investigates a model similar to MIORAN'S dam model."7"89' Let Un be the amount of water that flows into the dam during the nth period of time and let Tn be the demand during the same time. Furthermore, put Zn equal to the storage at the beginning of the nth period. As- sume that Zn remains constant until the end of the period, whereupon Un and -T are added simultaneously. If Z + U - Tn exceeds a, the capacity of the dam, the amount Zn+Un-Tn-a overflows, while there is a shortage if Zn+Un -Tn is less than zero. Using the general notation

ra, if x>a, (X)Oa= X, if O<x<a,

JO, if x < 0 wve can then write

Zn+l = (Zn JrUn-Tn ) o (2 )

This model differs from Moran's, which requires that the input be added to the storage before the end of the period. In Moran's model it is also assumed that Tn is a constant, say, m. It is well known that the recursive formula

7~~~U _ M) 1 ~ a--

966



A Simple Dam Process 967

holds in this case. Moran's model can thus be said to be a special case of (2). If, however, Tn is allowed to vary in a random manner, the process is no longer governed by such a simple recursive formula as (3) and the resemblance with (2) disappears.

Another reason for choosing the present model is that (2) defines a random walk with two reflecting barriers. The corresponding single-barrier random walk de- scribes a single-channel waiting-time process

W. +r1 = (W. + U. - T.) o X (4 ) where U, is the service time of the nth customer and T, is the length of the nth arrival interval. The correlation properties of (4) are fairly well known.'23'51 Proceeding from the queue model (4) to the dam model (2) should therefore make possible some interesting comparisons.

Section 1 investigates the correlation properties of the process {Zn } defined by (2) for the case when both the inflow Un, and the demand T, have negative exponential distributions. Section 2 contains a study of two limiting cases, (i) when the average inflow of water is equal to the average demand, and (ii) when the dam capacity is large and the inflow is nearly equal to the demand. The latter case is then investigated numerically in Section 3 and a comparison is made with the M/M/1 queuing system.

1. THE SERIAL CORRELATION IN A RANDOM WALK WITH TWO

REFLECTING BARRIERS: NEGATIVE EXPONENTIAL CASE

THIS SECTION CONSIDERS the random walk { Zn } defined by

[a, for Z +U,-T,, a, Zn,+l=t Zn+ U,-Tn, for O<Z.+U.-T.<a, (5)

t0, for Zn+Un-Tn-<O

during its steady state. Here, { Un } and { Tn } are independent sequences of independent random variables with negative exponential distributions, Un with parameter ,u and Tn with parameter X.

Define the bivariate Laplace-Stieltjes transform

M (01, 02, h) = E[exp (- 0Zn- 02Zn+h)]* (6) Then

11 (01, 02, h + 1) = E{ E[exp (-,Zfn-02Zn+h+l) IZn, Zn+hf I * (7)

Put X, = U- Tn and let D (x) be the distribution function of Xn. Combining (5) and (7), we get

M(01, 02, h+1)=E{exp(-,lZn) {exp(-02a)[1-D(a-Zn+h)]

+fa-Zn+ih exp[-02 (Zn+h+X)] dD (x) +D (-Zn+h) (8)

Since

D (x)={ -X/(X?,)]x, for x>O, (9)

(8) yields, after some calculations, the details of which will not be given here, the following recursive formula:



968 Nits Blomqvist

lf (61, 02, h++ 1) =[ (X -02) (i?+02)] * M (1 2, h)-

[M402/ (X + M) (x-602)] * M (01, X, h) (10)

+ [XO2/ (X +4) (+02)] exp a p+02)] -M (01, -p, h).

We now introduce the generating function

R (0i, 02, S) = Eh-0M(1 2 = ||< 1

Multiplying (10) by Sh+1 and adding from h = 0 to oo yields

R (6, 62, s)-R (6, 02, 0) = [XPS/ (X-02) (U+02)] R (62, 02, S)

- [A 02S/ (X +U ( - 02) ] * R (01, XA, s) (12)

+ [X 02S/ (X +M) (M+ 02)] * exp a (+ 02) R (01, -A, s) . Denote

A (0) = 6/ (X>-0), B (0) = [0/ (?+ 0) ]exp[-a (p + 0)]

and observe that R(01, 02, 0) =M(01, 02, 0) =S(61+02) ),where p (0) is the Laplace- Stieltjes transform of the steady-state distribution of Zn. Furthermore, let -y (0) Xju/ (A- 0) (m +0) be the transform of Xn. Then (12) may be written as

[1 - S(02)]R (01, 02, s) =(p(01+02) - [,sl(X+,u)] -A(02) R (0i, A, s) (13) +[X\s/ (X +u) ] B (02) R (01, -i, S).

