IRIA

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d'Informatique
et d'Automatique

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B. P. 105 78150 - Le Chesnay
France
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laboratoire de recherche
en informatique
et automatique

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

A DIFFUSION MODEL
FOR MULTIPLE CLASS
QUEUEING NETWORKS

Erol GELENBE
Guy PUJOLLE

12p.

Rapport de Recherche N° 242

Août 1977
A DIFFUSION MODEL FOR MULTIPLE CLASS
QUEUEING NETWORKS

Erol Gelenbe*, Guy Pujolle**

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

Résumé :

Un réseau de files d'attentes ouvert comprenant plusieurs classes de clients servis avec la discipline FIFO, est analysé. Une approximation par les processus de diffusion est utilisée, ce qui permet d'obtenir des résultats aisés à appliquer. Un modèle de réseau de communication par commutation de paquets est analysé par cette nouvelle technique. Des paquets longs et des courts sont autorisés. Ils correspondent respectivement à des transferts de fichiers et à des travaux en temps partagé.

Abstract :

We obtain a simple and analytically tractable diffusion process model for treating open networks of queues with FIFO service and multiple classes. The model, which provides approximate explicit stationary distributions for queue lengths of a case for which exact results are unavailable, is applied to a model of a packet switching computer network. In such networks short and long packets, corresponding to time-sharing usage and file transfers respectively, are frequent and we study this phenomenon with a diffusion model.

<footnote>* Dept. de Mathematiques - Université Paris-Nord, av. J.B. Clément - 93 Villetaneuse (France)</footnote>

<footnote>** IRIA/LABORIA - B.P. 105 - 78150 Le Chesnay (France)</footnote>
# A DIFFUSION MODEL FOR MULTIPLE CLASS

## QUEUEING NETWORKS

Erol GELENBE
Dept. de Mathématiques
Université Paris-Nord
Ave. J.B. Clément
93 Villetaneuse
FRANCE

Guy PUJOLLE
IRIA/LABORIA
78150 Le Chesnay
FRANCE

## 1 - INTRODUCTION

The panoply of existing mathematical tools for the modelling of computer systems does not include solution techniques for systems with FIFO service and multiple distinct classes of customers, even when service times are exponentially distributed and arrival processes are Poisson.

The purpose of this paper is to extend our previous work [4] to the case of general service times with multiple classes of customers, FIFO service at each service station and renewal processes representing the arrivals. Our approach uses a diffusion process model [3, 4, 5] for which we provide an analytical solution.

The theoretical results are then applied to a model of CIGALE [6], the packet-switching subnetwork of the CYCLADES computer network with two classes of traffic. The first class represents packets which arrive irregularly and which are of widely differing lengths : these represent packets originating from time-shared computation on the network. The second class arrives in long batches and in bursts and are of a constant length : they represent file transfers through the network.

We also present numerical results and validations (using simulation results) for a model of an interactive system.
## 2 - BEHAVIOUR OF A SINGLE QUEUE IN A NETWORK WITH MULTIPLE CLASSES

In this section we are concerned with an open network containing n stations. Each station contains one server. Customers of the network are divided into R classes; at each station, service is rendered in strict FIFO order with no priorities between classes.

The r-th class, $1 \le r \le R$, is characterized by:

i) a stream of arrivals to the network which is a renewal process: its rate is $\lambda_{0,r}$ and the squared coefficient of variation of interarrival times is $\kappa_{0,r}$

ii) a general service time distribution function $G_r^i(x)$ at the i-th service station, $1 \le i \le n$.

Furthermore, transitions of customers through the network are described by a Markov chain $(p_{i,r}; j, r')$, where $1 \le i \le n$, $1 \le r, r' \le R$, $1 \le j \le n+1$. $p_{i,r}; j, r'$ is the probability that a class r customers leaving station i enters station j in class r'. The fictitious stations $(n+1)$ denotes a departure from the network. We shall note $q_{i,r}$ the probability that an arriving customer of class r enters station i of the network.

The reader will notice that the QN (queueing network) we have thus defined cannot be solved by any of the available exact solution methods [1]. In particular, FIFO service with general service times cannot be treated by the classical BCMP [2] result though it can be handled by the approximate method of [3], [4]. Thus the model we are treating in this paper extends our previous work [4] to the case of multiple classes. Such models are of particular interest in performance evaluation studies of computer systems which take into account the existence of multiple job classes. In the area of computer networks they reflect well the presence of "short" and "long" packets: the former can represent interactive processing while the latter can represent the transfer of files.

