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NATIONAL BUREAU OF STANDARDS REPORT

4571

Probabilities of Non-Penetrat ion and Targets Downed for

Constant Characteristic Kill Probability

V 0 Memoranda

1 0 Combinatorial Analysis of Uneoordinated System, K. Goldberg

2 0 Statistical Analysis of Uncoordinated System, J. R. Rosenblatt

3c Analysis of Coordinated and Sector Coordinated Systems, J. R.

Rosenblatt

Remarks on Avg Pnp 5 K. Goldberg

5 o Tables of Avg Tgt Dn and Avg Pop in Coordinated System

6 b Tables of Avg Tgt Dn and Avg Pnp (approximate) in Sector

Coordinated System

7 o Tables of Avg Pnp in Uneoordinated System

8„ Tables of Avg Tgt Dn in Uncoordinated System

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1102-40-5126

12 March 1956

Probabilities of Non-Penetration and Targets Downed

Constant Characteristic Kill Probability

Semi-Annual Progress Report on MIPR 55-2175-SC-91

July 1 to December 3X # 1955

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication, re

unless permission I:

25, D.C. Such per

cally prepared If t

Approved for public release by the

director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

In part, Is prohibited

andards, Washington

port has been specif!-

sport for its own use.
.

.
Probabilities of Non-Penetration and Targets Downed for

Constant Characteristic Kill Probability

Semi-annual progress report on MIPR 55-21 75-SC-91

July 1 to December 31 j 1955

I. Introductions This summary and the attached memoranda

represent the analysis and computation done on the three

systems proposed by Dr. E. Biser in his letter to the Bureau

of Standards dated April 11, 1955.

In Section 2 the systems are defined in their general form.

Their common property is the assumption of a constant kill

probability without considerations of elapsed time and distance.

They differ in the degree of coordination supposed in each case.

In Section 3 the results of the analysis are listed in

closed form together with some observations on the behavior of

these results. The two basic functions used to rate the systems are the number of targets downed (Tgt Dn) and its average

(Avg Tgt Dn), and the probability of non-penetration (Pnp)

and its average (Avg Pnp). One of the observations is that

the behavior of the latter function is to a large extent determined by that of another function; the probability that

all targets are fired at (Pp). The variances of these functions

are also given.

In section b are listed further possibilities for analysis

and computations suggested by the results already obtained and

by Martin Orr of the Signal Corps in a recent conference. These

-2-

new investigations would be Aimed at completing the overall

view of the systems by introducing slight modifications in their

definitions e Section 5 is the compendium of memoranda sent to Dr 0 Biser

between November 1955? and March 1956 with some revisions.

They contain all the results listed in section 3 as well as the

tables computed from these results,. The memoranda containing

tables are at the very end 0

II „ Definition of the Systems

1 0 General Assumptions

The number of batteries is B? the number of targets

in a raid is T? and the total number of missiles available is m c The missiles are distributed equally among the batteries (m/B

is an integer)*

It is assumed that the single shot kill probability is a constant, This characteristic kill probability (CKP) is denoted by p,

2 . Assignment Systems

An assignment system describes how batteries

are assigned to targets in an engagement (including random

assignment). The system describes, moreover, how many missiles

are fired at each target in a given engagement (this may be

a random variable).

-

3-

The assignment system, together with the CKP p, determines

the probability distribution of the random variable

N = number of targets down in an engagement.

That is, in hypothetically repeated engagements with the same

B, T, m, and p, and with the same assignment system, N may take

any integer value from zero to T 0 Roughly speaking, the probability distribution of N gives the frequencies of these values

in the sequence of hypothetically repeated engagements*

In particular, the assignment system determines the average

number of targets down ( AVg Tqt Dn ) in such a sequence of

engagement

s

e It also determines the average probability of

non-penetration ( Avq Pnp ) , which is the frequency of engagements in which all targets are downed, i„e 0 , the probability

that N equals T„

A „ Coordinated System

The total number m of missiles is distributed as equally

as possible among the targets,. If m/T is an integer, each

target is fired on by exactly m/T missiles and the precise

scheme of assignment of batteries to targets is irrelevant,,

(The case where m/T is not an integer is slightly more

complicated,, It is, however, a special case of the sectorcoordinated system as described below)*
'

*
-kB„ Sector-Coordinated System

The set of B batteries is

each containing the same number

having the same number m/S of mi

targets is assigned to each sect

be overlapping),, The batteries

their m/S missiles as equally as

signed to them 0 If the m/S miss

exactly equally among the target

a randomly chosen subset of the

An illustrative example is

system involving 20 targets, 20

particular assignment scheme,, F

alternative method of distributi

The m/S missiles fired by a sect

as possible among the targets as

to the restriction that at least

target by a given sector if the

all.

partitioned into S sectors,

of batteries, and hence each

ssiles„ A subset of the T

or (where these subsets may

in a given sector distribute

possible among the targets asiles cannot be distributed

s, the "extras" are fired at

assigned targets,,

given of a sector-coordinated

batteries, 5 sectors, and a

or this example, the following

on of fire is also considered:

or are distributed as equally

signed to that sector subject

two missiles are fired at a sector fires at the target at

In sector-coordinated systems, the number of missiles

fired at a target in an engagement depends on the number of

sectors to which the target is assigned and on the distribution

of fire in those sectors.

-5-

C, Battery-Wise Uncoordinated System

Each battery chooses one target at random (independent

of choices made by other batteries) and fires all of its m/B

missiles at that target,, The number of missiles fired at a target in an engagement depends on the number of batteries which

happen to choose that target,,

XIIo Results,, A list of the formulas for the various functions defined in section 2 is given below. Generally B indicates the number of batteries? T the number of targets? p the

constant characteristic kill probability and m the number of

missiles. Other symbols are defined as they are used. The

values of these variables used in the computation of the tables

to be found in the final memoranda are indicated at each step,

A, Coordinated systems

(A, 1 ) Avg Tgt Dn = T(1-b)

(A, 2) Avg Pnp = ( 1 — b)

T

(A, 3) Var Tgt Dn = Tb (1-b)

where b = (1-p) m// ^, Avg Tgt Dn and Avg Pnp are computed for

T = 20? m= 20(20)160 and p = 0,2? 0,3? 0,5? 0,7? 0,9„

B, Sector-Coordinated System

Given T targets? m missiles? CKP p, and S sectors of batteries with m/S missiles per sector. Let q = 1-p. Suppose

m/S is an integer.

- 6-

Let 6 . . = 1 if i target assigned to j sector

1 J 0 otherwise,

i “ 1 5 o o o 5 T 5 j “ "J ? O O O 9 O

Le t d

j — J ~ ^?oao?S 0

Let Tj, kj, (j = 1, eoo ,S) be integers defined by

ra/S = k . d . + r . J J J

(B o 1 ) Avg Tgt Dn = T

- !i

q

6 ij (k

j

+1)

i=1 j=1 1 d

j

r . 6 . . k . + (1 - q

**

j

This formula applies when each sector distributes fire as

equally as possible among the targets assigned to it„

Avg Tat Dn, Special Case

Let T = 20, S = 5\ and suppose target.s one through ten are

assigned to sectors two and four, targets eleven through twenty

are assigned to sectors three and five, and targets eight

through thirteen are assigned to sector one 0 The formula above

specializes to the following,,

2k

(B.2) Avg Tgt Dn = 20 - yyy q

1 (10 - r^ + r^ q)

2

2k + k - JoO q

1 2

C l 0-r

1 -+-r

1 q)

2 (6-r

2

+r

2 q)

- 7-

where k.j ,k 2

,r., 9 r 2

are defined by

m/5 = lOk-j + r.j 9 < 10

m/5 = 6k

2 + r 2

? r 2 < 6

for any m which is an integer multiple of 5„

If the equal distribution of fire is restricted by the requirement that at least two missiles be fired at a target if

any are fired at it by a sector 9 then the formula above does

not hold for m < 100 o The correct formula for any m < 100 which

is an integer multiple of 20 is the following,,

(B.3) Avg Tgt Dn = 20 - j^q (10 - 1 + ’ q

2

) 2

2k 1 “ TOO q

2

^ 10“r

]

* + r

1

, q"; ') 2 (6-r 2

! + r 2

«q

2

)

where r^ * 9 r 2

' 9 k

2

* are integers defined by

ra = 10 r.j 1 TM = 6k 10 < r 2

* 9 r 2

< 6 ,

Useful exact formulas for Avg Pnp can be given only in a few special cases (values of m) of the above particular sectorcoordinated system,, These , together with various approximations

which were used for other values of m, are given below in

memorandum 3»

