lim _1_ ntlll n

n L foeT

k=1 k

~ 1 n E""N' [g] s lim _ L goeT

ntlll 'n k=1 k

13

P. a .s •

(3. 4. 4)

P. a.s.

with g = J foe N(dt) or g = N«TO',T1']). See ~8 for a discussion of ( T' T'] t

0' 1 ergodicity and the validity of (3.4.4). As a matter of fact, the "proof" of (3.4.3) by <3.4.4) is a faniliar exercise for quet.eing theorists in the oontext, for

instance, of busy cycles. See Part 2, §3.4. This is why we also call the exchange

formula the cycle formula. a

R emcrk 2 : F ormul a <3.4.3) can al so be vi ewed as a gener al izati on of Wal cf s

identity since it reads

N( 0, Ti ]

[ L f oeT ] - EPw [N( 0, Ti]J EPJ. f] • k=O k

(3.4.5)

4 FROM PALM PR03JBn.ITY TO STATIODRY PR03JBn.ITY

11.1 The inversion fonnulae

Let h be a non-negative measl.l"able function from (nxE, F x B) into CR,B) such

that for all w e 0

J h(w,t) N(w,dt) = 1 E

and let f be a non-negati ve randan variable on (0, F).

The general1zed Canpbell's formula (3.3.1) applied to v(w,t)

f(e_tw) yields

E[f] = A J h(e_tw,t) f(e_tw) P~(dw)dt. OxE

( 4. 1. 1 )

If for instance we ta ke h (w, t ) = 1 [TO(w) ,0)

, we obtain the first inversion formula

T

° f 1 foet dt] , E[ f] = A EN [ ( 4. 1. 2a )

° which in the special case f = 1A (A € F) reads III

p( A) = A J pOe T > t, et e A)dt. ( 4. 1 • 2b )

° N 1

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

14

o The next formulae are obvious oonsequences of the aT -invariance of PN and of ( 4. 1 • 2a) and (4. 1 • 2b) • 1

We obtain the seoond inversion formula

-T -1

E[f] = ~ E~ [J foa_t dtJ o

or, for f = 'A (A € F) ,

and

P(A) = ~ r P~(-T_, > t, 6_t e A)dt. o

More gener ally,

o E[ f] = ~ EN [

T j n+1

T n

11.2 Feller's paraoox

Taking f = 1 in (4.'.2a) yields

o A EN [T1 ] = , •

Let FO be t he cum ul ati ve di stri buti on fun cti on of T1 , under P~

o F (x) = PN (T ~ x). o 1

Letting A = (T, > v, -TO> w} in (4.'.2b), with v,w ell+, we obtain .,

P(T1 > v, -TO> w) = A f (1-Fo (u»dU. v+w

( 4. 1. 3a )

( 4. 1 • 3b )

( 4. 1 • 3c )

( 4. 1. 3d )

( 4. 2. , )

(4.2.2)

( 4. 2. 3)

In particular, with v=O, and observing that, by oonstruction of T,' P( T, >0) =, .,

P( -T > w) = A f ('-F 0 (u))d u o w

(4.2. 4a)

This shows in particular that P(-TO > 0) =, since A{( 1-FO(u»du = AE~T, J=1. Similarly, taking w = 0 in (4.2.3), we get 0

.,

P( T, > v) = A J ('-FO(U))du. ( 4. 2. 4b)

Thus -To and T1 are identically distributed under P.

15

The case of renewal processes

Let (N,8 t ,P) be a stationary point p"ooess with finite intensity A and P~ t:e its Palm probability. Suppose moreover that under P~ the inter-event sequenoe

(Sn,ne'lJ) defined by

(4.2.5)

is i.i.d. process.

Then is called a renewal p"ooess and (N, P) a delayed renewal

The existenoe of such a mathematical object is !?ranted by the results of the

forthcoming ~ 4.4.

(4. 2. 6) * The distribution of the sequenoe S o under P and P N!....

(S , n e- 'lJ - {O)) is the same 11

'lJ 7J Proof Let g: CR ,B)" (B,B) be an arbitrary non negative measurable function. It

suffices to show that

* 0 * E[ g (S ) J = E/I. g (S ) J.

By the inversion formula T

* 0 1 * E[g(S )] = A EN[f g(S (8 ))du]. o u

But if u e- [0, T1 ), S( eu ) = s, sot hat

T * of1 * 0 * E[g(S)] = A EN[ g(S )du] = A EN[T1g(S)]

o * 0 where we have used the independence of T1 = So and Sunder Pw

Ne xt we shall s how that

* ( 4. 2. 7) ~ and S are P-independent.