As in the approach we had taken earlier [4] our analysis proceeds in two parts: the first concerns the computation of the parameters of the arrival process to each queue while the second part uses the results of the first part in the queue length computations using diffusion approximations [3].

### 2.1 - The interarrival statistics to each queues

Let $Ks_{i,r}$ denote the squared coefficient of variation of the service time at queue i, and $\kappa_{i,r}$ the squared coefficient of variation of interarrival times to the queue i, for customers of class r. $\lambda_{i,r}$ will be the arrival rate of class r customers to queue i, and $\mu_{i,r}^{-1}$ will be their average service time.

From conservation of flow arguments we have the system of equations:

$$ (1) \qquad \lambda_{i,r} = \lambda_{0,r} q_{i,r} + \sum_{i=1}^{n} \sum_{r=1}^{R} \lambda_{j,r'} p_{j,r'}; i,r $$

for $1 \le i \le n$, $1 \le r \le R$. These equations may also be obtained by first considering $e_{i,r}$, the average number of times that a customer of class r passes through the station i before leaving the QN. Notice that $e_{i,r} = \lambda_{i,r} / \lambda_{0,r}$ so that (1) can be interpreted as a system of equations for the $e_{i,r}$. Clearly, using (1), we have a unique solution for the $\lambda_{i,r}$.
Let us denote $u_{i,r} = \lambda_{i,r}/\mu_{i,r}$ : it can be viewed as the load imposed by class r customers to station i. We shall define

$$ (2) \qquad u_i = \sum_{r=1}^{R} u_{i,r} $$

and

$$ (3) \qquad \lambda_i = \sum_{r=1}^{R} \lambda_{i,r}, \quad \Pi_i = \lambda_i / \lambda_i $$

where $u_i$ is the utilization (steady-state probability that the queue is busy), and $\lambda_i$ is the total arrival rate, associated with station i.

Having obtained the $\lambda_{i,r}$, from (1) we need to compute the $K_{i,r}$ (already defined as the squared coefficients of variation of the interarrival times of class r customers to station i). These are obtained by assuming that the arrival and departure processes of class r customers to and from each queue are renewal processes. Although this assumption is not always true, it has provided very accurate predictions of interarrival statistics in a previous study [4], as well as exact results in all known cases including for very light or very heavy traffic.

Let $\tau_i$ be the time separating two successive departures from station i. We shall write:

$$ (4) \qquad \tau_i = \begin{cases} S_i & \text{with probability } u_i \\ S_i + A_i & \text{with probability } (1-u_i) \end{cases} $$

where $A_i$ is an interarrival time to queue i, and $S_i$ is $S_{i,r}$ with probability $\lambda_{i,r}/\lambda_i$, $S_{i,r}$ being the service time of class r customer at station i. Therefore, as expected from the conservation of flow:

$$ \begin{aligned} E[\tau_i] &= E[S_i] + E[A_i] (1-u_i) \\ &= \sum_{r=1}^{R} \mu_{i,r}^{-1} \Pi_{i,r} + \lambda_i^{-1} (1-u_i) \\ &= \lambda_i^{-1} = \left( \sum_{i=1}^{R} \lambda_{i,r} \right)^{-1} \end{aligned} $$

Also, we obtain:

$$ \begin{aligned} E[\tau_i^2] &= u_i E[S_i^2] + (1-u_i) E[S_i^2 + 2A_i S_i + A_i^2] \\ &= E[S_i^2] + 2(1-u_i) \lambda_i^{-1} \sum_{r=1}^{R} \mu_{i,r}^{-1} \Pi_{i,r} + (1-u_i) E[A_i^2] \end{aligned} $$

Denote by $C_i = \lambda_i^2 \{E[\tau_i^2] - (E[\tau_i])^2\}$ the squared coefficient of variation of interdeparture times at the i-th queue, and let $K_i$ be the squared coefficient of variation of interarrival times to the i-th queue. We then have
$$
(5) \quad C_{i+1} = \lambda_i^2 E[S_i^2] + (1-u_i)(K_i+1) + 2u_i(1-u_i) \\
\qquad = \lambda_i \sum_{r=i}^{R} u_{i,r} \mu_{i,r}^{-1} (K_{i,r}+1) + (1-u_i)(K_i+1+2u_i)
$$

