Data communications in the mobile radio channel

Data communications in the mobile radio channel

Citation for published version (APA):

Mortel-Fronczak, van de, J. M. (1983). Data communications in the mobile radio channel. (EUT report. E, Fac. of Electrical Engineering; Vol. 83-E-142). Eindhoven University of Technology.

Document status and date: Published: 01/01/1983

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Electrical Engineering

Data communications in the mobile radio channel

By

Joanna Fronczak

EUT Report 83-E-142 ISBN 90-6144-142-0 ISSN 0167-9708 Coden: TEUEDE September 1983

Department of Electrical Engineering

Eindhoven The Netherlands

DATA COMMUNICATIONS IN THE MOBILE RADIO CHANNEL

By

Joanna Fronczak

EUT Report 83-E-142 ISBN 90-6144-142-0 ISSN 0167-9708 Coden: TEUEDE

Eindhoven

Data communications in the mobile radio channel/by

Joanna Fronczak. Eindhoven: University of Technology.

-Fig. - (Eindhoven University of Technology research report / Department of Electrical Engineering, ISSN 0167-9708, 83-E-142) Met lit. opg., reg.

ISBN 90-6144-142-0

S150668.5 UOC 621.396.037.37-182.3 UG1 650

SUMMARY

This report contains analytical and numerical approaches to a data communication problem in the mobile radio channel. Instead of the power calculations used in previous studies of mobile packet radio, the signal-to-interference envelope ratio is determined. As shown in this report, this results in a mathematical simplification of the problem and hence in faster computer algorithms. The influence of significant receiver thermal noise is also discussed for the first time in the literature.

Fronczak, Joanna

DATA COMMUNICATIONS IN THE MOBILE RADIO CHANNEL.

Department of Electrical Engineering, Eindhoven University of Technology (Netherlands), 1983.

EUT Report 83-E-142

Address of the author:

Mgr inz. Joanna Fronczak, Institute of Computer Science,

University of Mining and Metallurgy, ul. Mickiewicza 3D,

30-059 Krak6w, Poland

PREFACE

During the period 20th of December 1982 to 4th of March 1983 I stayed

as a visitor in the Telecommunications Division at the Eindhoven University of Technology. I dealt with packet data communication in the mobile radio channel.

The problem of the link throughput for users sharing a common channel was solved by means of an analytical and numerical method described in this rpport.

I would like to express my gratitude to prof.dr. J.e. Arnbak for his scientific guidance and to thank all the others who helped me.

CONTENTS Summary Preface List of symbols 1 . Introduction 2. Analytical approach 3. Numerical results

4. Comparison with previous approach 5. Extension to the case with significant

receiver thermal noise 6. Conclusions

Appendix 1: Behaviour of link throughput when

receiver threshold Ko tends to infinity and to zero, respectively Appendix 2: Computer programs - short description

and listings References i i i iv 1 3 7 15 16 18 20 23 32

LIST OF SYMOOLS

M - number of users T - packet duration

A - user activity

G - traffic offered to the channel

Pa -

probability of disturbance

S - successful traffic over the channel

p. (.) - distribution function

:r:rt) - signal (time function)

y(t) - interference (time function)

p - mean signal power s

P

n - mean interference power

f (r) - spatial distribution function of interferers

' r

ro

rmin - distance parameters defining the scenario r

max of interferers

:;, - powC"r thrt?sholo of the receiver

o

p - distance from the desired transmitter to its receiver

K - envelope threshold of the receiver

a

~ - exponent of the inverse-power law for ground-wave attenuation in the mobile channel

p - maximum interference power

n~max

P . - minimum interference power

n3mt.n

frp ) -

distribution function of P

n n

PT - transmitted power G

-T antenna gain of the transmitter

H

-T antenna height of the transmitter G

-R antenna gain of the receiver

H -_H antenna height of the receiver

E.(-w) - exponential-integral function

'/..

Si - link throughput

FTN - mean power of thermal noise

P - mean power of total noise (thermal noise plus interference)

c

CR - mean signal-to-mean thermal noise power ratio

I. INTRODUcrION

When considering data cormnunication in the mobile radio channel, a ring-structured n~twork modpl is often used. The model consists of M users offering data packets of duration T (time units) and with activity A

(packets per time unit, and per user) to the ring according to the Poisson distribution function. This network model corresponds to a desired transmitter-receiver pair of mobile users located in an area with other interfering users sharing the same channel. A packet collision happens i f (and only if) one or more other packets start within the

collision period of length 2T centered around the starting time of the desired test packet (Fig. I).

M

3

.1

T T

Fig. 1.a Ring network Fig. 1.b Collision period

The communication network defined in this way is of the well-known Aloha type

r

41 .

( 1 )

and the probability of disturbance (with the expoflPntial waiting times following from the poisson statistics) is

1 -exp{-2(M-l)AT} (2)

the successful traffic over the network is

2 G 1.5 o.s ---.1 ,2 1/2e .3 ,4 S

Fig. 2 Successful traffic in pure Aloha.

The "perfectn

Aloha network saturates at 18% of its theoretical capacity (Fig. 2), so users must be prepared to retransmit their data packets under heavy-traffic conditions. However, retransmissions should not occur immediately, since this will only limit the successful trafic S further (see Fig. 2).

Reference [1] considered an imperfect data channel in which the signal does not necessarily arrive with the same strength at the receiver as sent from the transmitter. It was demonstrated that this may lead to higher throughput than in a "perfect" Aloha conununication channel (in which a packet collision always leads to complete loss of data). Assuming a finite probability of suffering fading of the interfering packets, the probability of being able to communicate during packet collisions is non-zero. This was demonstrated for a shared channel with slow Rayleigh fading, i.e. a channel with slow random fluctuations of the signal x, according to the distribution function

2x 2 - }

- exp{-x

IP

p

S

(4 )

S

where P is the mean power of the signal x(t).

S

If the signals in the common channel arriv(~ from different distunces to the receiver, according to the spatial distribution function

f

(r)

]"

l

s

J , f (]") _Y' 5 where 4 1'0 5 - exp{- -4 !i ]" (5 )

r . .; r ~ r m1-n max

r . « r « r

m~n 0 max

where r ~ r . ~ r define the scenario of interferers (see Fig. 3), i t

o m'Ln max

is possible to calculate the total mean interference power from M-l

competing users and their total joint interference signal, which has an envelope which is again Rayleigh distributed.

Fig. 3

2

oL--L~ __ ~~~ ____ __

2 3 !:. 4

'.

The spatial distribution function l (ro). 1"

In r11 i t was assumed that, in the absence of thermal noise, destruction of the packet content during overlaps by n interferers occurs only if the

resulting signal-to-interference power ratio drops below a specified value, 2

0, which is dependent on the modulation type and receiver used.

