Analysis of Open Queueing Networks with Batch Services

Abstract

In this work we analyze an open queueing network with batch services. In more detail, the arrival process is Poissonian and each node consists of a single server and an infinite waiting queue. Arrivals are served in fixed-size batches: if the number of customers in a node is less than the predefined batch size, the server remains idle, otherwise he will select the required number of customers, which then will be served as a unique batch with exponentially distributed service time. In this paper we show that, under suitable conditions on the routing matrix, such queueing network is equivalent, in terms of stationary distribution, to a Jackson network with single-server nodes and state-dependent service rates. Finally, the goodness of the proposed approach is confirmed by comparing analytical and simulation results.

This work was supported by the Ministry of science and education of the Russian Federation in the framework of the basic part of the scientific research state task, project FSRR-2020-0006.

1 Introduction

Queueing systems and networks with batch services attract the interest of many researchers, since they permit to model and analyze various multi-user systems [1, 2], large scale semiconductor manufacturing systems [3], cloud computing systems [4] and wireless sensor networks [5].

The analysis of any queueing network is aimed at obtaining expressions for its stationary characteristics, the most important of which is the stationary probability distribution of the states of the system. Since the equilibrium equations for queuing networks with batch services have a high dimensionality, the calculation of the stationary distribution as a numerical solution of these equations is computationally difficult. Therefore, special attention has been devoted to the search for product-form solutions.

It is worth noticing that the fundamental works on queueing networks with batch services are relatively recent, as they were published in 1990 [6, 7]. In more detail, in [6] a continuous-time Markov chain is introduced to model queueing networks with simultaneous changes due to batch services, or discrete-time structure and clustering processes such as those arising in polymer chemistry. It is shown that if multiple instantaneous state transitions of the process are allowed and the Markov chain is reversible, then its stationary distribution has a product-form. In [7] a discrete-time closed queueing network with batch services is considered and the state of the network is defined by a vector with dimension equal to the number of customers. Each element of the status vector is associated with a specific customer and indicates the node occupied by that customer. So, customers transitions are reduced to changes of the corresponding labels, and it is assumed that the change of a label does not depend on the status of the labels of the other customers. It is shown that on an irreducible set of states and for arbitrary given functions of service and routing, there is a product-form for the stationary probability distribution of the queueing network states. Chao [8] and Economou [9] considered networks, for which the quasi-reversibility conditions are met and the groups of customers at the end of the service in one node always pass to another node together.

To analyze queuing networks with batch services and an arbitrary distribution of the service time that do not admit a product-form of the stationary distribution, in [3, 10] it was proposed to use the decomposition method. Finally, in [11, 12] the stationary distribution was calculated as the normalized solution of the system of equilibrium equations.

In this paper, we consider open queueing networks with service of fixed-size batches of customers and independent routing. It is assumed that the batch size is significantly smaller than the number of nodes to which the customers can be routed at the end of the service. Thus, the network nodes work independently and this consideration permits to simplify the analysis of the queueing network, which is reduced to the investigation of the individual queues in isolation. In more detail it is proposed to calculate the stationary state probability distribution of the open network in a product-form, similar to the case of birth-death processes after recalculating the transition rates. To the best of our knowledge, this approach is new. Until now, indeed, the probability generating function [4, 13,14,15], the Laplace-Stieltjes transform [16], and the direct calculation of the stationary distribution as a solution of the Kolmogorov equations [17] have been mainly used to calculate the stationary characteristics of the queueing network.

The rest of the paper is organized as follows. Section 2 introduces the model of the queueing network, while in Sect. 3 an equivalent (in terms of stationary distribution) Jackson network with single-server nodes is proposed. In more detail, for such equivalent system state-dependent service rates as well as expressions for the stationary probability distribution are derived. Then, Sect. 4 compares the values of the analytical expression with the simulation results, and analyses the dependence of the characteristics of open queueing networks on different system parameters (batch size, arrival rate, service rate).

2 Statement of the Problem

Consider a continuous-time open queueing network N consisting of L nodes $S_i$, $i \in I$, $I=\{1,\dots , L\}$. Customers arrive to the queueing network N from an outside source (denoted in the following as $S_0$) according to a Poisson stream of rate $\lambda _0$. Customer transitions between nodes and the source are defined by the routing matrix $\varTheta =(\theta _{ij})$, $i, j=0,\dots , L$, where $\theta _{ij}$ is the transition probability from node $S_i$ to node $S_j$. The state of the network is defined by a vector $s=(s_1, \dots , s_L)$, where $s_i$ is the number of customers at node $S_i$. Denote by $X=\{s: s_i \ge 0\}$ the state space of the queueing network N.