But the input process to the queue is obtained as the superposition of the process
of external arrivals plus the output process of the queues. In fact, assuming that
the output of the j-th queue is a renewal process, the arrival process to the i-th
queue from the j-th queue is characterized by a rate and a squared coefficient of
variation which are respectively

$$
(6) \quad \lambda_j P_{ij} = \left\lceil (C_j - 1) P_{ji} + 1 \right\rceil
$$

where P<sub>ji</sub>, the probability that a customer leaving station j enters queue i, is
given by

$$
(7) \quad P_{ji} = \sum_{r=1}^{R} \sum_{r'=1}^{R} (\lambda_j r / \lambda_j) p_{j,r;i,r'}
$$

Let $\alpha_i(t)$ be the total number of arrivals to the i-th queue in the interval $(0,t)$;
its variance will then be (asymptotically, for large $t$) [4]

$$
(8) \quad \sum_{j=0}^{n} [(C_j - 1) P_{ji} + 1] \lambda_j P_{ji,t} = \lambda_i^3 \{E[A_i^2] - (E[A_i])^2\}t
$$

where C₀ is obtained from the superposition of the R input processes as shown below.
From (8) and (5) we obtain the system of n equations for the Cᵢ.

$$
(9) \quad C_i = \lambda_i \sum_{r=1}^{R} u_{i,r} \mu_{i,r}^{-1} (K_{i,r} + 1) + (1-u_i)(K_i + 1 + 2u_i) - 1
$$

where

$$
(10) \quad K_i = \lambda_i^{-1} \sum_{j=0}^{n} [(C_j - 1) P_{ji} + 1] \lambda_j P_{ji,t}
$$

where $1 \le i \le n$. Concerning $C_0$ which is the squared coefficient of variation of
external interarrival times to the network, notice that the variance of the number
of arrivals in (0,t) is, asymptotically for large t,

$$
\left( \sum_{r=1}^{R} \lambda_{0,r} \right) C_0 t = \sum_{r=1}^{R} \lambda_{0,r} K a_{0,r}
$$

so that we compute $C_0$ from

$$
(11) \quad C_0 = \sum_{r=1}^{R} \lambda_{0,r} K a_{0,r} / \sum_{r=1}^{R} \lambda_{0,r}
$$

We can now solve for $C_1, \dots, C_n$ from (9). The $K_1, \dots, K_n$ will then be immediately
available from (10).

In the arrival process to queue i, $\Pi_{i,r} = \lambda_{i,r} / \lambda_i$ is the probability that an arriving customer is of class r. Therefore, the squared coefficient of variation of interarrival times of class r customers to queue i will be

$$
(12) \quad K_{i,r} = (K_i - 1) \Pi_{i,r} + 1
$$
Thus $\lambda_{i,r}$ obtained from (1) and $\kappa_{i,r}$ obtained from (12) (after solving (9) and using (11)) characterize the arrival stream of class r processes to station i, $1 \le i \le n$, $1 \le r \le R$.

## 2.2 - Diffusion approximation to the queue length process

We now consider the behaviour of any queue, say the i-th, in the network and we propose to approximate the queue length process by a diffusion process.

Just before a service begins at the queue, the probability that the customer at the head of the queue is of class r is $\Pi_{i,r}$. Therefore its service time distribution will be $\sum_{i=1}^{R} G_{r}^{i}(x)\Pi_{i,r}$: let $\mu_{i}^{-1}$ denote the mean of this distribution and $v_{i}$ its variance.

As in [3, 4] the queue length probability function $P_i(x, t)$ is assumed to satisfy the diffusion equations

$$ (13) \qquad - \frac{\partial f_i}{\partial t} - b_i \frac{\partial f_i}{\partial x} + \frac{1}{2} \alpha_i \frac{\partial^2 f_i}{\partial x^2} + \lambda_i P_i(t) \delta(x-1) = 0 $$

$$ (14) \qquad \frac{d}{dt} P_i(t) = - \lambda_i P_i(t) + \lim_{x \to 0^+} [-b_i f_i + \frac{1}{2} \alpha_i \frac{\partial f_i}{\partial x_i}] $$

where $P_i(t)$ is the empty queue probability at time t. The parameter $b_i$ is the "drift" of the diffusion model, while $\alpha_i$ is the rate of change of the variance :

$$ (15) \qquad b_i = \lambda_i - \mu_i, \quad \alpha_i = \lambda_i K_i + \mu_i^3 v_i $$

