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Near optimal behaviour of the packet switching
broadcast channel

Banh Tri-An, Erol Gelenbe

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Banh Tri-An, Erol Gelenbe. Near optimal behaviour of the packet switching broadcast channel. [Research Report] IRIA-RR-299, IRIA. 1978, pp.24. hal-04716515

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NEAR OPTIMAL BEHAVIOUR
OF THE PACKET SWITCHING
BROADCAST CHANNEL

BANH TRI AN
Erol GELENBE

Rapport de Recherche N° 299

Mai 1978

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
B. P. 105 78150 - Le Chesnay
France
Tél.: 954 90 20
NEAR OPTIMAL BEHAVIOUR OF THE PACKET SWITCHING BROADCAST
CHANNEL

Banh Tri An*, Erol Gelenbe**

Résumé :

Dans cet article nous étudions une procédure adaptative qui permet de stabiliser et d'optimiser les performances d'un canal à diffusion de paquets (du type ALOHA discretisé). Nous obtenons des conditions suffisantes pour que l'algorithme (qui n'utilise que des données obtenues en «écoutant» les périodes de silence du canal) stabilise le canal. Des résultats de simulation sont présentés pour mettre en évidence le comportement pratique du système sous l'influence de la politique de contrôle proposée.

Abstract :

The subject of this paper is an adaptive control procedure for stabilizing and optimizing the perfor-
mance of ALOHA-like slotted packet switching channels. We prove sufficient conditions, via a
mathematical model of the adaptive procedure, under which the procedure stabilizes the channel.
This procedure is based on «listening» to the silent periods of the channel and modifying the
retransmission probabilities of the blocked terminals as a function of this measurement. The opti-
mal choice of the parameters of the adaptive procedure is analysed and simulation results are given
in order to illustrate its practical effect.
# 1.- INTRODUCTION.

Both the slotted and unslotted variety of broadcast channels for packet transmission have received considerable attention [2, 4, 6, 8] and there have been several implementations of such systems [1, 10]. Since the operation of the ALOHA system [1] in the early seventies, many proposals have been made to improve its performance [9, 12, 13]. Firstly, it was pointed out that a "slotted" version of the access scheme would double the potential throughput and bring it to $1/e \approx 37\%$ [12]. Then a formal proof of the inherent instability of such systems was given in [5] while optimal control policies stabilizing the system and maximizing channel throughput (in number of packets successfully transmitted per unit time) were established in [6] and validated by simulation in [3]. Independently, various alternative channel control policies were discussed and simulated in [8] and the related papers.

The optimal control policy given in [6] requires a perfect knowledge of the state of the system; therefore it cannot be directly used since the state -which is the number of blocked terminals in the network - is not known to each one of the blocked terminals which are the entities having to apply the control. This is the reason why we had introduced in [3] an adaptive control policy requiring the blocked terminals to listen to the channel, make an estimate of the traffic rate and accordingly modify its retransmission probability (which is the control parameter).

The purpose of the present paper is to prove sufficient conditions for the convergence of adaptive policies such as the one given in [3] and to provide further evidence concerning the fact that performance achieved is very close to the one under optimal control. Experimental confirmation can be found in [11] where an implementation is described.
Although many other access scheme have been proposed and proved to achieve throughput well over 1/e, the adaptive control policy is still necessary somewhere in the improved schemes. As long as no other channel is provided to permit a perfect information exchange and concerted actions of the terminals, the random access principle is expected to remain at some level. And there, the throughput is upper-bounded by 1/e. As an example, we can point out the "ALOHA state" of the reservation scheme proposed by ROBERTS [13]. Although the overall throughput could be over 1/e, a random access technique is used when the reservation messages are transmitted. Precisely there, a control policy should be implemented to achieve a throughput rate close to 1/e.

Cases where improved performance is obtained via FM capture are discussed in [4] while a thorough study of multiple frequency channels using buffering techniques is presented in [14].

## 2.- MODEL DESCRIPTION AND ASSUMPTIONS.

The state transition diagram shown in Fig.1 represents transmissions and retransmissions of packets of constant length taking place in fixed-duration time slots.

The terminals are initially in the active state. As soon as some terminal has a packet completely formed, it starts a transmission at the beginning of a time slot. If no other terminal does the same simultaneously, the transmission is said to be successful and the corresponding terminal returns to the active state. If two packets or more are transmitted in the same time slot, they are said to collide with one another. The
corresponding terminals then enter the blocked state from where they will retransmit the same packet in a future slot according to Bernoulli trials with probability f. A blocked terminal joins the active state again as soon as its backlogged packet is transmitted successfully.