- 8-

A general formula for Avg Pup which applies t© all sectorcoordinated systems with equal distribution of fire is also

given below Q Because it is not in very useful form, and

because it requires a lengthy development of additional notation,

it is omitted here 0 Tables were computed for the particular sector-coordinated

system described above, for CKP = 0 o 2, 0 o 3? 0 o 5? 0„7? 0 o 9? and

m = 20 (20) 1 60 0

Uncoordinated

(C.1) Avg Tgt Dn = T[ 1 - ( 1 - ~p) B

]

(C.2) Avg Pnp =2 (-1

)

1

(?) (1 - i) B

i=0

(C.3)

(c„4)

/(T)

where J ^ are the Stirling numbers of the second kind and

a = ( 1 — p ) ^ ®

0 The formulas (C„2) and (C 0 ^) are special cases

of the formulas for the probability that exactly n targets

(p.j(n)) and the probability that exactly n targets are

shot down (p 2 (n))„ Clearly = p-j (0) and Avg Pnp = p 2 (T)„

(c 0 5)

(C.6)

Pi (n) XL _ ./(T-n)

n»TE ^ B

P2 (n) =

(J)

jn __ ("I

i=0

(T-n+i j'J®

-9-

We observe that Pp is that largest value that ean be taken

by Avg Pnp with a fixed number of batteries and targets e It

is also the approximate value taken by Avg Pnp when p is large

or m/B is large, that is when a is small 0 Thus the order of

magnitude of Avg Pnp for fixed number of batteries and targets

is Pp Q This observation is formalized and generalized in the memorandum "Some Observations on Pnp". It is proved that Avg Pnp/P„

F is a constant for all systems with similar engagement strategies,,

Thus strategies may be compared on the basis of their Pc>0 r Tables were computed for B = 10(10) 40 , T = 5(5)20, m/B = 1(1)9,

12, 15 and p = 0 o 2, 0.3, 0.5, 0.7, 0.9 o

IV, Proposed Extensions. Graphs will be made to present the

results in visual form. The nature of these graphs will be

decided on in conference with Dr. Biser and Martin Orr.

As suggested by Mr. Orr we will compute two functions which

will estimate the effect of the loss of information of the

number of targets on the number of targets downed. These funcB t

tions are T[ 1 — (-—;—

)

B

] - t[1- (““)**] *nd t[ 1 — ( 1 — ^) T] where

t is the supposed number of targets.

Because the Stirling numbers of the second kind enter so

frequently in our considerations of Pnp in the uncoordinated

case, and especially in the formula for Pp, we propose extending
,
- 10-

the known tobies of these numbers to cover the coses for which

they arise in this connection.

Because Avg Pnp/Pp is the basic function from which Avg Pnp

can be computed for various equivalent strategies by multiplying by Pp, for that strategy, we propose computing Avg Pnp/Pr

in the uncoordinated system.

Because in actual practice the number of batteries is likely

to be of the same order, and possibly less than, the number

of targets in which case Avg Pnp would be too small for comfort,

we propose computing Avg Pnp for a modified missile-uncoordinated

system in which the number of batteries would be essentially

extended and the number of missiles per battery would be essentailly reduced. This would have the effect of raising the value

of Pp and thus of Avg Pnp,

To Dr. E. Biser

From i K. Goldberg

Subject s Battery-Wise Uncoordinated System Based on Constant

Kill Probability

After learning of your formula for Tgt Dn for the above

case, we have derived additional formulas for this case. The

results are as follows:

( 1 ) Avg Tg Dn = T [1 — ( 1 — ) B

] T ^ Avg Pnp = 2 (-l) i

(^) (1-

~

i) B

1=0 1 T

where T is the number of targets, B the number of batteries,

a =' (1-P} m/® with p the constant kill probability and m/B the

number of shots fired by each battery.

We also have

(2) p^(n) = prob. exactly n targets are not assigned

p^Cn) = avg. prob. exactly n targets shot down

=

t! /(T-n)

n t

B

yB

n = (!) 2 (-1) 1

(?) (1- (T-n+i)) B

where

* i=0 T

y(k)

J

is the Stirling number of the second kind defined

by

oo

2

s=0

ki /

(k)

x

s = (ex-l)^

? s.

The formulas in (2) can be summed to give the probability

for at least, or at most, n targets missing fire or not shot

down. Note also that p (T) is Avg. Pnp.

2/
'
z

The derivations of these formulae follow from their

basic definitions for a single engagement:

(3) Tgt Dn * 2 (l-aKi)

1=1

'Hp it (1-a^I)

i=l

T

**i 1=1

B, k. > 0

Prob. first n targets, and no others, shot down = ... (l-akn) a

kn+l ... a

1'!

Number of unselected targets = number of k^ equal to 0*

In order to find Avg Tgt Dn, Avg Pnp and p^(n) we must

sum the values for a single engagement over all possible

distinct engagements and divide by the number of such engagements.

An engagement is defined by a vector (k^, kg, ••• k^,),

distinct engagements having distinct vectors. Given any set

C = -£cjJ of T non-negative integers there are

(?) =

Bi

_ i _ i . » c • c • • • . c . 1 - 2- T

L /

distinct vectors (k^k^, • • • ,k^) with the set K = *|kidentically equal to the set C = {°i} • The symbol (®) where K = w can also be defined by

a generating function which will prove useful:

•D k-i kp km

?

( K) *1 x

2 *T = (X

L + Xg+a..+ ^ ) B

The sum is taken over all partitions of B into T nonnegative parts k^jkg,...,!^ •
.

. .<71..
-3-

(k)

(5)

Thus the number of distinct engagements is

2(|) = T

B

.

The sum of Tgt Dn over all distinct engagements is

2(®) 2 (l-ak

i) = T

C

- ( a+T-1

)

T B iB

‘ K i=l

( 6 )

The sum of Pnp over all distinct engagements is Tv T

2(5) n (l-a~i) = 2 (-1 ) 1(?) (ai+T-i) * i=l i =0 1

i /T' , \B

The sum of the probabilities that only the first n

planes are shot down over all distinct engagements is

( 7 ) 2 ( B

) (l-a*!) ... (l-a

k

n) a

kr,+1

. . . a*^ =

n

2

i =0 = (-1) 1

(£) (a(T-n+i)+n-i)

B

Therefore Avg Tgt Dn is the value of the formula in

(5>) divided by that in (I4.), Avg Pnp is the formula in

( 6 ) divided by that in (ij.), and P2

(n) is the formula in ( 7 ) divided by that in ([{.) and multiplied by (^) to get all the

ways that exactly n, not just the first n, targets can be

chosen.

To find p^(n) we first find p^(0) . For each engagement

the probability that no targets miss fire is 1 if all the

are positive and 0 otherwise. „ The sum of these probabilities

over all distinct cases is 2 (§) summed over all partitions of

B into T positive integers k., ,k?

, . • ,1^. This is just Bl times

the coefficient of x® in

(

e

x

-l) Tor Tl Thus p1 (0) = Tl

-kNow suppose the first n targets miss fire and all the

others receive fire. The probability that no targets among

the last T-n miss fire from B batteries is p^CO) for T-n

targets and B batteries or

To get p^(n), the probability that exactly n targets miss

fire, we must multiply this by the number of ways of

choosing tne n lucky targets which is (^) » Mrs, Joan Rosenblatt has derived all these formulas,

except p]_(n), by independent means and her results, which

include a formula for the variance of Tgt Dn, will be

transmitted to you in a separate memorandum.

B

Memorandum Number 2

Statistical Analysis of Uncoordinated System

)

Uncoordinated System Based on

Constant Kill Probability

by

Joan Raup Rosenblatt

National Bureau of Standards

Given B batteries, T targets, m missiles, and constant

kill probability p.

teger, The number a is the probability that a target will

survive when fired on by one battery, since it is assumed that

each battery will fire all of its m/B missiles at one target.

Under the uncoordinated system, each battery selects its

target at random, independent of selections made by other batteries • An engagement is characterized by the numbers , k£,

k^, where k^ denotes the number of batteries which fire

on target i. B = k, + . . « + k^ «,

Quantities which may be used to describe the properties of

this system include the following:

(1) Average number of targets down.

