# Performance limits of the tmm protocol: modelling and measurement

Erol Gelenbe, Patrice Mussard, Jean-Louis Grange

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Erol Gelenbe, Patrice Mussard, Jean-Louis Grange. Performance limits of the tmm protocol: modelling and measurement. [Research Report] IRIA-RR-230, IRIA. 1977, pp.18. hal-04716470

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PERFORMANCE LIMITS
OF THE TMM PROTOCOL:
MODELLING AND MEASUREMENT

Erol GELENBE
Jean Louis GRANGE
Patrice MUSSARD

Rapport de Recherche N° 230

Avril 1977

18p
PERFORMANCE LIMITS OF THE TMM PROTOCOL :
MODELLING AND MEASUREMENT

Erol Gelenbe*, Jean-Louis Grangé**, Patrice Mussard**

Résumé :

TMM est une des procédures de transmission utilisées dans CIGALE et dont la principale caractéristique de fonctionnement est le contrôle de la transmission en cours par blocage des émissions suivantes jusqu'à réception d'un acquittement correct ou échéance d'un time-out. Ce rapport présente une évaluation de ce type de procédure permettant de rechercher les limites des performances de TMM compte tenu de la capacité de transmission de la ligne disponible et des caractéristiques du trafic. Des mesures effectuées sur CIGALE permettent de valider les prédictions d'un modèle mathématique de TMM que nous proposons. Ce modèle nous permet également de calculer la longueur moyenne des paquets qui maximise le débit de la procédure TMM.

Abstract :

The TMM protocol uses a stop-and-wait procedure with acknowledgements for error control.
The purpose of this paper is to investigate the basic limits of TMM with respect to its utilisation
of available line capacity. Measurement results are obtained and a mathematical model is presen-
ted in order to predict TMM's performance and to derive optimum average lengths for random
length or fixed length packets.

<footnote>* IRIA-LABORIA, B.P. 105 -78150 Le Chesnay.</footnote>

<footnote>** IRIA-Projet CYCLADES, B.P. 105 - 78150 Le Chesnay.</footnote>
<figure><img src="image_3.png" /></figure>

# 1. Introduction

CIGALE [1] is the packet switching network of CYCLADES [2]. The basic service offered by CIGALE simply consists of transporting independent packets to their destination : this type of service has been called Datagram service within CCITT*.

In order to reach their destination, packets are forwarded from node to node through point to point synchronous lines. Since lines introduce delays and errors, line handling protocols are needed to provide for efficient and reliable transmissions. Standard manufacturer's protocols are made available in CIGALE so as to avoid any special piece of hardware or software for host connections. The purpose of this paper is to present some results on performance measurement and evaluation of the TMM protocol.

TMM is a CII standard protocol [3]. It allows full duplex symmetrical transmission, i.e. physical transmission in both directions at the same time. It uses a stop-and-wait scheme to control the packet transmission : a new packet can be transmitted over the line only when the previous one has been acknowledged. A timeout is set before transmitting each packet. If it expires before receiving the acknowledgement, the packet is retransmitted. A simple packet numbering scheme allows to detect and discard duplicated packets. It is to be noted that, since the transmission of a new packet cannot occur before receiving the acknowledgment of the previous one, any delay in acknowledgment transmission leads to the waste of a portion of the bandwidth available for the reverse traffic. In particular, short packet traffic may be considerably slowed down by long packets transmitted in the reverse direction : short packet acknowledgments are delayed due to the transmission time of long packets.

<footnote>* Comité Consultatif International Télégraphique et Téléphonique</footnote>
Figure 1 shows how a part of the bandwidth is wasted if packets are not of the same length in both directions of the line.

<figure><img src="image_4.png" /><figcaption>FIGURE 1</figcaption></figure>

Transmission diagram in both directions for the TMM protocol

## 2. The measurements

In order to evaluate the performance of the TMM protocol various measurements have been performed. For that purpose, the configuration shown in Figure 2 has been set up.

<figure><img src="image_5.png" /><figcaption>FIGURE 2</figcaption></figure>
Two nodes are connected by a local telephone line at 19 200 bits/sec. Base band modulation modems are used and the line length is extremely short (about 2 meters) : propagation delays through the hardware are negligible compared to the transmission delays.

