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Estimation of service times
distribution for
operating systems modelling

Jacques Anne
J.Leroudier,A.Schroeder

Rapport de Recherche n° 93

Décembre 1974.
ESTIMATION OF SERVICE TIMES DISTRIBUTION
FOR OPERATING SYSTEMS MODELLING

Jacques Leroudier, Anne Schrøder

Ce papier se propose d'obtenir, à partir de mesures réelles, des
formules explicites pour les variables utilisées dans la modélisation
de systèmes informatiques. Les modèles utilisés sont souvent des modèles
à files d'attente dont les temps de service peuvent être supposés des
combinaisons linéaires de distributions d'Erlang et même des distributions
arbitraires dans le cas d'approximation par un processus de diffusion
(dans ce papier nous nous intéressons plus particulièrement à des
combinaisons linéaires de distributions gaussiennes).

+

The purpose of this paper is to obtain from real measurements explicit
formulas for different variables used in operating systems modelling. The
models used are often queueing networks related where the service times
have to be supposed linear combinations of Erlangian distributions or
arbitrary distributions in the case of the diffusion approximation
modelling (in this paper we will focus our interest on combinations of
Gaussian distributions).
ESTIMATION OF SERVICE TIMES DISTRIBUTION

FOR OPERATING SYSTEMS MODELLING

Jacques LEROUDIER - Anne SCHROEDER

IRIA - LABORIA
ROCQUENCOURT
78150 LE CHESNAY - FRANCE.

# I - Presentation of our purpose

Modelling is an abstraction to represent the reality which cannot be directly analyzed or understood. In the performance evaluation of operating systems, the models which are used, are essentially of two kinds : mathematical and simulation models.

These models are tools to understand the mutual influence of the different variables of the concerned problem. They provide results of different nature : equations for analytical models, numerical tables or curve nets in simulation models.

In both cases a main problem is to use realistic parameters. Therefore we must have measurements on the behaviour of existing operating systems. Generally these measurements are automatically sampled and classically analyzed : computation of means, variances, histograms... etc.

Such data may be used in simulation models if the samples are available, but difficulties arise to introduce them in mathematical models which need explicit formulas.

Up to now, experimental samples were used to give realistic means or variances, or to make adequation tests with given a priori distributions.

The purpose of this paper is to obtain from real measurements explicit formulas for the different variables which are used in mathematical or simulation models. Let us note these formulas allow to generate artificial samples for Monte Carlo simulations.

What variables and what types of formulas do we consider ? The answer to this
question is given by the nature of the models used in performance evaluation.

In view of the complexity of real operating systems the mathematical models are probabilistic and more specifically queueing networks related [2,14]. These models are characterized by the distributions of the service times and of the arrivals at the servers. In the oldest models, the arrival distributions had to be Poisson and the service time distributions exponential [10,11]; but great strides have been made to allow less restrictive assumptions [5,15]. Service times distributions may now be supposed linear combinations of convolution products of exponential distributions (any distribution having rational Laplace transform). This property comes from the decomposition of a non-exponential server into a network of exponential servers (stages method) [5].

<figure><img src="image_1.png" /></figure>

At the end of the exponential service of mean $1/\mu_i$, the service stops with the probability $b_i$ (absorbing state) or continues, with the probability $a_i = 1-b_i$, in another exponential server of mean service time $1/\mu_{i+1}$.

Let us note that this method has two important particular cases: combinations of exponential or erlangian distributions.

Moreover recent papers about the diffusion approximation [9,12] allow to consider arbitrary service times distributions (in this paper we will focus our interest on combinations of gaussian distributions).

To summarize this presentation, we are looking for linear combinations of erlangian and gaussian densities for the different service times distributions obtained from real data.

In section II, we present the measurements which were used for the estimations, in section III, we develop the statistical problem [16] and in section IV, we give some numerical results.

## II - The measurements

The measurements were made on the CP/CMS system at Grenoble University. A thorough description and an interpretation of these measurements can be found in [13].

Let us recall that the CP/CMS system is composed of the two systems CP and CMS.
CP generates virtual machines which are IBM 360 computers while CMS is running conversationally in an IBM 360 computer (therefore a virtual machine).

