## 1. Introduction

Since the early 1900th the Erlang multi-server queueing systems with losses (

By efforts of many mathematicians (A. Ya. Khinchin and B. I. Grigelionis first of all), it was proved that the superposition of a large number of independent flows having uniformly small intensity approaches to the stationary Poisson input when the number of the superposed inputs tends to infinity. It explains the fact that the flows in classic telephone networks (where flows are composed by small individual flows from independent subscribers) have the exponentially distributed inter-arrival times.

Concerning the service time distribution, situation was more complicated. The real-life measurements have shown that the service (conversation) time can not be well approximated by means of the exponentially distributed random variable. So, due to the good matching of results obtained for the

So, the question why the Erlang's models give very good results for a practice was highlighted. The special books containing the tables for a loss probability under the given values of the number

However, the flows in the modern telecommunication networks have lost the nice properties of their predecessors in the old classic networks. In opposite to the stationary Poisson input (stationary ordinary input with no aftereffect), the modern real life flows are non-stationary, group and correlated. The
*Batch Markovian Arrival Process*) arrival process was introduced as a versatile Markovian point process (

As it was mentioned above, the question why the inter-arrival times in the classical networks have the exponential distribution was answered in literature. However, Erlang's assumption that the service time distribution has the exponential distribution is not supported by the real networks measurements. In the case of the

It follows from discussion above that it is interesting to extend investigation of Erlang's models to the case of the

In paper [2], we investigated the

Even if one will use such general models of the arrival and service process as the

Importance of investigation of the queues operating in a random environment (

Mention that the
*RE* can be found there. In this chapter, we consider the models

## 2. The Mathematical Model

We consider the queueing system having

The input flow into the system is the following modification of the

while the correlation coefficient

At the epochs of the process

The service process is defined by the modification of the

The system under consideration has

Our aim is to calculate the stationary state distribution and main performance measures of the described queueing model.

For the use in the sequel, let us introduce the following notation:

## 3. Process of the System States

It is easy to see that operation of the considered queueing model is described in terms of the regular irreducible continuous-time Markov chain

where

Let us enumerate the states of the chain

It is well known that the vector

where

Structure of this generator and methods of system (1) solution vary depending on the admission discipline.

### 3.1. The Case of Partial Admission Discipline

Lemma 1. Infinitesimal generator(17) |

where

Proof of the Lemma follows from analysis of Markov chain

To solve system (1) with the matrix

*Theorem 1.* In case of partial admission, the stationary probability vectors

where the matrices

the matrices

the matrices

the vector

### 3.2. The Case of Complete Rejection Discipline

*Lemma 2.* Infinitesimal generator

(29) |

The proof of the Lemma is analogous to the proof of the previous Lemma and takes into account the fact that the number of customers in the system does not change when the number of customers in an arriving batch exceeds the number of free servers.

To solve system (1) with the matrix

### 3.3. The Case of Complete Admission Discipline

Lemma 3. Infinitesimal generator(31) |

Essential difference of complete admission discipline is that the state space of the Markov chain

*Theorem 2.* In case of complete admission discipline, the stationary probability vectors

where the matrices

the matrices

the matrix

the matrices

the vector

The proof of the Theorem follows from the theory of multi-dimensional Markov chains with continuous time, see [6]. It is worth to note that Neuts' matrix

### 3.4. The Case of an Infinite Size of a Buffer
( L = ∞ )

The system under consideration in this section has an infinite waiting space. If an arriving batch of customers sees idle servers, a part of the batch corresponding to the number of free servers occupy these servers while the rest of the batch joins the queue. If the system has all servers being busy at a batch arrival epoch, all customer of the batch go to the queue.

*Lemma 3.* Infinitesimal generator

(39) |

In what follows we perform the steady state analysis of the Markov chain having generator of form (2). To this end, we use the results for continuous time multi-dimensional Markov chain (

*Theorem 3.* The necessary and sufficient condition for existence of the Markov chain

where

the vectors

*Proof.* Using the results of [6], we directly obtain the desired condition in the form of inequality

where

It is easy to show that inequality (7) is reduced to the following inequality:

where

To get the equations for the row vectors

The value

To solve system (1) with the matrix

*Theorem 4.* The stationary probability vectors

where the matrices

the matrices

the matrix

the matrices

the vector

## 4. Performance Measures

Having the probability vector

*Theorem 5.* The loss probability

(i) in the case of

(ii) in the case of

(iii) in the case of

Proofs of formulae (10) - (12) are analogous. So, we will prove only formula (10). According to a formula of the total probability, the probability

where

It can be shown that

By substituting (14)-(16) into (13) after some algebra we get (10).

