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r AD AO7T 833 CENTER FOR NAVAL ANALYSES ALEXAIICRIA VA NAVAL WARFARE—ETC F/ t5~ 3 ~~ JA CLASS OF CO$4PRJTATIVE NARKOV NATRICES~~ U) N

NOV 79 QV SI .A SS . I$ S U . W R N % f t t $UNCLASSIFIED CNA—PP—259 NL

I.O~~~~

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I .1 I C’

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/ /) PROFESSIONAL PAPE5 259 Nova~~~ r W9 /

~~A CLASS OF COMMUTATIVE/

L~It~RKOV MATRICES r

IPavid Vbass# I/ Ih-Chir1g11su~ // Walter R.,f~unn

L~~~~ermn

D D~~iur~~~~~~~~~~

Approved for p~~~~ r~~

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CENTER FOR NAVAL ANALYSES2000 North Beauregard Street , Alexandria, Virginia 22311

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~_~~~___s. ~ rt~~~~_ —~

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The Ideas expressed in this piper are those of the authors.The paper does not necessarily represent the views of either theCenter for Naval Analyses or the Department of Defense.

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PROFESSIONAL PAPER 259 / November 1979

A CLASS OF COMMUTATIVEMARKOV MATRICES

David V. GlassIb-Ching HsuWalter R. NunnDavid A. Pen n

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Navel Warfare Analysis Group

CENTER FOR NAVAL ANALYSES

T°° NTth T~ITTIdnInhIThl

_ _ _ _ _ _ _ _ _ _ -~~— - -.-— ,.---~ --~~-~~~~~~~ -— .‘

Ir;~r~ ‘1’

~~~~~~~~~~~~ ~ cl~~ ;’-; c t ~~ t k o v m~i t t i c ~~:; w h i c h

~i r 1 ~ i n i imp i n ~~~~~ I d ~~~ t ; \’ :; ~~~~ ~~~~~~~~ ~ (FtJ F~~~M A~~

The I f l o d f ’ 1. ii I ~i~;t r.~ t m ‘t~ ~k ’ v c ! t i ~ w h i c! i t ;

n~ t t i me — ho mo qe n~’~~u s t )u t. i ; t i I 1 ami~ f l I h i e t ~

an~l 1vt i c t t e t t : ; ~ T h ’ i i . i t r i cv~; 1 t -~ • -~t t ~~~~ t i t o

• ~ i1 mu t it I V ’ ’ ~l u ~ t ~‘1 m ‘ I ;; hoW n t n

C ~ ~t~~~~t i U I l & I ’~ 1 t t ~~ t. I I X nm 1 ~ i ~ I i ~~ it i )f l . The 3 ~m , i t r i o :e t ; •~~re u l s~ ~;h oW n I ~) he i i~i o n a l i r i h l e

and t h e c i e f l vo ’ c t or ; ; h~~ve 1 ;; i m p l e I -~~rt i t , name —

l v compo i-;~’d i-- i t e 1 t i ~ n t ~; o t P n ; c a l ‘s ~ r i in e l e .

A de ; ;c r i pt i on of t h e ci ense • ;v ’~t em mode I

j i ye n an d t t i i m p 1 i c~j t i ~)n C) I t he ma I h ema t i ca 1

r~~~ t i lt t o r t h i • ; niod ’ I i re 1 i ;; ~~~~~~~~~~~~~ 1.

L ~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~

~ ~~~—

~~~- -—-

~~ --- - --

I!~~~~‘i tv r tor i

A do f .~n;; e s vs I ’ ‘in nod. I , I . ; ; c r i t ed 1 i t e t , I ea I ;; t o t I i

t o t i l a t on ~ •i c Urs ;; et n a t t - i ’ r ; w i t h n i t vit t v t ;

.ii iid n g r u t I w i t h ‘ l O n t i t ~~~~ A , w~ i ’• 1 )

(_) i • ~~~ tA = • I i — I 1

11 1 I ’ ‘ i

If o r q iv i ’ n ~ ~i n i ~ w i th ()~ p~ I and 1 = p+q . ~h i n i )

r e r ’ en t ’~ t h e p r o b a b i l i ty o t ] I i i l u t e ’ ; i n i

t I r i a l s , q i v e n q , t he n i e h i b i I i t y of a L i i l u r e

i n a s i nq i . t r i a l . I i ! ; i m p I i c i t ~’ , I h i ’ ; o1a ~~;; of

m a t r ic e s w~ 11 he r e f er r e d t o is typ e

~or typ e ~ m a t r i ce s , we ‘s t a h l i ; ;h a pr oo f of

mim i t ip i i c t t ive c o n m i i t r u t ‘v i t y by I h i t t i o n a l j~~a t i o n