Avg Tgt Dn

(2) Probability of non-penetration (i.e., probability that T

targets are down)

.

-2- -¥ l

)

B

(3) Variance of the number of targets down.

Var (Tgt Dn) = T Q. - B+ T(T-l) ^1-2 T

2

^

- 1 ~ a

J

2B

( i+) For each n (n = 0, 1, ..., T), the probability pn

that

exactly n targets are down.

P = Pnp (see (2)) , T

1 - (n 4- i) n 0 , 1 ,

• • • f T .

In particular.

P

0

_Ba

P

1 = * (* +

B

- TaB

.

These quantities are derived in the following.

1, Distribution of Number of Targets Down . The random variable with which we are concerned is N, the

number of targets down. Let the distribution of N be given

^(P , P , •••, P

T

> , where

p = Pr(N = n) n 0, 1, «««, T, 9

-3

-

Then, in particular,

Pnp = p

T

.

Prom the distribution of N, we also obtain

T

Avg Tgt Dn = EN = 2 np , n=0 n

and

Var (Tgt Dn) = E(N - EN) 2 = 2 n

2

p - (EN)

2

* n=0 n

The quantities p , «*., p are not convenient expressions,

0 i

however, so that we do not evaluate EN and Var N from the

relations given above®

2. Evaluation of p * n

Observe that there are T possible equally likely configurations of assignments of B batteries to T targets* Let

K = (k_^, k^) represent the configuration of assignments

in one engagement where k^ batteries fire at the i th target.

Corresponding to each K, there are configurations which

differ only in that a different set of batteries is assigned

to the ith target.

i

Thus, we have

p = T

“B

2 ( B

) p

®

n K

K *n

(K)

, n = 0, 1, d « ® j T,

where p

1 ' denotes the conditional probability that exactly

n

n targets are down in an engagement with configuration K * Furthermore, for any function f (n) , we have

T

Ef(N) = 2 f (n)p = T"B

2 (f)«J .2 f(n) p

(K)

n=0 n K

K' n~0 n

We will write

T (K)

Ejf(N) = 2 f (n) p K n=0 n

to denote the conditional expectation of f(N) when K is fixed,

(K)

3. Evaluation of p

n

Observe that N may be regarded as the sum of T random

variables, N - X + ... + X , 1 T

where X^ has the value one or zero according as the i th target is down or survives.

Now, when K is fixed, the X^ are mutually independent.

The conditional distribution of X^ is given by

Pr (X = l|K) = 1 - a *

Pr (X = 0|K) = a

k

i

We have, then,

9

(K) T k± B

p = it a = a

0 i=l

and for n = 2 , 3, • . . , T - 1

,

use in the relations given in the preceding section for calculating Ef (n) . Another method will be developed to carry out

those calculations* We are, however, in a position to complete

the evaluation of p (n=0, 1, T).

Ij.. Calculation of p , n

B

a) Evidently, p = a

0

(K)

Except for p^ 9 these are not convenient expressions to

n

-6-

b) ^

= T (a +l^-ay

T k •? g

2 it a - Ta

i=i

T-'z,?,,.TM

K

K 1

Now, in general.

T"

B

2 (®)a Xl

K

K

k : a = (1

1-a . r -rr~ ; T

B

Thus ,

c) The remaining expressions for p are calculated in similar

n

fashion.

5, Calculation of Avg Tgt Dn «,

EN = T

“ B

2 ( B

) E N

K

K K

T T

, k,

E N = 2EL = 2 (1-a 1

) K i=l K

1 i=l

- 7 -

The calculation of the expression given above for Avg Tgt

Dn Is straightforward «,

6, Calculation of Var(Tgt Dn) 0

E(N - EN) 2 = E(N - E N)

2

+ E(E N - EN) 2

K K

a) E(N - E N) 2 = T

”B

2 (g) E (N - E N) 2

K K K K

p But E (N - E N) is simply the variance of a sum of independent

K K

random variables, since N + a • o + Xfp as noted above

Now

ki kj

Var X = a

1

(1 - a

1

) , i = 1, „ 6 „ *

Hence

E(N - E N) 2

k

T k. = T"B

2 (£) 2 a

1

(1 - a

1

)

K

K

i=l

b) E(E N - EN) 2 = T"B

2 (|) (E N) 2

- (EN) 2

K K K

Var N is obtained by evaluating the two terms,

Memorandum Number 3

Analysis of Coordinated and Sector Coordinated Systems

Coordinated and Sector -Coordinated Systems

Based on Constant Kill Probability

by

Joan Raup Rosenblatt

* National Bureau of Standards

The statistical method used to study the uncoordinated system

is applied to the coordinated and sector -coordinated systems. The

following is a summary of the investigations reported in these notes.

(1) Coordinated System,, A formula is obtained for the

variance of the number of targets down. For n » 0, Ijo.., T, the

prcfe ability pn

that exactly n targets are down is given. These formula©

as well as the formulae for Avg Tgt Dn and Pnp are also obtained for the

case where m/T (number of missiles per target) is not an integer.

(2) Sector -Coordinated System. A formal method of solution is

found. Avg Tgt Dn is obtained in reasonably computable form. Pnp

would be extremely laborious to compute exactly! certain inequalities

and an approximation are given. Var (Tgt Dn) could be computed exactly,

but it would probably be sufficient to use an approximation for this

quantity.

The device employed in obtaining a solution for the sector

coordinated system - an incidence matrix describing assignment of

targets to sectors - could be used to analyze other possible forms of

coordinated systems.

2

The system considered is different in one respect from the

system described in the initial memorandum setting forth this problem.

It was intended that the following rule should be invoked? a sector

must fire at least two missiles at each target on which it fires. This

rule was ignored.

(3) Distribution of Fire . For the coordinated system, equal

distribution of fire is never worse than and can be better than distribution according to the u at least two u rule.

The two methods of distribution differ only when the number of

missiles per target is less than two. The following comparisons hold

when the two methods differ. Avg Tgt On is greater with equal distribution. Pnp is no less and sometimes greater with equal distribution.

The variance of the number of targets down, on the other hand,

is smaller with the “at least two” rule.

(U) Examples; Sector -Coordinated System. For the assignment

scheme outlined in the original memorandum on this problem (20 targets,

5 sectors), the exact formulas for Avg Tgt Dn have been worked out for

m/S = U(U)32. For m/S ® i4,( U)l6, the equal distribution of fire and the

t} at least two 1 * systems are different; Avg Tgt Dn formulas have been

obtained for both cases. The equal distribution of fire system always

gives a larger value for Avg Tgt Dn.

An approximation for Pnp is given for m/S = U(U)32. The method

of approximation does not distinguish between the two possible

systems of distribution of fire for m/s = i;(U)l6. Upper and lower

3

bounds for Pnp are given*

In two cases, the exact formula for Pnp is not too unwieldy,

(i) Pnp is given exactly for ra/S « 20 „ (ii) Bip is given exactly

for m/S =12, under the Hat least two M system.

Given T targets,, m missiles,, and a constant kill probability p

Under this system,, all engagements are alike. Each target receives ra/T missiles, (If we suppose m/T is not an integer 9 the situa

tion is slightly more complicated. The formulae for this case are

given at the end of this section,)

Let b - (l^p)®1/^ the probability of survival of a target/

Avg Tgt Dn - T(l=b)

Var (Tgt Qn)» Tb(l-fc)

Rip . (l-b)T

The derivation of these formulae follows.

Let Xj_ equal one or zero according as the i^ target is down

or survives, are independent and identically distributed

with

e 1 )35 1 $ U“1$ o o , $ I ,

Now the number of targets down is N « X^ +,,,4 X>p , It

follows at once that

Vti * Pr(N « n) - (£)bT ~n

(l=b)n

, n«4,l,...,T ,

Pnp n pT

= (l-b)T

,

E N = T(l«b) s

Var N = Tb ( 1 =b ) „

5

Now, suppose m/T is not an integer, say

m = k +

where k is an integer and 1^. s^T = 1 0 Using the argument developed

in Dr. Alt*s memorandum of 20 May, (T - s) targets will receive k

missiles and s targets will receive (k + 1) missiles. Thus,

Pr (*i - 1)

i - q

q

k+X

1 - ~ k

Xf XE1, o o o , S

if i«s+l, ...,T ,

where q « 1 - p

We obtain immediately g

Pnp * pT

- (1 - qk+l)S( 2. - q

k

) T “s

, ( l.U)

Arg Tgt Dn - EN« sQ.** q

k+1

) + (T - s) (1 - q

k

), ( 1.5 )

Var (Tgt » Var N & sq

k+1

(1 - q

k+1

) (T ~s)qk (l - q

k

) , ( 1.6)

and for n = 0,1, ,, OJ T ,

pn - 2 (?) (n I f) (1 - q**1

) 1

(qktl

) s *1 (l-q*)11 -1( q

k)T -s “n+i

i-0 w v (1.7)

2o Sector -Coordinated System.