The active terminals are assumed to form an infinite Poisson source with rate $\lambda = \sum_{i=1}^{\infty} ia_i$, where $a_i$ is the probability of i transmissions from all of them in each slot.

Denoting by $B(k)$ the number of blocked terminals in the system at the beginning of the k^th^ slot, by $Y(k)$ the number of retransmissions from them and $X(k)$ the number of transmissions from the active terminals in slot k, we can build a sequence of r.v. (random variables) $\{T(k)\}_{k \in \mathbb{N}}$ with the r.v. $X(k)$ and $Y(k)$ :

$$T(k) = X(k) + Y(k) \quad , \quad k \ge 0$$

$T(k)$ is the total traffic in the channel during slot k. The conditional traffic $T_n$ is defined as

$$T_n = E\{T(k) | B(k)=n\}$$

and is independent of k because of the stationarity of traffic arriving from the active terminals and of the assumptions concerning transmissions from the blocked terminals. Notice that $X(k)$ is independent of $Y(j)$, $0 \le j \le k$ and that $Y(k)$ depends only on $B(k)$.

We obtain :

$$T_n = \lambda + \sum_{i=1}^{n} ib_i(n)$$
where $b_j(n) \triangleq \text{Pr} [j \text{ retransmissions from the blocked terminals} | B(k)=n]$

This is essentially the traffic model used in [3,4,5].

The adaptive policy we are considering follows from the results restated below :

### THEOREM 1 [6]

For maximizing the throughput of the broadcast channel, the parameter $f$ of the Bernoulli trials must be set to :

$$f^*(n) = \frac{a_o - a_1}{na_o - a_1}$$

where $n$ is the value currently taken by $B(k)$.

### THEOREM 2 [3]

For the broadcast channel under optimal retransmission policy $f^*(n)$, we have :

$$T_n = 1 + \lambda - a_1/a_o + (a_1/na_o)(1-a_1/a_o) + O(1/n^2)$$

If we add the assumption that the active terminals form a Poisson source, we obtain :

$$T_n = 1 + (\lambda/n)(1-\lambda) + O(1/n^2)$$

This result can easily be derived from the definitions of $T_n$, $a_i$ and $b_i(n)$ so we give it without proof.
### 3.- THE CONTROL ALGORITHM.

The expressions of $T_n$ given above are very interesting, for they tell us that $T_n$ tends to a constant $T^*$ independent of $n$ as soon as the number $n$ of blocked terminals is high enough (practically when $n > 5$). If the active terminals form a Poisson source then $T^*= 1$. The following algorithm was introduced in [3].

We are aiming at estimating as accurately as possible $T(k)$ then regulating it around $T^*$.

By listening to the channel, we can distinguish silent slots from the others and define the "silence" r.v. $S(k)$ :

$$ S(k) = \begin{cases} 1 & \text{if } T(k) = 0 \\ 0 & \text{otherwise} \end{cases} \quad k \ge 0 $$

Dividing the channel time into "windows" of $L$ slots, we then define

$$ S'(k) = \sum_{j=0}^{L-1} S(k-j)/L \quad 0 < L < \infty, \quad L-1 < k < \infty $$

$$ \hat{T}(k) = -\ln S'(k) $$

If the process ${T(k)}_{k \in \mathbb{N}}$ is Poisson and stationary then $S'(k)$ is the maximum likelihood estimator of $\Pr[T(k) = 0]$ and $\hat{T}(k)$ is an estimator of $T(k)$.

Let ${f_k}_{k \in \mathbb{N}}$ be a sequence of r.v. taking values in $]0,1[$ and which will give the values of the retransmission probability in the adaptive control. The process ${f_k}_{k \in \mathbb{N}}$ depends on the process ${\hat{T}(k)}_{k \in \mathbb{N}}$ in the following way :
$$f_{k+1} = \begin{cases} f_k \frac{l}{l-1} & , \text{ if } \hat{T}(k) \le 1 - \epsilon_1 \\ f_k & , \text{ if } 1 - \epsilon_1 < \hat{T}(k) < 1 + \epsilon_2 \\ f_k \frac{l}{l+1} & , \text{ if } \hat{T}(k) \ge 1 + \epsilon_2 \end{cases}$$

where $\epsilon_1, \epsilon_2 > 0$ are parameters whose values are determined according to some criterion (cf. next paragraph) and $l \in N^+$ is an integer characterizing the control during slot k.