1)t ’~~ t~~’5 V i i C i q ’n v a l u e s arl( 1 ‘i qen ect a i s . I n d o i n g so ,

we d i ; ;c ov . ’r t h e ~n t r i q u i n i r ;u l ~ t h •~~ t i . ’ & omp l ’ t ~~ ;I

of L i n e a r N’ in d ep ’n d en t e i~~o ’n v ct e r ; l i t ti ; ; 1 low t

t r i a n g u l a r m a t r i x whose row s r u t ’ I I lo ;;e ot I i ’ a l ’ s

t r i i n .j l e . The oj i e n v e c l o r s a t ; in d ep e n d en t ot p c i i ~;.

A lso , we show t h a t the c i a ;s ot m a t r i ces of t~~pe .1 1

closed unde r matrix m u l t i p l ic a t i o n .

I t i e .b t I v a t i ’ )fl ( i t t Im. ’ ;. nd t I i i ’lfld t j c i i t . ’ s i i Its , t he

i . ’st ol h . p a } ’ r t s dev ~~t i d t a ~~~~ ex p l a n a t t o i l ot t h i ’

t I . ot t y p. ’ n m I r ic.’ ; 1 n t h e mode I , and I o t !n’

j n t c m m . ’t i ’ t o n ~~t t h e n i t h i m i t i c i l i n s u l t s t a r t h i ; ;

tn ~ I ’ I

4 ‘IAT H l : M A T I C A E ~ R l i S t I I T S

l h ’ot i ’ m i

Eot any posit iv e i n tt ’clvr ml , let A he an (n+l ) t v ( n + l

nat r ix ot typ~ i t W i t h p~~O , q ~o and p+q = 1

Then A i s d i a 1o n ~t 1 i~~i I lt’ with A = , w h e r e

I) = D i a i 1., g , . . . , 1~ } and , i n d e p en d e n t ~ f r and ‘j

S is simply a lower triangular matrix whose rows

coinc ide with those of Pascal ‘s triangle ( if presented

i n t he f o r m of a r i ght t r ianqic).

Proof.

Since I , ~~~

, q~~~’ , . .. , , wi t h , 0 q ~ 1 , a m . ’

the n+l distin ct eiqenvalues 01 matr ix A , A can he

d iaqonal ized to I) = P i l l 1 , q , . . . , q”.}

Ma th e mi t ica 1 induct ion on n will he used t o pr ove AS SD .

Prom h er e on , approp r iate subscripts wil l be used to

i n d i ca t e s izes of nat r i ce s u n d e r i n v e s t i q a t i o n ;

t h e r e f o r e :

( ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ _~ -_ — • — —

— ~~~~~~~~ _ _ . J L - .__.._-~

_ -~‘

n n n ~~I

E E E~~

i p ( i ) ! i N ) •! — —~~~~~~~~~~~

— — — 1

1 1) 0 0 — - - — — 0

1 1 0 0 — — — — 0

S = 1 2 1 (1 — — — — 0n + l

I n 1 , n \ / n 1 — •— — — —I l l 1 2 r ’

When n = l , A 2 = and S~, 1~ ~I

It can be easily verified tha t A 2S , = S 2 D 2

~~~~~ 3

p.- - _ _ _

We W a n t t0 C~~t~~ ) 1j h A ~

f l 4 1

S ~~~~~ “ I t r i x k ) l

~~ck ~flu l ~ i pi

~~ Cat ion Y1e1 d~;:

I 0lPfl

P (ç ) P~-1

~ • .. I S ) + q fl( ( ~~) , ( ç ) ,

0‘ 0

S [ ) Ifl~~~

sn÷l nn+l

I‘ 0

~~~ (~~)q ,

• • • ,

— ~~~~~~~~ - -—~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- - -- ~~~~~~~~~ ~~~~~~~~~~~~~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~

~. 1 r~~’ — - — . - ~~~~~~ -.- ~~~~~~~~~ ~~~~~~~ • _ _

S i ”c e A S S P , i i i t i t r i w i ins to ~~~~~~~ t i t e rIi II fl n

c om p a r is o n °~ ~~~~~~~~ w t t h n+ 1~~n + l ’ I S I L it :

