Mathematical Computer Performance and Reliability

Mathematical Computer Performance and Reliability G. Iazeolla, P.J. Courtois and A. Hordijk (editors) ©Elsevier Science Publishers B.V. (North-Holland), 1984

INVITED PAPER

On Functional Equations of One and Two Complex Variables Arising in the Analysis of Stochastic Models

Guy Fayolle INRIA Rocquencourt, France Part 1 On a functional equation of one complex variable

We consider a variety of the Capetanakis-Tsybakov-Mikhailov (C.T.M.) collision resolution algorithm (CRA) and study in detail the mean collision resolution interval (CRI). The expected durations of these CRI's, defined in terms of the specific CRA, are shown to satisfy a difference equation which we convert to a functional equation for their exponential generating fonction, which is nonlocal with a noncommutative iteration semi-group. Using Mellin transform techniques and properties of the iteration semi-group, we show that, for ·any arrivai rate À below a certain bound À.max> the mean CRI for n colliders is asymptotically proportional to n. Moreover the system proves ergodic iff À < À.max- Ali proofs are omitted here and can be found in Fayolle, Flajolet and Hofri [ 1]. 1.1 Specification of C.T.M.-C.R.A. with continuous input

(a) A single error-free channel is shared among many users (sources, nodes, stations ... ) which transmit packetized messages. Time is slotted. Users are synchronized with respect to slots, so that packets are transmitted at the beginning of slots only. Each slot is equal to the time required to transmit a packet [see ALOHA network conceptl. · .. (b) Each transmission is within range of every user. When more than one user transmit simultaneously, packets will collide (interfere) and none is received correctly. These collisions are treated as transmission errors and each user must strive to retransmit his colliding packet until it is received correctly. The users ail employ the same algorithm for this purpose. (c)

Each user monitoring the channel knows, by the end of the slot, if that slot produced a collision or not.

(d) Each active user maintains a conceptual stack. At each slot end he determines his position in the stack according to the following procedure: When an inactive user becomes active, he enters Jevel 0 in the stack. He will transmit at the nearest slot and will always do so when at stack Jevel O. At slot end, if it was not a collision slot, a user in stack level 0 (there is at most one such user) becomes inactive and ail users decrease their stack Jevel by 1. At slot end, if it was a collision slot, ail users at level i, i ~ 1 change to level i + !. The users at level 0 split into two groups; one remains at Jevel 0, while the other pushes himself into level 1. (No such user is cognizant of how many users are at each level.) The partition is made on the basis of a random variable (tlipping a coin each user independently of the other active users has a probability p of staying at level 0 and q = 1-p of having to wait at level 1. The time it takes to dispose, under this algorithm, of a group of n colliders initially at level 0 is denoted by Ln. 1.2 Functional equation for the generating function of the mean (R.I.).

Assumptions: H 1: There is an infini te number of sources.

H2: The number of new packets appearing in the system in one slot is a random variable, independent of the time t and of the history of the channel up to time t, having a Poisson distribution of parameter À. The random variables Ln satisfy the following recursive relationship:

L0 = L 1 = 1

|

55

G. Fayolle

56

where

I: number of messages immediately retransmitted X: number of new arrivais in that immediate slot

Y: number of new arrivais in the slot following the resolution of the I +X messages. I is obtained from a Bcrnonlli trial with parameter p; X and Y arc Poisson random variables with mean À. Introducing

z E G, the complex plane ,

we get (z)-(Hpz)-(Hqz)

1-2(À)e-=(l+Kz),

(!)

where À

K

À

e P-e

=

q ---,.----,.- '

À

-

e

--

À

q __

q

e

cjJ(O)

1,

'(O)

=

0.

-r

p

1.3 Iterative scheme for the functional equation

From now on p )o q (p+q

=

1) and

CT1(z) def À+pz To solve (!) we need the introduction of a noncommutative iteration semi-group H of linear substitutions generated by Œ1, Œz where the semi-group operation is the composition of fonctions. The identity of H is denoted by 0

(5)

which appear in the approximation (3) of the an. Denoting f* (s)

=

fo~J (x)x 5 -

1

dx the Mellin transform of a fonction f (x ), we have

lemma 3. The Mellin transform of {J(x) is {J* (s) = 0(-s)[r(s)+Kr(s+l)+K~(-s)r(s)], where

r(s)

is

the

gamma

fonction

and

the

integral

defining

{J*

is

(6)

absolutely

convergent

for

-2 < Re(s) < -1. Now by the inversion theorem for Mellin transforms,

1 ·Jc+i~ . {J*(x)x- 5 ds, for any c: -2

{J(x) = - . 21 'lr

C-l

< c < -1

(7)