So the probability of disturbance is the probability that this ratio is

below Zo' multiplied by the probability of collision having the Aloha form (2); i t gives the loss of traffic. That approach led to rather complicated mathematics and slow numerical programs for calculating the

link throughput.

An alternative approach to this problem is to avoid power-based cal-culations and consider instead the signal envelopes which are Rayleigh distributed. This is the aim of the present paper.

2. ANALYTICAL APPROACH

Assume the traffic offered to the channel by M transmitters to be Poisson distributed with packet duration T and user activity A. An

arbit.rary selected packet can be destroyed by contention during overlaps by n interferers if and only if the resulting receiver signal-to-inter-fprpnce power ratio

P IF

drops below a certain value Z (receiver

s n - 0

M-l prob{loss; Z M, p/r } = o 0

I

n=l prob{n, 2T} prob{P IP < Z } s n 0 (6)

where p is the distance from the desired transmitter to its receiver. In order to avoid power calculations, the power ratio in (6) is now replaced by the envelope ratio x/y, where both the signal x and the interference yare Rayleigh distributed (see (4». The corresponding probability densities are

P (x) x 2x 2 -= - e.xp{ -x

IP }

P

S S and 2y 2 -= - exp{-y

IP }

P

n n

Since x and yare independent Rayleigh-distributed variables, the

probability density of z = xly is given by (see [61)

P (z) = z 2P S P n z

Now, the probability that x/y drops below a certain value Ko is

(7 )

prob{xly < K }

=

prob{xly < K

Ip

=

canst}

o 0 n prob{P n = const} (8)

since P is a random variable as well, influenced, e.g., by the spatial

n

distribution and number of interferers.

The conditional probability prob{xly < K

IF

= const} is directly

cal-o n

culated from Eq. (7)

K

o

prob{xly <

Kip

_{() n} = const} =

J

p'''' (z) dz = o

Ii'

0

~

0 ('J) / ' + :; p n

The probability of receiving a certain mean interference power P is

n

determined by the spatial distribution of interferers around the receiver, which in [1] was given by (5). This expression was a con-sequence of applying the inverse-fourth-power law for ground-wave attenuation in [1]. For an inverse-beta-power law i t is generalized to

f (p) p (8+1) B Po B+ 1 f',+ 1 exp{ - -8-P

The following results apply for the general case, i.e., with

B

as a parameter.

( 10)

The distribution function of the mean received interference power P

n

is a gamma distribution, since i t is a sum of n exponentially distributed, independent interference powers [7]

In-l}! =-1 -JcO exp{-h P } n n (11 ) where

,

F

T-, GT~ HT are the transmitted power, antenna gain and antenna height of the transmitter, respectively, G

W HR are the antenna gain and antenna

height of the receiver. From Eqs. 8 and 9

P

_J?

n~max prob{x/y < K }

_f

0 f(P)

dP

(12 ) 0 P n n P _{n~min K2 +} s 0 P n

The upper interference-power boundary P applies for r . « r ,

n~max m~~ 0

so i t can be replaced by infinity, while the lower boundary P •

n~m~n

applies for r » r , so i t can be replaced by zero. As evident from

max 0

Hence

00

J?

hn

prob{X/Y < Ko} =

f

0 _{(n-lJ !}

pt-l

_n exp{-h

P } dP

_{n n}

0

J?

P s + -0 P n Ps ~ a

Define now

x2

=

a, which is constant, because P

s = pS' where a is given

following 0 (11), and p is the distance from the desired transmitter to its corresponding receiver. The integral has the form

-h u

e du

which has the following solution [2]

ah e n Ei(-ah) +

L

(k_l}!(_a)n-k where u (;, p n ' Ei(-ah) (;, 00

f

ah -t ~dt t

i:; I Ill' l'xp<lIll'td i ,11-i III t'qriJl fllllCL.ioll

I II.

Thus prob{xly < K } o hn =

71-n

-"l")"! n

L

(k-l)! (_a)n-k k=l

The final result can, after some transformations, be written in the following form: prob{xly < K } o n

L

(k-l)!(-W)n-k₁ k=l with 1

J!3

o

Since the conditional probability that the signal-to-interference envelope ratio (due to n interferers) is below K is given by (13),

o

while the probability of an overlap by precisely n interferers is given

by the Poisson expression

prob{n,2T}

=

exp{-2AT(M-l)} {2AT(M-l )}n

n!

the total probability of packet destruction becomes:

prob{ loss; K , M, p/r } = o 0 M-1

L

=1 prob{n,2T} prob{xly < K } o (14 ) ( 15) (13 )

The throughput of an individual link established over the fading channel

is

fll

s.

= AT[1 - prob{loss; K, M, p/r }j

~ 0 0 (16)

3. NUMERICAL RESULTS

The final expression for the link throughput (16) has been programmed for computer calculations (in FORTRAN). The exponential-integral function

P.i(-w) appearing in this expression becomes very small for large argu-ments (say, for W > 20; this happens for small K ), and i t becomes very

-9

large for small argument values (say, for W < 210 ; this happens for large K ), see Fig. 4. Computer calculations require care with very

o

small and very large values, if we want to have proper numerical results. Thprefore the limits of the throughput for Ko tending to infinity and Ko tending to zero arA of great importance. Appendix 1 shows how these limits have been theoretically derived.

y

- 3

- 2

Ei(-w)

o

~~====~='----

.2 .4 .6 .8 1.0 1.2 1,4 1.£ W

Fig. 4 Exponential-integral function y ~ Ei(-w).

Figures Sa to f show the link throughput as a function of the traffic offered to each of M = 10 equivalent links sharing the common radio channel, with the receiver threshold Ko as a parameter, for different values of the ratio

pi

P

o giving the relative position of the desired

transmitter and the undesired cluster of interferers distributed as in Fig. 3 for

S

=

4.

In each figure, curves are enclosed between the two limits:

and

K -+ 0 o

AT exp{-2AT(M-l)} > -+ 00

For moderate values of the value of the ratio

K , the throughput values increase depending on o

pip . For larger K i t tends to the pure Aloha

o 0

throughput, the faster the larger the ratio r/r is. o

0 .2 · \92 • \84 .! }6 ",: · 1 GA .16 · I S2 • 1 44 · I J!i .Illl .12 . \ J 2 .104 .095 .088 .OS .072 .064 .056 .048 O' .032 . 024 .016 .nn8 Fig. 5 .02 .04 .06 .OS .1 .12 .14 .16 ·1 , .2

/

/ / , /

/'

/ /

/

K

·'7//

0 .' .-/ K -::O,f/i "

'/

/

K - lodR

•

0

•

K ~ -3d" Ko ::: 0, 10, 20, 3 .02 .04 _·06 .OS .1 .12 .14 .16 .18 .2

Link throughput (Eq. 16) with M = 10 links in a

common channel, with receiver threshold Ko as a parameter.