The characteristic equation 1= s-y (0) has here two (real) roots. Let v be the positive and w the negative root. Putting 2= v and w in (13), we get two equations to solve for R (01, X, s) and R (01, -a4, s). If these solutions are inserted into (13), the following result is obtained:

0(61+02) p(6i+v) sp61+w) A(02) A (v) A (w)

R (1, 02, 8)= B(02) BI) B (w) (14)

We note in passing that, if in (14) a is permitted to tend to infinity while 02 is kept positive, B (02), B (v), and B (w) tend to zero. Since w is negative, while 02 and v are positive,,the terms containing B (w) dominate. Hence,

(P(01+02) ((01+v)

R (Oi) 02, 8)-- [1~(02)] A (v) ( am oo ) ( 15 ) [I1- ST(62)] .A (v)

This is the generating function of the bivariate transform of the steady-state waiting times in an MI/MI/1 queuing system [compare formula (5) in reference 21.

Formula (14) can now be used to derive (i) the steady-state distribution of Zn

and (ii) the serial correlation of the random walk { Zn.} during its steady state.

(i) The Stationary Distribution of Zn

Multiply formula (14) for R (01, 02, s) by 1-s and let s tend to 1, observing that the roots v and w are functions of s, one of which tends to 0 as s tends to 1. With-




out loss of generality, we assume that X <,?, in which case v tends to 0 and w tends to, say, w, = X-A. The left-hand side of (14) then approaches p (01)O (02), while the right-hand side tends to

{p (01) [A (02)B (wi)-A (wi)B (02)]}/

{v' (1)[1-y (02)] [A' (0)B (wi)-A (wi)B' (0)]} Hence,

p (0) = {A (0)B (wi) -A (wi)B (0) }/

{v'(1)[1--y (0)][A'(0)B (wi)-A (wi)B'(0)]} ( )

It is easily seen that v' (1) = Xj,/ (X -,4). Inserting the expressions for A ) and B(.), we get the known result [4]

p (0) = [ (8-X)/ (M-X+0)1 { [,u (gu+0)e-x _ (17) - X (X - 0)eG +)a]/[.2e Xa _X2e-pa] I )

The transform (17) corresponds to a distribution with discrete probabilities

PO= [ih (-)e )ea]/[,,2eXa _2e-pa] Pa =[X (tu - X) e a1] /[A2e a _ X2 e-iaI

and a negative exponential density with intensity ,u - X between the barriers.

(ii) The serial correlation of the stationary process {Zn

From formula (14) we may obtain the generating function of the covariances

C(h) = cov (Zn, Zn+h) (h=O, 1, . by differentiation, since

Zh:O C (h)sh= (82R/501502)01=02=0[-E (Z)/ ( -s)1. (18) We get

(1S) ?h=01 C Shs =(PI' 0)-_sp2 (O)-_S(p1 (O)_y (0)/ (1-_s)

-p' (v) . [A' (O)B (w) -B' (0)A (w)]/[A (v)B (w) -B (v)A (w)] (19)

+(' (w) [A' (0)B (v) -B' (0)A (v)]/ [A (v)B (w) -B (v)A (w)],

where the functions A (-) and B (-) are defined above and (v, w) are the roots of the characteristic equation.

Our main goal is to find an explicit expression for E C (h). This is accomplished by letting s tend to 1 from below in (19), observing that, if X <A, v then tends to 0 and w tends to w, = X -j. Some heavy but straightforward algebra yields the following formula:

Zho? C (h) = [X// (A.v-))] { {X[) 2? 2-X A]/[ (,u-X ) ]-E (Z) [E(Z2)

-E2 (Z) ] _ 1 [E (Z) E (Z2) - E (Z3 ) I- [X/A/ (I~A -) ] (20) -E (Z) I { 1[,4+)Xa (A - ) ]/[Xv (A- ) )] E (Z)}

{[a (1 +?a/2)]/[,A+ \+ua]-E (Z) } }.