Each node is loaded with the regular CIGALE node system including the TMM protocol software (TMMS) and an artificial traffic generator (ATG).

In all experiments, ATG is used to generate fixed length packets at the maximum rate in both directions, i.e. as fast as TMMS can accept them.

A typewritten is connected to each node and is used to control the ATG (e.g, start, stop) and to extract measurement data from the node system.

A number of experiments have been conducted with various combinations of packet length at each end. Throughput results obtained are summarized in Figure 3 and 4.

In the following sections of this paper we present a mathematical model of TMM and validate it with the measurement data obtained. We then obtain some basic properties of this protocol. In particular we compute the maximum throughput of TMM in the presence of packets of variable lengths.

A study of the time-out optimization for stop-and-wait schemes is presented elsewhere [6]. In this study we assume that error rates can be neglected.
throughput (Kb/sec)

Line utilization (%)

<figure><img src="image_6.png" /><figcaption>Figure 3 : TMM Performances (*) (Reverse Traffic Packet Length : 37 octets)</figcaption></figure>

throughput (Kb/sec)

Line utilization (%)

<figure><img src="image_7.png" /><figcaption>Figure 4 : TMM Performances (*) (Reverse Traffic Packet Length : 67 octets)</figcaption></figure>

<footnote>(*) The plotted throughput and line utilization include overhead (i.e. synchronization characters, packet header, acknowledgements).</footnote>
### 3. The model and its validation

The behaviour of the protocol is modelled by examining the state of each of the two nodes : let $S_1$ and $S_2$ be the state node 1 and 2 respectively. $S_1$ can take one of three values :

- state P corresponds to a packet transmission from 1 to 2

- state A corresponds to the transmission of an acknowledgement from 1 to 2

- state W is the state in which the sender of node 1 waits for an acknowledgement to arrive from node 2.

The states P, A, W of node 2 are defined in similar fashion.

We are assuming here that pure software times between state transitions of the senders are negligible. We shall see via our measurement results that this assumption is justified.

If both nodes are considered we obtain the state transition diagram of Figure 5.

<figure><img src="image_8.png" /><figcaption>FIGURE 5</figcaption></figure>
In state PP both senders are transmitting user packets, and a packet is
travelling on each of the two lines. Depending on which packet transmission
terminates first, state PW or state WP is entered. For instance, in state
PW the packet transmission of sender 2 is complete and it enters the wait
state (waiting for an acknowledgement to arrive from node 1). From state PW
the system moves to state AA when the packet transmission from 1 to 2 is
complete, since receiver 1 has received a packet and now proceeds to acknowledge
it and so does receiver 2. Finally, after this last acknowledgement is
transmitted the system will once more enter state PP.

Notice that the system cannot be in state PA or in state AP : if an
acknowledgement is being transmitted from 1 to 2 then 2 _must_ be waiting for it
in state and vice-versa. Similarly, we do not allow state WW since we have
excluded the possibility of software waiting times (such times have been
assumed to be negligibly short compared to packet or acknowledgement
transmission times; an estimate is obtained in Section 5).

The time spent in state AA is α (i.e. the transmission time of an
acknowledgement). In the computations everything will be normalized so that
α will also be the length in bytes of the acknowledgement packet. Let π₁ and
π₂ be the average length of user packets sent from 1 to 2, and from 2 to 1,
respectively. We shall assume in this section that user packets are exponen-
tially distributed ; this restriction, which simplifies the analysis is not
essential and is removed in the computations given in the Appendix. Our
purpose in keeping this restriction in this section is to avoid complications
which may confuse the reader.