00

To evaluate (6), we shift the line of integration to the right taking residues of the integrand into account. The first residues give the dominant term in the asymptotic expansion of {J(x), as x- > oo. This requires some deeper properties of the iteration semi-group H and more detailed analytic information on IJ(s) and ~(s). Formally, consider the Dirichlet series given by a sum of the form ~

w(s) =

r (:: lt seems convenient to introduce the following notations (valid until the end of the paper). b 1 (x) =p(l - _!_)+11 2(1- - 1 -) X h (x) a 1 (x)

(4.1)

=

q(l - - 1 -)+11 (1 - _!_) h (x) I X

2(

b y)= p(l -

1

k(y) )+112:; C(-Jf:)

if b 1 (x) has no zero in B ( in

B(-Jf:)

continuous on

and satisfying the boundary condition (5.1). This is a particular

case of a Dirichlet problem for a circle.

/FI)

">:;

if b 1 (x) has a zero in B ( F(x, 0).

say x 0

Provided that b 1(x) has no root in B(

,

we have still a Dirichlet problem for the function (x-x 0 )

">:;ffi),

FCx, O) is determined from (5.1), in B(

a constant, by Schwarz's formula (Muskhelishvili [li] Section 41).

">:;ffi),

up to

lzl < 1

(5.2)

where D is a real constant:

u'(p)

and

=

-

µ;(I - h (~) )F(O, O) 1- lm ----~---l-~ µ *(1 - _!_)-µ 2 (1 - - 1-) I X h (x)

-,\

u(p)

(5.3)

where:

H h(-Jf: (p)

=

sin p H (p)

1 =-_o.,.----------------

[p; (µ;-µ;)H2(p) +(µ;-µ; H1 H2)H (p)-µ;] (1-~)

eiP)

À2+µ2+a -

.j[(jÇ+J;;;) 2+aH(jÇ-J;;;) 2+al 2,\2

and a = ,\ 1+µ 1 -2~ cos p. Note that H (p) is real. From (5.3), we deduce easily that u (p) is an odd fonction of p. This implies:

F(

(5.4)

[ii{z,O)=_!_J;"

"\/ J:;

A similar formula holds for F (O,

a 2 (y)

"?:;

{Fi z)

;>:! 0 in C(l) (whatever the position of

+F(O,O)

lzl:;

{7i;]

are the eventuai'zeros of a 2 (y) in [O, y 1] U [y 2 , I]; is analytic in B (

Upon setting: (6.3)

">:;/Ti').

U(z)

b'(z) . [z-k(a 2)];'[z-k((3 2)];'

=

a 1 (z)

(z--y,r'

It follows: (6.4)

Re[i U(z)G(z)]

From [ 11] formula (40-10 - 40),

G(z)

(6.5)

is given by:

G(z)

=

Der(z)

=

0

for lzl

=

">:;

{7i;

66

G. Fayolle

where D is a constant, nonzero and:

1 r(z) = - 2i7r

(6.6)

J

(og[t-XJ(t))dt

C(

{E) ,../À,

iU(t) iU(t)

and:

t-z

where: J(t)

(6.7)

=

x=

-

x=

-

1

2i7r

iJ (t) log[--] or: U(t)

_!__ [arg U(t)]

7r

C(

-Jf: , !:'._) À,

denoting by arg (z) the fonction "argument of z ".

Lemma 6.2. (6.8)

(denoting by sgn (x) the fonction "sign of x" with sgn (0)

* NP * Nz *

is the number of zeros of a 2 (y) on [ 1, 1

=

0) where:

">:;/Fil

is the number of zeros of b (x) on [O, x 1] U [x 2, l]

argj b:(x)] -

a ~) x~,

is the variation of the argument of b:(x) along the "contour" [x 1, x 21, starting from x 1

a ~)

above x 1x 2 1, going to x 2 and coming back to x 1 below [x 1x 2 1.

Lemma 6.3. (6.9)

Lemma 6.4. For

!)

x~

-2, the homogeneous Riemann-Hilbert problem has no solutions different

2) For x ~ 0, the homogeneous Riemann-Hilbert problem has exactly the general solution is given by:

where c 0 , c 1,. . ., ex are constants subject to ex= cx-k> k =a, !,. ..,

~rom

zero.

x+ 1 linearly independent solutions;

x but otherwise arbitrary.

Demonstration. See Muskhelivshvili [li) S 40-p. 100. Theorem 6.2. !)

the homogeneous Hilbert problem satisfying the boundary condition (7.4) has, at most; one solution. In other words, x ~ O.