The simple function $f^*(n) = \frac{a_o - a_1}{na_o - a_1}$ of the optimal control is then replaced by a more complicated one which depends on the set of parameters {$\epsilon_1, \epsilon_2, \hat{T}(k), l, \delta$}, where :

* $\delta$ is the step of the control algorithm and is chosen in terms of the window width $L$

* $c$ is a constant determining the contribution of the blocked terminals to the channel traffic and is chosen in terms of $\lambda$ and $\delta$.

We see that $\delta$ must be chosen somewhat larger than $\lambda$ so that the system is capable of recovery from the worst cases. This is a consequence of the following argument. During a sudden traffic peak, the number of blocked terminals increases quickly in the system and the control algorithm does not react promptly enough. The result is that the retransmission rate is much too high, even when the input rate drops back to $\lambda$ again. Due to this very fact, the number of blocked terminals keeps on growing at the rate $\lambda L$ per window. And the retransmission probability must decrease fast enough to compensate the increase in number of blocked terminals.
Instead of computing $S'(k)$ as a moving average, we may also consider, with a view to saving processing time, the possibility of jumping from window to window and keeping the same control throughout a given window. A refinement can also be considered, which consists in exponentially smoothing the computation of $S'(k)$: we can then decide at what speed the influence of the past history should fade. These two modifications have not been examined in the present paper but are reported on elsewhere [16].

### 4.- PARAMETER SETTINGS.

The behaviour of the control algorithm just described can be modelled as a birth and death process whose transition diagram is given in Fig.2. The state of this process is the parameter $\ell$ of the algorithm.

Assuming that the total channel traffic is Poisson, we can derive its transition probabilities $\Pi_{\ell, \ell+1}$, $\Pi_{\ell, \ell}$, $\Pi_{\ell, \ell-1}$:

$$ 
\begin{aligned}
\Pi_{\ell, \ell+1} &= \operatorname{Pr} [\hat{T}(k) \ge 1+\varepsilon_2 | \ell] \\
&= \sum_{j=L(1+\varepsilon_2)}^{\infty} \frac{(LT_{\ell})^j}{j!} e^{-LT_{\ell}}
\end{aligned} 
$$

$$ 
\begin{aligned}
\Pi_{\ell, \ell-1} &= \operatorname{Pr} [\hat{T}(k) \le 1-\varepsilon_1 | \ell] \\
&= \sum_{j=0}^{L(1-\varepsilon_1)} \frac{(LT_{\ell})^j}{j!} e^{-LT_{\ell}}
\end{aligned} 
$$

$$ \Pi_{\ell, \ell} = 1 - \Pi_{\ell, \ell+1} - \Pi_{\ell, \ell-1} $$

$$ \text{where : } T_{\ell} \triangleq \lambda + \frac{nc}{\ell\delta} $$

is the total channel traffic for a given value of $\ell$.
Denote by {$P_\ell, \ell \ge 1$} the steady-state solution of
this birth and death process. It is easily obtained as

$$
P_{\ell} = \frac{\displaystyle\sum_{\substack{\ell-1 \\ \Pi}}^{\infty} \frac{(LT_{\cdot j})^i}{j} e^{-LT_{\cdot j}}}{\displaystyle\sum_{\substack{j=1 \\ \Sigma}}^{\ell-1} L(1-\epsilon_1) \frac{(LT_{\cdot j})^i}{j} e^{-LT_{\cdot j}}} \quad \begin{array}{c} i=L(1+\epsilon_2) \\ i=0 \end{array} \quad \begin{array}{c} i! \\ i! \end{array} \\
= \frac{\displaystyle\sum_{\substack{1 + \ell=0 \\ \Sigma}}^{\infty} \frac{(LT_{\cdot i})^j}{j} e^{-LT_{\cdot i}}}{\displaystyle\sum_{\substack{m=0 \\ \Pi}}^{\ell-1} \frac{L(1+\epsilon_2)}{j=1} \frac{(LT_{\cdot i})^j}{j!} \quad \begin{array}{c} m-1 \\ i=1 \end{array}} \quad \begin{array}{c} j=L(1+\epsilon_2) \\ j=0 \end{array} \quad \begin{array}{c} j! \\ j! \end{array}
$$

and it obviously depends on the parameters $\epsilon_1$ and $\epsilon_2$.
$P_\ell$ represents the steady-state probability of finding the
adaptive control algorithm in its state $\ell$, i.e. in the state
at which the retransmission probability has the value $f = c/\ell\delta$.