L ( ~~’ n1t , (1 t t ~~ ! ,

11~~1 tx 1

hi_~~ l L;~~l ~ ~~~~~~~~~~ (

II = I (~~)~ (

li) ‘ ~n-.l

~~ri-I (1

l’o es tab l ish t h e e~~i u a L it y , we I i r s t ex p an d a t y n i c a l ,

3~i\’ ( r + I I s t , t .5r m ot I i t R ,H . S . ot .;guat ian 1i

r n‘~~ ~r 1 wher e r = 0, 1 , , . . . n — 1 ,

ii” ( ) = q ( ) ( p+q n — r i nce p+q = l . P xpan d nq ( p+ ’-~

) ~~~~~

r n n—r n — r 0 n — r n— r—1 1 n—i 1 n — r — 1= q

~~~~~~ ~~ ) p

~ ~ ) i q + . .. ( 1 )p ~

n — r it+

~~~~~~~~~~~~ ~i I

r n n — r n — r 0 n n—r n— r—i 1 n n--i 0 i i — t= q I ‘~r~ ~ 0 ~~ q ~ ~ r~ ~ i -1 + . • . ( ) ( )i r]

r n r n— i - I) n r+ i . n — r — l 1 n ri 0 n — r= q L ~ ( 1 )P ~ r+ 1 ~ r ~ + ~

(~~) ‘~ r~~

S U - . (fl

) ( fl~~~I)

~ t + k )( r#k ) or ~ ~ mu- i

Ni m s , i t ( ’H as t h e f a t [owl;; m

r ii n r r i—r r ii r~~1 n — i — I r i - I fl ~~ 0 ~

~~~~~~~ ~ 1 +

T h e I a ; m t t i e r ) ot the . It • S Of •~~~~~

tj i ~ i on I i t i d i ’ l t i ; a

nt h a t its (r + l ) s t term is t h e sum of 1~~~~) ar i d a dot

Wh i le q °~ (fl

) is imply the (r~~1 )~~t element in

[ ( ~~~) , (~~ ) , .. . ,(~~~~1 H , the dot product is that of the

first n elenents of the (n+1)st rt~~ i ;~ A an d th e—- n + l

(ri-i )st column i n S , t h i s co lumn being equal to

[0 , . . . , 0 , (c

) , ( r+ 1 ) , (~~~~

) . Af te r e x p a n s i on ,

m m r n—r r nthe dot produce 1;~ e q u a l to ~~

C 1~)p ~ ~ r+i

r*l n—r ÷l r+l n n— i n—I -

r q + . . + n—i~ r ~ • t’~ ’t i~~~ S u m

; 1 ;m s q fl (

fl ) is indeed id . ’nt i c al to t i e e a r l j ar ex p a n —

s ion I or the ( r + i ) st t e rm of t h e R.}1 .S . of e q u a t i o n 1

The i’iu ality A~ ÷ 1S~~~1 = S~~~1%~~1 is now establ ished ,

and this completes the proof.

-~~~ —-

~~~~~~~~~~~~~~~~~~~~~~~ - —.5’ - - “--“ — -

I’’ -~~ t ’ ! n ~. “ t a t ’ I j i t c t t t t ’ . - .‘o(’t tm mt ~ m t I v L ~~y L~~~;: N) r

;;. t m ~~ t i . mn~ t r ic i-s m )t t’~’p&’ ‘t

[‘ r i l t . T (~ I i~~~r j Pr I . w ri ( nI l ) x ( r + l I ‘~~~~t t l . . - ~~; -~~~ !

‘‘ ~~~ ‘ U ’ ~t , ~r i :~~~ ~l ‘ ,

~n th A = an I 1~ t 1 0 5 ’ w ; ‘ s P a n d

ii ’ m r .- ~~~~~~~~~ ,u a t r ; c ‘ ; ~m r i . I S is i : ; . 1 w .r ‘ i i n m m l m m

ma i x l e sc i i d in ‘‘h ’ i ; 1 . “he c m : - ’ ; ; i t iv I t V ( i t A

it T m w m ; ~ v i den , s i t — n

AR ~~~~~~~~~~~

I ., , , , — I 1 • 1 S

• We com m e n t h a t t he t i t r i c e ; S , ~~~ I h i , I t h e~

e q ual t o (1 1 , i n ; see’u i n comi i na t r r i c s et t m c ; , .‘. i . ,

I , ’ ’ e ren c i ? 1 , t ’ t q ’~ 15. ~~~‘ ‘~~ et Ice 1 a l - S T) f ’t ’ ~) \ c I I J~ h . ’