Each node $S_i$, $i=1,\dots , L$, operates as an infinite capacity single-server queue. Arriving customers are placed in the waiting queue if the server is busy. Customers are served in batches, and let $b_i$ be the customer batch size for node $S_i$. The server remains idle until the required number $b_i$ of customers arrives at the node and then the service of the batch starts immediately; otherwise, $b_i$ customers are selected in any order for service, while the others remain in the queue. The service times of batches at node $S_i$ are exponentially distributed with parameter $\mu _i$, $i = 1,\dots ,L$. After a batch finishes its service at node $S_i$, each customer will go, independently of the others, to node $S_j$ with probability $\theta _{ij}$, $i,j=0,1,\dots ,L$.

Our aim is to find the stationary distribution $\pi (s) = (\pi _1(s_1), \dots , \pi _L(s_L))$, $s \in X$, for the queueing network N, where $\pi _i (s_i)$ represents the stationary distribution for node $S_i$, $s_i = 0, 1, \dots$, $i = 1, \dots , L$, starting from the analysis of a single node.

3 Analysis of the Model

In this paper we analyze large scale networks with individual routing of the customers, assuming that the number of possible destinations is significantly larger than the batch size. Hence the probability of the simultaneous arrival of two or more customers in a node can be neglected. Therefore, we will assume that each node in N is fed by a Poisson stream of customers.

First we will study the isolated node $S_i$, $i=1,\dots , L$. It is known that the equilibrium equations for this node have the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \lambda _i \pi _i(n)=\mu _i \pi _i(b_i), \; n=0, \\ \lambda _i \pi _i(n)=\lambda _i \pi _i(n-1)+\mu _i \pi _i(b_i+n), \; 1 \le n \le b_i-1 ,\\ (\lambda _i + \mu _i) \pi _i(n)=\lambda _i \pi _i(n-1)+\mu _i \pi _i(b_i+n), \; n \ge b_i .\\ \end{array}\right. } \end{aligned}$$

(1)

where $\lambda _i$ denotes the arrival rate to node $S_i$, $i=1, \dots , L$.

We define a birth-death process $\xi _i$, which will be equivalent in steady-state probabilities to the Markov process describing the node $S_i$. Let the process $\xi _i$ be defined on a set of states $\{ 0,1, \dots \}$, let $\lambda _i=\lambda _i(n)$ be the transition rate of the process $\xi _i$ from state n to state $n+1$, which does not depend on the state n, $n \in \{ 0,1, \dots \}$, and let $\widetilde{\mu }_i(n)$ be the transition rate of the process $\xi _i$ from state n to state $n-1$, where $n \in \{ 1, 2, \dots \}$. The states $\{ 0,1, \dots \}$ and the parameter $\lambda _i$ of the process $\xi _i$ correspond to the states $\{ 0,1, \dots \}$ and the parameter $\lambda _i$ of node $S_i$. Let us find the rates $\widetilde{\mu }_i(n)$, $n=1,2,\dots$. To this aim, note that the steady-state probabilities of the birth–death process $\xi _i$ are given by [18]

$$\begin{aligned} \pi _i(k)=\pi _i(0)\prod \limits _{n = 1}^k{\frac{\lambda _i}{\widetilde{\mu }_i(n)}}, \; k=1,2,\dots , \end{aligned}$$

(2)

where

$$ \pi _i(0)= \left( 1+ \sum _{k=1}^{\infty } \prod \limits _{n = 1}^k {\frac{\lambda _i}{\widetilde{\mu }_i(n)}} \right) ^{-1}, \; i=1,\dots , L. $$

By substituting (2) in (1), we get the expressions that define $\widetilde{\mu }_i(n)$, $n=1,2,\dots$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \widetilde{\mu }_i(n)=\lambda _i-\mu _i \frac{\lambda _i^{b_i}}{\widetilde{\mu }_i(n+1) \cdot {\dots } \cdot \widetilde{\mu }_i(b_i+n)}, \; 1 \le n \le b_i-1 ,\\ \displaystyle \widetilde{\mu }_i(n)=\lambda _i+\mu _i-\mu _i \frac{\lambda _i^{b_i}}{\widetilde{\mu }_i(n+1) \cdot {\dots } \cdot \widetilde{\mu }_i(b_i+n)}, \; n \ge b_i .\\ \end{array}\right. } \end{aligned}$$

