84

10 - POISSON STREAMS

10.1 Privileged transitions

The situation is that of sections 6,7,8 and 9. The main features of interest in the lJ"esent section are the existence of a stationary poi nt process

N = I cST and of a sequence of marks (U , n e ZI) where U ~(X l' nf'V n n nn-

~N,et'p) ~here 0n-1' Xn,Yn ) e

GxSxGx( n M). Recall that X • X(T ) is ses s n n the state at time Tn' a~d an-1 is the

o 0 site of A(X 1) whi ch triggers the transi tion X 1 + X , i.e. X 1 --!!:l.)X • If PN n- . n- n n- n is the Palm probability associated with (N,at,p), we have (Eq. (8.1.2» :

Let

H C GxScG

s be a subset of lJ"i vileged transi tions g + ~.

Let now NH be the point p-ocess associated with H, Le.

N ( .) • I 1 {( X ~ X)eM I cST (.) --H nf'V n-1' n-1' n n

Fran (10.1.1)

Calling"'H the right-hand side of (10.1.5) and applying (10.1.4), we have

P~(X_l·g, ~-ls, XO"'8', YO € ii> •

1 H(g,S,g')

and therefore

ACs ,g) A

----"- peg ,s ,g') PA(g)-{ s}( B)I "'H A(g)

(10.1 .2)

(10.1.3)

( 1 O. 1 .4)

( 10.1 .5)

(10.1.6)

pON (X o"'8' ,ioeS) • I 1H(g,s,g') ACs,g) p(g,s,g') (A ( PA(g)-{s} B)/L. 10.1.7) H (g ,s )eGxS A(g) °H

F. Baccelli et al., Palm Probabilities and Stationary Queues© Springer-Verlag Berlin Heidelberg 1987

Let l.5 recall that

= x g'

85

P A( g' ) (8) •

10.2 Sufficient conditions for Poissonian streCIIS

( 1 O. 1. 8)

Suppose that for all ~ e G, 8 € ( 0 (H n M »), the right-hand sides of aes s s

(10.1.7) and (10.1.8) are equal. It is then clear that the restrictions of P and

P~ to a( ~ ; n <: 1) are equal, where (~, n e Z/) is the sequence of p:>ints of NW H

From the characterization of Poisson processes of 87, Part 1, it follows that NH is a Poisson process.

The following condition is sufficient for this to happen

(g ,s ,g' ) € H ~ A( g') n s' = (A( g) - {s} n S'),

x = g'

L 1 H( g ,s ,g') x ).( s ,g) p (g ,s ,g' ) (s,g)eSxG g

L x ).(s,g) p(g,s,g') (g,s,g' )eH g

Indeed (10.2.1a) implies that for all B € ~ (Hs n Ms)

PA(g)_{s}(fl) & PA(g' )(8) , if (g,s,g') e H

and (10.2. lb) concl udes the proof. a

( 1 O. 2. 1 a)

(10.2.1b)

(10.2.2)

APPmroIX

Appendix 1. Change of scale

With each macrostate g e G, we associate a measurable flow (T:, tell) on

(n M, ® Ms) by s€S s f£S

s Note that if s r;. A(g), c(s,g) - 0, so that Ttc(s,g) ms ~ ms.

Define the following r.v. on (n , ® M) s€S f£S s

x (m ) inf { 1 s ; s e A( g) /,

c (s ,g)

where x (m ) is the first lXlint of ms strictly to the right of O. 1 s

Lanma

Proof

PAC ) is Tg-invariant. - g --,

Def ine the mappi ng k s,g

k (m) ( .) - L Ii ( ) (. ) • s,g s i£l/ Xi ms

C (s ,g)

(A.1.1)

(A.1.2)

( A. 1 • 3)