Given T targets, m missiles, CKP p, and S sectors of batteries

with m/S missiles per sector. Let q

= 1 - p. We suppose that m/S

is an integer.

The assignment pattern for the sect or coordinated system is

given by the T X S incidence matrix (£ji)$ where

I

Sid

1 if the i^*1 target is assigned to tl*® j*'*1 sector

0 otherwise j,

/ i 35 1, « o e ,T | J

53 lj 0005S a

-T

Let » 2 . be the number of targets assigned to the

J i«l

sector*

The following discussion does not include the requirement that

% sector fire at least two missiles at a target if it fires on the

target at all during an engagement e This modification could be made.

Let

iiy^S ® k.d. + r. , J J J

where k^ « jla/S’d^J is the greatest integer contained in

m/S dj 9 and r^ (o^r^ ^ » l) is an integer.

Again using Dr, Alt's argument, we have in any engagement that

) of the targets assigned to the j^h sector receive kj missiles

the remaining r^ targets receive (k^ +1) missiles.

Now there are

( 20)

( 2 . 2 )

D Tr

(

dj

) .i-l

y.3)

equally likely configurations for particular engagements, A

7

configuration may be described by a matrix L » (1^ ) which is related

to (<£..) in the following ways For j = 1,<,..,S, (d.-r.) of the d^

0 J

J

ones in the column of (S .) are replaced by zeros,, That is, a

matrix L has a one in its jTM1 column for every target which receives

(k^ 4- 1) missiles from the j’k*1 sector in this engagement.

Now the number of targets down, N, may again be represented by

the sum N *= +<,.o+ . In any engagement, with fixed,

X^, 0 oo,Xj> are usually independent. Thus, using the conditionalexpectation notation developed in the discussion of the uncoordinated

system, we have

Pr = 1/L) - 1 - q^"1

2 Sjj (v V , i=l , o o ®T (2.U)

where the exponent is the number of missiles fired at the l

1"*1 target

in the engagement described by L.

From these quantities, we may obtain formally

N « Z Pr (X±

= 1/L) , i~l

E N 2 El n , L

(2*5) Avg Tgt Dn S

8

T

= IT Pr (X±

» l/L) , i-1

Pnp = pT (2o6)

wher© 2 denotes summation over all possible configuration matrices L.

L

The variance of N arid Pr (N = n) may also be obtained formally*

The computational difficulty arises from the fact that in

order to evaluate these formulae, we must compute all the T X D

values of the exponent

S

* hy *

Calculation of Avg Tgt Dn '

The following formula is derived, which does not depend on

the individual configuration matrices L.

T _S_

E N « T *=*2 1=1 j=l

The following derivation calls attention explicitly to certain

assumptions which have been made implicitly above* Let

d

J

' Sk

3 (2o7)

Y. . = y

if the sector fails to kill the

i^k target

otherwise.

I

9 .

In a given engagement, (Y^ , • .„,Y^

S

) are mutually independent and have

the distributions defined by

Pr (Tjj- 1/L) q

^LJ Ck

J + 1

ij) i=l *1 , »

0 , S

We write

E N » E N) ,

where N denotes the conditional expectation of the number of targets

down for fixed engagement pattern L, and £(•) denotes the average

(i„e., expectation) over all possible equally likely Engagements E( 8

)

is equivalent to D“^ 2 (*)* Now

L

T

E

l

N « 2 Pr (Xj* 1/L) , i»l

and, since (Y^, »,.,Yj_

g

) are independent.

Pr (X±-l/L)

S

1 - M Pr (Yy 1/L).

Hence

T S

- 2 E 7T Pr (Y±1 * 1/L) . i«l j-1 J E N « T

10

Now, for fixed target (i), the random variables

Pr (Y^ « 1/L) and Pr (I i * 1/L)

are independent (j «f j')* This follows from the assumption that all

engagement patterns L are equally likely, which implies that two

sectors make independent random (equi-probable) selections of the

targets which are to receive (k.+ 1) missiles. It follows that

J S S

E IT Pr (Y.,= 1/L) = 7T E Pr (Y, .= 1/L) . j=l 3 j=l 3

But

E Pr (Y

±j= 1/L)

^

ZPr (Y^* 1/L)

D

ij

a - L.)

since the proportion of engagements in which the sector fires

(kj-fr 1) missiles at the 1^ target is r^/dy This completes the

derivation of the expression for E N given above.

Calculation of Var (Tgt Hi) .

A similar argument may be employed to derive

T S * 2 ]fEYu - 2 TT (S V,)

i=l j=l ^ i=l j=l ^

T S

T

j'

+ 2 2 IT Cov (Y, , , \ .)

lSi-^i^T J«l

Xl

3

^

(2.8)

Var N

11

where

E X,^

q

^(k

5 +1)

+ (l ,

and

Cor . %2j)

.

~r

j

(dr

r

3

) ^

s h^)k

i

d

j

2(dr

1)

(l - q )(i

Calculation of Pnp 0

To evaluate Rip exactly would require enormous computations*,

Th© following inequalities may prove to be useful:

(1) Pnp i. (Avg Tgt On)

, T

S

T

(2) Pnp > ~]T

i»l

1 « q*

2 <T. .k, \

,=i 3

I

T f j

(3) fnp^TT U - q*1 ' i-l \

Equality will hold in (2) if and only if the numbers m/Sd^

ar© integers (j=l,...,S). (It is assumed that the CKP p«l-q is

between zero and one. If p=0 or 1, then equality will hold in all

three relations.)

The following formula is suggested as a possible approximation

12

S

Pnp « j f 1 -

q«3“^

i=l

Z ^(kj^/d.)

(2,9)

at the i^*1 target.

Approximation for Var (Tgt Ita) .

The approximation given here is, like the approximation for

fhp, based on the average number of missiles fired at each target.

3o Distribution of Fire,

A comparison is made between equal distribution of fire and

distribution according to the "at least two" rule, for the coordinated system,

'

. i Given m missiles, T targets, and CKP p

= 1 - q.

First, observe that if the number of missiles is at least

2T, then every target will receive at least two missiles by equal

distribution of fire. Hence, the two methods of distribution will

differ only if 2 T .

Var N = Z q

&

^ (1 - q

&

^) ,

T

( 2 . 10 ) i=l

13

Next* it is easy to see that if m ^ 2 T, the "at least two"

rule implies that at least one target will not be fired on. Hence

Pnp =* 0* For equal distribution of fire, Pnp is greater than zero

for T^.m < 2 T and Pnp = 0 only if m< T. In summary: with respect

to Pnp, the two methods of distribution of fire are alike except

when T<Lm < 2 T$ in this case, equal distribution is better.

Now, consider the comparison of the two methods with respect

to Avg Tgt Dn and Var (Tgt On). Let m = 2 r + a , a = 0 or 1,

'•At least two” rule.

' - a)(l - q

2

) + a(l - q

3

)

(r - a) q

2

(1 - q

2

) 4- a q

3 (l - q

3

)

a) (1 - q) if l^m <CT

a - T) (1 - q

2

) + (2T - 2r - a)(l - q)

if I< m ^L2T

a) q (1 - q) if 1 <Lm < T

a - T) q

2 (1 - q2

) + (2T - 2r - a) q (1 - q)

if T-^m < 2T

It is readily verified that Avg Tgt Dn and Var (Tgt Dn) are always

greater for the equal distribution case.

Avg Tgt Dn = (r

Var (Tgt Dn) =

Avg Tgt Dn = i'

(2r +

(2r +

Var (Tgt Dn) = *

(2r +

(2r +
v

' : : vv=

; - -

''

' *

t

. - * v .»• ih ‘

: " >'

; . i ,

' r \

.

. I

‘ " ' • "

«: -5
-14-

4, Example: Sector-Coordinated System.