AT

(a) normalized distance from desired transmitter to its receiver p/r = 5. o .2 .\92 .\84 .}76 .\6B .16 .152 · \44 I Jr, · \28 .12 · \12 · 104 .096 .088 .08 .072 .064 .056 .048 .04 .032 .024 .016 .008 0

0 .2 .192 .184 .176 S. t .168 .16 .152 · 144 · 136 .128 .12 .112 · 104 .096 .068 .08 .072 .064 .056 ,048 .0' .032 .024 .016 .008 Fig. 5 .02 . D. .06 .08 .1 .12 .14 .16 .i8 .2

o;~

K 0 K ;;;: -2odR 0 Ko ;;;: -lOdB Ko ;;;: -;JdR K

o

0 dB K 10, 0 0 ₂ 0 .02 D. 06 .08 .1 .12 ·14 .16 .i8 _{.2 AT} continued.

(b) normalized distance from desired transmitter

to its receiver p/r _o = 4. .2 .192 .184 .176 .168 .16 .152 .144 .136 . 12B .12 .1\2 ·104 .096 .08a .08 .072 .064 .056 .048 .0' .032 .024 .016 .ooa 0

0 .2 · 192 · 184 · 176 '\ .1£38 .16 .152 .144 .136 .128 .12 .112 .104 .096 .088 .OB .n7? .IJG4 .056 .048 .04 .f! II' • 111'4 .016 .008 n 0 '-"1 ~. 5 .06 .08

/

~

/ . con t 1 nuod. .1 .14 .16 .18 .2

(c) normalized distnnce from desired transmitter to t tH t"(1Cf! I vor pi;' ,".

"

.2 .192 .184 .176 .\68 .16 .152 .144 .136 .128 .12 .1\2 .104 .096 .088 .OB .072 .064 .056 .048 .0' .n1i'

a .2 .192 .184 .176 S. _~ .168 .16 .152 . 144 .136 .128 .12 .112 . 104 .096 .088 .08 .072 .064 .056 .048 .04 .032 .024 . 016 .008 Fig. 5 .02 .04 .06 .08 .1 .12 .14 .16 18 .2 KO :::: -30, -20 dB K :::: -10 d 0 Ko :::: -J dB ---+ Ko :::: 0 dB K = 10, 20, 3 .02 04 .06 .08 1 .12 14 .16 .18 2 h continued.

(d) normalized distance from desired transmitter to its receiver p/r = 2. o .2 .192 • 184 ·176 ·166 .16 .152 · 14'1 .136 • L28 .12 .112 .104 .096 ·088 .08 .072 .06<1-.056 .048 .04 .032 .02 • .016 .008 0

0 .2 · 192 • 184 .176 e · \58 o.

,

.16 .152 · I 44 · )36 .128 .12 .112 .104 .096 .088 .08 .072 .054 .056 .048 .04 .032 .024 .016 .008 Fig. 5 .02 04 .06 .08

.,

·12

. I'

.16 .\8 .2

::/

K ~ -30, -20 d 0 K ~ -10 dB 0 ~ K ::-3 dR 0 Ko = 0 dB .->< K 10 dR 0 Ko :: 20, 30 .02 .04 .06 .08 .1 .12 .14 .16 .\8 .2 continued.

(e) normalized distance from desired transmitter to its receiver p/r

=

1. o ·2 .192 . 184 .176 .168 .16 .\52 . 144 .136 .126 .12 ·112 .104 .096 .08e .08 .072 .064 .056 .048 .04 .032 .024 .016 .OOB 0

"

0 .2 .192 .184 .176 S.

,

.168 .• 6 .152 · 144 .136 .128

."

· 112 · 104 .096 .08B .08 .072 .064 .056 .048 .0' .032 .024 .016 .008 Fig. 5 02 .04 .06 .08

..

."

...

.is . .8 2

;/

/'X ~-;Mfl KO ~ -30~ _{-20, -lod}

~

_.

/~KO=OdB

~ K 10 dB 0 ~ r. = 20, 30 dB .02 .04 .06 oa

..

.il

...

.is . .8 2 >r continued.

(f) normalized distan~e from desired transmitter

to its receiver p/r

=

0.5. o .2 .192 . I 84 .17G .168 .is .152 .\44 .136 .128

."

.112 . 104 .096 .088 .08 .072 .064 .056 .048 .0< .032 .024 .016 .008 0

4. COMPARISON WITH PREVIOUS APPROACH [1]

Results obtained from the method described above are qualitatively close

to that presented in [1], but not idp.ntical because of the different

assumptions: here the receiver is described by the envelope ratio (K ),

o whereas in [1] the power ratio (Zo) was adopted.

For dominant interferers (piTa = 5 - Fig. Sa) and for high Ka' each

overlap destroys the test packeti thus the pure Aloha channel is recovered. For morp distant interferers (piT = 0.5

o Fig. Sf), and for lower K , o an

overlap no longer necessarily results in packet contention. An imperfect channel may thus increase the throughput i f a certain value of the

rr~c;ni.vE'r th.rE'shold can be assured.

Comparison of previous results from [1] with those obtained here may be easied by Fig. 6.

,

10

,

10

,

, 0

,

, , Fig. 6

,

.,

,

10 10 10 I TTTTTI , II _I , , II _I 10' , , , ,

,

_"I 10' I~"lll

,

10 , ",-~~-~".1 0 p n , 10

.,

10

Receiver threshold K with p as a parameter.

For given conditional probabilities p of not exceeding the receiver threshold Ko' Fig. 5 relates this threshold to the mean signal-to-interference power ratio

FgiFn

as derived from (9)

K

o (17)

where p = prob{x/y < K

Ip

o n = canst} is related to the error probability.

5. EXTENSION TO THE CASE WITH SIGNIFICANT RECEIVER THERMAL NOISE

The above results apply in the event that the receiver thermal noise can be neglected. In the case with significant thermal noise, a few corrections should be introduced. The total mean noise power is now a

sum of the mean interference power and mean thermal noise power

( 18)

Hence, the probability of not exceeding the receiver threshold KG is given by 00

K

hn prob{x/y < K } =

_f

0 p'-1 exp{-hP } dP (19) In-V! 0 _p c - c c P TN

K+

s 0 P c Define 00 /:;

f

Al _{IA '} 0 P TN _/:;

f

Al = IE 0

Eq. 19 can be written as follows

The first integral, LA' has exactly the same solution as in the absence of thermal noise, (13). The second integral, IE' becomes after sub-stituting P by (18) and after some transformations

c n - n-1 h P_TN (n ] ) ! P P n - - + ['TN

After substituting _n by v, IE is given by

FTN where h

P

= 8+1 TN 8 1 CR n (v+1) {h CR exp -+ 1 +

x!

