A generalization of a problem on transfers

A CE.NERALlZATiON OF A PROBLEM ON TRANSFERS* A. M. TOMASHPOL’SKII Moscow (Received

A RECURRENCE interval

of one of the signals

We consider

“not transfer’

Snot transfer”

of which is a multiple

an adder of sm stages

without cyclic

be formed and let the propagation

marked) used for the formation

be generated

of each marked place. instant of beginning =transfeS,

transfer,

of these

from the outputs of the m-th, 2m-th,. . . , sm-th places addition,

of the

at the output of all the

working in a

of both terms advance

of such an adder let the signals

place of the adder occur after time r. Let the signals will be called

function

of the given number m.

with base r, in which all the digits

At each place

simultaneously.

the distribution

of a summation to the instant of the %ransfer”,

of an adder, the subscript

system of computation

1967)

is found, describing

of time from the beginning

appearance stages

relation

7 Februav

“transfer”,

signals

through one

“transfer”,

“not transfer”

of the adder (these

of a signal

if and only if there is one of these

places

about the ending of the signals

at the output

We denote by t (r, s, m, r) the interval of time from the the summation to the appearance

‘not transfer”

of one of the signals

at the outputs of all the marked places.

The problem we are considering

is the discovery

of the distribution

function

of this quantity. This problem arises an addition

because

real circuits

often have considerable

number of circuit

inputs,

forming signals

delay 0, which generally

that is, with increase

about the ending of increases

with the

in the number of marked places.

Since the time required to obtain a signal about the ending of a summation, n-th place of an asynchronous

adder is estimated

T=t(r,[ $],m,T)+@([i]) l

Zh. vy’chisl.

Mat. mat. Fiz. 8, 1, X7!-244, 1968.

347

in the

by the expression

(1)

A. hf. Tomwkpal’sk.ii

848

( C8/‘naldenotes the integral

part of n/A,

for small valacs

of m, for example,

if

m = 1, the signal about the ending uf the actual instant when the enmmation ends. In these cases it is advisable to choose m in such a way that the quantity @ ( L&/ml 1 exceed instant

of the appearance

by a negligible of the signals

amount the interval

ltransfe?,

of time from the

=tlot transfer”

at the output

af all the marked places to the actual endiag of the summation; in the worst case this interval equals
choice

of the qnautity m and of an estimate

in asynchronous adders of this type, knowledge the quantity t (r, s, m,r) is necessary. We denote by F Ck, r, s, m 1 the probability reasoning similar to that of IIlI, we obtain

for the quantity

of the distribution

function

T

of

that t(r, s, m, r) < k; r. Then by

where P (xc, r, s, m) is the probability of the event U, consisting of the fact that in the snr-place adder the quantity t (t; ‘s, m r) ‘> k r, while in the (s - 1) mmplace adder obtained s

-

from the *sn-place

adder by deleting

the m leading

places,

of the incompatible

events

c (t,

1, m, rI,
Ui tk,~i.$min(k+m-l,sml)), each of which if i 4 .sm - 1 consists of the following: in the E’first places of tfre sm*place adder the partial sums equal r - 1; in the place preceding this sequence the pmial sum differs frum r - 1; in the adder obtained from the original one by discarding the ( film1 + f 1 m leading places, t (P, s - [i/ml - 1, m, r) 4 ‘kr. CIhe event U,,, consists only of the fact that the value of the partial sums in the (ms - 1) leading places equals r f 1.1 Putting that all the digits values

of both addends

0, 1, 2, . . . , (t - 11, we find

are independent

and assume

the

349

where 1

From (21, (3) we finally

,

if

F(k,

r, st m) =

i=sm-*.

obtain. miri

F(k,

i
If

lo

S(i, 8,m)=

P, s-

1,

sm-1)

(kfm-1,

2

m)--

izk

(

>c:F k, r, s -

[$-]-1,

i

I \

m)

1

TAGLE

I:$ 3108 2.63

if

k>,sm.

(4)

-

-I 4.15

k
I

-

3.70 3.57 3.24 3.02 2.60 2.63

if

4.51 4.29 3.79 3.67 3.44 3.11

40

44

48

5.58 5.26 4.87 4.61 4.36 4.00

5.72 5.39 4.98

5.84 5.52 5.17

4*74 4.36 4.21

:*E 4139

4.80 4.55 4.19

5.03 4.76

3.91

b-E 3173 3.47

3.44 3.47

5.42 5.11 4.76 4.46 4.18 4.00

5.24 4.95 4.60 4.30 3.97 3.76

Table 1 gives the values, computed from the recurrence relation (4, of the mathematical expectation of the quantity ‘1 (R, m) = d2, [n/ml, m, r) / 1, proportional (see (1) ) to the mean value of the time interval from the beginning the summation

to the appearance

of one of the signals

at the outputs of all the marked digits without cyclic transfer.

of an n-digit

“transfer”, binary

asynchronous

Translated

1.

MIKHELEY, V.N. maximsI transfers

Determination

Zh. vfchisl.

of the distribution Mot.

mat,

function

Fir. 3, 5. 942949,

of

“not transfer”

of the length 1963.

adder

by J. Rerry

of