Given m/S = 4(4)32, T = 20, 8=5, and the assignment

scheme under which sector one fires on targets 8-13, sectors two

and four fire on targets 1-10, sectors three and five fire on

targets 11-20.

The general formula for Avg Tgt Dn may be specialized to

the following.

14 2k..

Avg Tgt Dn = 20 - q (10 - r

± + ^q) 2

2 k +k

- TUS q

1 2 (1° - r

l + ri q > 2 (6 - r2

+r2

q)

where the integers k^, kg, r^, are defined by

5

= 10 k

x + r , 0 < r

1 < 10

' - 6 kg + r

2 , 0 < r

2 < 6.

For m/5 < 16, the above formula applies only if each sector

distributes fire ns equally as possible among the targets assigned

to it. If the "at least two" rule is applied, then the following

formula is correct,

Avg Tgt Dn - 20 - ^(10 - r£ + r£q2

) 2

2k * - q

2

<10 " r

i

+ r

i q2 > < 6 - r

2

+ r

2 q2 >

where the integers kg, r^, r£ are defined by

m

TU r

1

m

TJ5

6 k

2 + r2

t 0 < r ’ < 6
-

'

T ,

'•

: . *

.
-15-

It is not always possible to obtain convenient exact expressions for Pnp 0 The following list gives, for each values of m/S,

the exact expression for Pnp where possible, and otherwise an

approximate expression. A discussion of the methods of approximation is added below.

m/S = 4

Pnp = (l-q)^^/15 (210) 2 with equal distribution.

Pnp 35 0 with "at least two" systems.

m/S ~ 8„ 9 .20 2 with "at least two" systems.

Pnp - (l=q 2

) /15 (210 )

Approximately, with equal distribution,

„ . „ 8/5.14 44/15.6

Pnp - P

F (l-q 7

) (1-q )

where ?

F

= .533441,

and Pnp < P

F (l-q 2

) 20

.

m/S - 12

Pnp - (1050)

2

(1-q 2

) 18

(1-q

4

) 2

4.

15 (l+q 2 +q

4

) 2

+ 110 (1+q 2

) 2 (l+q 2 +q^) + 101 (l+q 2 )*]

with "at least two" systems

Approximately, with equal distribution,

„ . ,, 12/5 14 22/5.6

Pnp - (1-q ) (1-q ) ,

(1-q 2

) 14

(1-q

4

) 6

< Pnp < (1-q 3) 20

.

and
.
-16-

m/S = 16

Approximately,

Pnp = (l-q 46^5

) 14 (l~q88//15

) 6 with equal distribution,

Pnp - P

F

(l-q 16//5

) 14 (l-q

88/?15

) 6 with "at least two" systems,

where P^, = 0 533441 0 With equal distribution,

(1-q 2

) 14

(1-q

4

) 6

< Pnp < (1-q4

) 20

.

m/S - 20

4 v 14 /. 7 x 4 /. 8. 2 Pnp - (l=q ) (1-q ) (1-q ) 2

(The equal distribution and "at least two" systems are the

same for m/S > 20 o )

m/S 24

Approximately

„ . ,, 24/5.14 ,, 44/5,6 Pnp - (1-q ) (1-q )

(1-q

4

) 14

(1-q

8

) 6

< Pnp < (1-q

6

) 20

,

m/S - 28

Approximately,

Pnp - (l-q28/V4 d-q154/15

) 6

.

(1-q

4

) 14

(1-q

8

) 6 < Pnp < (1-q

6

) 14

(1-q

11

) 6

.

m/S *= 32

Approximately,

„ . ,, 32/5,14 Pnp 53 (1-q )

,, 6,14 ,, 11,6 (1-q ) (1-q )

(l-q176/15

) 6

.

< Pnp < (1-q

8

) 20

.

- 17-

The approximations are obtained by finding the average number of missiles fired at each target, as described in section two

of this memorandum. For small values of m, however, these approxi

mations have been improved by taking account of the fact that in

some engagements some targets are not fired on at all* This is

reflected in the use of the factor P

F

which is the probability

that all targets are fired on.

In some cases, it is possible to improve on the upper bounds

for Pnp which were listed in section two. The following bound is

used when it is better,

Pnp < (l-qm^T

) T

.

Memorandum Number 4

Remarks on Avg Pnp

TO; Dr, E 0 Biser February 8, 1956

Evans Signal Laboratories

FROM; K, Goldberg

National Bureau of Standards

SUBJECT; Some Remarks on Pnp

We have proved the following lemma which is relevant to

our considerations of Avg Pnp;

Lemma ; Let S be a strategy for firing missiles at targets.

With S associate two functions; the average probability of

non-penetration (Avg Pnp) and the probability that all targets

are fired on (P„) „ Also associate the set C of all possible

different eases affected by S in which the probability of nonpenetration (Pap) is positive.

If S

1 is another strategy (with associated Avg Prtp>, Pp

and C 1 ) such that Pnp 1 is equal to Pnp for each particular*

case, then C 8 = C implies

(L) P

! P (Avg Pnp) = Pp (Avg Flip 8 )

In other words if Pp ^ 0 then (Avg Pnp)/Pp is the same for

all strategies in which* the Pnp are equal for* equal cases and C

is a constant. If Pp = 0 then Avg Pnp = 0,

The implication of this lemma is that the order of magnitude

of Pnp is determined primarily by Pp, Thus, it is reasonable to

compare two strategies only if they* have equal Pp, just as it

is reasonable to compare two strategies only if they have equal

CKP,

This has a direct bearing on our considerations of the

uncoordinated system. This system represents a minimal strategy

and should be useful in determining a minimum Avg Pnp for all

strategies. However, the Avg Pnp in this system is actually

too small, it provides an unreasonable minimum. The reason

for this is that Pp is usually too small. As in our memorandum

of November 1955;

y(T> B

P

F

= Pl (0) = t; j b /i

Thus if B = T we hsve Pp = Ti/T^ or

—T J-- e

A (2 tjT) 2

F

-2-

which becomes very small very rapidly c On the other hand

(Avg Pnp)/Pp = (1-a)T when B = T and this is of the proper

order of magnitude for comparison purposes*

Thus we propose computing (Avg Pnp)/Pp for all cases in

the uncoordinated system* This will provide a minimum for this

function for all strategies with the same CKP and the same

number of missiles and targets,,

The proof of the lemma is straight-forward; Avg Pnp is

the sum of the Pnp over all possible distinct cases ( ^Pnp)

divided by the total number of distinct cases (N) * P^ is the

number of distinct cases in which Pnp is positive (Nr ; divided

by N s

On the other hand ^1L Pnp is equal to the sum of the Pnp

over those eases in which Pnp is positive (2E r

Pnp) since all

other Pnp vanish. Thus if does not vanish we have

(Avg Pnp)/Pp = (2LC Pnp)/N

c

,

Clearly this is a constant if C is constant and Pnp is

derived in the same way for the same case. In fact both

Pnp and are constant under these conditions.

Memorandum Number 5

Tables of Avg Tgt Dn and Avg Pnp

in Coordinated System

H

(\J

o"\_rd-U\MD

C^-oO

H

CM

r^^tlAvO

0-00

COORDINATED SYSTEM

Pnp

0 »

0.3 0.5 0,7 0*9

.(10)349 .(6)954 . .(3)798 .122

.(0) 134 .(5) 142 .(2)317 .152 ,818

.(6) 587 .(3) 225 .(1)692 .578 .980

.(4) 265 .(2) 412 .275 .850 .998

.(3) 356 .0252 .530 .953 1.000

.(2) 229 .0818 .730 .986 1.000

.(2) 903 .179 .855 .996 1*000

.(1) 254 .305 .925 .999 1.000

Tgt Dn

0.2 0.3 0.5 0.7 0.9

4.00 6.00 10.00 14.OO 18.00

7.20 10.20 15.00 18.20 19.80

9.76 13.14 17.50 19.46 19.98

11.81 15.20 13.75 19.84 20.00

13.45 16.64 19.38 19.95 20.00

14.76 17.65 19.69 19.99 20.00

15.81 18.35 19.84 20.00 20.00

16.64 18.85 19.92 20.00 20.00

Memorandum Number 6

Tables of Avg Tgt Dn and Avg Pnp (approximate)

in Sector Coordinated System

Sector-Coordinated System

Tables of Avq Tqt Dn and Avq Pnp

The illustrative sector-coordinated system for which

these computations were made is described in section four

of the memorandum on sector-coordinated systems. There

are twenty targets and five sectors of four batteries each.