Some performance measures for the case

The probability to see

The mean number of customers in the system

The probability to see

The mean number of busy servers

The mean number

The vector

The vector of conditional means of the number of busy servers under the fixed states of the random environment

The vector

The probability

The vector

where

The probability

The vector

The probability

The probability

## 5. Actual Sojourn Time

Let
*Theorem 6.* The Laplace-Stieltjes transform

where

*Proof.* We derive the expression for the

Analogously, the entries of the matrix

*Theorem 7.* The mean sojourn time

(86) |

where

Proof. To get expression (18) for

## 6. Numerical Examples

The goal of the numerical experiments is to demonstrate the feasibility of the proposed algorithms for computing the stationary distributions of the number of customers and the sojourn time in the system and to give some insight into behavior of the considered queueing systems. In particular, the following issues are addressed:

Comparison of the mean sojourn time of an arbitrary customer and the probability of immediate access to the servers in the systems with varying traffic intensities and different coefficients of correlation in the

Comparison of the mean sojourn time of an arbitrary customer and the probability of immediate access to the servers in the original system in a

Demonstration of possible positive effect of redistribution of traffic between the peak traffic periods and normal traffic periods (experiment

Comparison of the exact value of performance measures of the system in a

Investigation of the rate of convergence of the mean sojourn time and the probability of immediate access in the system with the finite buffer to corresponding performance measures of the system with an infinite buffer when the buffer size increases (experiment

Demonstration of the possibility to apply the presented results for optimization of the number of servers in the system (experiment

In numerical examples, we consider the systems operating in the

In the presented examples, we will use several different

We consider four arrival processes

All these

Based on these

Following this way, we construct the

The mean rates of service are

In the first experiment, we compare the dependence of

In the experiment we use service processes defined by

We consider three queueing systems which have different combinations of the

The input flow in the first system is defined by

The input flow in the second system is defined by

In the third system the input is defined by

*Figures*
*1*
and *2* show the dependence of the mean sojourn time

*In the second experiment* we compare the values

Input flow is described by

*Figures*
*3*
and *4* show the dependence of the the mean sojourn time

It can be seen from Figures *3*
and *4* that an approximation of the mean sojourn time and the probability that an arbitrary call reaches the server immediately by means of their values in some specially constructed more simple queueing system can be rather bad.

The idea of the third experiment is the following. Let us assume that the

We assume that the averaged arrival rate

It can be seen from
*Figures*
*5*
and *6* that the smoothing of the peak rates can cause essential decrease of the mean sojourn time and the increase of the probability that an arbitrary call reaches the server immediately upon arrival in the system.

In the second experiment, we have seen that an approximation of the system performance measures by means of their values in more simple queueing system can be bad. However, it is intuitively clear the following. If the random environment is "very slow" (the rate of the
*RE* and their averaging by the *RE* distribution. If the random environment is "very fast", approximation called below as "mixed parameters" can be successfully applied. This approximation consists of averaging parameters of the arrival and service processes by the distribution of the

In the fourth experiment, we show numerically that sometimes the described approximations make sense. However, in situations when environment is neither "very slow" nor "very fast", these approximations can be very poor. We consider the

We vary the parameter
*
Figures 7, 8* and *9*. In application of "mixed system" approximation, the averaged arrival rates under both states of the RE are equal to 3.488. The averaged service rate is equal to 2 at the first state of the

*
Figures 7, 8* and *9* confirm the hypothesis that the first type approximation ("mixed system") is good in case of "very slow"
*
Figures 8
*) because it is not quite clear how to make averaging of service intensity. Simple averaging of service rates under the different states of the

In the *fifth experiment* we compare the mean number of customers, probability of immediate access to the servers and loss probability in the

Looking at *
Figures 10-12*, it should be noted that the rate of convergence of the curves corresponding to the disciplines

Finally, in the *sixth experiment* we consider the next optimization problem:

where

It is clear that this problem is not trivial. When the number of servers is small, the cost of servers maintenance is also small, but the mean sojourn time is large. If we increase the number of servers, the mean sojourn time decreases while the cost of servers maintenance increases.

Let us assume that the cost coefficients be fixed as

On Figure 13, dependence of the cost criterion

Based on *
Figure 13
*, one can conclude that our analysis allows effectively solve the problems of the system design and that the optimal value of the cost criterion (in this example it is provided by