The total probability of packet destruction is now

M-1 prob{loss; K , CR, M, pip} = o 0

I

n=1 (20) (21) (22)

wlll'rp the s(~cond tf>nn rnpre~,,;ents t:.lj(! probabi IJty of dist.urhance by

thermal noise (TN) only, prob{n,2T} is given by (14) and prob{xly < K }

o

prob{O, 2T} = exp{-2AT(M-l)}

II'

prob{X/Y TN < Ko} = _--,0:...-._ P

II'

°

s + -P TN (23) (24)

where we have used (9) after replacing P

n by PTN

noise envelope is also Rayleigh distributed.

since the thermal

Numerical results

The results obtained numerically are shown in Fig. 7.

The link throughput (16) has been calculated by using Eqs. (22) and (24)

with the parameters

CR

=

20 dB

p/l'o = 0.5

For small values of K (-30 dB to -3 dB), the corresponding

character-°

istics of S. remain unchanged after the introduction of the thermal

1-noise (see Fig. 5f). For larger values of K , the less robust-receiver

°

will be adversely affected by the thermal noise. Nevertheless, the critical saturation effects encountered in the pure Aloha channel are

still softened, as evidenced by the curve K 10 dB. Rayleigh fading

°

thus reduces overload effects in the presence of thermal noise, too.

6. CONCLUSIONS

A model has been formulated for packet transfer between users accessing a common radio channel with a slow Rayleigh fading. Numerical results show that fading may result in better link throughput, compared to the throughput of pure Aloha. The formulas apply in two cases: a) when the transmitter powers are such that the receiver thermal noise can be omitted and b) when the latter becomes significant.

~----0 2 _{r r} -· j 92 l · j 84 ! · 176 · 168 " · I S2 · 144 .136

I

· 128

."

· I 12 · 104

I

.096 ·088 .08 .on ·Df>4 .056 .048

"'

. 11 j ) 1l/1 ,016 .1)08 0 0 Fig. 7 III d/?

.,

.192 .184 ·168 .16 .152 · 144 .136 · 126 · 12 · I 12 · [04 .096 .088 .08 .072 .064 .056 ,048 .04 • .0"17 .024

Link throughput with M

=

10 links in a common channel, with K as a parameter, in the presence of thermal noise,

a

for p/"o = 0.5.

APPENDIX 1: Behaviour of link throughput when receiver threshold K o

tends to infinity and to zero, respectively

As pointed out in Sect. 3, it is necessary to establish analytical limits for the link throughput when Ko tends to infinity and to zero.

a. First case: Ko .... ro

Define

_P1

=

prob{n, 2T}

,

P 2

=

prob{xly < Ko}

Pi

=

prob{loss; Ko' M, plY' } 0

For K .... ro (w"" 0), Eq. 13 looks as follows

o = 1 [(_l)n-l wn w Ei(-w) + In-l}! e n

L

K=l

lim w n Ei(-w) = lim Ei(-w)

w+0 w+0 1

n

W

-w

e

=

lim w (L'Hospital's rule)

w+0 -n _n+l w l. 1 -w n :: _{"'m -}

_e

_w w+0 n

=

0 n (K_l)!(_w)n-K lim

_L

₌

(n-l)! w+0 K=l Thus

1

['2 = (n J)! [0 + (n-1)!l = 1

and Eq. 15 reduces to

M-1

I

n=1 where ['1 is given by (14). Consequently M-1 Pi = exp{-2AT(M-1)}

I

n=l

=

exp{-2AT(M-l)} [exp{2AT(M-l)}-1]

=

1 - exp{-2AT(M-1)}

Hence the throughput of the i-th link (16) is given by:

+ 00

(25)

(26)

From (26) we can see that for large values of K the link through9ut

o takes the form of the Aloha expression (3).

b. Second case: K 4 0

o

For K + 0 (w + 00) we start again with (13)

o

n

I

K=l

In this case we use for calculation of Ei(-w) the expressions [3]

Ei(-w) = -E 1 (W) -w e + -W Thus we find: L .,

[I

..!:.:.-.

+ i=O (-w)" for

= 1 [(_l)n-l wn w (n-l)! e --w e

-w

L .,

L

1- • • + i=O (_w)1-n

L

(K-l)! ( - w ) ! n-K K=l L-terms + { (_,,)n-l+l.' (_,)n-2+2.' ( w w --w )n-:3:3' ()n-4 +. -w + ... }] n-terms

L

=

n + 10, this number turns out to be large enough for a good

numerical approximation for W + 00. As a final result of the expression

only the 10 last terms of the first sum in the squared brackets remain, and the rest cancels out for any value of n.

The remaining part looks as follows

+ •.. + (n+8)(n+7) ... n

w9

(n+9)(n+8) •.• n +

10

w

This asymptotic expression can be used in (15), if W is

large enough (> 20). Thus, if K is very close to zero, we can

a

assume P2 - O. Then Pi is zero as well, so

S. = AT •

1-(27)

APPENDIX 2: Computer programs - short description and listings

In order to obtain numerical results for calculation of the link

throughput, five programs have been written. The first (PROGRAM LINK) calculates the link throughput

various values of K ~ pip and

a a

S.

1-AT

for definite values of M, and for

(the traffic offered to the channel

per user). Every run of this program gives one value of S. for a

1-particular input. Calculation of prob{xly < K } for K + 0 is made by

a a

the asymptotic expansion for Ei(-w) (27). Function MMDEI from IMSL is

used.

The second (PROGRAM PREPARE) prepares the results for the plotting procedure MCURVD (PLOTSYSTEEM FORTRAN).

The third (PROGRAM PLOT) makes plot drawings (Figs. Sa to f and 7).

Each run of this program gives one figure for one value of the ratio

p/~o (or for one value of CR).

The fourth (PROGRAM CHECK) calculates and plots the relation between

K and

F IF

ratio (Fig. 6).

a s n

The fifth (PROGRAM LINK 1) calculates the link throughput S. in the

1-presence of thermal noise, for definite value of M, and for various values of K , p/r , CR and AT. This is the extention of the first

a a

program. Subroutine DOIACF from the NAG-library is used.

Each program contains comments. Separate listings are included in this appendix.

<ELECNA5)JOANNA ON U S E R 6

DATE & TIME PRINTED: THURSDAY, MARCI~ 3, 1983 ~ 09: 58: 58.