-1 This mapping performs a change of scale, so that if we denote n ~ P ok ,

s,g s s ,g (Ms,HS,nS,g) is a stationary point !l"ocess with the intensity >,(s,g) - >'sc(s,g). If

s e A(g), >.(s,g) > 0, and the Palm !l"obability nO associated with t ,n -1

ok. s ,g

s ,g s s,g is

Define the !l"obability Q on ( n Ms ' ® Hs) by seA ( g) f£A( g)

-Q = ® f£A( g)

n s,g

The Palm probability QO of Q with respect to

N = L seA(g)

is gi yen by

m s

(A. 1.11)

(A.1.5)

13 0 -

I A( g) I L

1-1

87

(A.l.6)

where {u 1' ••• ,ul A(g) I } is an enuneration of A(g). Clearly QO is T T' -invariant,

where'f' is the r.v.definedon ( II M, ® H)by seA(g) s £A(g) s

( A. 1. 7)

Denote by k the mappi ng from II Minto itself def ined by k «m , s eAt g») g seA(g) s g s

• (k (m), s e A(g». If T is the restriction of T to ( II Ms' ® H), and s,g s seA(g) £A(g) s

if T~ is the restriction of T: to this space, then T~ is the image of T t , by kg'

Therefore the probability

o IA(g)1 Q - L

1-1

is 18- invar i ant. 'f

The lanma is now established since QO is the restriction of P A(g) to

( II M, ® H), and since T:m & ms when s fI A(g). 0 seA(g) S £A(g) S s

(A. 1. 8)

Rancrk 1 : The point p"ocess N defined by (A.l.5) admits the intensity A(g) under = --Q, and the inversion formula giving Q in terms of QO is

'f' Q( A) • A (g) E':Qo [f 1( (T m , seA ( g » e A) d u ] , o u s

-1 - -1 Therefore, since Q • QOkg and QO • QO 0 kg ,

T Q(A) • A(g) EQo [f l((18m ,s e A(g» e A) du), o u s

R ancrk 2 : We have

EA [t] = E- [t']. P A(g) %

(A € ® £A(g)

H ). s

(A € ® H). £A(g) s

( A. 1. 9)

( A. 1 .10)

(A.l.l1)

88

Therefore in view of the construction of (Z(t), t G- R ) in !!6, and of the

definition ofT,

(A.1.'3)

where the 1 ast i nequal1ty f oll ows from ass un pt ion A.2 of !!7. U si ng (7.'.4) we obtain

E O[ T, ] c K LX- K P g£G g

(A.,.,4)

Appendix 2. Proof' of Insensl t1Ylty (Proof' of' (8.1.1»

We aoopt the notation A(g) - (u" ••• u/ A(g) / }. Let g e G and B € ~ 'DIs be

fixed. Ck is the event

C~=(Tgr' (B n (xO(m ) -O}, t uk

where xO(m s ) is the_ first point of ms less or equal to zero. Considering all the

possible values for °0 ' we get

pO [ X (0) • g, X (T,) • g', Y (T,) e B J

/A(g)/O = L p [X (0) - g, X (T,) • gf, Y( T,) e B n {x (m ). 0 J]

k _, 0 uk

/ A(g) / 0 g • L p [X(O) -g, Y(O) e Ck, X(T,) - gf J.

k -,

Using the conditional independence of X( T,) and Y(O) knowing (X (0) ,~O) (see ~6. 4), we can rewrite the last expression as

where the R.H.S. is obtained fran the very definition of pO (see (6.3.6».

But in view of the lemma of Append1x "

89

Since

we finally get

( A. 2. 1)

For any u 6- A(g) such that g ~ g' is allaied, we have

PA(g)_{U}(B) - PA(g')_{v}(B) if (A(g')-A(g» n S'· {v}. (A. 2. 2)

To show this, use the birth restriction property of h.5, which implies that if u -

u is immediately reactivated after the transition g + g', then necessarily u e- s' since v -£ S' i also use the JToperty that for Poisson JTocesses stationary

probab1lity and+ Palm JTobab1lity coincide on all events implying only the

restriction toB - {O} of the point JTocess (see h of Part 1). We have similarly

PA(g)-{u}( B) K P A(g') (B) if A(g') n S' c A(g) and u fI A(g') ( A. 2. 3)

and

PA(g)_{U}(B) KPA(g')_{u}(B) ifA(g') n S'C A(g) and ueA(g'). ( A. 2. 4)

