Magnetic tape utilization viewed as an inventory control problem

Comput.& Indus EngngVol 5, No. 2, pp 8q-9~.1981 Prinled in Great Britain

0360-8352/81020(~q-07502.0010 Per~monPressLid

MAGNETIC TAPE UTILIZATION VIEWED AS AN INVENTORY CONTROL PROBLEM KENNETH L. SLEPICKA and KHALU. F. MATTA Department of Aerospace and Mechanical Engineering. College of Engineering, University of Notre Dame. Notre Dame. IN 46556, U.S.A. (Receiled December 1980, receired for publicatiml 5 January 1981) Abstract--Magnetic tapes are extensively used to store information. However, their low cost leads to problems in the el~cient use of them as a storage media. There exists no scientific method to control the space utilization and the addition of new tapes to a tape library. A method is developed which provides control on the use of tapes, reduces the number of tapes in the library and minimizes the cost of storing information tapes. An inventory model is developed with the unused space on tapes as inventory. Replenishment can be done in one of two steps. The first deletes the expired datasets from the tapes and if the new inventory level is less than S', the least order level, a second step adds new tapes tapes to the library bringing the level up to S. the maximum order level. The level of inventory after replenishment can be anywhere between S' and S, The inventory model determines (1) When to delete expired datasets. (2) When to add new tapes to the library. No analytical solution is developed for the general cost equation for this model. However, an iterati~e procedure is ofi.ered which can be used to minimize the total cost of having unused space in the tape librao. 1, I N T R O D U C T I O N

In the operation of computing centers, magnetic tapes because of their low initial cost are used as the principal means to store large volumes of information[l]. With the increased use of electronic data processing, the amount of information that needs to be stored on tapes is growing as is the associated storage costs. In many cases this leads to the situation where steps must be implemented to minimize tape storage costs while still meeting the demand for storage of information on tapes [2--4]. Incremental tape storage space can be provided in two ways. One is to purchase new tapes; the other is to improve the utilization of the present tapes. This second method would involve storing information at a higher density-large blocks- and/or deleting information which is no longer of value. Since either approach for providing space can be used, a rule or procedure must be developed to decide when to use each method. A model which can be used to describe this situation and used to decide how to provide additional tape space is presented in Section 2. Section 3 contains the development of a general cost equation for this model. The results of applying this approach to a particular tape library and final conclusions are presented in Sections 4 and 5 respectively. 2 I N V E N T O R Y MODEL[5]

The demand for space on magnetic tape to store information can be described as an inventory system with unused space as the quantity stored. The operation of the system is illustrated in Fig. 1. The amount of unused space on tapes is reviewed on a periodic basis and if it is below a prescribed level s,-reorder point-then additional tape storage space must be provided. It is assumed that any demand for tape storage space will be met because the cost of other magnetic storage devices--drums or disks--can be considered infinite relative to tapes. Therefore sp is set at a point which will always provide enough unused space to satisfy any demand. This level is equal to the maximum possible demand during any reviewing period--%,. Replenishment can be accomplished in 2 ways. The first is by utilizing denser storage and deleting non-essential data; the second is by the addition of new tapes. The first replenishment approach would be desirable in most situations because it would minimize the total number of tapes in the system. However, this process cannot solely be used because at some point there will be only a few tapes to delete and the storage density will already be high. Therefore, a combination of both methods will be required starting with the improved utilization of current tape space. The replenishment procedure will be initialized when the level of unused space in inventory is below s,, at the end of a review period. If the first step of replenishment raises the

90

KF,,~,'f IH L. SIrr, ll K~

and

KH xl I1 1 M ~ll

Deman~ for topes

Inventory of unused space

e Iova lable

Need more space

Delete expired datasets

t Sufficient space

gained Need more Space Oraer ne~, ~aPes

]

Fig. I. Tape system opera/ion

amount of unused space on tape to an adequate level, minimum order level S', then the procedure would be terminated. Otherwise, new tapes are added to the system bringing the level to S, maximum order level. The cost of having space available for storing data on tape, C~, includes building, racks, and tape handling expenses. Because all requirements for tape space are met in this system, there are no shortage costs. The cost of replenishing unused tape space is divided into two parts: (1) Cn the cost of ordering new tapes and (2) C2,_ the cost of deleting and increasing the density of information on tapes. The C22 cost is a function of the number of tapes which effects the setup and execution of the routine to increase storage density and delete information. Also, if there is an incremental cost for CPU time, the number of retained datasets, their size and position on the tapes will contribute to this replenishment cost. The actual cost of tape, C , must also be included in this analysis. If replenishment is done by deleting information, then a savings has been achieved by not buying a new tape. Thus, the problem is one of determining the two order levels, S' and S, which will minimize the cost of unused space on the magnetic tapes. This problem is complicated by the fact that not only is the demand probabilitistic but the level of unused space in inventory after replenishment is a random variate between S' and S. Figure 2 shows a graphical representation of the model's operation.