10 C---·---20 C- PROGRAM LINK

30

C---40 C- AUTHOR : JOANNA FRONCZAK 50 C- DATE: FEBRUARY 1983

60

C---70 FILE 2(TITLE='RESULTS',NEWFILE=FALSE,PROTECTION=SAVE,

80 * KIND=DISK,UPDATEFILE=TRUE) 90 $ SET AUTOBIND

100 $ BIND=FROM IMSL/=ON APPL

110 C---120 C- THIS IS A NUMERICAL PROGRAM WHICH CALCULATES THE PROBABILITY 130 C- OF LOOSING PACKET DURING OVERLAP BY N INTERFERERS

140 C- AND THE THROUGHPUT OF AN LINK IN THE ABSENCE OF THERMAL 150 C- NOISE

160 C---·---170 C- INPUT DATA - NAMELIST 'DATA'

180 C- NR= ••••• NUMBER OF THE RECORD TO BE WRITTEN IN THIS RUN 190 C- KO= ••••• RECEIVER THRESHOLD

200 C- ROR= •••• DISTANCE FROM DESIRED TRANSMITTER

210 C- TO ITS RECEIVER TO DISTANCE PARAMETER RATIO 220 C- M= •••••• NUMBER OF TRANSMITTERS

230 C- LT= ••••• USER ACTIVITY (PACKETS PER UNIT OF TIME AND PER 240 C- TRANSMITTER) TIMES PACKET DURATION (LAMBDA*TAU) 250 C- BETA= ••• PARAMETER FOR POWER LAW

260 C---~---270 C- OUTPUT DATA - FILE 'RESULTS'

280 C- EACH RECORD IN THIS FILE CONSISTS OF FOUR VALUES : 290 C- KO,ROR,LT,SI ( FORMAT: 1X,4F15.10 )

300

C---310 REAL LICZ1,LICZ2,MIAN1,MIAN2,LT,Kl,KO 320 INTEGER IOPT,IER

330 DOUBLE PRECISION XMIAN

340 NAMELIST IDATAI NR,KO,ROR,M,LT,BETA 350 READ DATA 360 Ml=M-l 370 PROBL=O. 380 CONS2=(BETA+l)/(BETA*ROR**a*KO**2) 390 CONS1=2*LT*Ml 400 IOPT=1 410 LICZ1=1. 420 LICZ2=1. 430 MIAN1=1.· 440 MIAN2=1. 450 UN1=-1. 460 DO 10 N=l,Ml 470 LICZ1=LICZ1*CONSl 480 MIAN1=MIAN1*N 490 PROB1=EXP(-CONS1)*LICZ1/MIAN1 500 C---_---510 C- FOR SMALL CONS2 EI(-CONS2)=INTEGRAL(EXP(-T)/T DT) ;

520 C- LOWER BOUNDARY=CONS2 , UPPER BOUNDARY=INFINITY, 530 C- SUBROUTINE MMDEI FROM IMSL IS USED

540 C---550 C- FOR LARGE CONS2 EIC-CONS2)=-EXP(-CONS2)/CONS2*Cl-1/CONS2+

560 C- 2/CONS2**2-6/CONS2**3+ ••• )

570

c---580 IFCCONS2.GT.20) GO TO 40

---590 EXPC=EXP(CONS2)*MMDEI(IOPT,-CONS2,IER) 600 LICZ2=LICZ2*CONS2 610 UN1=UN1*(-1) 620 SUMK=O. 630 K1=1 640 DO 20 K=l,N 650 HELP=(-CONS2)**(N-K) 660 SUMK=SUMK+K1*HELP 670 Kl=K1*K 680 20 CONTINUE 690 PROB2=(UN1*LICZ2*EXPC+SUMK)/MIAN2 700 MIAN2=MIAN2*N 710 GO TO 60 720 40 PROB2=0.0 730 XMN=N 740 XMIAN=l. 750 UN2=-1. 760 DO 50 11=1,10 770 UN2=UN2*(-1) 780 XMIAN=XMIAN*CONS2 790 PROB2=PROB2+UN2*XMN/XMIAN 800 50 XMN=XMN*(N+II) 810 60 CONTINUE 820 PROBL=PROBL+PROB1*PROB2 830 10 CONTINUE 840 SI=LT*(l.-PROBL) 850 WRITE(6,100) PROBL 860 WRITE(2=NR,200) KO,ROR,LT,SI 870 ENDFILE 2 880 100 FORMAT(10X,6HPROBL=,F20.10) 890 200 FORMAT(lX,4F15.10) 900 STOP 910 END

PREPAR:E DN U S E R 6

DATE & TIME PRINTED: THURSDAY, MARCH 3, 1983 e 12:44:42.

10

C---20 C- PROGRAM PREPARE

30

C---40 C- AUTHOR : JOANNA FRONCZAK 50 C- DATE = FEBRUARY 1983

60

C---.

70 FILE 2(TITLE='RESULTS',KIND-DISK.PROTECTION-SAVE)

80 FILE 3(TITLE='A'.KIND-DISK.PROTECTION-SAVE,NEWFILE-TRUE)

90

C---100 C- THIS PROGRAM PREPARES THE XAR AND YAR ARRAYS FOk THE PROCEDUr 110 C- 'MCURVD' TO PLOT AND SAVE THEM IN THE FILE 'A'

120

C---130 C- INPUT DATA - NAMELIST "INPUT' AND CONTENTS OF THE FILE "RESUl

140

C---150 C- NAMELIST 'INPUT'