Proof: Canputations similar to the alxlve yield

* o[ E [f(SO) g(S )] = A EN f(SO) T1 * g (S ) ]

= A E~[f(SO) T1 ] ~[g(s*)] a So far the following was provep: under P, S = (Sn' n e 'lJ) is a sequenoe of

independent randan variables, and S = (Sn' n e 'lJ-{0}) is an Ll.d. sequenoe. Can we expect that S = (S , n e 'lJ) is Ll.d. under P, i.e. that S has the same

n 0 0 distribution under P and underP W The answer is negative in general: from

16

E[f(SO)] = A E~[f(SO) T1 ] = A ~[f(SO) SO],

we obtain (by taking f(SO) = SO' and recalling that A = 11E~SoJ)

E[SO] = 1 E~[S~]. ~[So ]

Therefore, if E[SO] E~[SO],

~[SO]2 = ~[S~].

o 0 The vari ance of So has hence to be zero under P N' i.e. Sn = cons tant, PN-a.s.

Indeed, one can check fran (4.2.3) that in this particular case, So has the same

deterministic distribution under P and P~.

11.3 The mean value fonnulae

o Let (N,St'P) be a stationary point jTocess with finite intensity, and let PN

be the associated Palm probability.

Let (Zt' teE) be a stochastic p:'ocess with values in a meastrable space (E,£) and such that

and

Then, for any non-negative measurable function g: (E,E) ... (E,B)

T 1

E~[fg(Zt)dtJ o

E[ Jv g(Zn) 1 (Tne(O, 1])]

E [I 1 {T e (0 1]) ] n€ZI n '

( 4. 3. 1)

( 4. 3.2)

( 4. 3. 3)

These formulae just rephrase the inversion formula (4.1. 2a) and the defini tion o formula (3.1.1) of Pw

11.4 The inverse construction

Define on (O,F) a flow (St' teE) and a point jTocess N such that N(St)=TtN.

Let pO be a jTobabil ity measure on (0, F) such that

o P (00 ) = 1, ( 4. 4. 1)

where .00 = {TO = o}, and

o 0 p (aT e.) = p (.), (n e V). n

Moreover we will assune that

o 0<E[T1 J<",

17

( 4. 4. 2)

(4.4.3)

We will see that pO is the Palm Il"obability P~ associated with the stationary point process (N,at,p) for sane p"obability P that is at-invariant for all t e R. Moreover, in view of the inversion fonnulae, P will be uniqte.

As the inversion formula requires, if such a P exists, it smuld satisfy

(4.4.4)

Clearly (4.4.4) defines a probability on (n,F). We mI.Bt smw that P is at-invariant for all t e R, and that

o 0 PN = P •

We first remark that, on nO'

T j+1

J Tj

and therefore, since pO is aT -stationary n

T 1 EO [ J

n P( A) = 1 Aoes n Ef\ T1 ] 0

Also, for any t&R

T

CIs J.

EO [ J n 1 a A oa CIs J = 1 ~' __ o t s n fl[T, J

Therefore

p( A)-P( a A) I;s _' __ ::-' __ t n fl[T, J

ill T

J n T -t

n

Since thi s is tr ue for all n, P( A) = p( etA) •

(4.4.5)

Fran (4.4.4) T,

ECf] = EOcf foet dtJ.

EPc T, J ° In particular I

18

E[ N« 0, e:J) J =

T, , _'_EO [f N« t ,t+ e:J) dt J

T, = _, __ '_EO [f N( t ,t+ e:Jdt 1 -

EPCTJ£ T-£ £ EP[T, J £ ° , , and ther efor e lim E[ N( 0, £J) J

£ .. 0 £

--::-' __ • Since E[ N« 0, e:J) J = ~, we get

EPc T, J £

and therefore, in view of assunption (4.4.3), ° < ~ < "'. We can hence define the Palm pro ba bility P~ associ at ed wi th (N, et , P). By the i nversi on form ul a

T

° ' p(eT €A) = ~ EN[ f (, Aoer +t)dt 1 ° ° ° T,

~ E~ [f ' A dt 1 = ~ E~ T , AJ , ° '

where we have observed that 'A oeT +t = , A when t e (0, r, J.

° Using now the definition of P in terms of pO, i.e. (4.4.4)

Therefore

° ° E C T, , AJ = ENC T, , AJ, (A 1') F) •

Since T, > ° by construction, pO =: P~ •

5 EXAIf'LES

5.1 Palm p=obability and superposition of independent point p=ocesses

The si tlBti on is that des cr i be din Exampl e '.2.2 wi th the addi ti onal

ass unption

(':£i:£k), (5. ,.1)