3. COST EQUATION

The general form of the inventory cost equation is given in eqn I I) C(S', S) = CIII + (C_,1 + C2,)1~ 4 C~I~

(1)

where I~ is the amount of unused space in inventory. I_, is the number of replenishments, and I~ is the number of new tapes added to the system. To find the S and S' that will minimize eqn (1), the equation must be written in known terms. Let

Magnetic tape utilization viewed as an inventory control problem

91

Amount of unused spoce

5 S-o

m

S t +~ S'! St-u

\ sp

Time

6

q~

-

First stoge replenishment

q2 = Second stoge replenishment

Fig.

]

w~ :

Review period

t~ :

Replenishment period

Tape inventory ',ystem

Q be the amount of unused space on hand at the beginning of each review period after replenishment, if it has occurred. H ( Q ) is the probability distribution of Q, P ( D = x) is the probability distribution of the demand during a review period, P ( R = Q) is the probability distribution of the final replenishment level and P( r = Q) is the probability distribution of the inventory level after the deletion process. The amount Q = S - u can be reached in three ways. (I) The level was S and during the review period a demand equal to u occurred. (2) The level was S - u and a demand equal to 0 occurred and (3) a replenishment was required at the end of the review period which raised the level of S - u. Mathematically this can be expressed as:

H(S-

u) = H ( S - u ) P ( D = O)+ H ( S ) P ( D = u) ÷ P ( R = S - u)R'

(2)

where R' is the probability that a replenishment occurs at t~).This term is fixed and will be calculated later. Rearranging eqn (2) H(S - u)= H(S)

P(D=u)

I

P(D=O)

*

P(R=S-u)

I

PID=O)"

(3)

Similarly if Q = S - 2u H(S-

2u) = H ( S - 2 u ) P ( D = 0 ) + H ( S + P ( R = S - 2u)R'

u ) P ( D = u)+ H ( S - u ) P ( D = u)+ H ( S ) P ( D - - 2u)

(4)

which can be arranged as: H(S

2u)

H(S-

u ) P ( D = u) * H ( S ' ) P ( D = 2u) + P ( R = S - 2U)_R," 1-P(D=O) I P(D=OI

(5)

92

KENt',I-IH [.. S l t P I ( ' K ~, a n d

K H ~ I II F. M A I l.~

This can be put in the form

[ P(D-2u) P2(D = u) ,] H(S - 2u) = H(S) [ 1 - P(D = O) + (I Z p-~)--=-~))2j .,[P(R=S-u)P(D=u) (I-P(D=O)):

+K [

P(R=S-2u)]

--+ i--P]D-~

J"

(6)

If a function A is defined as y =0

A(y) =

A(y - x ) x =u

P(D=x) I - P ( D =0)

y =//,21l . . . . .

(7)

And a function B is defined as:

B(y)=

0

y=O

P(R = S - y ) ~ y) ~--p~/~-~ + B(y-x) P(D= ~=. I-P(D=O)

v =u,2u.,

(8)

Then

H(S

-

(9)

2u)= H ( S ) A ( 2 u ) + R'B(2u).

By induction it can be shown that

(1o)

H ( S - y) = H ( S ) A ( y ) + R'B(y). Hence

H(Q)

=

H ( S ) A ( S - Q)

+

(11)

R'B(S - Q).

Where

P(R=Q)=O

for s p - < Q < S '

(12)

because if the first stage of replenishment is less than S', the level of tapes will be raised to S in all cases. And S

~_,

H ( Q ) = 1.

(13)

Therefore S

H(S)A(S-Q)+

R'B(S-Q)=

I.

(14)

Q = ~p + u

Rewriting this by taking the constants out of the summations and moving the second term to the right hand side S

H(s) ~,

S

AIS-O)=I--R'

Y. B~S-O~

~1.~

Magnetic tape utilization viewed as an inventory control problem

93

Then S

~,

1-R'

B(S-Q)

Q=sp+u S

H(S) =

(16)

A(S - Q)

Q=sp*u

It is also true that H(S) = R ' x P(R = S)+ H(S)P(D =0).

(17)

Therefore: H ( S ) - R ' x P ( R = S)

(18)

l - P ( D = O)

where S-I

P(R=S)=I-

~. P ( R = Q ) .

(19)

Q=S'

Combining eqns (16) and (18) yield S

R'xP(R = S)_

l-

R'

~,

B(S = Q)

Q=sp+u S

1 - P ( D = O)

~,

(20)

A ( S - Q)

Q=so+u

Solving for R' R'=

P(R = S)

l

~

I--ND-0)

s ~_, B ( S - Q )

A(S-Q)+

Q=st,+u

(21)

Q=sp *u

Since R', H(Q) and H(S) are all defined in known terms, I~, the average amount of inventory on hand is equal to: s

I,= ~

Q=~v +u

(22)

QH(Q).

The number of replenishments during a period of time can be derived by: 12 -

R' wp

-

s ~,

[ P(R = S)

I A(S-Q)+

s ~,

B(S-Q)]

.