160 C- NREC- ••••••• NUMBER OF DATA RECORDS IN FILE "RESULTS' 170 C- NPOINT- ••••• NUMBER OF POINTS FOR EVERY CURVE

180 C-· NROR- ••••••• NUMBER OF DIFFERENT ROR VALUES <THE NUMBER 190 C- OF PLOT DRAWINGS)

200 C- M- .••••••••• NUMBER OF TRANSMITTERS

210

C---220 C-- OUTPUT DATA - FILE "A" WHICH CONTAINS THE XAR AND YAR ARRAYS 230 C- WRITTEN IN FORMAT lX,4F15.10

240 C- XAR - CONTAINS NPOINT VALUES OF X (U)

250 C- YAR - CONTAINS NROR+NREC/NPOINT GROUPS WITH NPOINT VALUES 260 C- OF Y (SI) IN EACH GROUP

270 C---280 REAL AR(350,4),XAR(20),YAR(420)

290 DATA AR.XAR.YAR/1840*0/

300 NAMELIST /INPUT/ NREC,NPOINT,NROR,M 310 READ INPUT 320 IG-NREC/NROR 330 IP-NREC/NPOINT 340 DO 10 ID-l,NREC 350 10 READ (2,100) (AR(ID,IC),IC-l,4) 360 CALL SORT(l,NREC,2,AR) 370 1'18=1 380 IS-I 390 ISS-IG '100 410 420 430 '140 4~'jO 460 470 4BO 490 500 :;.l0 520 5~~() 5'lO 5~5() ~)60 :,i"70 ~,BO 20 50 :l :10 CALL SORTCIS,ISS,l,AR) IF(NS.EG.NROR) GO TO 20 NS-NS+:l IS=18S+1 ISB-ISB+I1l GO TO ~~O NS:::i IS-1 ISS-NPOINT CALL SORT(I8,I88,3,AR) IFCNB.EQ.IP) GO TO 40 NS=NS+l I!3"ISS+:1 IS8=ISS+NPOJ:NT 00 TD ~jO CONTINUe: l·,(~~i'. DO 110 I=l,NPOINT X?\I~ ( I ) "?)I": ( I ,~,)

590 IPP=IP/NROR 600 DO 140 IR=l,NROR 610 DO 120 J=l,IPP 620 DO 120 I=l,NPOINT 630 IJ=I+(J-l)*NPOINT 640 ID-IPP*NPOINT*(IR-l) 650 rC=(IPP+l)*NPOINT*(IR-l) 660 120 YAR(IJ+IC)=AR(IJ+ID,4) 670 DO 130 I=I,NPOINT 680 130 YAR(IJ+IC+I)=XAR(I)*EXP(-2*(M-l)*XAR(I» 690 140 CONTINUE 700 IYAR=NREC+NROR*NPOINT 710 WRITE(3,100) (XAR(I),I=I,NPOINT) 720 WRITE(3,100) (YAR(J),J=I,IYAR) 730 ENDFILE 3 740 100 FORMAT(lX,4F15.10) 750 STOP 760 END 770 SUBROUTINE SORT(IS,ISS,NW,AR) 780

C---790 C- THIS SUBROUTINE ARRANGES RECORDS IN THE DEFINITE FRAGMENT

800 C- OF TWO-DIMENSIONAL ARRAY AR ACCORDING TO THE CONTENTS

810 C- OF CHOSEN FIELD

820

C---830 C- PARAMETERS:

840 C- 15,155 - DEFINE THE FRAGMENT OF ARRAY AR

850 C- NW - CHOOSES THE FIELD

860 C- AR - TWO-DIMENSIONAL ARRAY 870 C---880 REAL AR(350,4),HELP 890 LOGICAL LOG 900 60 1=15 910 LOG=.FALSE. 920 40 IF(AR(I,NW).LE.AR(I+l,NW» GO TO 20 930 DO 10 J=I,4 940 HELP=AR(I,J) 950 AR(I,J)=AR(I+l,J) 960 10 AR(I+l,J)=HELP 970 LOG=.TRUE. 980 20 1=1+1 990 IF(I.EQ.ISS) GO TO 30 1000 GO TO 40 1010 30 IF(.NOT.LOG) GO TO 50 1020 GO TO 60 1030 50 CONTINUE 1040 RETURN 1050 END

<ELECNA5>PLOT ON USER6

DATE ~ TIME PRINTED: THURSDAY, MARCH 3, 1983

e

10:05:05.

10 . C---20 C- PROGRAM PLOT

30

C---40 C- AUTHOR : JOANNA FRONCZAK 50 C- DATE: FEBRUARY 1983

60 C---70 $ SET AUTOBIND

80 $ HOST IS SERVICE/PLOTTERHOST ON APPL

90 $ BINIIER RESET LIST

100 $ SET LIBRARY

110 FILE 3(TITLE='A',KIND=DISK,PROTECTION=SAVE)

120 C---130 C- THIS PROCEDURE MAKES THE SEPARATE PLOT DRAWINGS FOR EACH 140 C- VALUE OF PARAMETER ROR (DISTANCE FROM DESIRED TRANSMITTER 150 C- TO DISTANCE PARAMETER RATIO)

160 C- X-AXIS - USER ACTIVITY (LT)

170· C- Y-AXIS - THROUGHPUT OF AN LINK (SI)

180 C- EACH DRAWING SHOWS SEVERAL CURVES MADE FOR DIFFERENT 190 C- VALUES OF KO (RECEIVER THRESHOLD)

200 C---··---210 C- INPUT DATA - NAMELIST 'DRAWIN' AND FILE 'A' WITH XAR AND YAF 220 C- ARRAYS

230 C---240 C- NAMELIST 'DRAWIN'

250 C- NPOINT= ••••• NUMBER OF POINTS FOR EVERY CURVE 260 C- NROR= •.• , ••• NUMBER OF PLOT DRAWINGS

270 C- IYAR= ••••••. NUMBER OF ELEMENTS IN THE YAR ARRAY WHICH

280 C- CONSISTS OF :

290 C- - Y VALUES

300 C- - ONE ADDITIONAL GROUP OF VALUES

310 C- FOR EACH PICTURE CONTAINING

320 C- VALUES OF FUNCTION :

330 C- LT*EXP(-2*(M-1)*LT), WHICH

340 C- IS THE LIMIT OF SI FOR KO

350 C- TENDING TO INFINITY

360 C- (IYAR=NREC+NROR*NPOINT)

370 C- NPLOT= •••••• NUMBER OF THE ~ICTURE TO BE PLOTED IN

380 C- THIS RUN (NUMBER OF THE SUCCESIVE VALUE OF ROR 390 C---400 SUBROUTINE YOUTOO

410 REAL XAR(20),YAR(500) 420 DATA XAR,YAR/520*01

430 NAMELIST JDRAWINI NPOINT,NROR,IYAR,NPLOT 440 READ DRAWIN 450 READ(3,200) (XARII),I=1,NPOINT) 460 READ(3,200) IYARIJ),J=l,IYAR) 470 IH=IYAR/NROR 480 NY1=INPLOT-1)*IH+l 490 NY2=NPLOT*IH 500 CALL MCURVD(90,O.O,O.O,21.0,21.0,1,NPOINT,NY1,NY2, 510 - XAR,YAR,.FALSE.,.FALSE.) 520 200 FORMATllX,4F15.10) 530 RETURN 540 END

(ELECNA5>CHECK ON U S E R 6

DATE

&

TIME PRINTED: THURSDAY, MARCH 3, 1983

e

10:08:01.