The stationary solution is readily obtained from (13), (14)

$$ P_i = \lim_{t \to \infty} P_i(t) \quad , \quad f_i(x) = \lim_{t \to \infty} f_i(x, t) $$

$$ (16) \qquad P_i = 1 - \lambda_i / \mu_i = 1 - u_i $$

$$ (17) \qquad f_i(x) = \begin{cases} u_i [1 - e^{-\gamma_i x}] & , \quad 0 \le x \le 1 \\ -\gamma_i u_i e^{-\gamma_i x} & , \quad x \ge 1 \end{cases} $$

where $\gamma_i = 2b_i/\alpha_i$ for $\gamma_i < 0$ (which is true and only if $\lambda_i < \mu_i$).

The average length $L_i$ of queue i is given by

$$ (18) \qquad L_i = u_i [1 + \frac{u_i K_i + Ks_i}{2(1-u_i)}] $$

where $Ks_i = \mu_i^2 v_i$ is the squared coefficient of variation of service time at station i.

A formula for the average transit time $T$ of a customer through the network is particularly useful. We then have
$$ (19) \qquad \begin{aligned} T &= \sum_{i=1}^{n} L_i / \lambda_i \\ &= \sum_{i=1}^{n} \mu_i^{-1} \left[ 1 + \frac{u_i K_i + K s_i}{2(1-u_i)} \right] \end{aligned} $$

## 2.3 - The number in queue for each class

From the distribution for the total number in queue we will now work back to the distribution of the number of customers of each class in queue. We proceed as follows. Discretize the probability density function $p_i(x)$ by defining

$$ p_i(n) = \int_{(n-1)}^{n} f_i(x) dx, \quad n = 1, 2, \dots $$

and

$$ p_i(0) = 1 - \mu_i $$

$p_i(n)$ will be the discrete approximation to the stationary queue length distribution at station i. Let $p_{i,r}(\ell)$ be the probability of finding $\ell$ customers of class r at station i; we take

$$ (20) \qquad p_{i,r}(\ell) = \sum_{n \ge \ell} \binom{n}{\ell} \Pi_{i,r}^{\ell} (1 - \Pi_{i,r})^{n-\ell} p_i(n) $$

since each customer in queue i belongs to class r with probability $\Pi_{i,r}$.

## 2.4 - The average response time for a class r customer

A quantity of interest is the average response (or transit) time through the network for customers of each class. Denote this quantity by $T_r$ for class r : $T_r$ is the average time spent by a customer of class r between the instant at which it enters the network and when it leaves it. Clearly, a customer's waiting time at each queue does not depend on its class ; let $T_{i,r}$ be the response (or transit) time of a class r customer through station i and $W_{i,r}$ the waiting time. We have

$$ W_{i,r} = L_i / \lambda_i - \mu_i^{-1} $$

and

$$ (21) \qquad T_{i,r} = L_i / \lambda_i - \mu_i^{-1} + \mu_{i,r}^{-1} $$

Therefore

$$ (22) \qquad T_r = \sum_{i=1}^{n} T_{i,r} \frac{\lambda_i}{\lambda_0} $$

since a class r customer will visit station i on the average $\lambda_{i,r}/\lambda_{0,r}$ times.

# 3 - APPLICATION TO A PACKET SWITCHING NETWORK

In this section we apply the diffusion approximation method with classes of customers to the analysis of the packet switching sub-network CIGALE of the computer network CYCLADES [6]. We take the 7-node network described on Figure 1. Nodes A through G are minicomputers used for packet buffering and switching. Packet routing
is assumed fixed and summarized in Table 1. The routes which are not indicated on Table 1 are chosen by the smallest number of nodes which are traversed. We assume two classes of customers : a class of short packets or interactive packets and a class of large packets (transfer of files, for example). The characteristics of these classes are given in Table 1 and 2. These assumptions are chosen so that large packets have all the same length and arrive in bursts of a large group of packets. Short packets arrive singly and irregularly (Poisson traffic) and their length is exponentially distributed. This traffic is reported as being typical of computer network applications [7]. We assume that packets (short or long) arriving from the outside of the network have identical routing probabilities.

The packet switching network of Figure 1 can be represented as a queuing network with 16 queues. Routing probabilities can be computed from external arrivals and the routing table.