The measurements were picked up in CMS, consequently at the level of the virtual machines use, following the scheme :

<figure><img src="image_2.png" /></figure>

The following variables are collected when a CMS command is issued :

- the name of the virtual machine concerned,

- the name of the command,

- eventually, the name, the type and the size of the file concerned by the command,

- the virtual time to perform the command (CMS time),

- the virtual time + the management time by CP to perform the command,

- the real time for the user in front of his terminal,

- the time of the day,

- the number of terminal I/O,

- the number of disk I/O,

- the recursion level of the command (0 for a primitive CMS command sent from the terminal, 1 for a command issued from a macro-command,...).

The samples analyzed in this paper were picked up during a time interval of 25 days and included 59,301 CMS commands.

The CP/CMS system under measurements was composed of the system CP - 67 (version 3.0) and CMS (version 3.0) and the hardware configuration was the following :

- one CPU IBM 360/67 (this model has the translation addresses mechanism),

- 1024 K bytes of core,
- 3 selector channels

- one connected to two paging drums

- two others each connected to one unit of 8 2314 IBM disks

- 1 multiplexor channel connected to

- 50 terminals

- 4 magnetic tapes drivers

- 2 printers

- 2 readers-punchers.

CP/CMS was running from 7 AM to 12 AM and from 3 PM to 7 PM; during the other periods of the day, the running system was OS-MVT of IBM. The users included researchers in different areas: physics, numerical analysis, languages, computer science and students, on Saturday.

Using this sample we want to give explicit formulas for the service time distributions. In that aim, we will study the histograms of service times and restrict ourselves to the analysis of 3 variables : the virtual time, the virtual time + the management time by CP and the real time used by some CMS commands.

## Histograms under consideration

<figure><img src="image_3.png" /></figure>
-5- ASSEMBLER COMMAND

<figure><img src="image_4.png" /></figure>

FORTRAN COMMAND

<figure><img src="image_5.png" /></figure>

PL/1 COMMAND

<figure><img src="image_6.png" /></figure>
### III - The statistical problem

The statistical problem to be solved is the estimation, using an experimental sample, of the unknown parameters of a probability density of the following type :

$$ (1) \qquad f(x) = \sum_{1 \le i \le k} c_i f_{\lambda_i}(x) \quad (x \in \mathbb{R}) \quad \text{with} \quad \sum_{i=1}^{k} c_i = 1 $$

where $\{f_{\lambda_i}/\lambda_i \in \mathcal{L} \subseteq \mathbb{R}^s\}$ is a distribution family of some known type, depending on an unknown parameter $\lambda$ (which may be vectorial).

A density of type (1) is a mixture of probability densities and the $c_i$ represent the proportions of the distribution $f_{\lambda_i}$ in the mixture.

The problem of solving mixtures is a usual one which presents links with classical statistics and pattern recognition.

Because of the precise problem presented here, we shall limit ourselves to the cases of samples of real variables and of Gaussian, exponential or Erlang mixtures; however the method we propose could be applied as well to any family of distributions, even multivariate.

Knowing $k$ and $(f_{\lambda}/\lambda \in \mathcal{L})$, this technique computes from the observed sample : the $(c_i/1 \le i \le k)$, the $(\lambda_i/1 \le i \le k)$ and a partition $(P_i/1 \le i \le k)$ of the sample in $k$ classes such that the likelihoods of the $P_i$ are the largest, when considered as samples of unknown components of distributions $f_{\lambda_i}$.

These results give a theoretical density satisfying a likelihood criterion which can be compared to the observed distribution by comparison of histograms or by adequation tests.

**N.B.** : It must be noted that it is not an adjustment of the histogram function by some known function which is achieved ; by the way, it is not the pointwise approximation of this function which is of some interest but the individual behaviour of the observations

When the number $k$ of components of the mixture is a priori unknown, several trials may be performed for different values of $k$ and a comparison of the results can be done with the help, for instance, of an adequation test.

There exist several techniques to estimate the unknown parameters of real Gaussian mixtures (for instance [6, 1, 4, 17]). Methods that may theoretically be applied to other types of distributions are less frequent and practically used only in the Gaussian case.
The method we propose presents the advantage to be general since it can be applied to any type of mixtures (which allows to deal with Erlang mixtures) whatever the dimension of the sample is, and for any number of components in the mixture. We shall present here a variant that tends to optimize a likelihood criterion, but other criteria may be optimized in the same general scheme, associated with various estimation methods. The algorithm is presented in its most general case in [16].