(3)

Let $M_i=\lim \limits _{n\rightarrow \infty }{\widetilde{\mu }_i(n)}$; if the limit exists, then:

$$ \mu _i \lambda ^{b_i}_i=(\lambda _i+\mu _i-M_i) M^{b_i}_i $$

or

$$\begin{aligned} M^{b_i+1}_i - (\lambda _i + \mu _i) M^{b_i}_i + \lambda ^{b_i}_i \mu _i=0 . \end{aligned}$$

(4)

The existence of the equivalent birth-death process $\xi _i$ requires that the previous equation has a positive solution, fulfilling the stability condition for each node $S_i$.

The answer is provided by the following theorem (without loss of generality we denote the generic $M_i$, for $i\in I$ by x).

Theorem 1

The equation

$$\begin{aligned} x^{b+1} - (\lambda + \mu ) x^{b} + \lambda ^{b} \mu =0 \end{aligned}$$

(5)

has two positive roots, the largest of which belongs to the interval

$$\left( \frac{b(\lambda +\mu )}{b+1}, \, \frac{(\lambda +\mu )^{b+1} -\lambda ^b \mu }{(\lambda +\mu )^b} \right) .$$

Proof

Consider the function

$$f(x)=x^{b+1}-(\lambda +\mu )x^b+\lambda ^b \mu $$

for $\lambda <b \mu$ and $b \ge 1$.

It is easy to verify that f(x) is continuous for any $x \in R$ and $x_1=\lambda$ is a root of f(x). To determine the existence of other roots let us consider the first derivative of f(x):

$$\begin{aligned} f'{(x)}=(b+1)x^{b-1}\left( x- \frac{b}{b+1}(\lambda +\mu )\right) . \end{aligned}$$

(6)

The equation $f'{(x)}=0$ has only one positive root

$$x^*=\frac{b(\lambda +\mu )}{b+1},$$

with $x^*>x_1$. Indeed,

$$x^*-x_1=\frac{b}{b+1}(\lambda +\mu )-\lambda =\frac{b\mu -\lambda }{b+1}>0,$$

since $\lambda <b\mu$ and $b \ge 1$. Since $f'(x)>0$ for

$$x \in \left( \frac{b(\lambda +\mu )}{b+1}, \lambda + \mu \right) ,$$

then the function f(x) is increasing in such interval. Moreover,

$$f\left( \frac{b(\lambda +\mu )}{b+1}\right) <0$$

and $f(\lambda +\mu )>0$, hence in the interval $\left( \frac{b(\lambda +\mu )}{b+1}, \lambda + \mu \right)$ there is a value of x such that $f(x)=0$.

To further refine the estimation of the root, let us note that in the above-mentioned interval the function f(x) is convex, since

$$ f''(x)=bx^{b-2}((b+1)x-(b-1)(\lambda +\mu ))>0 $$

for

$$ x>\frac{b(\lambda +\mu )}{b+1}>\frac{(b-1)(\lambda +\mu )}{b+1}. $$

The tangent line to f(x) at the point $x=\lambda +\mu$ is

$$ y(x)=\lambda ^b\mu +(\lambda +\mu )^b(x-(\lambda +\mu )) $$

and its intersection with the horizontal axis is

$$x_0=(\lambda +\mu )-\frac{\lambda ^b\mu }{(\lambda +\mu )^b}.$$

Since the function f(x) is convex, $x_0$ is an upper bound for the roots of f(x), and this implies that the largest root of Eq. (5) belongs to the interval

$$\left( \frac{b(\lambda +\mu )}{b+1} , \, \frac{(\lambda +\mu )^{b+1} -\lambda ^b \mu }{(\lambda +\mu )^b} \right) .$$

Taking into account the previous theorem and the stability condition of the equivalent birth-death process, Eq. (4) has a unique root, located in the interval $(\lambda _i, \lambda _i+\mu _i)$, which can be determined numerically (explicit closed-for solutions can be easily derived only for $b=1$ and $b=2$). From the system of Eqs. (3) it follows

$$ \widetilde{\mu }_i(b_i)=\widetilde{\mu }_i(b_i+1)=\widetilde{\mu }_i(b_i+2)= \dots = M_i, $$

and then the service rates $\widetilde{\mu }_i(b_i-1)$, $\widetilde{\mu }_i(b_i-2)$,$\dots$, $\widetilde{\mu }_i(1)$ can be easily calculated. Thus, the rates $\widetilde{\mu }_i(n)$ are determined for each state n of process $\xi _ i$.