The moments of Z are derived from the transform sp (0) of (17):



970 Nils Blomqvist

W2ex_2) (-X)3 . E (Z3) =6X (ex_ 1-x-x2/2)-X x,

where x = a ( -X X).

2. TWO LIMITING CASES

IN THE SEQUEL we shall restrict ourselves to considering the case when A and X are nearly equal, i.e., when the supply of water almost balances the demand. Two separate cases will receive special attention: the case when tz - X-*0, while a remains fixed, and the case when tz - X-*0 and a-> oo in such a way that the product a ( - X) stays bounded.

(i) The case of A=X

The following steady-state transform of Z for tz=X is obtained from (17) by letting ju-X---0:

sp (0) = [X+0- (X - )e0a?/[ (2+Xa)]. (22)

The corresponding distribution consists of two (equal) discrete probabilities in z = 0 and z = a and a uniform continuous distribution between the barriers.

The following expression for the generating function of the covariances is obtained from (19):

(1-s) E~h= C(h)sh=a (Xa+6)/12(Xa+2)

+ (X2_v2) . {2X (1-e-a) -av[X+v+ (X-v)e-a]}/ (23)

{X (Xa+2)V3. [X+V+ (X V)e-av]

where v=XVi -s. In this formula let s-1l; then we get

Eh=_ C (h) = a2[X3a3+ 10X2a2+4OMa+60]/[120 (Xa+2)]. (24)

(ii) The case of A - X->O and a->o

In this case, as follows readily from (17), the random variables (z- X)Z tend in probability to a limiting random variable 2 with the following transform:

, (0) = E (e-' ) = [1/ (1 + 0) I{- I [1 - e-(1+6)1]1 (1 - e-x)}' 1(25)

where x = a ( - X). This corresponds to a negative exponential distribution trun- cated at Z= x.

Similarly, putting k = X/ju, one obtains from (20), 2 Eh( C(h)-(l-k) 4[2x3 ex(ex+1)/ (e -1) ]' (26)

and, from (21), X 2var (Z ) - ( 1- k) -'[ 1_ 2 ex / (ex - 1) 1 (27 )




3. SOME NUMERICAL COMPARISONS

As MENTIONED ABOVE, if the upper barrier a in the random walk (2) is allowed to increase indefinitely, the single-barrier random walk thus obtained is identical to the sequence of waiting times { W. in the M/M/1 queue. For this system, we have (see reference 2)

2Zh-o C (h) = 2k (1-k)-4, A2var (W) =k (2-k) (1-k (28)

TABLE I ASYMPTOTIC VALUES FOR THE SUM OF SERIAL CORRELATIONS

[As a function of x=a(o-X), where a=dam capacity, M=inflow intensity, and X= demand intensity.]

x (u-_))2var (Z) (1-k)2Eh=- p(h)

0.0 0.0000 0.000

0.5 0.0205 0.025 1.0 0.0793 0.097 1.5 0.1682 0.211 2.0 0.2759 0.357

2.5 0.3911 0.525 3.0 0.5037 0.705 3.5 0.6067 0.886 4.0 0.6959 1.061 4.5 0.7700 1.223

5.0 0.8293 1.368 5.5 0.8754 1.495 6.0 0.9103 1.603 6.5 0.9363 1.692 7.0 0.9552 1.765

7.5 0.9689 1.823 8.0 0.9785 1.868 8.5 0.9853 1.903 9.0 0.9900 1.929 9.5 0.9932 1.949

10.0 0.9955 1.964 00 1 2

These formulas may be compared with the corresponding formulas (20) for the storage system. We shall here, however, restrict ourselves to a few special cases. Consider first the case when supply nearly balances demand and the dam capacity is large, or, when A -X is close to zero, a is large and x= a (,A-X) is constant. This is case (ii) in the previous section. We observe in passing that a direct comparison for Ah exactly equal to X is not possible, since no steady state then exists for the queue. Table I contains asymptotic values of var (Z) and E p (h) for different values of x. The case of x= oc corresponds to the M/M/1 queuing system.