### 3.1. The algorithm

Let $E$, a finite subset of $R$, be a sample of $N$ observed measurements; ($f_λ/λ \in \mathcal{L}$) is the family of probability densities to which the components of the mixture are supposed to belong, and $k$ is the supposed number of these components. The algorithm runs as follows:

Starting from any initial partition ($P_1^0, P_2^0, ... P_k^0$) of $E$, or from any initial $k$-tuple of values for the unknown parameters ($λ_1^0, λ_2^0, ... λ_k^0$), the following functions $f$ and $g$ are successively applied:

$f: \mathcal{L}_k \to \mathbb{P}_k$ and $g: \mathcal{L}_k \times \mathbb{P}_k \to \mathcal{L}_k$ where $\mathbb{P}_k$ is the set of all partitions of $E$ in $k$ classes, and $\mathcal{L}_k$ that of all $k$-tuples of elements of $\mathcal{L}$:

$$ (1) \qquad \begin{aligned} \mathcal{L}_k \xrightarrow{f} \mathbb{P}_k \\ L^0 \longmapsto \mathbb{P}^0 \end{aligned} $$

$$ \begin{array}{rcl} \mathcal{L}_k \times \mathbb{P}_k & \xrightarrow{g} & \mathcal{L}_k \xrightarrow{f} \mathbb{P}_k \\ (L^0, \mathbb{P}^0) & \longmapsto & L^1 \longmapsto \mathbb{P}^1 \\ & & \mathcal{L}_k \times \mathbb{P}_k \xrightarrow{g} \mathcal{L}_k \xrightarrow{f} \mathbb{P}_k \\ (L^1, \mathbb{P}^1) & \longmapsto & L^2 \longmapsto \mathbb{P}^2 \\ & & \text{etc...} \end{array} $$

More precisely, $f$ and $g$ are defined as follows:

$$ \begin{array}{rcl} f: \mathcal{L}_k & \longrightarrow & \mathbb{P}_k \\ \mathcal{L} = (\lambda_1 \dots \lambda_k) & \longmapsto & f(\mathcal{L}) = (P_1 \dots P_k) \end{array} $$

such that for all $i$, $1 \le i \le k$:

$$ P_i = \{ x \in E / f_{\lambda_i}(x) \ge f_{\lambda_j}(x), \forall j \ne i \text{ with } j > i \text{ in case of equality} \} $$

$$ \begin{array}{rcl} g: \mathcal{L}_k \times \mathbb{P}_k & \longrightarrow & \mathcal{L}_k \\ ((\lambda_1 \dots \lambda_k), (P_1 \dots P_k)) & \longmapsto & g(L, P) = M = (\mu_1 \dots \mu_k) \end{array} $$

such that, for all $i$, $1 \le i \le k$, $\mu_i$ "improves" the likelihood of $P_i$ with respect to the distribution $f_{\lambda_i}$:
in other words,

$$V_{\mu_i}(P_i) = \prod_{x \in P_i} f_{\mu_i}(x) > V_{\lambda_i}(P_i) = \prod_{x \in P_i} f_{\lambda_i}(x)$$

If there is not any $\mu_i$ such that this strict inequality is true, the old value is kept: $\mu_i = \lambda_i$.

**N.B.** : In the particular -but most common- case where $(f_\lambda/\lambda \in \mathcal{L})$ is such that there exists for $\lambda$ a maximum likelihood estimator, a possible function $g$ is :

$g: P_k \to \mathbb{L}_k$ 
$P_i \to L$ where, for all $i$, $1 \le i \le k$, $\lambda_i$ is the maximum likelihood estimate of $\lambda$ calculated on the sample $P_i$.

The general scheme (1) of the algorithm may then take a simpler form :

$$(2) \qquad \mathbb{L}_k \underset{g}{\stackrel{f}{\rightleftharpoons}} P_k$$

The proof of the following theorem, which ensures the convergence of the method, can be found in [16].