10 C---20 C- PROGRAM CHECK

30

C---40 C- AUTHOR : JOANNA FRONCZAK

SO C- DATE: FEBRUARY 1983

60 C---~---70 $ SET AUTOBIND

80 $ HOST IS SERVICE/PLOTTERHOST ON APPL

90 $ BINDER RESET LIST

100 $ SET LIBRARY

110 C---120 C- THIS SUBROUTINE COMPARES TWO RATIOS : FIRST OF SIGNAL

130 C- ENVELOPES (Y-AXIS),SECOND OF SIGNAL POWERS (X-AXIS),

140 C- AND MAKES A PLOT-DRAWING FOR SEVERAL VALUES OF PARAMETER P 150 C---160 C- INPUT DATA - NAMELIST "CHECIN", X VALUES, P VALUES

170 C---180 C- NAMELIST "CHECIN"

190 C- NPOINT= ..•• NUMBER OF X VALUES

200 C- NCURV= •••.• NUMBER OF P VALUES I NUMBER OF CURVES

210 C---220 SUBROUTINE YOUTOO

230 REAL XAR(20),YAR(200),P(20) 240 DATA XAR,YAR/220*01

250 NAMELIST ICHECINI NPOINT,NCURV 260 READ" CHECIN 270 READ (5,110) (XAR(IX),IX=l,NPOINT) 280 READ (S,110) (P(IP),IP=l,NCURV) 290 DO 10 I=l,NCURV 300 DO 10 J=1,NPOINT 310 YAR(J+(I-l)*NPOINT)=SGRT(-P(I)/(P(I)-l)*XAR(J» 320 10 CONTINUE 330 NY=NCURV*NPOINT 340 CALL MCURVD(90,0.0,0.0,21.0,21.0,1,NPOINT,1,NY, 350 - XAR,YAR,.TRUE.,.TRUE.) 360 110 FORMAT(7F10.4) 370 RETURN 380 END

e

10

C---20 C- PROGRAM LINKl

30

C---40 C- AUTHOR , JOANNA FRONCZAK

50 C- DATE. FEBRUARY 1983

60

C---70 FILE 2(TITLE="RESULTS',NEWFILE-FALSE,PROTECTIONaSAVE,

BO * KIND-DISK,UPDATEFILE=TRUE)

90 $ SET AUTOBIND

100 $ BIND-FROM IMSL/-ON APPL,NAGF/-ON APPL

110

C---120 C- THIS IS A NUMERICBL PROGRAM WHICH CALCULATES THE PROBABILITY

130 C- OF LOOSING PACKET DURING OVERLAP BY N INTERFERERS

140 C- AND THE THROUGHPUT OF AN LINK IN THE PRESENCE OF THERMAL

150 C- NOISE

160

C---170 C- INPUT DATA - NAMELIST "DATA'

180 C- NR- ••••• NUHBER OF THE RECORD TO BE WRITTEN IN THIS RUN

190 C- KO= ••.•• RECEIVER THRESHOLD

200 C- ROR- •••• DISTANCE FROM DESIRED TRANSMITTER

210 C- TO ITS RECEIVER TO DISTANCE PARAMETER RATIO

220 C- H= ••••.• NUMBER OF TRANSMI1TERS

230 C- LT- •••.. USER ACTIVITY (PACKETS PER UNIT OF TIME AND PER

240 C- TRANSMITTER) TIMES PACKET DURATION (LAMBDA*TAU)

250 C- BETA= ••• PARAMETER FOR POWER LAW

260 C- CR= ••••• MEAN SIGNAL TO THERMAL NOISE POWER RATIO

270

C---2BO C- OUTPUT DATA - FILE "RESULTS"

290 C- EACH RECORD IN THIS FILE CONSISTS OF FOUR VALUES :

300 C- KO,ROR,LT,SI ( FORMAT: lX,4F15.10 )

310

C---320 COMMON IPARI N,HP,CR,KO

330 EXTERNAL FUN

340 REAL LICZ1,LICZ2,HIANl,MIAN2,LT,Kl,KO

350 INTEGER IOPT,IER

360 ,DOUBLE PRECISION XMIAN

370 NAMELIST IDATAI NR,KO,ROR,M.LT.BETA,CR

3BO READ DATA

390 M1=M-l 400 PROBL~O. 410 CONS2-(BETA+l)/(BETA*ROR**S*KO**2) 420 CONSl-2*LT*Mi 430 HP=(BETA+l)/(BETA*ROR**BETA*CR) 440 HPCON~EXP(-HP) 450 IOPT=l 460 LICll=l. 470 LICZ2=1. 480 MIANi=l. 490 MIAN2=1. 500 HPPOW-l. 510 UN1=-1. 520 DO 10 N=l,MI 530 LICZ1=LICZ1*CONSl 540 MIAN1-HIAN1*N 550 PROB1=EXP(-CONS1)*LICZl/HIANl 560

c--·---····--·--·-·-·--·--·-·---···--·----···-·--··,.--.-.---.... ,.- ... - ... ---.--.--... ---... -- ....

-.-.-.-510 C- FOR SMALL CONS2 EI(-CONS2)=INTEGRAL(EXP(-T)/T DT)

590 C- SUBROUTINE MMDEI FROM IMSL IS USED

600 C---610 C- FOR LARGE CONS2 EI(-CONS2)=-EXP(-CONS2)/CONS2*(1-1/CONS2+

620 C- 2/CDNS2**2-6/CONS2**3+ •.. ) 630 C---640 IF(CONS2.GT.20) GO TO 40 650 EXPC=EXP(CONS2)*MMDEI(IOPT,-CONS2,IER) 660 lICZ2~lICZ2*CONS2 680 UN1-UN1*(-I) 690 SUMK=O. 700 K1=1 710 DO 20 K=1,N 720 HELP=(-CONS2)**(N-K) 730 SUMK=SUMK+K1*HELP 740 Kl=Kl*K 750 20 CONTINUE-760 PROB2=(UN1*LICZ2*EXPC+SUMK)/MIAN2 770 GO TO 60 780 40 PROB2=0.0 790 XMN=N 800 XMIAN=l. 810 UN2--1. B20 DO 50 Il~1,10 830 UN2-UN2*(-1) 840 XMIAN=XMIAN*CONS2 850 PROB2=PROB2+UN2*XMN/XMIAN 860 50 XMN-XMN*(N+II) 870 6() C(]NTINUE 875 HPP(]W-HPPOW*HP B80 A-O.O 890 8=1.0 900 RELACC=O.O 910 ABSACC=1.()E-4 920 ACC~().O 930 ANS-O.O 940 NN-O 950 IFAIl=O 960 CALL DOlACF(A,B,FUN,RElACC,ABSACC,ACC,ANS,NN,IFAIL) 970 PROB2=PROB2-ANS*HPCON*HPPOW/MIAN2 980 MIAN2=MIAN2*N 990 PROBL-PR08L+PROB1*PROB2 1000 10 CONTINUE

1010 PROBL=F'ROF.'L+EXP (-CONSI ) * KO**2/ (CR+KO**2) 1020 BI=LT.Il.-PROBl) 1030 WRITE(6,100) PROHl 1040 WRITE(2=NR,200) KO,ROR,LT,SI 1050 ENDFILE 2 1060 100 FORMAT(10X,6HPROBL=,F20.10) 1070 200 FORMAT(lX,4F15.10) lOBO STOP 1090 END

1100 REAL FUNCTION FUN(XI 1110 COMMON IPARI N,HP,CR,KO

:l1 ~;O I~E?)L >:, KG

1130 FUN-IX+ll.*N*EXPI-HP*X)/(X+1+CR/KO*.21

1140 RETURN

References

[1] Kuperus, F. and J. Arnbak

PACKET RADIO IN A RAYLEIGH CHANNEL.