To each value of $\ell \in \mathbb{N}^+$ corresponds a conditional
average throughput $D_\ell = T_\ell e^{-T_\ell \ell}$, so that the average throughput
$\bar{D}$ for the adaptive control policy is given by

$$
\bar{D}(\epsilon_1, \epsilon_2) = \sum_{\ell=1}^{\infty} D_{\ell} \cdot P_{\ell}
$$

The parameters $\epsilon_1, \epsilon_2$ should be chosen so as to
maximize this throughput. We have used a direct search method
for finding the values of $\epsilon_1^*$ and $\epsilon_2^*$ such that :

$$
\bar{D}(\epsilon_1^*, \epsilon_2^*) = \max_{\{\epsilon_1, \epsilon_2\}} \left[ \sum_{\ell=1}^{\infty} D_{\ell} P_{\ell} \right]
$$
We solved a numerical example by the HOOKE and JEEVES method [7]; with $\lambda = 0.3$, $L = 60$, $\delta = 20$ and $c = 14$. In accordance with the underlying idea of theorem 2, we found for different values of $n$ essentially the same optimal couple :

$$ (\epsilon_1^*, \epsilon_2^*) = (0.2, 1) $$

that is, these results do not seem to depend significantly on $n$.

Fig.3 plots $\bar{D}$ vs $\epsilon_1$ with $\epsilon_2 = 1$, while Fig.4 is obtained by setting $\epsilon_1$ to 0.2 and varying $\epsilon_2$. Due to the complex form of $P_\ell$, roundoff errors are significant in the numerical computations which lack precision when $\epsilon_2 > 1$, but this is the less interesting part of the curve because of its flatness in this region.

Fig.5 to 9 show the simulation results of the adaptive policy with different values of $(\epsilon_1, \epsilon_2)$. As $B$ represents the current number of blocked terminals in the system, theorem 2 states that the total traffic rate $T = \lambda + Bf$ is equal to 1, i.e. the product $B.f$ is equal to $1 - \lambda$ under optimal control. Given an input rate $\lambda$, when we set the value of $f$ to $\frac{c}{\ell\delta}$ in the adaptive policy, we might as well say that we assume the number of blocked terminals is equal to $\hat{B} = (1 - \lambda)/f$. The evolution in time of the real number of blocked terminals as well as that of $\hat{B}$ are plotted side by side to facilitate comparison. For $(\epsilon_1, \epsilon_2) = (0.2, 1)$, the throughput is obviously higher; this is proved by the fact that the 100 initially blocked terminals reenter the active state more quickly. The difference between the real number of blocked terminals and $\hat{B}$ is also lesser on the average, in that case.
Our simulation experiments also confirm the salient features displayed in Fig.3 and 4 : the result of the adaptive control is much less sensitive to the variation of $\epsilon_2$ than to that of $\epsilon_1$, especially when $\epsilon_2$ is greater than one.

## 5.- CONVERGENCE OF THE ALGORITHM.

Our aim is now to show that for reasonable values of $\lambda$ ($\lambda = 0.3$ in our numerical example), the adaptive control algorithm stabilizes the channel.

We shall carry out a proof for the special case where the window width $L$ is set to one, the proof in the general case will follow the same line. We shall see that the mathematical expectation

$$ \Delta(n) \triangleq E\{B(k+1)-B(k-1) \mid B(k-1)=n\} $$

is always negative and decreases when $n$ increases. And the proof will be completed by invoking a result of PAKES [17], restated below :

### PAKES's Lemma.-

Let $\{X(k), k \in N\}$ be an irreducible and aperiodic Markov chain whose state space is the set of the non negative integers. The following conditions are sufficient for its ergodicity :
$$1) |E\{X(k+1)-X(k) | X(k)=n\}| < \infty \quad \forall n$$

$$2) \lim_{n \to \infty} \sup E\{X(k+1)-X(k) | X(k)=n\} < 0$$

Starting from a situation characterized by :

$$B(k-1) = n, \ell_{k-1} = \ell \quad \text{and} \quad f_{k-1} = f = \frac{(1-\lambda)}{\ell \delta},$$

the adaptive control acts with $(\epsilon_1, \epsilon_2) = (0^+, 0)$ and gives rise to the following transitions :