**Theorem** : The algorithm defined above converges and increases at each step the following criterion :

$$W(L,P) = \prod_{1 \le i \le k} V_{\lambda_i}(P_i)$$

In the particular case of the simplified scheme (2), the couple $(L^*, P^*)$ obtained at the point of convergence is such that the $\lambda_i^*$ are the maximum likelihood estimates of $\lambda$ for the sample $P_i^*$ and these $P_i^*$ themselves are the measurements that are most likely to have been drawn from the distribution $f_{\lambda_i^*}$.

The couple $(L^*, P^*)$ obtained is a local optimum depending on the initial partition $P^0$. Practically, it will be useful to perform several trials for $P^0$ and to choose the solution that gives the best likelihood criterion.
## 3.2 Application to univariate Gaussian mixtures

(The multivariate Gaussian case is studied in [16]).

In this particular case, the distributions of the components are to be found in the family ($f_λ/λεL$) where :

$$ \begin{cases} \forall x \in \mathbb{R} \\ \forall \lambda = (\mu, \sigma^2) \in \mathbb{R} \times \mathbb{R}^+ \end{cases} \qquad f_{\lambda}(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left[-\frac{(x-\mu)^2}{2\sigma^2}\right] $$

This family is known to admit a maximum likelihood estimator for $(\mu, \sigma^2)$ (see, [8] for instance) which is :

$\hat{\mu}$ = empirical mean,

and $\hat{\sigma}^2$ = empirical variance.

The simplified form (2) of the algorithm can then be used with the following functions f and g :

f : L₁ → P such that, for all i, 1 ≤ i ≤ k :

$$ P_i = \{x \in E / \frac{(x-\hat{\mu}_i)^2}{\sigma_i^2} + \text{Log} \sigma_i^2 \le \frac{(x-\hat{\mu}_j)^2}{\sigma_j^2} + \text{Log} \sigma_j^2, \forall j \ne i \text{ with } j>i \text{ in case of equality} \} $$

and

g : P₁ → L where, for all i, if the cardinal of Pᵢ is denoted

by $|P_i|$ :

$$ \lambda_i = (\hat{\mu}_i, \hat{\sigma}_i^2) = \left( \frac{1}{|P_i|} \sum_{x \in P_i} x, \frac{1}{|P_i|} \sum_{x \in P_i} (x - \hat{\mu}_i)^2 \right) $$

The maximized criterion function then is :

$$ W(L,P) = \frac{1}{(\sqrt{2\pi})^N} \quad \prod_{1 \le i \le k} \sigma_i^{|P_i|} \exp \left[ - \frac{\sum_{x \in P_i} (x-\mu_i)^2}{2\sigma_i^2} \right] $$

## 3.3. Application to mixtures of Gamma distributions

The family ($f_λ/λεL$) now is :

$$ \forall \lambda = (\alpha, \beta, \omega) \in \mathbb{R}^+ \times \mathbb{R}^+ \times \mathbb{R} $$

$$ \forall x \in \mathbb{R} / x \ge \omega $$

$$ f_{\lambda}(x) = \frac{\alpha^{\beta}}{\Gamma(\beta)} (x-\omega)^{\beta-1} \exp \left[ - \alpha(x-\omega) \right] $$

Since it is not possible to calculate a triplet $(\hat{\alpha}, \hat{\beta}, \hat{\omega})$ which is a maximum likelihood estimator of $(\alpha, \beta, \omega)$, the simplified scheme (2) cannot be used and a function g : $L_k \times P_k \to L_k$ improving the likelihood is to be found.
We propose the following one :

$$
\begin{array}{r@{\quad}l}
g : \mathcal{L}_k \times \mathbb{P}_k \xrightarrow{g_1} \mathcal{L}_k \times \mathbb{P}_k \xrightarrow{g_2} \mathcal{L}_k & \\
\text{((A,B,O),P)} \longrightarrow \text{((A,B,O'),P)} \longrightarrow \text{(A',B',O')} &
\end{array}
$$

$$
\text{where } A = (\alpha_1 \dots \alpha_k), \quad B = (\beta_1 \dots \beta_k), \quad \emptyset = (\omega_1 \dots \omega_k)
$$

$$
\text{and } A' = (\alpha'_1 \dots \alpha'_k), \quad B' = (\beta'_1 \dots \beta'_k), \quad \emptyset' = (\omega'_1 \dots \omega'_k)
$$