2)

The system is ergodic iff x

- 0, which is equivalent to:

Functional Equations

db 1(x) (6.10)

da 2 (y)







o-µ;

67

µ2-À2

if

1L1À2-µ2À1

µ 2 ~ À2

IL1-À1

µ1À2-µ2À1 IL1-À1

.

if

1L2 ~ À2

Theorem 6.3. Assuming (6.11) or (6.10), F(z, O) is given by: F(z, 0) F(O, O) =

(6.12)

where F(O, O)

Problem B

=

1

qµI

+

µ1µ2-pq

µ1µ2

µ1µ2-pq

1L2À1-µ1.À2+~;(µ1-À1) ~ ~~~

G(z) G (O) '

· G (z) is derived from (7 .2) and (7 .3) using:

µ1µ2

r(z) = _I 27!"

J C(

fE)

-v>::·

Mt)dt t-z

"JOINING THE SHORTER QUEUE"

BI. - Problem Formulation and Assumptions Let us consider two parallel M/M/l queues with infinite capacities and exponential service time distributions with means J_ for queue 1 and J_ for queue 2 under the following assumptions. a (3 a)

the arrivais form a Poisson stream with mean À.

b)

a customer, upon arrivai, is assigned to the shorter queue.

c)

if the queues have equal length, the arriving customer joins queue i with probability 1.

11";,

i

=

1, 2 and

7l"1+7l"2 =

This problem has been studied (only in the symmetric case, i.e., 11" 1 = 11" 2 = 1/2 and a= {3) formerly by Kingman [12) and more recently by Flatto and MacKean [JO), who improved Kingman's results and obtained the stationary distribution. Ali proofs of this section can be found in (5) and (6). To study the behaviour of the system al the steady state, we need the following fonctions: (analytic w.r.t. x and y whenever lx 1, IY 1< 1):

F 1 (x,y)

=

= ~ p(i, i+j)xiyj i,j-0

F2(x,y) = i,

~ p(i+j, i)xiyj j-0

P 1 (x) = ~ p(i+I, i)x;; i-0

(!.!)

P 2 (x) = ~ p(i, i+l)xi;

Az(x) = ((3Hx)P 1 (x)+aF(O, O)

i-0

p(i, i)xi;

G;(y) = F;(O, y),

i = 1, 2

68

G. Fayolle X

.!'.._

y

X

T 1 (x, y)= ;\(l - -)+a(l -

1 y

)+(W - -)

X V 1 T (x, y)= ;\(l - -)+(:l(l - L...)+a(l - -) 2 y X y

S = À+a+/3 The Komogorov's forward equations for the p (m, n) yield

F 1 (x,y)T 1 (x,y) =a(l - ..!'...) G 1 (y) + (

À7riJ 2-Àx-(:l

y

X

F 2 (x,y)T 2 (x,y) =(:l(l - ..!'...) Gi

(4.6) dt:i.(x) dxlx ~

0

l

which is equivalent to (4.7)

X'o n 3=o

Knowledge of this generating fonction is sufficient to determine the Laplace transform >/;Cs), of the distribution of the total sojourn time T. Indeed we have the

Lemma 1.1. i/;(s)

73

Functional Equations Our task is ta find the gcnerating function (J).

lemma 1.2. G (x, y, z, s) satisfies the following functional equation RG (x, y, z, s)

=

AG (0, y, z, s)

+ BG (x,

y, 0, s)

+C

.

(2)

where R, A, B, C are known functions of x, y, z ,s R

=

;\(! - _!__) X

c=

+ µ, 1(1-px-q ~) + µ 2 (1

- ):'._)

Z

Z

+ µ, 3 (1-z)+s

=

0

~~~-/J-~2/J-_3:...._~~­ ( J - x )(s +µ3- µ3z)

Moreover, G (x, y, z, s) must be analytic inside the region

lx 1
-J(µ 1+s))) is determined first, either from (JO) (if z (Àf(µ 1+s )) is inside L) or from (5) (if it is between L and Le). The r.h.s of (2) then yields a value for G(À/(µ 1+s), O).

x

=

Conclusion of part II The mathematical analysis of models involving random walks with two-dimensional state space tends to be equivalent to the general boundary value problem: Find a fonction (z) analytic inside a simply connected domain bounded by a closed curve !t' and satisfying on !!'the condition a(t)+(t)+b(t)+(a(t))+c(t)-(t)+d(t)-(a(t)) = f(t),

t E

!t'

where i)

a (t), b (t), c(t), d (t) and f (t) are known fonction, usually verifying a Hiilder condition

ii)

a(t) is an automorphism of the curve !t'onto itself. References

[!)

Fayolle, G., Flajolet, P., Hofri, M: "On a fonction equation arising in the analysis of a protocol for a multi-access broadcase channel," INRIA report No. 131, April 1982.

[2)

Fayolle, G., Hofri, M: "On the capacity of a collision channel undt