The method described in section 2 is used to compute the expected number of customers with Jackson's [1] formula which assumes Poisson traffic for a single class of packets whose length is exponentially distributed. If $\lambda_i^{-1}$ is the average service time for a customer at station i in Jackson's model, we have taken :

$$ \lambda_i^{-1} = \frac{\lambda_i}{\lambda_{i,1} \lambda_{i,1}^{-1} + \lambda_i}{\lambda_{i,2} \lambda_{i,2}^{-1}} $$

Numerical results are given in Table 3.

## 4 - APPLICATION TO A MODEL OF AN INTERACTIVE SYSTEM

The model of an interactive computer system examined by Anderson and Sargent is described in Figure 2.

It has already been used by Reiser and Kobayashi [8] and Gelenbe and Pujolle [4] to examine the accuracy of diffusion approximations for one customer class.

We shall apply our method to this model with two classes of customers. Jobs of class 1 and 2 arrive to the system according to Poisson processes of parameters $\lambda_1 = 0.5$ and $\lambda_2 = 0.25$, respectively. After passing through server 1, they leave the system with probability 0.5 or enter the queue of server 2 with probability 0.5. At the output of server 2 the customers feed-back into queue 2 with probability 0.5 or they go to queue 1.

We shall study this model assuming several sets of service time distributions server 1 and 2. These are indicated in Table 4 with the results. In this table we present average queue lengths which are compared with those obtained from simulations which have been carried out with a package developed by Merle and Veran [9]. We give only the estimated average length obtained from the simulations.

The results described in Table 4 provide an indication of accuracy of the approximation by diffusion process with several classes. We also tabulate the relative error in each case, under the assumption that the "correct" result is obtained from the simulation. Together with a number of other authors [8,10] we notice that the queue with feed-back is where the largest errors seem to occur. A new treatment of this case will be given so as to obtain more accurate results (following an idea of Kühn [10] but with diffusion approximation).
<figure><img src="image_3.png" /><figcaption>Figure 1</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2</figcaption></figure>
<table><caption>Table 1</caption><thead><tr><th>Source</th><th>Destination</th><th>Via Node</th><th>Input rate class short packets</th><th>Input large packets</th></tr></thead><tbody><tr><td>A</td><td>F</td><td>C</td><td>1.5</td><td>1.</td></tr><tr><td>B</td><td>F</td><td>C</td><td>1.</td><td>1.</td></tr><tr><td>C</td><td>G</td><td>F</td><td>1.</td><td>1.</td></tr><tr><td>D</td><td>G</td><td>F</td><td>2.</td><td>0.5</td></tr><tr><td>E</td><td></td><td></td><td>1.5</td><td>0.5</td></tr><tr><td>F</td><td>C,D</td><td>B</td><td>1.</td><td>2.</td></tr><tr><td>G</td><td>A,B</td><td>C</td><td>0.5</td><td>2.</td></tr></tbody></table>

<table><caption>Table 2</caption><thead><tr><th></th><th>Squared coefficient of var. external arrival</th><th>Squared coefficient of service times</th><th>Service rate link 1,2,3,5</th><th>Others</th></tr></thead><tbody><tr><td>Class 1 short packets</td><td>1.</td><td>1. (exponential)</td><td>15.</td><td>10.</td></tr><tr><td>Class 2 large packets</td><td>5.</td><td>0. (constant)</td><td>3.</td><td>2.</td></tr></tbody></table>