Let now u 6- A(g') n S'. Write the local balance equation for (g' ,u),

multiply both sides by PA(g')_{u}(B). Using (A.2.2), (A.2.3), (A.2.4) and

o relation (7.1.4) linking Xg and xg ' we obtain

and

the

o A ( u ,g') P ( B) L ° L Xg' A(g') A(g')-{u} - _ Xg

gE:Gu sfA ( g )

peg ,s ,g' ) Hs ,g) ( A(g) PA(g)-{ s} B)

t X 0 P ( ') AC u ,g) P ( B) + L g g,u,g A(g) A(g)-{u} •

gE:Gu

(A. 2.5)

Similarly, multiplying both sides of (7.5.2) by "A(g,)(B), and using (A.2.2),

( A. 2. 3), (A. 2. 4), we get

o t ACs ,g' ) xg' L - --- PA(g' )_( s}( B) •

sfA(g')nS' A(g')

~ xO ~ A(S,g) g I A(g)::>A(g')n S' g sfA(g)nS' A(g) p(g,s,g') PA(g)_{s}(B). (A.2.6)

90

The aumnation of thia relation and of relation (A.2.5) when u apans A(g')nS'

yields (uae the birth restriction p"0perty)

o xg' ~

BfA(g' )

A (a ,g' ) t t 0 A( a ,g) PA(g' )_{ a}( B) - L L Xgp(g ,a ,g') PA(g)-{ a}( B)

A(g') g£G BfA(g) A(g)

that ia to Bay in view of (A.2.1)

o A

Xg' P A( g' ) (B) - P( X (T,) = g', Y (T,) e B), Q.e.d.

Appendix 3. The trans1tion II .. Jcs (Proof' c:L (8.1.2»

Reproducing canputations already done at the begining of the preaent appendix,

we get

Appendix II. Proof c:L (8. 3. 5)

and

By construction, ~

o A

xeo) -Xo' YeO) -(T_T yo). o

Denote by Zo the R. H. S. of (A. 4.1). The Palm inveraion formula yields

p[X(O) - g, yeO) e B] = ptxo=g, ~ € B]

T,

, ~[f , (XOOTu·g, ZOOTueB) du).

E~T1J 0

But for u e- [0, T, ) ,

&. ZOOT - (T OY ) u u 0 •

(A. 4.1)

eA.4.2)

91

Hence

[A ...D [JT1 ~'::--.. A ]

P X(O)a g, Y(O) e B] - ---E""N 1(X 0"'8, TU Yo € B)du.

t'tJ: T1 ] 0

(A. 4. 3)

In view of the definition of rP, the R.H.S. of (A.4.3) reduces to

K.A(g)

where we have used (A. 1 .1 0) and (A. 1 .14) to get the last expr essi ons. Equation (8.3.5) follows fran equation (7.'.4).

!ppendix 5. Proof of (9.1.3)

Let

d(g)- r xg' r p(g',s.g)A(s,g') g ilG seA(g' )

50

and for g,g e G , let 50

A(g' ,g) • 6(g' ,g) O(so,g) - A(g» + ~ p(g' ,s,g) A(s,g'), sf'A( g' )

s.'o50

where 6 denotes the Kronecker sl'llbol. With these notations, for g e global balance equation of (r,A) reads

+

( A. 5.1)

(A. 5.2)

the

If p(.,s ,.) : G x G +lR is an arbitrary substochastic matrlx, the state o So So

space G can be partitioned into one inessential class KO and a fanily of So

essential cl~ses of irreduclbility K"K 2, ... Clearly, a(g) ·0 (see (9.3.5» for geK.LetK -

o be the subset of Gs where a(g) > 0 and K be the subset where a(g) < O. N oU ce 0 + •

that transi tions between K and K are Impossi bl e. In vi ew of equation (9.3.4) the substraction 0,.(9.2.8) - A (A.5.3» yields the relation