Formula (1) tells us that the (approximate) variance of the mean z of a steady- state realization zj, Z2, *. * , z, consists of two parts, one showing the variance in the



972 Nils Blomqvist

hypothetical case of no serial correlation in the sequence IZn , and the other, E p(h), showing the influence of this correlation. In a comparison between a single-barrier random walk (queuing system) and a double-barrier random walk (dam system) one might perhaps expect that the difference in the precision of 2 should mainly depend upon the difference in var(Z). Table I shows that this is not the case; also, the factor E p (h) exhibits large variations with x. However, the ratio E p (h)/var (Z) is an increasing function of x; i.e., the influence of the serial correlation on the precision of 2 increases with the capacity of the dam, and is thus always less for the dam than for the corresponding queue.

After multiplying the two columns of Table I with each other, one observes a considerable influence of x on the values of var (Z). As an example, the precision of 2 is about 260 times larger for a dam with x= 1 than for the corresponding queue (X=oo).

TABLE 11 THE PRECISION OF AN OBSERVED AVERAGE OF WATER LEVELS {Z.) IN A FINITE DAM

WITH A=X AND OF WAITING TIMES {W,} IN THE MI/M1 QUEUE

Dam Queue

Dam capacity in units of period nCV2(Z) Load k nCV2(W)

demand Xa

0 1.00 0.1 30.4 1 1.69 0.2 22.2 2 2.47 0.3 21.5 3 3.36 0.4 23.8 4 4.38 0.5 29.0 5 5.52 0.6 39.3

. . . . . . 0.7 61.6 10 13.22 0.8 124

. .. . . 0.9 443 00 00 1.0 00

Another way of comparing the two systems is the following. Consider a dam where the average inflow is equal to the average demand, i.e., 4 = X. The corresponding (exact) formulas are given in Section 2, case (i). The left half of Table II gives the relative precision of an observed average z (CV = coefficient of variation) for different values of Xa, which is the dam capacity measured in units of the average period demand. The right half of the table contains corresponding precision values for the waiting times of differently loaded M/M/1 queuing systems. These values have been taken from an earlier paper by the author.121

The two series of values in Table II are not directly comparable, but one notices the difference in order of magnitude. The very large sample sizes (run lengths), necessary in order to obtain a given precision, that have been observed for queuing systems are in general not found for the finite dam, but they are still large. As an example, if the dam capacity is a= 9 units and the average demand is A-' =3 units, about 1340 observations are needed in order to get CV (Z) = 5 per cent. Some comparisons may be made with the simulation of a discrete dam carried out by GANI

AND MORAN.161




REFERENCES

1. V. E. BENE', "The Covariance Function of a Simple Trunk Group, with Applications to Traffic Measurement," Bell Syst. Tech. J. 40, 117-148 (1961).

2. NILS BLOMQVIST, "The Covariance Function of the M/G/1 Queuing System," Skandi- navisk Aktuarietidskrift, 157-174 (1967).

3. , "Estimation of Waiting-Time Parameters in the GI/G1l Queuing System; Part I, General Results," Skandinavisk Aktuarietidskrift, 178-197 (1968); "Part II, Heavy-Traffic Approximations," Skandinavisk Aktuarietidskrift, 125-136 (1969).

4. D. R. Cox AND H. D. MILLER, The Theory of Stochastic Processes, Methuen, London, 1965.

5. D. J. DALEY, "The Serial Correlation Coefficients of Waiting Times in a Stationary Single-Server Queue," J. Australian Math. Soc. 8, 683-699 (1968).

6. J. GANI AND P. A. P. MORAN, "The Solution of Dam Equations by Monte Carlo Meth- ods," Australian J. Appl. Sci. 6, 267-273 (1955).

7. P. A. P. MORAN, "A Probability Theory of a Dam with a Continuous Release," Quart J. Math. (Oxford) 7, 130-137 (1956).

8. ,The Theory of Storage, Methuen, London, 1959. 9. N. U. PRABHU, Queues and Inventories, Wiley, New York, 1965.

10. J. F. REYNOLDS, "On the Autocorrelation and Spectral Functions of Queues," J. Appl. Prob. 5, 467-475 (1968).

11. J. RIORDAN, "Telephone-Traffic Time Averages," Bell Syst. Tech. J. 30, 1129-1144 (1951).



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Serial Correlation in a Simple Dam Process