The results obtained for the process $\xi _i$ can be applied to any node, and so we can create an open queueing network $\widetilde{N}$ with nodes $\widetilde{S}_i$ and service rates $\widetilde{\mu }_i(n)$, where n is the number of customers in the node $\widetilde{S}_i$, $n=1,2, \dots$, $i=1, \dots , L$. The other parameters of $\widetilde{N}$ coincide with the corresponding parameters of the original queueing network N.

$\widetilde{N}$ is equivalent in stationary distribution to the queueing network N with batch services and is a Jackson network.

The arrival rates in nodes $S_i$ are determined by the following equations

$$ \lambda _i=\frac{\omega _i}{\omega _0} \lambda _0, \; i=1, \dots , L, $$

where the vector of visitation rates $\omega =(\omega _1, \dots , \omega _L)$ is the solution of the equation $\omega \varTheta = \omega$ with the normalization condition $\sum _{i=0}^L \omega _i = 1$.

The queueing network N and its equivalent network $\widetilde{N}$ are stable if the utilization coefficient in the node $S_i$, $i=1, \dots , L$,

$$\rho _i=\frac{\lambda _i}{b_i \mu _i} <1, $$

and, under such conditions, we can compute the stationary distribution for $\widetilde{N}$. We obtain

$$ \pi (s)=\prod \limits _{i = 1}^L \pi _i(s_i), \; s \in X, $$

where

$$ \pi _i(s_i)=\pi _i(0) \prod \limits _{n = 1}^{s_i} \frac{\lambda _i}{\widetilde{\mu }_i(n)}. $$

Then, the average number of customers in the node $S_i$, $i=1,\dots ,L$, is given by

$$ \bar{s}_i=\sum _{n=1}^{\infty } n \pi _i(n), $$

the average sojourn time in the node $S_i$, $i=1,\dots ,L$, is

$$ \bar{u}_i= \frac{\bar{s}_i}{\lambda _i}, $$

and the average response time of the queueing network is

$$ \bar{\tau }= \frac{1}{\lambda _0} \sum _{i=1}^L \lambda _i \bar{u}_i. $$

4 Numerical Examples

Numerical examples are reported in this section to verify the goodness of the product-form approximation for complex networks and investigate the dependence of their characteristics on different system parameters (batch size, arrival rate, service rate). Although different topologies have been investigated, for sake of brevity just one network topology is considered, focusing on overall system performance parameters as well as on characteristics of single queues.

Consider the queueing network N with the following parameters (unless otherwise stated): $L=14$, $b=(3,\;2,\;2,\;3,\;2,\;2,\;3,\;3,\;2,\;2,\;3,\;2,\;2,\;3)$, $\mu =(0.8,\;0.6,\;0.9,\;0.6,\;0.8,\;0.8,\;0.9,\;0.6,\;0.7,\;0.8,\;0.9,\;1.0,\;0.7,\;0.7)$, and

$$ \varTheta = \begin{pmatrix} 0.0 \;\; 0.3 \;\; 0.4 \;\; 0.3 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0\\ 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0\\ 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.2 \;\; 0.2 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0\\ 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.2 \;\; 0.1 \;\; 0.2 \;\; 0.1 \;\; 0.1 \;\; 0.0\\ 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.2 \;\; 0.2 \;\; 0.1\\ 0.2 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.2\\ 0.2 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.2 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1\\ 0.2 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1\\ 0.2 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.1\\ 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.2 \;\; 0.1 \;\; 0.1\\ 0.3 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.0\\ 0.3 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1\\ 0.3 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.0\\ 0.3 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.0 \;\; 0.1\\ 0.4 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.0 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.1 \;\; 0.0\\ \end{pmatrix}. $$

The considered network satisfies the assumptions introduced above. Indeed, the network consists of a relatively large number of nodes, the size of the batches that are served together is significantly less than the number of possible output nodes and the routing probabilities are of the same order of magnitude (there is no privileged path through the network). Hence, the Poissonian assumption can be reasonably assumed for any node of the network.

The first two sets of tests investigated the accuracy of the developed method by comparing the analytical values with the results of discrete-event simulation. In more detail, in the first experiment we analysed the (overall) average response time as a function of the input rate $\lambda _0$.