i , t h 1 . n;. n - ‘t S~~ i ;~ ~ j y n l i by ( — 1 ) ~~ ) , so t

~~— I

~ ~ i i n ’ I d i rn~ - I I~ ’ I ~-

~ b i ;;e in i “ ‘ a

app~~o ; r ; m t e l o c a t i o ns . Uence , th e ; r ’ c I t m I

de’ comno - ; i’ i on I A can he wr t ‘- . - ru d ~wn a I ;m s; S -

ins pection , i.e.,

A~~~1 =

~!j~ i j + q~~2~~ + . . . + ~~~~~~~~~~~~ I ’

- A~-~ — _..~~~~~~~~~~~~~~

_ _ _ _ _ _ _ _ _ _ _ _ _ • .-~~~~~ ,--~~~~~-• -•- - -~~ -.- --- • -—_---—

-s’~~~’? ’ ‘h .’ r ’ i I ’ i~~’i V. ’t i r ( 1 , 15 t h . k t h L’r ) l ~ f l l ; i l ot

• t ; ; t t L ’ t O W v’ c t o r V I S •_ H ’ k t h row f—k — n + l

~ I i r ’ ’t ‘~‘ - l t t j C i~~~~i ’ ; i of T i m i _ o t - ‘~~~~~~~ is i ) ; ! ’ . - t j : .

; ‘ 11! t o;;. t h 5 i t I j ; ~~ i , -;t h ‘lp : ; ; . - ’ it Of A is (~~1 ~~

a n d t h e ~~, k t h “ I - - n t 01 n i s ( 12 ~~~~~~~ . n t h e

i , k t h r ’ j , ’ f l O f l t at Mt i s t i : . ;; h y

~~~( ~) (~~) (l~~

q 2k e l

i_ i .) i k

~~~ 1 i~~2~~ ~~ ~~~~~~~~~~~~ ( 1 1 ) J~ k ~~ 1-J

k i—k=

~~~ (q 1~~;, ) ( 1 — q 1

i - i i — i - -w ;; - r e wi h a n ’ u a e l t h e i d e n t i t y (J

) C~~) =

~~~ ~~~~~~~~~~~~~~

and r . conn i ‘e l t h i bi no m i a l exp ans ion of ~~~

+ ~i n ) i—k s

Clea rl y , t h e steps r e m a i n v a l i d w i t h q 1 and q~ in ter—

chan l e -j , an d t h e c om mu t at i v i tv of A and B f o l l o ws .

~~~~~~

— -- ~~~—“ —— - - - ~~~—‘ ‘- —--—‘— ——-~~~~~~‘~~~~—— ~~ —“-—~~~ —--—~ --- _— -----~ --‘•-•-, ‘—--—‘ “—-- -_‘--— -— --_ _ - - — - —- —-.—

Fur H i er ,~~:ncn i n a t i o n of t he p roof shows t h at we havr_-

m iso shown t h a t t h e c lass of m a t r ic e s of typ c~ is

~ 1 ( ) s e I u n d e r ma t - n i x m u l ’ ipl. i c a t ion . Mor e or : c i s e. Iy

Th~-o rem 3. I f A and B a re m a t r i er s of type a t t h e

same or h - r m l w i t h parame t r~ , r esp - r : t i v e l y ,2

t h e n AB is of type (~~ w i t h p a r a m e t e r q 1q 2 .

Theorem 3 will be m sefui for providing computationalp

simplification in our model.

MODEL DESCRIPTION

The f o r e g o i n g c lass of m a t r i c e s a r i s e in the c o n t e x t

of a defense system model in which a cluster of n

missiles attemp ts to penetrate several lines of

defense. The number of missiles surviving the first

l ine of defense is Binomially distribu ted with

parameters n , q1, i.e. , there are i survivors with

n i n—i -probahility (.)q1 p 1 , i = 0 , l , . . ., n , and q 1 i—p t .

Suppose tha t n 1 m i s s il e s s u r v i v e the fi rs t l i n e of

defense. We assume that the number of missile s

surv iving the second line of defense is Rinomially

d j ; ; t m i i i m ~~, r I w j t h j m r ~in~~~~~~t 1) 1, ‘ l - . ~ and n t c . a

I n sh i o- I , we i s sume t h a t I h i - 1 m et ; ot dot enso arc

i ri I . t i ’m i d o m ’mt. a nd p r a m h i c . - . i t t n i t io n in ac. ’o rdan i ’t ’ W I tIi a

~ i m i o m n i i I ‘ l i st i — t i s i t 1°;; e~-ict i t i m i . ’ n t l e t en se h a v i n r i i t ; ;