Each sector has the same number m/5 of missiles.

The tables were computed for m = 20 ( 20 ) 1 60 , and for CKP

p = c 2 ? c3( o2) „9o

Table I ; Avg Tgt Dn is given for both the equal distribution of fire system and the "at least two" system. (There

is no difference between the two systems for m >. 100 .)

Table II s Either Pnp or an approximate value for Pnp is

given for both the equal distribution of fire system and the

"at least two" system.
.
Table I: Avq Tqt Dn

072 o~3 o3 077 079

m

P (a) (b) (a) (b) (a) (b) (a) (b) (a) (b)

20 3.40, 3.75 4.69, 5.44 6.63, 8.48

40 6.46, 6.96 8.73, 9.73 11.9, 14.1

60 9.03, 9.48 11.8, 12.7 15.3, 16.9

80 11,2, 11.4 14.3, 14.6 17.6, 18.2

100 13.1 1 6 0 2 19.1

120 14.3 17J 19.4

7.84 , 11.1 8.4l, 13.3

13.8, 17.0 14.7, 18,9

17.0, 19.0 1 7.7, 19.9

18.9, 19.6 19.4, 20,0

19.9 20 .0

19.9 20 .0

140 15.3 17.9 19.7

160 16.2 CO

e

va 19.8

20.0 20 .0

20 o 0 20.0

(a) For the "at least two" system.

(b) For equal distribution

Table II; Avq Pnp

P

m

0.2

(a) (b)

0,3

(a) (b)

0.5

(») (b)

0,7

(a) (b)

0.9

(a) (b)

20 o,.ooo 0,„000 0,.000 o,.ooo 0,,000

40 . 000 , . 000A o 000, s 000A . 000, ,001

• 000 , . 049A ,000, .371 A

60 .000, , 000A .000, . 000A . 001

0 Q39A .01 5,. 436a .041 ,.945a

80 . 000A , . 000A .001 A, .002A . 096a, .180A „393A, 737A .?29A,.991A

100 .000 ,013 ,390 .891 .999

120 .001

.047A .591A .957A 1 . 000A

140 ,oo5a . 1 1 1 A .743A ,984a 1 . 000A

1

,f6o .01 4a . 202A ;-845a ;994a 1 . 000A

"A"

"0"

(a) For "at least two" system

(b) For equal distribution

denotes approximate values. Others

denotes identically zero.

are exact to three places.

Memorandum Number 7

Tables of Avg Pnp in Uncoordinated System

MEMORANDUM TO; Dr. E. Biser

FROM ; K. Goldberg

SUBJECT : Tables of Average Pnp in Uncoordinated System

The average probability of non-penetration (Average P ) has

been computed for B = 10 (10) 40, T = 5 (5) 20, p = 0.2, 0.3 (0.2)

0.9 and m/B - 1 (1) 9, 12, 15. The tabulation of this computation

is attached.

The formula used is

T /t) / l-a\B m/B

Avg P

qP = ( — 1

1

v.i/ (l - ~T~

)

where a = (1 -p)

that

The probability/all targets are fired on, P„ , is the number

found under p = 0.9 and m/B » 15 for any particular B and T. This

number determines ooth the maximum for Avg P and the order of

magnitude for Avg Pnp except for small CKP ana small numbers of

missiles. Because of the importance we tabulate it separately below:

t\b 10 20 30 40

5 ,522 .942 .993 .999

10 .000 . .214 .629 .858

15 0 .001 .087 .338

20 0 .000 .001 .035

»' .000" indicates a value computed to three decimal ]

is actually T*/TT . "0" indicates absolute zero 0

0.9

AVERAGE P

0.2

np

IN BATTERYWISE UNCOORDINATED CASE

0.3 0.5 0.7

B = 10, T = 5

1 .001 .010 .077 .226 .423

2 .022 .082 .273 .433 .512

3 .070 .188 .398 .496 .521

4 .135 .282 .461 .514 .522

5 .201 .354 .492 .520 .522

6 .261 .405 .507 .521 .522

7 ,312 .441 .515 .522 .522

8 .355 .466 .518 .522 .522

9 .389 .483 .520 .522 .522

12 .455 .509 .522 .522 .522

15 .488 .518 .522 .522 .522

B = 20, T - 5

1 .042 .156 .499 .768 .906

2 .252 .516 .814 .911 .939

3 .479 .723 .895 — .934 .942

4 .639 .822 .922 .940 .942

5 .740 .871 .933 .942 .942

6 .803 .898 .938 .942 .942

7 .844 .914 .940 .942 .942

8 .871 .924 .941 .942 .942

9 .890 .930 .942 .942 .942

12 .919 .938 .942 .942 .942

15 .931 .941 .942 .942 .942

B = 30, T = 5

1 .159 .510 .800 .946 .987

2 .556 .812 .962 .987 .993

3 .784 .928 .984 .992 .993

4 .887 .964 .990 .993 .993

5 .935 .978 .992 .993 .993

6 .958 .985 .993 .993 .993

7 .971 .988 .993 .993 .993

8 .978 .990 .993 .993 .993

9 .983 .991 .993 .993 .993

12 .989 .993 .993 .993 .993

15 .991 .993 .993 .993 .993

2 - 1 .323 .635 .927 .988 .998 2 .767 .933 .992 .998 .999 3 .919 .982 .997 .999 .999 4 .967 .993 .998 .999 .999 5 .984 .996 .999 .999 .999 6 .991 .997 .999 .999 .999 7 .994 .998 .999 .99,9 .999 8 .996 .998 .999 .999 .999 9 .997 .999 .999 .999 .999

12 .998 .999 .999 . 999 .999

15 .999 .999 .999 .999 .999 B = 10, T = 10

Average P

np = 0 (to three places) B = 20, T = 10 1 .000 .000 .004 .038 .138 2 .000 .004 .056 .145 .206 3 .003 .026 .121 .192 .213 4 .013 .060 .165 .208 .214 5 .030 .095 .189 .212 .214 6 .051 .126 .201 .214 .214 7 .073 .150 .208 .214 .214 8 .095 .168 .211 .214 .214 9 .115 .182 .213 .214 .214

12 ,160 .203 .214 .214 .214

15 .186 .210 .214 .214 .214 B - 30, T = 10 1 .000 .003 .064 2 .010 .071 .326 3 .057 .210 .486 4 .138 .339 .561 5 .228 .432 .596 6 .310 .495 .612 7 .378 .538 .621 8 .432 .566 .625 9 .475 .586 .627

12 .554 .614 .628

15 .591 ,624 .629

262 .517

529 .618

600 .628

620 .629

626 .629

628 .629

628 .629

629 .629

629 .629

629 .629

629 .629

3

m/B\P 0 .2 0.3 0.5 0.7 0 o 9 B = 40 , T = 10 1 .001 .021 .225 .550 .785 2 .056 .240 .621 .794 .851 3 .206 .483 .763 .841 .857 4 .373 .634 .815 .853 .858 5 .507 .719 .838 .856 .858 6 .604 .770 .848 .857 .858 7 .672 .800 .853 .857 .858 8 .720 .819 .855 .858 .858 9 .754 .832 .856 .858 .858

12 .811 .849 .857 .858 .858

15 .835 .855 .858 .858 .858 B = 20 , T = 15 1 .000 .000 .000 .000 .000 2 .000 .000 .000 „000 .001 3 .000 .000 .000 .001 .001 4 .000 .000 .000 .001 .001 5 .000 .000 .001 .001 .001 6 .000 .000 .001 .001 .001 7 .000 .000 .001 .001 .001 8 .000 .000 .001 .001 .001 9 .000 .001 .001 .001 .001

12 .000 .001 .001 .001 .001

15 .001 .001 .001 .001 .001 a ii CO o T = 15 1 .000 .000 .000 .006 .045 2 .000 .000 .011 .048 .082 3 .000 .003 .037 .074 .087 4 ,001 .012 .058 .083 .087 5 .004 .025 .072 .086 .087 6 .010 .039 .079 .087 .087 7 .017 .051 .083 .087 .087 8 .025 .060 .085 .087 .087 9 .034 .068 .086 .087 .087