$$B_{k-1} = n \underset{f_{k-1} = f}{\longrightarrow} B_k \underset{f_k = \left\{ \begin{array}{ll} f^+ = f \quad \frac{\ell}{\ell-1} \\ f^- = f \quad \frac{\ell}{\ell+1} \end{array \right.}{\longrightarrow} B_{k+1}$$

$$B_k = n + \left\{ \begin{array}{ll} -1, & f_k = f^- \quad \text{with probability } a_o b_l(n) \\ 0, & f_k = f^+ \quad \text{and } a_o b_o(n) \\ 0, & f_k = f^- \quad \text{and } a_o[1-b_o(n)-b_l(n)] \\ 0, & f_k = f^- \quad \text{and } a_l b_o(n) \\ 1, & f_k = f^- \quad \text{and } a_l[1-b_o(n)] \\ i, i \ge 2, & f_k = f^- \quad \text{and } a_i \end{array} \right.$$
For the passage from slot k to slot k+1, the transition probabilities are easily obtained :

$$ \text{Pr} [B_{k+1}=j \mid B_k=n_k] = \begin{cases} a_o b_l(n_k) & \text{if } j = n_k - 1 \\ a_o[1-b_l(n_k)] + a_l b_o(n_k) & \text{if } j = n_k \\ a_l[1-b_o(n_k)] & \text{if } j = n_k + 1 \\ a_i, & \text{if } j = n_k + i \end{cases} \quad i \ge 2 $$

It follows that the mathematical expectation

$$ \Delta_{\ell} \triangleq \mathbb{E}[B_{k+1} - B_{k-1} \mid B_{k-1}=n, \ell_{k-1}=\ell] $$

is given by the expression :

$$ \begin{aligned} & - 2a_o^2 b_l(n)b_l^{-}(n-1) - a_o b_l(n)[a_o - a_o b_l^{-}(n-1)] + a_l^2 b_l^{-}(n) \\ & + a_o^2 b_o(n)b_l^{+}(n) - a_o^2[1-b_o(n)-b_l(n)]b_l^{-}(n) - a_o a_l b_o(n)b_l^{-}(n) \\ & + a_o b_o(n)a_l[1-b_o^{+}(n)] + a_l[1-b_o^{-}(n)][a_o - a_o b_o(n) - a_o b_l(n) + a_l b_o(n)] \\ & + a_l[1-b_o(n)][a_o - a_o b_l^{-}(n+1) + a_l b_o^{-}(n+1)] + a_o a_l b_l(n) + a_o a_l b_l^{-}(n+2) \\ & + \sum_{i=2}^{\infty} i \{a_i[2a_o - a_o[b_l(n)+b_l^{-}(n+1)]] + a_l[b_o(n)+b_o^{-}(n+\ell)]\} \\ & + a_{i+1} a_o[b_l(n)+b_l^{-}(n+\ell+1)] \\ & + 2 \sum_{j=1}^{i-1} a_j a_{i-j} - a_l a_{i-1}[b_o(n)+b_o^{-}(n+i-1)] \} , \end{aligned} $$
with the following notations :

$$b_o^+(n) = [1 - f \frac{l}{l-1}]^n$$

$$b_1^+(n) = nf \frac{l}{l-1} [1 - f \frac{l}{l-1}]^{n-1} = nf \frac{l}{l-1} b_o^+(n-1)$$

$$b_o^-(n) = [1 - f \frac{l}{l+1}]^n$$

$$b_1^-(n) = nf \frac{l}{l+1} [1 - f \frac{l}{l+1}]^{n-1} = nf \frac{l}{l+1} b_o^-(n-1)$$

$$b_o(n) = [1 - f]^n$$

$$b_1(n) = nf [1 - f]^{n-1} = nfb_o(n-1)$$

Recalling the definition of $P_\ell \triangleq \text{Pr}[l_k = l]$, we can apply the law of total probabilities to

$$\bar{\Delta}(n) \triangleq E\{B_{k+1} - B_{k-1} \mid B_{k-1} = n\}$$

and obtain :

$$\bar{\Delta}(n) = \sum_{\ell=1}^{\infty} \Delta_{\ell} P_{\ell}$$

A numerical program computes the value of $\bar{\Delta}(n)$ and shows that $\bar{\Delta}(n)$ takes its values in $\mathbb{R}^-$ and decreases when $n$ increases.
Denote by $N^e \triangleq \{i : i=2k, k \in N\}$ the set of all even-valued integers and by $N^o \triangleq \{i : i=2k+1, k \in N\}$ the set of all odd-valued integers. $\{B(k), k \in N^e\}$ and $\{B(k), k \in N^o\}$ are two imbedded and mutually disjoint Markov chains. The above computations, in association with PAKES's lemma, constitute then a numerical proof that these two Markov chains are ergodic. This means that the channel is stable or that the adaptive control algorithm is convergent.