Such that, for all i, 1 ≤ i ≤ k :

$$
V_{(\alpha_i, \beta_i)}(\omega_i^!)^{(P_i)} = \max_{\omega \in \mathbb{R}} V_{(\alpha_i, \beta_i)}(\omega)^{(P_i)}
$$

and

$$
V_{(\alpha_i! \beta_i!)}(\omega_i!)^{(P_i)} = \max_{(\alpha, \beta) \in \mathbb{R}^+} \sum_{i=1}^{k} V_{(\alpha \beta)}(\omega_i!)^{(P_i)}
$$

In other words, $g_1$ defines $\omega_i!$ as the maximum likelihood estimator of $\omega$, knowing
$\alpha_i$ and $\beta_i$, while $g_2$ estimates the couple $(\alpha_i!, \beta_i!)$ by the maximum likelihood, with $\omega_i!$
known. (For further details, see [16])

f is once more defined in the same way and we are therefore placed in the right conditions to verify the convergence theorem.

V - Numerical results

For each histogram we give the estimated distribution, the corresponding vali-
dity criteria and a figure to visualize the approximation.

Since the estimated densities are in the form

$$
f(x) = \sum_{i=1}^{k} c_i f_{\lambda_i}(x) \quad \text{where } c_i = \frac{|P_i|}{N}
$$

the results tables will give k and for all i (i=1,2,...,k) the |Pᵢ| and the λᵢ. In
gaussian approximations the λᵢ are the means μᵢ and the standard deviations σᵢ.
while in the Gamma approximations the λᵢ are the parameters : αᵢ, βᵢ and ωᵢ.

The validity criteria are the following :

- the likelihood criterion optimized by the algorithm,

- the adequation $\chi^2$ between the observed and the estimated distributions which allows a verification of the approximation though the method does not try to optimize this criterion.

The histograms presented in this section are different in their aspect from those
in section II where the number of classes is 50 while here this number varies accor-
ding to the population in each class and gives the degree of freedom of the $\chi^2$ in-
volved (see figures).
## 5.1. Gaussian approximations

### 5.1.1. Assembler command - CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>116</td><td>1.21</td><td>0.43</td></tr><tr><td>2</td><td>66</td><td>35.57</td><td>14.38</td></tr><tr><td>3</td><td>41</td><td>9.55</td><td>0.96</td></tr><tr><td>4</td><td>102</td><td>20.56</td><td>1.78</td></tr><tr><td>5</td><td>56</td><td>3.24</td><td>0.52</td></tr><tr><td>6</td><td>91</td><td>2.02</td><td>0.20</td></tr><tr><td>7</td><td>66</td><td>6.99</td><td>0.49</td></tr><tr><td>8</td><td>36</td><td>4.90</td><td>0.53</td></tr><tr><td>9</td><td>96</td><td>14.24</td><td>1.28</td></tr></tbody></table>

$$ egin{cases} W = 6652.0 \\ \chi_{12}^2 = 6.6 \end{cases} $$

<figure><img src="image_7.png" /></figure>

### 5.1.2. Assembler command - CP+CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>174</td><td>10.30</td><td>3.68</td></tr><tr><td>2</td><td>48</td><td>67.58</td><td>25.12</td></tr><tr><td>3</td><td>74</td><td>40.33</td><td>2.98</td></tr><tr><td>4</td><td>57</td><td>31.89</td><td>1.59</td></tr><tr><td>5</td><td>100</td><td>23.66</td><td>2.78</td></tr><tr><td>6</td><td>217</td><td>3.64</td><td>0.83</td></tr></tbody></table>

$$ egin{cases} W = 7942.0 \\ \chi_{10}^2 = 5.2 \end{cases} $$

<figure><img src="image_8.png" /></figure>
5.1.3. Fortran command -CMS time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>135</td><td>2.08</td><td>0.34</td></tr><tr><td>2</td><td>100</td><td>15.02</td><td>3.19</td></tr><tr><td>3</td><td>128</td><td>3.54</td><td>0.49</td></tr><tr><td>4</td><td>197</td><td>1.08</td><td>0.22</td></tr><tr><td>5</td><td>96</td><td>0.48</td><td>0.10</td></tr><tr><td>6</td><td>40</td><td>0.26</td><td>0.07</td></tr><tr><td>7</td><td>32</td><td>0.03</td><td>0.01</td></tr><tr><td>8</td><td>33</td><td>33.37</td><td>16.13</td></tr><tr><td>9</td><td>86</td><td>8.35</td><td>0.74</td></tr><tr><td>10</td><td>122</td><td>5.86</td><td>0.69</td></tr></tbody></table>