With this assumption we show (in the Appendix) that the average time
spent by the system in state PP at each passage through that state is
$(\pi_1^{-1} + \pi_2^{-1})^{-1}$. The average time spent in state PW and WP is $\pi_1$ and $\pi_2$, respectively. Similarly we show that from state PP the system will next enter state PW with probability $\pi_1/(\pi_1 + \pi_2)$ or state WP with probability $\pi_2/(\pi_1 + \pi_2)$.
We notice that the model shown on Figure 5 is a semi-Markov process [4]. The steady-state probability that it will be in state S is denoted by $p(S)$: we shall need $p(PP)$, $p(PW)$, $p(WP)$ since these states are the only ones which contribute to the throughput. We define $D_1$ the throughput of sender 1 as

$$D_1 = p(PP) + p(PW)$$

and $D_2$ the throughput of sender 2

$$D_2 = p(PP) + p(WP)$$

Notice that in state PP both senders are busy. The total throughput is

$$(1) \quad D = D_1 + D_2$$

These quantities are easily obtained as [3]

$$(2) \quad D_1 = \frac{1 + \pi_1 / \pi_2}{1 + \frac{\pi_1}{\pi_2} + \frac{\pi_2}{\pi_1} + \frac{\alpha}{\pi_1} \left(1 + \frac{\pi_1}{\pi_2}\right)}$$

$$(3) \quad D_2 = \frac{1 + \pi_2 / \pi_1}{1 + \frac{\pi_1}{\pi_2} + \frac{\pi_2}{\pi_1} + \frac{\alpha}{\pi_1} \left(1 + \frac{\pi_1}{\pi_2}\right)}$$

To simplify the notation we shall use $x = \pi_1 / \pi_2$, $y = \alpha / \pi_1$. We notice the following after some simple algebra :

- For fixed $\pi_2$, $D_1$ is a monotone increasing function of $\pi_1$ which becomes 1 as $\pi_1 \to \infty$. Similarly, for fixed $\pi_2$ it is monotone decreasing in $\pi_2$ tending to 0 as $\pi_2 \to \infty$.

- The above statements can be made also for $D_2$ if we interchange $\pi_1$ and $\pi_2$ (see Figure 6).
These theoretical results are illustrated on Figure 6. We have taken three values of $\pi_1$ (10, 20 and 100 bytes) and have set $\alpha = 7$ bytes (this is the length of an acknowledgement packet in CIGALE [1]). We show $D_1$ and $D_2$, the forward and reverse traffic rate as a function of $\pi_2$ (the average length of packets in the reverse traffic. We also show $D$ the total throughput (sum of forward and reverse traffic rate). Notice that equations (1), (2), (3) give the normalized traffic rate ; the true traffic is obtained by multiplying the normalized traffic rate by the link capacity in bits per second.

On Figure 7, 8, we validate the analytical model by providing a comparison with measurements on the CIGALE packet switching network.

Consider first the measurement results presented on Figure 5. The values are $\pi_1 = 37$ bytes (27 bytes of header, plus 10 bytes of text), and $\pi_2$ taking the values 27, 37, 67, 117, 227, 282 bytes. What is shown is the total traffic in one direction. $\hat{D}_1$ and in the other $\hat{D}_2$ in bits per second. $\hat{D}_1$, $\hat{D}_2$ include the acknowledgements.

We also show the measured number of acknowledgements per second $A_m$, and the theoretical values $A_t$ computed from the following formula :

$$ (4) \qquad A_t = \frac{\hat{D}_1 - 56 A_t}{8 \pi_1} $$

where 56 is the number of bits in an acknowledgement. Notice that we could also use the formula

$$ (5) \qquad A_t = \frac{\hat{D}_2 - 56 A_t}{8 \pi_2} $$

The small discrepancy between $A_t$ and $A_m$ indicates the existence of measurement errors. In Figure 6 we show the useful traffic $D_1'$, $D_2'$ given in bits/second, where