12 .056 .080 .087 .087 .087

15 .070 .085 .087 .087 .087

4

m/By* 0.2 0.3 0.5 o o

B = 40 , T = 15

1 .000 .000 .005 .063

2 .000 .006 .094 .238

3 .004 .042 .202 .307

4 .020 .102 .268 .329

5 .049 .161 .303 .335

6 .086 .210 .320 .337

7 .125 .247 .329 .337

8 .161 .273 .333 .338

9 .193 .292 .335 .338

12 .261 .322 .337 .338

15 .298 .332 .338 .338

B = 20 , T - 20

Average P

np = 0 (to three decimal places )

B = 30 , T = 20

1 .000 .000 .000 .000

2 .000 .000 .000 .000

3 .000 .000 .000 .000

4 .000 .000 .000 .001

5 .000 .000 .000 .001

6 .000 .000 .001 .001

7 .000 .000 .001 .001

8 .000 .000 .001 ,001

9 .000 .001 .001 .001

12 .000 .001 .001 ,001

15 .000 .001 .001 .001

B = 40 , T = 20

1 .000 .000 .000 .001

2 .000 .000 .002 .016

3 .000 .000 .011 .028

4 .000 .002 .021 .033

5 .000 .006 .027 .035

6 .001 .012 .031 .035

7 .004 .017 .033 .035

8 .006 .021 .034 .035

9 .010 .025 .035 .035

12 .019 .032 .035 .035

15 .026 .034 .035 .035

0 . 9

„ 228

c 326

o 337

. 338

.338

.338

.338

.338

.338

.338

.338

.000

.001

.001

.001

.001

.001

.001

.001

.001

.001

.001

.014

.033

.035

.035

.035

.035

.035

.035

.035

.035

.035

Memorandum Number 8

Tables of Ayg Tgt Dn in Uncoordinated System

TO : Dr. E. Biser February 16, 1956

FROM : K. Goldberg

SUBJECT : Avg Tgt Dn in the Uncoordinated System

The table of values of the average number of targets downed is

attached. It was computed from your formula

Avg Tgt Dn = T^l-(l-^5-)

B

j

, a = (l-p

)“/B

m

You will note that for large p or large /B the values tend to a

limit. In these cases a^O so that

< l B ®

(Avg Tgt Dn)/T^l-(l-4 ) ^1 - e" T

and in such cases Avg Tgt Dn may be approximated from this formula.

We tabulate the limiting values for Avg Tgt Dn below:

T\ 10 20 30 40

5 4.5 4.9 5.0 5.0

10 6.5 oo •

oo 9.6 9.9

15 7.5 11.2 13.1 14.1

20 CO

• o 12.8 15.7 17.4

-2 -

Avg Tgt Dn in the Uncoordinated System

T = 5

B =10 B = 20

m/B 0.2 0.3 0.5 0.7 0.9 0.2 0.3 0.5 0.7 0.9

1 1.7 2.3 3.3 3.9 4.3 2.8 3.5 4.4 4.8 4.9

2 2.6 3.3 4.0 4.3 4.4 3.9 4.4 4.8 4.9 4.9

3 3.2 3.8 4.3 4.4 4.5 4.4 4.7 4.9 4.9 4.9

4 3.6 4.0 4.4 4.5 4.5 4.6 4.8 4.9 4.9 4.9

5 3.8 4.2 4.4 4.5 4.5 4.7 4.9 4.9 4.9 4.9

6 4.0 4.3 4.4 4.5 4.5 4.8 4.9 4.9 4.9 4.9

7 4.1 4.3 4.5 4.5 4.5 4.8 4.9 4.9 4.9 4.9

8 4.2 4.4 4.5 4.5 4.5 4.9 4.9 4.9 4.9 4.9

9 4.3 4.4 4.5 4.5 4.5 4.9 4.9 4.9 4.9 4.9

12 4.4 4.4 4.5 4.5 4.5 4.9 4.9 4.9 4.9 4.9

15 4.4 4.5 4.5 4.5 4.5 4.9 4.9 4.9 4.9 4.9

B = 30 B = 40

1 3.5 4.2 4.8 4.9 5.0 4.0 4.6 4.9 5.0 5.0

2 4.5 4.8 5.0 5.0 5.0 4.7 4.9 5.0 5.0 5.0

3 4.8 4.9 5.0 5.0 5.0 4.9 5.0 5.0 5.0 5.0

4 4.9 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

5 4.9 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

6 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

7 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

8 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

9 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

12 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

13 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0

T = 10

B = 10 B = 20

1 1.8 2.6 4.0 5.2 6.1 3.3 4.6 6.4 7.7 8.5

2 3.1 4.1 5.4 6.1 6.5 5.2 6.5 7.9 8.5 8.8

3 3.9 4.9 6.0 6.4 6.5 6.3 7.4 8.4 8.7 8.8

4 4.6 5.5 6.3 6.5 6.5 7.0 7.9 8.6 8.8 8.8

5 5.0 5.8 6.4 6.5 6.5 7.5 8.2 8.7 8.8 8.8

6 5.4 6.0 6.5 6.5 6.5 7.8 8.4 8.7 8.8 8.8

7 5.6 6.2 6.5 6.5 6.5 8.1 8.5 8.8 8.8 8.8

8 5.8 6.3 6.5 6.5 6.5 8.2 8.6 8.8 8.8 8.8

9 6.0 6 . 4 6.5 6.5 6.5 8.4 8.7 8.8 8.8 8.8

12 6.2 6.5 6.5 6.5 6.5 8.6 8.7 8.8 8.8 8.8

15 6.4 6.5 6.5 6.5 6.5 8.7 8.8 8.8 8.8 8.8

m/B 0.2 0.3 in o © 0.7 0.9

B = 30

1 4.5 6.0 7.9 8.9 9.4

2 6.7 7.9 9.0 9.4 9.6

3 7.8 8.7 9.4 9.5 9.6

4 8.4 9.1 9.5 9.6 9.6

5 8.8 9.3 9.5 9.6 9.6

l * 6 9.0 9.4 9.6 9.6 9.6

^ 7 9.2 9.4 9.6 9.6 9.6

I 8 9.3 9.5 9.6 9.6 9.6

r 9 9.3 9.5 9.6 9.6 9.6

12 9.5 9.6 9.6 9.6 9.6

15 9.5 9.6 9.6 9.6 9.6

B = 10

1 1.9 2.7 4.3 5.7 6.9

2 3.2 4.4 6.0 7.0 7.4

3 4.2 5.4 6.8 7.3 7.5

4 5.0 6.9 7.1 7.4 7.5

5 5.5 6.5 7.3 7.5 7.5

6 5.9 6.8 7.4 7.5 7.5

7 6.3 7.0 7.4 7.5 7.5

8 6.5 7.2 7.5 7.5 7.5

9 6.7 7.3 7.5 7.5 7.5

12 7.1 7.4 7.5 7.5 7.5

15 7.3 7.5 7.5 7.5 7.5

B = 30

1 5.0 6.8 9.6 11.2 12.7

2 7.8 9.7 11.8 12.7 13.1

3 9.4 11.1 12.5 13.0 13.1

4 10,5 11.8 12.8 13.1 13.1

5 11.2 12.3 13.0 13.1 13.1

6 11.7 12.6 13.0 13.1 13.1

7 12.0 12.7 13.1 13.1 13.1

8 12.3 12.9 13.1 13.1 13.1

9 12.5 12.9 13.1 13.1 13.1

12 12.8 13.0 13.1 13.1 13.1

15 13.0 13.1 13.1 13.1 13.1

X

0.2 0.3 0.5 0.7 0.9

B = 40

X

5.5 7.0 8.7 9.5 9.8

7.7 8.8 9.6 9.8 9.8

8.6 9.3 9.7 9.8 9.9

9.1 9.6 9.8 9.8 9.9

9.4 9.7 9.8 9.9 9.9

9.5 9.8 9.8 9.9 9.9

9.6 9.8 9.8 9.9 9.9

9.7 9.8 9.8 9.9 9.9

9.7 9.8 9.9 9.9 9.9

9.8 9.8 9.9 9.9 9.9

9.8 9.8 9.9 9.9 9.9

B = 20

3.5 5.0 7.4 9.2 10.6

5.8 7.5 9.6 10.7 11.2

7.3 8.9 10.5 11.1 11.2

8.3 9.7 10.9 11.2 11.2

9.0 10.2 11.1 11.2 11.2

9.5 10.5 11.1 11.2 11.2

9.9 10.8 11.2 11.2 11.2

10.2 10.9 11.2 11.2 11.2

10.4 11.0 11.2 11.2 11.2

10.8 11.2 11.2 11.2 11.2

11.0 11.2 11.2 11.2 11.2

B = 40

6.2 8.3 11.1 12.8 13.7

9.3 11.2 13.1 13.8 14.0

11.0 12.5 13.6 14.0 14.0

12.0 13.1 13.9 14.0 14.1

12.6 13.5 14.0 14.0 14.1

13.0 13.7 14.0 14,0 14.1

13.3 13.8 14.0 14.0 14.1

13.5 13.9 14.0 14.1 14.1

13.6 13.9 14.0 14.1 14.1

13.8 14.0 14.0 14.1 14.1

14.0 14.0 14.1 14.1 14.1

-4-

m/B 0.2 0.3 0.5 0.7 0.9 0.2 0.3 0.5 0.7 0.9

1 B = 10 B = 20

1 1.9 2.8 4.5 6.0 7.4 3.6 5.2 7.9 10.2 12.0

2 3.3 4.6 6.4 7.4 8.0 6.1 8.1 10.7 12.1 12.8

3 4.4 5.7 7.2 7.9 8.0 7.8 9.7 11.8 12.6 12.8

4 5.2 6.4 7.6 8.0 8.0 9.0 10.8 12.3 12.8 12.8

5 5.8 6.9 7.8 8.0 8.0 9.9 11.4 12.6 12.8 12.8

6 6.3 7.3 7.9 8.0 8.0 10.6 11.9 12.7 12.8 12.8

7 6.6 7.5 8.0 8.0 8.0 11.1 12.1 12.8 12.8 12.8

8 6.9 7.7 8.0 8.0 8.0 11.5 12.4 12.8 12.8 12.8

9 7.2 7.8 8.0 8.0 8.0 11.7 12.5 12.8 12.8 12.8

12 7.6 7.9 8.0 8.0 8.0 12.3 12.7 12.8 12.8 12.8

15 7.8 8.0 8.0 8.0 8.0 12.6 12.8 12.8 12.8 12.8

B = 30 B = 40

1 5.2 7.3 10.6 13.1 15.0 6.6 9.1 12.7 15.2 16.8

2 8.4 10.8 13.6 15.1 15.6 10.3 12.9 15.7 16.9 17.4

3 10.5 12.7 14.8 15.5 15.7 12.6 14.7 16;7 17.3 17.4

4 11.9 13.7 15.3 15.7 15.7 14.0 15.8 17.1 17.4 17.4

5 12.8 14.4 15.5 15.7 15.7 14.9 16.3 17.3 17.4 17.4

6 13.5 14.8 15.6 15.7 15.7 15.6 16.7 17.3 17.4 17.4

7 14.0 15.1 15.7 15.7 15.7 16.0 16.9 17.4 17.4 17.4

8 14.4 15.3 15.7 15.7 15.7 16.3 17.1 17.4 17.4 17.4

9 14.7 15.4 15.7 15.7 15.7 16.6 17.2 17.4 17.4 17.4

12 15.2 15.6 15.7 15.7 15.7 17.0 17.4 17.4 17.4 17.4

15 15.5 15.7 15.7 15.7 15.7 17.2 17.4 17.4 17.4 17.4

Memor<ZY\au,m /Vo.Z

°-f fi 5 A^o.*/5"7l

Uncoordinated System Based on

Constant Kill Probability

by

Joan Raup Rosenblatt

National Bureau of Standards

Given B batteries, T targets, m missiles, and constant

kill probability p.

Let a = (1 - p)

/

, where m/B is assumed to be an integer* The number a is the probability that a target will

survive when fired on by one battery, since it is assumed that

each battery will fire all of its m/B missiles at one target.

Under the uncoordinated system, each battery selects its

target at random, independent of selections made by other batteries . An engagement is characterized by the numbers k^ , k^,

. . „ , k^ , where k. denotes the number of batteries which fire

on target i. B = k, + ... + k^ .

Quantities which may be used to describe the properties of

this system include the following:

(1) Average number of targets down.

Avg Tgt £>n = T

(2) Probability of non-penetration (i.e., probability that T

targets are down)

.

-2 -

(3) Variance of the number of targets down.

Var (Tgt Dn) = T (\ - ^rj

B

+ T(T-l) (\ - 2 T

2^

1 - 1 ~ a ^f

B

(4) For each n (n = 0, 1, . .., T), the probability pn

that

exactly n targets are down.

p = Pnp (see (2) )

,

T

(n + i)

1 -a

1 B

T J

n = 0, 1, o d 5 T

In particular,

P

1 = T

B

- TaB

These quantities are derived in the following.

1. Distribution of Number of Targets Down . The random variable with which we are concerned is N, the

number of targets down. Let the distribution of N be given

by (p , P , P

T

> , where

P

n

=Pr(N=n) , n=0,l, . .., T*

-3

-

Then, in particular.

From the distribution of N, we also obtain

T

Avg Tgt Dn = EN = Z np , n=0 n

and

T

Var (Tgt Dn) = E(N - EN) 2 = Z n

2

p - (EN)

2

. n=0 n :

The quantities • • • ; P,Tare not convenient expressions.

however, so that we do not evaluate EN and Var N from the

relations given above.

2. Evaluation of p . n

*D Observe that there are T possible equally likely config

urations of assignments of B batteries to T targets. Let

K = (k^, . .., kp) represent the configuration of assignments

in one engagement where k_^ batteries fire at the i th target.

Corresponding to each K, there are (®) configurations which

K

differ only in that a different set of k

i

batteries is assigned

to the ith target.

»

-kThus, we have

p = T

-B

2 (|) p

(K)

, n = 0, 1, .... T,

n n

where p denotes the conditional probability that exactly

n

n targets are down in an engagement with configuration K . Furthermore , for any function f (n)

,

we have

Ef(N) = 2 f (n)p = T“B

2 (?) 1 2 f(n) p

(K)

n=0 n K

K

L n=0 n

We will write

T (K)

E f(N) = 2 f(n) p K n=0 n

to denote the conditional expectation of f(N) when K is fixed.

(K)

3. Evaluation of p

n

Observe that N may be regarded as the sum of T random

variables, N=X + ... + X , 1 T

where X^ has the value one or zero according as the i th target is down or survives.

Now, when K is fixed, the X^ are mutually independent.

The conditional distribution of X is given by

-5

-

, k

Pr (X

i = l|K) = 1 - a 1

Pr (X

i = 0|K) = a

k

i

, i = 1, ..., T.

We have, then,

(K) -

T

,, ku

T i=l

(K) T k± B

p = it a = a- 0 i=l

and for n = 2, 3, . .., T - 1,

j + ...

1 2

B

a

(K)

Except for p^ , these are not convenient expressions to

use in the relations given in the preceding section for calculating Ef(n). Another method will be developed to carry out

those calculations. We are, however, in a position to complete

the evaluation of p (n=0, 1, ..., T).

n

Ll. Calculation of p . n

B

a) Evidently, p = a

- 6 -

b)

?i = T (a + - Ta

B

.

T k ,• g = 2 tc a - Ta

i=l JH

| <i^

( °

Now, in general.

-B r

^

T

D

2 (®)a 1 ... 2 K

K

r _= ( 1 - r

1-a B

Thus

,

/ i _ a \

® ®

p = T Ql - ( T—1) - Ta .

c) The remaining expressions for are calculated in similar

fashion.

f>. Calculation of Avg Tgt Dn .

EN = T

E N = K

'B

2 ( B

) E N

K

K K

T T

2 E X, = 2 (1

1=1 K

1 3=1

- 7 -

The calculation of the expression given above for Avg Tgt

Dn is straightforward.

6, Calculation of Var(Tgt Dn) .

E(N - EN) 2 = E(N - E N)

2

+ E(E N - EN) 2

a) E( N - E N)'

K

But E (N - E N)

K K

random variables

K K

= T

“B

2 (?) E (N - E N) 2

K K K

p

is simply the variance of a sum of independent

, since N = X_ + ...+}(_ as noted above.

ki ki

Now Var X. = a (1 - a ) , i = 1, . .., T .

Hence

P R r

T ki k i E(N - E N) - T“b

2 (®) 2 a

1

(1 - a

1

). k K

K

i=l

b) E(E N - EN) 2 = T -B

2 (?) (E N)

2

- (EN) 2

K K K

Var N is obtained by evaluating the two terms

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