## 6.- CONCLUSION.

In Fig.10 and 11, we make a performance comparison between the optimal control using BREMAUD-SEGALL's estimator [4] and the adaptive policy . Under identical initial conditions (all of the 100 terminals of the system are assumed blocked at instant $t = 0$; each of the terminals generates packets at rate $\sigma = 0.003$ in the active state), the channel behaves differently depending on the control policy but the final results regarding throughput and recovery time are much the same. By providing essentially the same type of performance, the adaptive control, thanks to its simplicity, is much more economical in processing time.
<figure><img src="image_3.png" /><figcaption>Fig.1</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 5</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 7</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11</figcaption></figure>
REFERENCES

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[2] ABRAMSON, N. "Packet Switching with Satellite", National Computer Conference, New York, June 4-8, 1973, AFIPS Conference Proceedings, 1973, Vol. 42, pp. 695-702.

[3] BANH, T.A. and GELENBE, E. "Performance Evaluation of Adaptive Policies for the Broadcast Channel", Proceedings of the Conference "Workshop on Data Communications", pp. 17-31. 15-18 September 1975. IIASA, Vienne, Autriche.

[4] BANH, T.A. and GELENBE, E. "Etude et Simulation de Réseaux de Diffusion sous Contrôle", R.A.I.R.O. Informatique / Computer Science, Vol. 11, N° 3, 1977, pp. 301-321.

[5] FAYOLLE, G., GELENBE, E. and LABETOULLE, J. "The Stability Problem of Broadcast Packet Switching Computer Network", Acta Informatica 4, 1974, pp. 49-53.

[6] FAYOLLE, G., GELENBE, E. and LABETOULLE, J. "Stability and Optimal Control of the Packet Switching Broadcast Channel", JACM, Vol. 24, N° 3, July 1977, pp. 375-386.

[7] HOOKE, R. and JEEVES, T.A. "Direct Search Solution of Numerical and Statistical Problems", JACM, Vol. 8, N° 2, April 1961, pp. 212-229.

[8] KLEINROCK, L. and LAM, S.S. "Packet Switching in a Multi-access Broadcast Channel: Performance Evaluation", IEEE Trans. Commun., Vol. COM-23, pp. 410-423, April 1975.

[9] KLEINROCK, L. and TOBAGI, F.A. "Packet Switching in Radio Channels: Part I -Carrier sense multiple access modes and their throughput delay characteristics", IEEE Trans. Commun. Vol. COM-23, pp. 1400-1416, December 1975.
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[11] MIEGE, S., MERVAN, D. and MUSSAT, C. "An Experimental Facility Simulating a Slotted ALOHA System", Proceedings of the "Conference on Satellite Communications", IRISA et Ecole Supérieure d'Electricité. Rennes, June 1976.

[12] ROBERTS, L.G. "ALOHA Packet System With and Without Slots and Capture", ARPA Network Information Center, Stanford Research Institute, Menlo Park, California, ASS Note 8 (NIC 11290), June 1972.

[13] ROBERTS, L.G. "Dynamic Allocation of Satellite Capacity through Packet Reservation", National Computer Conference, New York, June 4-8, 1973, AFIPS Conference Proceedings, 1973, Vol. 42, pp. 711-716.

[14] SPANIOL, O. "Mehrfrequenz-ALOHA-Netzwerke", Informatik-Fachberichte N° 9, Springer-Verlag, pp. 277-295. Proc. of the Conf. "Workshop über Modelle für Rechensysteme", 31/3-1/4/1977. Bonn, R.F.A.

[15] TOBAGI, F.A. and KLEINROCK, L. "Packet Switching in Radio Channels : Part III - Polling and (Dynamic) Split-Channel Reservation Multiple Access", IEEE Trans. on Communications, Vol. COM-24, N° 8, pp. 832-845, August 1976.

[16] BANH, T.A. "Réseaux d'ordinateurs à commutation de paquets : modélisation mathématique et simulation en vue de l'optimisation des performances". Thèse de Doctorat en Sciences Appliquées, 1977-1978. Liège University, Belgium.

[17] PAKES, A.G. "Some Conditions for Ergodicity and Recurrence of Markov Chains", Operations Research 17, 1969, pp. 1058-1061.