$$ (6) \qquad D_1' = \hat{D}_1 - 56 A_m $$
<figure><img src="image_9.png" /><figcaption>FIGURE 6</figcaption></figure>
<table><thead><tr><th>π₂ bytes</th><th>27</th><th>37</th><th>67</th><th>117</th><th>177</th><th>227</th><th>282</th></tr></thead><tbody><tr><td>A<sub>m</sub> ACKS/sec.</td><td>41.6</td><td>41.6</td><td>27.3</td><td>17.1</td><td>12</td><td>9.7</td><td>7.9</td></tr><tr><td>A<sub>t</sub> ACKS/sec.</td><td>41.5</td><td>40.6</td><td>26.2</td><td>17.33</td><td>11.8</td><td>9.66</td><td>8.09</td></tr><tr><td>D'<sub>1</sub> bits/sec.</td><td>14320</td><td>12000</td><td>8000</td><td>5100</td><td>3600</td><td>2800</td><td>2300</td></tr><tr><td>D<sub>1</sub> bits/sec.</td><td>12590</td><td>11366</td><td>7400</td><td>5345</td><td>3733</td><td>2968</td><td>2422</td></tr><tr><td>D'<sub>2</sub> bits/sec.</td><td>10720</td><td>12000</td><td>14700</td><td>16250</td><td>16800</td><td>17500</td><td>17800</td></tr><tr><td>D<sub>2</sub> bits/sec.</td><td>9185</td><td>11366</td><td>15401</td><td>16906</td><td>17849</td><td>18225</td><td>18462</td></tr><tr><td>D' bits/sec.</td><td>25040</td><td>24000</td><td>22700</td><td>21350</td><td>20400</td><td>20300</td><td>20100</td></tr><tr><td>D bits/sec.</td><td>21775</td><td>20442</td><td>22801</td><td>22251</td><td>21582</td><td>21193</td><td>20884</td></tr><tr><td>D̂<sub>1</sub> bits/sec.</td><td>14600</td><td>14300</td><td>16200</td><td>17200</td><td>17400</td><td>18093</td><td>18270</td></tr><tr><td>D̂<sub>2</sub> bits/sec.</td><td>11000</td><td>14300</td><td>9500</td><td>6040</td><td>4212</td><td>3342</td><td>2731</td></tr></tbody></table>

$$ \pi_1 = 37 \text{ bytes}, \pi = 7 \text{ bytes} $$

<table><caption>Figure 7</caption><thead><tr><th>π₂ bytes</th><th>27</th><th>37</th><th>67</th><th>117</th><th>177</th><th>227</th><th>282</th></tr></thead><tbody><tr><td>A<sub>m</sub> ACKS/sec.</td><td>27.3</td><td>27.3</td><td>27.3</td><td>17.1</td><td>12</td><td>9.7</td><td>7.9</td></tr><tr><td>A<sub>t</sub> ACKS/sec.</td><td>27.35</td><td>26.7</td><td>26.72</td><td>17.36</td><td>12.07</td><td>9.55</td><td>7.84</td></tr><tr><td>D'<sub>1</sub> bits/sec.</td><td>14665</td><td>14631</td><td>14297</td><td>10143</td><td>7054</td><td>5580</td><td>4578</td></tr><tr><td>D<sub>1</sub> bits/sec.</td><td>15740</td><td>14764</td><td>11970</td><td>8659</td><td>6361</td><td>5159</td><td>4269</td></tr><tr><td>D'<sub>2</sub> bits/sec.</td><td>5711</td><td>7845</td><td>14282</td><td>17167</td><td>17809</td><td>17969</td><td>18231</td></tr><tr><td>D<sub>2</sub> bits/sec.</td><td>6345</td><td>8154</td><td>11970</td><td>15148</td><td>16790</td><td>17488</td><td>17935</td></tr><tr><td>D' bits/sec.</td><td>20375</td><td>22476</td><td>28576</td><td>27310</td><td>24863</td><td>23549</td><td>22809</td></tr><tr><td>D bits/sec.</td><td>22085</td><td>22918</td><td>23940</td><td>23807</td><td>23151</td><td>22647</td><td>22204</td></tr><tr><td>D̂<sub>1</sub> bits/sec.</td><td>16194</td><td>16160</td><td>15823</td><td>10280</td><td>7150</td><td>5658</td><td>4641</td></tr><tr><td>D̂<sub>2</sub> bits/sec.</td><td>7240</td><td>9374</td><td>15811</td><td>17304</td><td>17905</td><td>18047</td><td>18294</td></tr></tbody></table>

$$ \pi_1 = 67 \text{ bytes}, \alpha = 7 \text{ bytes} $$

<caption>FIGURE 8</caption>
$$ (7) \qquad D_2' = \hat{D}_2 - 56 A_m $$

For the links being measured, the maximum capacity is 19 200 bits/second. In Figure 7 we also show the theoretical traffic obtained from equations (1), (2), (3). We see that the maximum error in the model prediction is 14 % (for the case $\pi_1 = 27$), while generally the error is of the order of 5 %. Similar results for other sets of measurements are given on Figure 8, where the largest error is 16 %.