So

92

L b (g') A( g' ,g) g' £G

So

where b (g) C A, x ( )-A x , g e G • Let a, g So g So

{g e G I b(g) > a I and So

{g e G I b(g) < a I. So

+ + Let g6-J. Since transitions from K toJ are imjXlssible, (A.6.4) reads

L + b (g') A( g' ,g) g'£J

a (g) • L _ b(g') A( g' ,g) g'£J

+ (A II - A ) {d(g) + A, L fa (g') peg' ,So,g) A(So,g') , So g' £KO ,

+ A, L + fa (g') (p(g' ,So,g) - 6(g,g'» A(So,g') I. g' £K ,

(A.5.4)

(A.5.5)

Assune that A,II - A < 0, which is always jXlssible. &unmation of the ri@l1t So

hand side of equation (A.5.5) for all g e J+ yields a non-jXlsitive result (use the relation 1: + peg' ,So,g) .,). Hence the sunmation of the left hand sides has to

g€K

yield a norrpositive result too, that is

A, A2 __ L + a(g) + L + beg') (- L + A(g',g»:S; o. (A.5.6) AO g€K g' £J gEa

Fran the very definition, A(g,g') i: 0, g f, g' and A(g' ,g) - 0, so that

1: + A( g' ,g) i: o. g€J

+ Hence, the left hand side of (A.5.6) is zero which implies that K..(1.

Let us show that K is empty too. The suretraction «9.2.8)-II.(A.5.3) yields

A2 _ a(g) ." (ft

\ l. x -IIx) A(' ) A g' £G a, (g' ) g' g ,g • So So

( A. 5.7)

93

Summation of (A. 5.7) entails

L a (g) - 0 g£G

So

(use the convergence of the series l: g€G

So which conci udes the proof of (9.3.1).

x I A( g ,g) I ). Hen ce K g

APPENDIX 6. Proof of the converse theorEIII in the general case

( A. 5. 8)

+ .q entails K a(J,

Let C = {g e G / c(so,g) ~ OJ. For g e c, the immediate activity assunption So

entails that roth d(g) and the second term in the right hand side of (A.5.3)

vanish. Hence the i nsensi ti vity bal ance equation is tri vi all Y sati sf! ed for all g in C. Furthermore, if g e c, equation (A.5.4) together with the relation a(g) = 0

entail that

L b(g') A( g' , g) • 0, g e C.

~-£C

We can r ewr i te (A. 6. 1) as

b(g) M (g) = L b(g') u (g' , g ), g e C, ~-£C

(A. 6.1)

(A.6.2)

where (M(g), u(g,.), g e C) is the substochastic infinitesimal generator on C

defined by

and

u(~ ,g) - I peg' ,s,g) a(s,g') sf.A( g' )

s.*so

M(g) = A(g) - u(g,g).

(A.6.3)

( A. 6. 4)

Hence, beg) - 0, at least for all g such that this infinitesimal generator on C

is strictly substochastic or equi valently such that

L A ( g ,g') < O. ~-£C

Let g & C. Using this fact in equations (9.2.8) and (9.2.9) together with the

propertya(g) = 0, g e G , we get respectively (A. 6.5) and (A. 6.6) belCAol. So

and

- (1

94

Xg, A(g' ,g) + A x C(So,g) - A (l-A IA) Xg c(so,g) So g 2 So 1

A

-~) L Al t -£G

So

a (l-Il) (d(g) + L Xg' p(g',so,g) A C(So,g'», t -£G So

So

(A.6.5)

where we have used the property "A(g' ,g) > 0, g i C implies L A(g' ,g) < 0".

Elimination of the term L t -£G

So

gee X, A(g',g) yields the desired result. The Il"oof of g

the converse theorem is thus concluded.