$$ egin{cases} W = 8124.9 \\ \chi^2_6 = 8.6 \end{cases} $$

<figure><img src="image_9.png" /></figure>

5.1.4. Fortran command - CP+CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>127</td><td>2.81</td><td>0.32</td></tr><tr><td>2</td><td>106</td><td>16.47</td><td>3.58</td></tr><tr><td>3</td><td>137</td><td>4.46</td><td>0.62</td></tr><tr><td>4</td><td>131</td><td>1.88</td><td>0.16</td></tr><tr><td>5</td><td>99</td><td>1.38</td><td>0.13</td></tr><tr><td>6</td><td>36</td><td>0.32</td><td>0.08</td></tr><tr><td>7</td><td>100</td><td>0.92</td><td>0.11</td></tr><tr><td>8</td><td>35</td><td>36.02</td><td>17.17</td></tr><tr><td>9</td><td>67</td><td>9.63</td><td>0.56</td></tr><tr><td>10</td><td>131</td><td>7.10</td><td>0.74</td></tr></tbody></table>

$$ egin{cases} W = 8285.2 \\ \chi^2_6 = 6.1 \end{cases} $$

<figure><img src="image_10.png" /></figure>
5.1.5. PL/1 command -CMS time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>33</td><td>5.98</td><td>0.10</td></tr><tr><td>2</td><td>80</td><td>4.00</td><td>0.42</td></tr><tr><td>3</td><td>121</td><td>21.23</td><td>4.84</td></tr><tr><td>4</td><td>68</td><td>8.12</td><td>0.44</td></tr><tr><td>5</td><td>53</td><td>5.30</td><td>0.25</td></tr><tr><td>6</td><td>47</td><td>56.74</td><td>27.56</td></tr><tr><td>7</td><td>113</td><td>2.19</td><td>0.82</td></tr><tr><td>8</td><td>30</td><td>6.69</td><td>0.24</td></tr><tr><td>9</td><td>59</td><td>13.30</td><td>0.60</td></tr><tr><td>10</td><td>63</td><td>10.26</td><td>0.70</td></tr></tbody></table>

$$ 
\begin{cases} 
W = 6731.8 \\ 
\chi^2 = 23.5 
\end{cases} 
$$

<figure><img src="image_11.png" /></figure>

5.1.6. PL/1 command -CP+CMS time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>104</td><td>12.24</td><td>0.88</td></tr><tr><td>2</td><td>126</td><td>43.96</td><td>30.20</td></tr><tr><td>3</td><td>134</td><td>9.07</td><td>0.81</td></tr><tr><td>4</td><td>131</td><td>5.87</td><td>0.96</td></tr><tr><td>5</td><td>73</td><td>15.91</td><td>1.32</td></tr><tr><td>6</td><td>99</td><td>21.92</td><td>2.72</td></tr></tbody></table>

$$ 
\begin{cases} 
W = 7812.3 \\ 
\chi^2 = 34.4 
\end{cases} 
$$

<figure><img src="image_12.png" /></figure>
5.1.7. Editing command -CMS time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>3512</td><td>0.11</td><td>0.05</td></tr><tr><td>2</td><td>2406</td><td>0.37</td><td>0.13</td></tr><tr><td>3</td><td>1205</td><td>1.84</td><td>2.39</td></tr></tbody></table>

$$ egin{cases} W = 43726.1 \\ \chi^2 = 727.9 \end{cases} $$

<figure><img src="image_13.png" /></figure>

5.1.8. Editing command -CP+CMS time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>634</td><td>6.18</td><td>6.35</td></tr><tr><td>2</td><td>2650</td><td>0.40</td><td>0.14</td></tr><tr><td>3</td><td>1702</td><td>1.85</td><td>0.49</td></tr><tr><td>4</td><td>2137</td><td>0.88</td><td>0.16</td></tr></tbody></table>

$$ egin{cases} W = 54285.2 \\ \chi^2 = 72.8 \end{cases} $$

<figure><img src="image_14.png" /></figure>
### 5.1.9. Editing command -Real time-