The high degree of accuracy of our model predictions confirm the validity of our simple mathematical model as well as its robustness : in the measurements all packet lengths are constant (and not exponential), and software wait times exist. The agreement between the theory and the measurements may be due to the fact that due to lack of synchronisation between nodes : a jitter is introduced which is perceived by the "other" node as a random packet length.

If we examine the data of Figure 5 we obtain that $D$, the total two-way throughput has a definite maximum value which is in the neighbourhood of $\pi_2 = 67$ bytes. This indicates that if we have $\pi_1 = 37$ we should set $\pi_2 = 77$ in order to maximize the total throughput. This does not seem to have an intuitive explanation and motivates our optimization study of the following section. We also notice that the maximum value of $D$, which is 22 801 bits/second, is well below the 38 400 bits/second which is the theoretical upper bound of the two-way link in this case. This point merits some analysis.

## 4. Packet size for throughput maximization

We have already noticed that $D_1$ is an increasing function of $x$, while $D_2$ is a decreasing function of $x(= \pi_1/\pi_2)$. We shall presently see that $D$ given by (1) :

$$ (8) \qquad D = \frac{2 + x + x^{-1}}{1 + x + x^{-1}} = y(1 + x) $$

has a maximum as a function of $x$. It is easily obtained by taking the derivative of $D$ with respect to $x$ and setting it to zero. $D$ is maximized by

$$ (9) \qquad x^* = \frac{1 - y}{1 + y} $$
and its maximum value is

$$ (10) \qquad D^* = \frac{4}{3 + y(2 - y)} $$

assuming $y < 1$, i.e. that the length of the acknowledgement is shorter than the average length of a packet. From (9) and (10) we have that :

- the maximum throughput cannot exceed 2/3 of two-way channel capacity (obtained when $y = 0$) ; this is similar to Pouzin's predictions [5].

- when $y > 0$, the optimum value of $\pi_2$ is larger than $\pi_1$. That is, the system is optimized when the traffic is not identical in both directions.

Let us apply these results to the example of Figure 7 ; we see that the optimum value of $\pi_2$ computed from (9) is 54.35 bytes, and $D^* = 22 978$ bits/second. On Figure 8, the optimum value of $\pi_2$ is close to 67 bytes and the corresponding value of the traffic is 22 700 bits/second.

On Figures 9 and 10 we illustrate the theoretical results of this optimization study.

## 5. Long constant packet lengths

In the previous analysis we have examined the case of asymmetric traffic with exponentially distributed packet lengths. This model seems to be satisfactory when either :

(i) packets in each direction are of different length (most of the measurements given pertain to this case), or

(ii) packets in both directions are of equal length, but are small enough so that the random fluctuations introduced by the system become significant.

Our previous model does not seem to account accurately for the case when both nodes are transmitting long packets of equal length. This is the case we will treat here.
<figure><img src="image_10.png" /><figcaption>FIGURE 9</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 10</figcaption></figure>
Suppose that $\pi \equiv \pi_1 = \pi_2$, and that packets transmitted in each direction are of constant length. If $\pi$ is sufficiently large, it is reasonable to suppose that the time spent by the system in states PW or WP (see Figure 1) can be neglected. The state transition diagram is now given on Figure 11.

<figure><img src="image_12.png" /><figcaption>FIGURE 11</figcaption></figure>

In this case

$$D_1 = D_2 = \frac{\pi}{\alpha + \pi}, \quad D = D_1 + D_2 = \frac{2\pi}{\alpha + \pi}$$

For instance, for $\pi = 282$ we have $D_1 = D_2 = 18735$ bits/sec. (if channel capacity is 19200 bits/sec.). A measurement experiment yielded $D'_1 = D'_2 = 17200$ bits/sec. The difference between the theoretical prediction and the measurement result can be explained by the existence of software wait times or of propagation delays. This phenomenon can be represented by a W state in the transition PP to AA and a W' state in the transition AA to PP; let $\omega$ and $\omega'$ be the times spent in W and W', respectively. We shall provide an estimate of $\omega + \omega'$ by comparing the theoretical and experimental results. We now must have

$$D'_1 \approx \frac{\pi}{\alpha + \pi + (\omega + \omega')} \times 19.200 \text{ bits/sec.}$$

so that $(\omega + \omega') = 25.8$ bytes. That is, $(\omega + \omega')$ is the time equivalent to the transmission time of 25.8 bytes along the line. In time units we have $(\omega + \omega') = 25.8 \times 8/19.200 = 10.75$ milliseconds. This time would include all software times in the nodes as well as propagation delays.