~~~ ‘ t ; i cc . ’ - - ; - ii ~i t i I m I y i r i d a ;t ;oc i a I n i

I t i s con’ ’i ; n i er i t. to i a ’ q i r d t i u t ’ m i ;;s i 1.’s • passaqe

t h r o t m c ! a p u t i cmi i i n Ii ne o~ t - ’ f i ’ t i t -a ’ is a t t-ons i t ion i n

~m ‘ i m r k~~ ’ oh u n ( i . - t i ’ n ’ - v m c . - 7 ) . i-’o r - x a m p i i ’ , oj v’s ii n =

an d = 0..., t h e u - i 1 m I x b e l o w i s t i ; . t m ’ i u : ; i t ion matrix

I or p e f l et u - m t ion it the I’ inst 1 inc at defense.

0 1 0 0

Tnt 1 . f .4 0

7 . i t . . . i i ’ -

It i t ; a st a n d a r d t ’esul t in pr~~ i~Th il i ty theory t h ati t t h e t he t a n d em v - i t i oh n , -~~t v on N , i d i , ; t m i hut ed asP i nap; i i i w i t h s a l a m i . - t . e r -; N , p , 1 nd i t N m d m t , ihu t e d as

• f l i n o m i i 1 w i t h j i ar a m e t . ’r s N* ,p * , t h e n the t m n c o u d i t i o n a l• -l i - ; t i i 1)11 t ion of X i U i norn i m l Wi I ii p a r a rne t i -r i ; N * ,

The proof is v i r t u a l l y I d e n t i c a l w i t h t h a t o~ th eorem ;;iI en i -’.~ we have .1 ~~T o L r i h j I i s t - ~~ proo f t h a t t h e n u m b er

of t _ m m i v i v a m s a t the second l i i i . ol d c t • ‘m m s e i ; ; d i s t r i h m i t e diS B i n o m i a l w i t h p i i - i m m m e t e t - s m~, ~~~~~~~ and an i n d u c t ‘mo r n

a r gu m e n t qj v e s t h e g e m m n r a l i , - s u l Ii I or s e v er i l l i n es a tdefens e

• 10

• • • _ _ _ • ~~~~ _~~1__ __

-- ~~~~•-

Thu s , i f 7 n i t ‘ ; - 1 I - - m i t - i I i i i t ; 1 i m~~’ at ib - f en s i - , I I mm 1

p1 ~l i t i Ii I .v t t i - i ’ I mj;;’; I I , ’ , ‘m * - p l os i ; ; 1 . 4 H . - t o r i -

t I i ; ’ f l i t - : i m . m t n i x i i ; imi st A d i t ‘m m ; . - d i l l I l i e ! h t ~~~~~~~~~ , d i f l 1

5 0- 1 I n i ( ~ m ‘ I ; p = 0 . . t i m - m i e n - i I t I m - m m m - ~ - m

m r i —~~-’ i s - i t n t ~~; i I ‘ ; m ii i ih I ‘ a d i ‘;- ‘~~~e n , i e n ~~~~ ‘ i a

l i I V 1 i l l i 1 ’~ h i t ~ i I I ty i-i m s t - m i l i m i t i on r , w I i o i r i i t ; r ’i ’i • ;,

n t ;i I S -~ i t e n : 1 ~ i V I I h , - 1 i f in I - I n l i t ’ I ; ; - i r id I

a - - m m 1 t i ~ I c 1 t h . ’ f t - n - t m I i am p t 1 1 x 1 ~ i . ~ ~~ \~~~~~~~‘ a i:

‘1’ . 1 fl OIl I i ‘ X d U~~) 1 ” , i t I ~ - r i n , i I ) i i I I ic’; 01 Ii l v i i i ;

I n c o m i n g m i s s i l e s w o n ’- ~‘m •r~ , •

r ; , t - n I T h ’ ( ’t _ iv i - - l\ ’ ,

th .-’n the ( ( i m j ) u l t d t i ( i u m W ( ) t l I . 1 t m , ’I -I

(0 . 5 . 5 ) ~l (1

.( ~ .4 0 = (.4~i . 4 4 . 0 8)

