40

n the hypothesis EO[o] < EO[l], the ergodic asslIIIption entails that lim r (0 -to ).-. Thus, T < a> pO _ a.s. i-O i 1

Consider now the sequence (w~, n e IN) obtained for Y - 0. One gets by induction

that w~ ~ w~, n e lN, so that (1.1.5) entails that w~ - w~ for all n ~ T (W~~w~.O).

° ° If E [0] < E [1], then, for any finite random variable Y, there exists a finite integer valued randcxn variable T such that for n ~ T WI _ WOo

(1.5.2)

- 'n n

Notice that the sequence w~ is simply related to Mn of (1.1.4) by wO _ M en n nO.

This together with (1.5.2) entails

(1.5.3)

( 1.5.4) If ~[o] infinity.

< EO[l], w~ converges in law to the law 'of M .. when n goes to

In other words, when the queue is stable, w~ will coincide with w~ in a finite time

and will converge in law to M ...

2 - FORMULAE FOR THE GlGlll- OJEUE

2.1 Construction of the time stationary workload

Let Ul, F, P) be a probability space endowed with a measurable semi-group of au1romorphisms (e t , t e R ) and a point pr~cess N verifying N~t - \N (t e R ).

° - e (e-e ) on a ). Suppose Tl

Suppol;le moreover that (p,e t ) is ergodic. Define eT - T that a pon-negative r.v. 0 is also defined on (O,F») 0

ASSLIlIe th8t the intens! ty of N is fin! te, 50 that there probability P associated wi th (N, P, et ). Suppose moreover that

. 1 ° 0 ° positive that 0,1 e L (P ) and finally that E 0 < E T.

For !t.! III e a construct W(IIl) by

~ W(IIl) - M (Ill) - lim M (w)

(2.1.1) a> ni-a> k n

M (w) - ( I «ooe -i)(w) -i +

max - (,oe )(w)}) n OSkSn i-l

exi sts a Palm T is PO-a.s.

As we saw in ~1, W is the pO-a.s. unique, pO-a.s. finite r.v. which verifies

+ ° woe" (W+o-,). P -a.s.

Define, for ~ w e a,

w (Ill) - (Woen )(lII) n

The event

A - {W < a>, W - (W + 0 - 1 ) +} n n+l n n n

(2.1.2)

(2.1.3)

(2.1.4)

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

41

is, by the construction (2.1. 3), e-invariant, and moreover pOi A) = 1. Therefore by the result (8.1.1) of Part 1, P( A) ~ 1, i.e.

W < '" P-a.s. (2.1.5) n

W n>1 ~ (W .. a - Tn) +, n n P-a.s.

In particular, if we define for all t & [Tn' Tn> 1 )

W (t) - (W n .. on .. Tn - Z ..

we find fran (2.1.5), (2.1.6) and the finiteness of on that

W(t) < "'. P-a.s. (2.1.7)

W(Tii) c Wn ' P-a.s.

Also fran the very construction of Wit)

(2.1.6)

W(t+s) - Wit )oes for all III eo,s ,t e E (2.1.8)

so that (W(t) , teE) is a Bt-stationary left-continuous !Tocess on (0, F, P) (see Fig. ~).

time

Figure ~

2.2 Li tUe' s Connula : the FIFO case

Following the lines of section (1.~), in the FIFO case, we can construct a left-continuous stochastic process (X(t), teE) on (0, F, P) re!Tesenting the nunber of custaners in the queue at time t and such that X (t .. s) - X(t)oes for all t.s eE.

For x,y (7 E, let f(x,y) - 1 and let W-(w+o)oe • Owing to the FIFO (x~O, y~-x) n n

assunption,

x (0) - 1: f (T , W ). r£ZI n n

(2.2.1)

Therefore. the original Canpbell'sfonnula (3.3.2), Part 1 andfonnula (3.1.~), Part 1 entail

i.e.

E[X(O)) - A f f .. f(f",y)dt pO[Wo € dy) 1R 1R

- A f (1 - pO(Wo ~ t)dt, o

E[X (0)) - ).E°[Wo) - A( i'[W) .. :p[ 0)). ( 2. 2. 2)

This is Little's fonnula. Since (p,e ) and (PO,6) are ergodic. the cross ergodic theorems of section 8.3 (part 1) yield'

42

1 n

E[X (0)] A lim 1: W( T ) P - a.s. (2.2.2') n k=1 n n ....

and

Ail[wo] lim 1- ft X (s )ds pO a.s. ( 2. 2. 2") t.... t 0

2.3 ProbabilIty of emptiness

The following "evolution formula" ooIds for the workload in GIGl1/" queues

t W (t) W(O) + 1: on 1{0(T St} -f (1 - 1{W(s) =O})dS.

n€ZI n 0 ( 2. 3. 1)

Indeed, the workload decreases at rate 1 as long as the queue is not empty. Assune

stabil i ty i.e.

p (2.3.2)

wet) is then P - a.s. finite so that owing to the lemma of ~ 1.3,

t

E[ 1: 1 {O(T St} an] n€ZI n

E[~ (1 - 1{X(s)=o})dS.

F urt hermore

so that

P[W(t) = 0] = P[X(t) = 0] = 1 - p. (2.3.3)

Remark. Formula (2.3.3) is true whatever the servioe discipline (FIFO, LIFO, RANDOM,

etc.). The onl y re quir ement is that of a wor k cons er ving dis ci pline i.e. the ser v€r

cannot be idle if the system is not empty and the servioe rate is always 1.

2.4 Takacs' formulae

The evolution equation of the ]:reoeding section yields the follOWing formula,

valid for all canplex nunber s with a positive real part:

43

-sW(Tn) -so e - sW (t) = e - sW ( 0) + L e ( e n - 1) 1 (0 <T H) +

n€ZI n t

+ s J e-SW(u) (1 - 1 (X(u) = O})dU.

°

( 2. 4. 1)

Ther ef ore, t aki ng e xpe ct at i on wi th respect to P, otis er vi ng that - sW (s )

e 1 (X(s)=O) = 1 (X(s)=O) and using the jTecedi ng resul t (2.3.3), we get :

).,il[e -s(W+a)] _ EO[e -sW] = s(p - E[e -SW(O)]). (2.4.2)

This general relation ret ween the Laplace transforms of the time and

customer-stationary workload was obtained by L. Takacs. In the particular case of

GIGIll I., queue, Le. a queue where (an' n e 21) is an LLd. sequence independent of

the arri val process, formula (2.4.2) recomes

(2.4.3)

3 - THE GlGIsI., QJElE

3.1 The ordered workload vector

The input jTocess is defined on the Palm space of a stationary point jTocess as

for G/ Gil I., que ues. Ther e ar e s ~ 1 s er vers at tendi ng cus tomers and the all ocati on

rule is that an arri ving cust omer is assi gned the server wi th the snall est wor kload •

Once assigned, this customer will wait and then re served at unit rate until

com pI eti on.

.. 21) We will cons truct the s tati onar y workload jTocess (W n ' n e where ..

(W 1 Ws ) is perm ut at ion in i ncr easi ng of the workload found W = ... , a order in n n' n each ser ver by the n-th customer upon arri val : W1 :> W2 :> ... ::; WS for all n e ZI. n n n

T his or dered vector s ati sfi es the recurrence reI ation

(3. 1.1)

.. + where e = (1, 0, ••• , 0), i = (1, ••• , 1), R is the operator arranging vectors of B S

in increasing order, and (Xl' ... ,

The equation to be solved on this Palm space therefore reads