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>μ<sub>i</sub></th><th>σ<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>2684</td><td>0.37</td><td>0.20</td></tr><tr><td>2</td><td>1981</td><td>1.20</td><td>0.33</td></tr><tr><td>3</td><td>906</td><td>12.30</td><td>9.54</td></tr><tr><td>4</td><td>1552</td><td>3.15</td><td>0.96</td></tr></tbody></table>

$$ \begin{cases} W = 63650.0 \\ \chi_{12}^2 = 3954.7 \end{cases} $$

<figure><img src="image_15.png" /></figure>

## 5.2. Gamma approximations.

Gamma approximations have been computed for the six samples related to the assembler, FORTRAN and PL/1 commands.

As for the editing command, it could not be treated because of the large size (7123) of its samples.
5.2.1. Assembler command

- CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>296</td><td>1.5</td><td>2.4</td><td>.7</td></tr><tr><td>2</td><td>86</td><td>1.9</td><td>2.9</td><td>5.7</td></tr><tr><td>3</td><td>288</td><td>.1</td><td>1.5</td><td>9.3</td></tr></tbody></table>

$$ egin{cases} W = -1485.7 \\ \chi_q^2 = 29.7 \end{cases} $$

<figure><img src="image_16.png" /></figure>

- CP+CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>242</td><td>2.6</td><td>3.8</td><td>2.3</td></tr><tr><td>2</td><td>428</td><td>.0</td><td>1.1</td><td>6.7</td></tr></tbody></table>

$$ egin{cases} W = -2036.8 \\ \chi_q^2 = 30.6 \end{cases} $$

<figure><img src="image_17.png" /></figure>
5.2.2. FORTRAN command

- CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>293</td><td>.1</td><td>1.9</td><td>-4.7</td></tr><tr><td>2</td><td>464</td><td>1.2</td><td>1.5</td><td>.5</td></tr><tr><td>3</td><td>212</td><td>.8</td><td>1.9</td><td>4.0</td></tr></tbody></table>

$$
\begin{cases}
W = -1984.4 \\
\chi_6^2 = 4.6
\end{cases}
$$

- CP+CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>289</td><td>.1</td><td>1.0</td><td>.9</td></tr><tr><td>2</td><td>402</td><td>1.6</td><td>1.8</td><td>.9</td></tr><tr><td>3</td><td>278</td><td>.5</td><td>1.5</td><td>4.3</td></tr></tbody></table>

$$
\begin{cases}
W = -1806.1 \\
\chi_6^2 = 4.8
\end{cases}
$$

<figure><img src="image_18.png" /></figure>

FORTRAN
test command
- CMS Time -
(sample size: 969)

{ curve - estimated
  curve ... observed

<figure><img src="image_19.png" /></figure>

FORTRAN
test command
- CP + CMS Time -
(sample size: 969)

{ curve - estimated
  curve ... observed
5.2.3. PL/1 command

- CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>563</td><td>.2</td><td>1.2</td><td>2.9</td></tr><tr><td>2</td><td>104</td><td>.0</td><td>1.0</td><td>19.6</td></tr></tbody></table>

$$ egin{cases} W = -1718.2 \\ \chi_{10}^2 = 44.5 \end{cases} $$

<figure><img src="image_20.png" /></figure>

- CP+CMS time -

<table><thead><tr><th></th><th>|P<sub>i</sub>|</th><th>α<sub>i</sub></th><th>β<sub>i</sub></th><th>ω<sub>i</sub></th></tr></thead><tbody><tr><td>1</td><td>578</td><td>.1</td><td>1.1</td><td>5.2</td></tr><tr><td>2</td><td>89</td><td>.0</td><td>1.0</td><td>30.8</td></tr></tbody></table>

$$ egin{cases} W = -2033.6 \\ \chi_{10}^2 = 21.4 \end{cases} $$

<figure><img src="image_21.png" /></figure>
## VI - Conclusion

We only presented some results to justify our approach. Our main interest, in this paper, was more to give a method than a directory of formulas. It seems this method should be used by the people doing measurements to put their results in a form usable by the persons dealing with modelling. To have explicit formulas allows in analytical modelling to express the relationships which are found, and in simulation modelling to generate statistical distributions close to the reality without having to use corresponding real samples.
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