—. i i ’ . 40 .1 (~

r ;a t h a t t he p r o b i t i i I i t V ol l i a v i nq 0 ,1 , ’ I i i ‘ ; - : 3 i f ’ ;

penet n i t c t he I in;; t 1 ; no of d c l i t t l U ’ W o t i i~i I i . - .4 0 , . 4 4

.08 , respectiv e iy. This m i ’ t ; m i l t ~m n t v ” c t or ~‘‘! \‘ m -s is t i n ’

i n p u t to thc second I i m m e ot dcl ens’’, and l.-y ;~ i m ~~1 - , n , ; 1 \‘c

su c h v e c t o r — m a t r i x m u l t i p i . i c . i t i O n ;~ I i i . ’ ~i r o b a I i i l i t v .~ t.

h a v i n g 0 , 1 , a m ’ 2 mt s’; i l e t ; P~~n et I l I i~ a 11 t h e d ot . i ; : es

can he ca icul .itcd . Tn p a t t i c m i i •u , i t I el I o w ; - t h _ i I

- _~~~~ i _ _ ~~~~~~~~~~~~~~~~~~~~~~~~ — S ~~

t tm ir r e are I l eve is of do 1 ’nse and the t r an s it ion matr ix

for each leve l is the m a t r i x A 3( k ) , k= 1 , 2 , . . . , L , t h e n

the distribution of the number of penetremtors is given

L(0 .5 .5) II A

3( k )

More generally , if there may be as ma ny as n—i incoming

missiles then the vector T will need to have n ele-

men ts and the transition matrices will have to he of

size n x n. The reader should verify that the transi-

tion matrices for this general case are precisely the

matrices A~ defined in the previou s section , with p

as appropr iate. Thus, we have a Markov chain in

• discrete time which is not time—homogeneous , i.e., the

transition matrices are not identical , and for which —

is we shall see — closed form results are available.

DISC U SSE ON AND CONCLUS IONS

A non—tr ivial question is “Does the ordering of the

lines of defense make any difference?” . Mathe—

matica lly, this is the same as asking if the ordering

of the A~~(k) makes any difference in evalu ating the

____ _____________ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1 , n - i i m c t iT ‘~r m (k) . it i s wi 11 known t h a t m at m i x

mmi i t n i icat ton is n o n —ç o mm u t - it ive in n m i ’ n~’ r m 1., and s i n c e

0’ ’ A~~( k ) 1’ i t m i c e s wil l n o t u s u a l l y he i d e n t : i c a l

(linleSS t h e [‘k ’ ’~ . i m r m d . mil ical ) it. is n et - ) P V I n i s

;n~~f h e m m t j o , m l i v t h a t t h e ; . ’ mat r i c o t - wi~ 1 c om m u t e .

However , the results in the previou s section show t h a t

t h e A ,-1 ( k ) mat r i eon in comnm i i I~ ‘~‘ and t i m e o rder ing of the

lines of defense makes no lit 1 e renc ’ ’ i n th it ; m odel

The naive approach to eva luating the product ‘fl’ A~~( k )

is to do it by b r u t e fo rce , p e r f o r m i n g the i n d i c a t e d

m a t r i x m u l t i p l i ca t ions one a t a t ime . A more •

s o p h i s t i c a t e d approach is to use theorem 1 and

recognize t h a t the product can he w r it t e n as SDS 1

where 1) is the diagona l matrix whose i th d iag o n a l e n t r y

is t!q~1 , requiring only two matrix multiplication s . A

still more s o p h i s t i c al - r d a p p r o a c h involves theor~ mn 3 ,

seeing that the p r o d u c t can be written down im m e d i a t e l y

using no matrix muit ipljcationn at all. This illustrates

the practica l utility of increased mathematical insiqht

in analys is of our model.

- ‘--—~~- - — -• •— - ----

F i n a l l y , we observe that our results for this model find

app licat ion in a more genera l setting . For example ,

instead of mi ssiles let the survivors be ships in a task

fo rce , and i n s t e a d of l eve l s of d e f e n s e le t the

a t t r i t i o n occur because of a sequence of a t t acks on the

task force. Intuitive ly , one feels tha t an optima l

defense should he insensitive to the sequencing of

a t t a c k s , eve n in e l a b o r a t e mode ls , so t h a t t h e r e is some

possihiiity of the kind of computational simplifications

t h a t ~~ h~m - .’e se~~ i in o~~r su ~p 1e mode l . k

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