Electron. Lett., Vol. 18(1982), p. 506-507. [2] Gradshteyn, I.S. and I.M. Ryzhik

TABLE OF INTEGRALS, SERIES, AND PRODUCTS. Corrected and enlarged ed.

New York: Academic Press, 1980. P. 312 and 925.

[3] HANDBOOK OF MATHEMATICAL FUNCTIONS WITH FORMULAS, GRAPHS, AND MATHEMATICAL TABLES. Ed. by M. Abramowitz and I.A. Stegun. New York: Dover, 1965. P. 228 and 231.

[4] Abramson, N.

THE THROUGHPUT OF PACKET BROADCASTING CHANNELS. IEEE Trans. Commun., Vol. COM-25(1977), p. 117-128. [5] ~, S.A. and W. Wasylkiwskyj

CO-CHANNEL INTERFERENCE OF SPREAD SPECTRUM SYSTEMS IN A MULTIPLE USER ENVIRONMENT.

IEEE Trans. Commun., Vol. COM-26 (1978) , p. 1405-1413. [6] Papoulis, A.

PROBABILITY, RANDOM VARIABLES, AND STOCHASTIC PROCESSES. New York: McGraw-Hill, 1965.

McGraw-Hill series in systems science. P. 229. [7] Kuperus, F.

PACKET RADIO BIJ MOBIELE VERBINDINGEN OVER KORTE AFSTAND.

M.Sco Thesis. Department of Electrical Engineering, Eindhoven

Eindhoven University of Technology Research Reports (ISSN OJ67-9708) (127) Darnen, A .. \.,H., P.M.J. \!an den Hof a.ud A.K. Bajd<lsinski

THEPAGE ~IRIX; An excellent tool for noise filtering of ~arkvv parameters, order testing and realization.

EDT RepQrt 82-[-127. \982. ISB~ 90-0144-\27-7 (128) Nicola, V.F.

MARKOVUL, MODELS OF A TRANSACTIO~AL SYSTEM SUPPORTED BY CHECKPOI~iING

AND RECO',fRY STRATEGIES. Part I; A model with state-dependent parameters.

EDT Report 82-E-128. 1982. ISBN 90-6144-128-5 (129) Nicola, r.F.

MliRK'OVIA:; ~ODELS OF A TRANSACTIONAL SYSTE~ SUPPORTED BY CHECKPOI~IING AND RECO\~ERY STRATEGIES. Part 2: A model with a specified number of completed transactions between checkpoints.

EUr Report 82-E-129. 1982. ISBN 90-6144-129-3 (130) Lemmens, W.J.M.

{Ul)

~ PREPROCESSOR: A precompiler for a language for concurrent processing on a multiprocessor system.

EUT Report 82-E~130. 1982. ISBN 90-6144-130-7

Eijnden, ?M.C.M, van den, H.M.J.M, Dortmans, J.P. Kemper and M.p.J. Stevens

JOBHANDLING IN A NETWORK OF DISTRIBUTED PROCESSORS. EUT Beport 82-E-131. 1982. ISBN 90-6144-131-5 (132) Verlijsdonk, A.P.

ON THE AP?LICATION OF BIPHASE CODING IN DATA COfiMUNICATION SYSTEMS.

EUT R~port B2-E-132. 1982. ISBN 90-6144-132-3

(13)) Heijnen, C.J.H. en B.H. van Roy

METEN EN 3EREKENEN VAN PARN~ETERS BIJ HET SILOX-DIFFUSIEPROCES. EUT Report 83-E-133. 1983. ISBN 90-6144-133-1

(134) ~, Th.G. van de and S.C. van Someren Greve

( 135)

A METHOD FOR SOLVING BOLTZMANN'S EQUATION IN SEMICONDUCTORS BY EXPANSION IN LEGENDRE POLYNOMIALS.

EUT Report 83-E-134. 1983. ISBN 90-61 44-134-X Ven. H.H. van de

TIME-OPTIMAL CONTROL EUT Report 8)-E-J)5.

OF A CRANE.

1983. ISBN 90-6144-J35-8 (136) Huber, C. and W.J. Bogers

(137)

THE SCHULER PRINCIPLE: A discussion of some facts and misconceptions. EUT Report 83-£-136. 1983. ISBN 90-6144-136-6

Daalder. J.E. and E.F. Sch~eu~s ~PHENOMENA IN HIGH VOLTAGE FUSES. EUT Report 83-E-137. 1983. ISBN 90-6144-137-4

Eindhoven University of Technology Research ReDarts (ISSN 0167-9708) :

(138) Nicola, V.F.

A SINGLE SERVER QUEUE WITH MIXED TYPES OF INTERRUPTIONS: Application to the modelling of check painting and recovery in a transactional system.

EUT Report 83-E-138. 1983. ISBN 90-6144-138-2 (139) Arts, J.G,A. and W.F.H. Merck

~DIMENSIONAL MHD BOUN~LAYERS IN ARGON-CESIUM PLASMAS.

EUT Report 83-E-139. 1983. ISBN 90-6144-139-0 (140) Willems, F.M,J.

COMPUTATION OF THE WYNER-ZIV RATE-DISTORTION FUNCTION.

EUT Repor~ 83-E-140. 1983. ISBN 90-6144-140-4

(141) Heuvel, W.M.C. van den and J.E. Daalder, M.J.M. Boone I L.A.H. Wilmes

INTERRUPTION OF A DRy-TYPE TRANS~IN NO-LOAD BY A VACUUM CIRCUIT-BREAKER.

EUT Report 83-E-141. 1983. ISBN 90-6144-141-2 (142) Fronczak, J.

DATA COMMUNICATIONS IN THE MOBILE RADIO CHANNEL. EUT Report S3-E-142. 1983. ISBN 90-6144-142-0