Table 1. Average response time of the queueing network.

Table 1 shows that the largest difference in the values of $\bar{\tau }$ is observed for $\lambda _0=0.1$ and does not exceed 10.2%, while for the other values of $\lambda _0$, the deviation is no more than 5%. Note that the intensity of the flow $\lambda _0=2.7$ is almost the maximum for the network under consideration, since for such value the stability condition for node $S_9$ is still met.

In the second example we focused on a specific node (the queue $S_7$), considering the average number of customers (Tables 2) as well as the average sojourn time in the node (Tables 3) for different values of the service rate $\mu _7$ with fixed arrival rate $\lambda _0=1.5$.

Table 2. Average number of customers in the node $S_7$.

Table 3. Average sojourn time in the node $S_7$.

The characteristics of the node $S_7$, derived by discrete-event simulation, were calculated in stationary conditions with a confidence interval of 0.001 and a confidence level higher than 0.95.

In the third experiment we investigated the dependence of the stationary characteristics of the nodes $S_2$, $S_6$ and $S_{11}$ on the intensity of the incoming flow $\lambda _0$ (see Fig. 1 and 2). The characteristics of the other nodes are not shown in the graphs for sake of clarity, since their behavior does not differ qualitatively form the reported ones.

Fig. 1.

Average number of customers in the nodes $S_2$, $S_6$ and $S_{11}$.

Fig. 2.

Average sojourn time in the nodes $S_2$, $S_6$ and $S_{11}$.

Figure 1 shows that the average number of customers in all systems monotonically increases with $\lambda _0$. Instead, the average (node) sojourn time reaches a minimum for some value of $\lambda _0$ as highlighted by Fig. 2. This can be explained as follows. When $\lambda _0$ is close to zero, the device is idle for a long time, and the customers forming an “incomplete” batch have to wait in the buffer until the last element of the batch enters the system. Instead, when the arrival rate into the considered system approaches its service rate, the average waiting time increases significantly. Thus, there is an optimal value of the arrival rate, at which the average sojourn time in the node is minimal.

The fourth experiment is devoted to the study of stationary characteristics of the nodes $S_7$ and $S_9$ for different sizes b of the batch in these systems. The input rate in this experiment is again $\lambda _0=1.5$.

Fig. 3.

Average number of customers in the node $S_7$.

Fig. 4.

Utilization coefficient of the node $S_7$.

Fig. 5.

Average number of customers in the node $S_9$.

Fig. 6.

Utilization coefficient of the node $S_9$.

The minimum value of the average number of customers in both nodes is achieved when the batch size is two (Fig. 3 and 5), while the utilization coefficient is a monotone decreasing function of b (Fig. 4 and 6), but its numerical value depends on the arrival rate at the considered node (in our example $\rho _9$ is almost twice $\rho _7$). It is worth noticing (see Fig. 6) that for $b_9=1$, the utilization $\rho _9$ of the node $S_9$ is close to 1 and this is confirmed by the high value of the number of customers in the system ($\bar{s}_9 \approx 20$ as shown in Fig. 5). When $b_9=2$, then $\bar{s}_9 \approx 2$, while the increment of $b_9$ leads to a slight increase in $\bar{s}_9$. Thus, the increase of the batch size can significantly improve the basic average characteristics of service systems. Actually, as shown by numerical experiments, the minimum value of both the average sojourn time and average number of customers in the system can be assumed at different values b, depending on the network topology and the routing matrix.

Finally, we calculated the stationary characteristics of the nodes $S_5$ and $S_9$ for different values of the service rate in these systems (assuming, as before, $\lambda _0=1.5$).

Fig. 7.

Average number of customers in the node $S_5$.

Fig. 8.

Average sojourn time in the node $S_5$.

Fig. 9.

Average number of customers in the node $S_9$.

Fig. 10.

Average sojourn time in the node $S_9$.

The graphs shown in Fig. 7, 8, 9 and 10 decrease monotonically with the growth of $\mu$ and asymptotically tend to their limit values.

5 Conclusions

In this paper large-size open queueing networks with batch services are considered. Under the assumption that the number of output nodes is significantly more than the batch size, it is shown that the stationary distribution of the queueing network can be expressed in product-form. Then, the parameters of the equivalent queueing network are derived and the goodness of the approximation is verified by means of discrete-event simulation.