<table><caption>Table 3</caption><thead><tr><th></th><th>Diffusion method</th><th>Jackson</th></tr></thead><tbody><tr><td>1</td><td>0.81</td><td>0.76</td></tr><tr><td>2</td><td>0.77</td><td>0.88</td></tr><tr><td>3</td><td>3.57</td><td>3.73</td></tr><tr><td>4</td><td>0.24</td><td>0.27</td></tr><tr><td>5</td><td>1.21</td><td>1.30</td></tr><tr><td>6</td><td>2.81</td><td>3.25</td></tr><tr><td>7</td><td>0.53</td><td>0.57</td></tr><tr><td>8</td><td>1.02</td><td>1.07</td></tr><tr><td>9</td><td>0.67</td><td>0.56</td></tr><tr><td>10</td><td>0.49</td><td>0.50</td></tr><tr><td>11</td><td>0.19</td><td>0.20</td></tr><tr><td>12</td><td>3.35</td><td>2.77</td></tr><tr><td>13</td><td>0.43</td><td>0.56</td></tr><tr><td>14</td><td>0.69</td><td>0.74</td></tr><tr><td>15</td><td>2.98</td><td>2.33</td></tr><tr><td>16</td><td>0.45</td><td>0.54</td></tr></tbody></table>
<table><thead><tr><th colspan="2"></th><th colspan="2">DIFFUSION</th><th colspan="2">SIMULATION</th></tr><tr><th></th><th></th><th>Mean number<br>in queue 1</th><th>Mean number<br>in queue 2</th><th>Mean number<br>in queue 1</th><th>Mean number<br>in queue 2</th></tr></thead><tbody><tr><td rowspan="4">server 1</td><td>EXP</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>2.54</td><td>2.86</td><td>2.66</td><td>2.79</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>21</sub><sup>-1</sup> = 0.5</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td>5.6 %</td><td>2.5 %</td><td></td><td></td></tr><tr><td rowspan="4">server 2</td><td>EXP</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>2.61</td><td>0.95</td><td>2.63</td><td>.96</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>21</sub><sup>-1</sup> = 0.3</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td>0.76 %</td><td>1.04 %</td><td></td><td></td></tr><tr><td rowspan="4">server 1</td><td>EXP</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>2.39</td><td>8.86</td><td>2.22</td><td>8.68</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>21</sub><sup>-1</sup> = 0.7</td><td></td><td></td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td>7.6 %</td><td>2.07 %</td><td></td><td></td></tr><tr><td rowspan="4">server 2</td><td>ERLANG 2</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>2.12</td><td>0.77</td><td>2.29</td><td>0.91</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>CONSTANT</td><td>μ<sub>11</sub><sup>-1</sup> = 0.3</td><td>7.4 %</td><td>15 %</td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">server 1</td><td>ERLANG 2</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>1.89</td><td>1.56</td><td>2.03</td><td>2.08</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>CONSTANT</td><td>μ<sub>21</sub><sup>-1</sup> = 0.5</td><td>6.9 %</td><td>25 %</td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="4">server 2</td><td>ERLANG 2</td><td>μ<sub>11</sub><sup>-1</sup> = 0.5</td><td>1.76</td><td>7.55</td><td>1.92</td><td>8.19</td></tr><tr><td>EXP</td><td>μ<sub>12</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr><tr><td>CONSTANT</td><td>μ<sub>21</sub><sup>-1</sup> = 0.7</td><td>8.3 %</td><td>7.8 %</td><td></td><td></td></tr><tr><td>EXP</td><td>μ<sub>22</sub><sup>-1</sup> = 0.4</td><td></td><td></td><td></td><td></td></tr></tbody></table>
REFERENCES

[1] Gelenbe E., Muntz R.R. - "Probabilistic models of computer systems, Part I" - (Exact Results) - Acta Informatica 7 - 35-60 - 1976.

[2] Baskett F., Chandy K.M., Muntz R.R., Palacios F.G. - "Open, closed, and mixed networks of queues with different classes of customers" - J.ACM 22 - 248-260 - 1975.

[3] Gelenbe E. - "On approximate computer system models" - J.ACM 22 - 261-263 - 1975.

[4] Gelenbe E., Pujolle G. - "The behaviour of a single queue in a general queueing network" - Acta Informatica 7 - 123-160 - 1976.

[5] Kobayashi H. - "Application of the diffusion approximation to queueing networks : Part I and II" - J.ACM 21 - 316-328 - 459-469 - 1974.

[6] Pouzin L. - "CIGALE, the packet-switching machine of the Cyclade computer network" - Proc. IFIP Congress 74 - North-Holland - 155-159 - 1974.

[7] Kleinrock L., Opderbeck H. - "Throughput in the Arpanet-protocols and measurement" - Proc. Data Comm. Symp. Quebec City - Canada - 1975.

[8] Reiser M., Kobayashi H. - "Accuracy of the diffusion approximation for some queueing systems" - IBM J.R. and D. 18 - 110-124 - 1974.

[9] Merle D., Véran M., Stewart W. - "A queueing network analysis package (ONAP)" Research Report IRIA - 1977.

[10] Kühn P. - "Analysis of complex queueing networks by decomposition" - Proc. of International Teletraffic Congress - Melbourne - 1976.