## 6. Conclusions

In this paper we have given a model of the stop and wait protocol (TMM in CIGALE [2]) and have validated it by using measurements on the actually implemented system.
We then show that the maximum throughput of this protocol cannot exceed 2/3 of total two-way throughput for exponentially distributed packet lengths. We also obtain the optimum value of relative packet sizes.

The case of long constant packets of equal length has been presented, and we have provided an estimate of software wait times.

In the Appendix we have presented a general model of the protocol, in which the exponential assumptions concerning packet lengths are not made.

This analysis shows clearly that the send and wait protocol is unable to exploit fully the capacity of the communication links if packets are not of fixed and equal length in both directions, and long enough to render the randomizing effect of software wait times and propagation delays negligible. Our study also shows the effectiveness of simple mathematical models for the study of network protocols.
# APPENDIX

We shall treat the case where the packet lengths are distributed according to some general distribution. Let $F_1(t)$, $F_2(t)$ be the probability distribution function for packets transmitted from 1 to 2, or 2 to 1, respectively. Then the probability, in the state transition diagram of Figure 1, of the transition "PP to PW" is :

$$p_1 \equiv \int_{0}^{\infty} dF_2(t) [1 - F_1(t)]$$

while the transition probability "PP to WP" is :

$$p_2 \equiv \int_{0}^{\infty} dF_1(t) [1 - F_2(t)]$$

The average time spent in state PP is :

$$\tau \equiv \int_{0}^{\infty} t[dF_2(t) (1 - F_1(t)) + dF_1(t) (1 - F_2(t))]$$

The average time spent in state PW is :

$$\tau_1 \equiv \int_{0}^{\infty} dx [1 - \int_{0}^{\infty} dF_2(t) (F_1(t+x) - F_1(t))]$$

while the average time spent in state WP is :

$$\tau_2 \equiv \int_{0}^{\infty} dx [1 - \int_{0}^{\infty} dF_1(t) (F_2(t+x) - F_2(t))]$$

Therefore

$$D_1 = \frac{\tau + p_1 \tau_1}{\tau + p_1 \tau_1 + p_2 \tau_2 + \alpha}$$

$$D_2 = \frac{\tau + p_2 \tau_2}{\tau + p_1 \tau_1 + p_2 \tau_2 + \alpha}$$

The results of Section 3 are obtained if we assume that

$$F_1(t) = 1 - e^{-t/\pi_1},$$

$$F_2(t) = 1 - e^{-t/\pi_2}$$
REFERENCES

[1] J.L. GRANGE - "CIGALE the packet switching subnetwork of CYCLADES" -
Journées Informatique Médicale - Toulouse - Mars 1976 - Réseau CYCLADES,
MIT 617.

[2] L. POUZIN - "Presentation and major design acpects of the CYCLADES computer network" - 3rd Data Comm. Symp. TAMPA - Nov. 1973 - 80-85 - Réseau CYCLADES SCH 511.

[3] N. NAFFAH - "Spécification d'interface TMM avec le réseau CIGALE" -
Déc. 1975 - Réseau CYCLADES TRA 521.

[4] S. ROSS - "Applied probability models with optimization applications" -
Holden-Day - San Francisco - 1970.

[5] L. POUZIN - "Efficiency of full-duplex synchronous data burk procedures" -
IFIP WG6.1 - INW 35 - June 1973 - Réseau CYCLADES IRA 510.

[6] G. FAYOLLE, E. GELENBE - G. PUJOLLE - "An analytic evaluation of the
performance of the stop-and-wait prptocol" - Rapport de Recherche n° 207 -
IRIA-LABORIA - Dec. 1976.