(23)

And the expected number of tapes ordered during a replenishment (I~) is: S' I

QP(r = Q) 13 =

S-

Q=o s'-I

Re

P(r < S') - - . P(r = Q)

14;p

(24)

O=o

Where P(r = Q) is the probability of having a level Q after the first stage of replenishment. Equations (22)-(24) can then be placed in eqn (I) yielding an expression which is a function of S, S', and the probability distributions of demand and replenishment. An iterative technique CAIE Vol. 5. No. 2--C

KENNETH L..~I.EPICKA and KHAI.I[. F. MMqA

94

can then be employed to determine the minimum cost of this equation. A computer routine can be used to systematically vary S' and S for a specific given demand distribution and replenishment distribution. The total costs can then be plotted as a function of S' and S and the minimum easily identified. 4. A N E X A M P L E

This inventory procedure was applied to the tape library of the Academic Applications Group at the Notre Dame University Computing Center. The demand was studied for a 4 year period. This provided the monthly demand distribution shown in Table I. The number of tapes replenished by the deletion process was assumed to follow the probability distribution in Table 2. The costs associated with providing unused space on tapes at the Computing Center were taken as: inventory cost C~ = $0.025/month-tape, ordering cost C2~ = $25/order, and the cost of tapes C3 = $1 l/tape. The exact cost of deleting tapes was not known due to the lack of operating experience with the procedure and whether there would be a charge for CPU time. Therefore a range of deletion costs were investigated. The results from this study are in Table 3 along with the optimum policy if deletion is not used [6] and the current practice. The primary observation from these runs is that the optimum cost for the two stage replenishment policy is a function of the deletion cost. The decision to use this approach instead of only buying tapes is dependent on what is charged to delete datasets. When the cost of deletion is the same as ordering, the minimum cost is nearly the same for both policies-deleting datasets or only buying new tapes. However it should be pointed out that although the inventory costs are the same, the total number of tapes in a library will be smaller when deletion is used. This would be an additional cost saving not reflected in the inventory model. It should also be mentioned that replenishment distribution probably will not be the same for each of the cases investigated. As the cost of deletion increases the frequency of the process will decrease. This means that most likely more datasets will be deleted when the tape deletion program is run. The net effect would be a reduction in the overall inventory cost because the deletion process would be more cost effective.

Table 1. Distribution of demand (monthly) for computer tapes

Fable 2. Distribution of replenishment by deletion Number of tapes replenished by depletion Frequency

Demand for tapes

Frequency

P ( D = 3) P ( D = 2) P ( D = 1) P ( D =01

0.146 0.146 0.270 0.438

. . . . . . . . . . . . .

P ( r ' 3! Per ::~ P(r 4) P e r : 51

Per - 6 )

H4(I I).l~

015 II.l~

0.15

Table 3. Inventory results Run number

Inventory

Ordering

Deletion

cost-C~

cost-C2~

cost-C22

Tape cost-C~

/S/order)

($/delet.)

(S/tape)

$/tp.-mo.)

Minimum

order leveI-S' (Fapes)

1 2 3 4 5* 6;

0025 0.025 0.025 0.025 0.025 0.025

25 25 25 25 25 25

Optimum policy without deletion. 'Current computing center policy

25

II II

50 150 --

~1 II It

--

II

10

4

Maximum order leveI-S' ('rapes)

.\~ erage m'.entor}-l, (Tapes)

Number of replenishments I: (replenishmentslmo.)

44 76

12711 21e,33 31488

0042 0.024 1~l)16 0.009 .',-, 00__

4 4

Ill

4

195

<5.014

47 12

.,-z . . . :~ . ~,z; "-4.qS0

0094

Magnetic tape utilization viewed as an inventory control problem

95

5. CONCLUSIONS

The deletion of expired datasets as a means to generate additional tape storage space can be an attractive approach if an optimum policy is followed. Questions that remain to be answered are (1) What is the sensitivity of this system with regards to changes in the relevant cost and the probability distributions of demand and replenishment, and (2) What would happen to the minimum cost if a choice was permitted as to whether the deletion algorithm was run when replenishment was required. This would provide a method which would delete datasets only when a significant increase in space would be gained. If this was not the case then new tapes would be ordered. This could possibly be a better policy depending on the cost of deleting and ordering. R'EFERENCES I. C. Holder, Caring for magnetic media. Data Management (Oct. 1977). 2. D. Grossman & H. Silverman, Placement of records on a secondary storage device to minimize access time. J. Association of Computing Machinery (1973). 3. C. P. Browne and D. F. Ford, Cost analysis and simulations procedures for the evaluation of large information systems. American Documentation (1964). 4. Gecsei & J. A. Lubes, A model for the evaluation of storage hierarchies. IBM Systems J. February 11974). 5. K. R. Matta, A two stage replenishment inventory control policy for computer tape management. Dissertation, University of Notre Dame (1980). 6. E. Naddor, Inventory Systems. Wiley, New York 11966).