This article presents an approximate convolution model of a multiservice queueing system with the continuous FIFO (cFIFO) service discipline. The model makes it possible to service calls sequentially with variable bit rate, determined by unoccupied (free) resources of the multiservice server. As compared to the FIFO discipline, the cFIFO queue utilizes the resources of a multiservice server more effectively. The assumption in the model is that the queueing system is offered a mixture of independent multiservice Bernoulli-Poisson-Pascal (BPP) call streams. The article also discusses the results of modelling a number of queueing systems to which different, non-Poissonian, call streams are offered. To verify the accuracy of the model, the results of the analytical calculations are compared with the results of simulation experiments for a number of selected queueing systems. The study has confirmed the accuracy of all adopted theoretical assumptions for the proposed analytical model.

In recent years there has been a rapid increase in development of networks, in particular of mobile networks. While the evidence indicates that increasing competition is bringing down the cost of mobile services and user equipment, this implicates that more and more of the percentage of total network traffic is generated by mobile devices. According to the report [

The dynamic development of telecommunications (mobile) networks, the growing number of offered online services with strictly defined Quality of Service (QoS) parameters, and the ever-increasing number of network users cause network operators to introduce a number of different traffic management mechanisms that increase the effectiveness of the network. Good examples of the above mechanisms are threshold compression mechanisms [

This article proposes an approximate model of a queueing system with continuous FIFO (cFIFO) service discipline for a mixture of multiservice Bernoulli-Poisson-Pascal (BPP) call streams. In general, the cFIFO discipline assumes that calls that are in the queue are serviced according to the FIFO discipline. However, when the server has a lower amount of resources than demanded by the first call in the queue, then it can start servicing this call with a lower bitrate than the one demanded by the call. In the proposed model, a convolution algorithm is used to determine the occupancy distribution in the queueing system. Appropriate convolution algorithms to model full-availability multiservice systems with losses and multiservice access network systems (the so-called multiservice tree network) are proposed in [

The article is organized as follows. Section

The multiservice queueing system with the SD-FIFO queue service discipline is described in [

Queueing system with the SD-FIFO service discipline.

The system is composed of a multirate server with the capacity

Er denotes the Erlang traffic (Poisson call stream),

En denotes the Engset traffic (Bernoulli call stream),

Pa denotes the Pascal traffic (Pascal call stream),

Let us consider now a queueing system with the SD-FIFO discipline composed of a multiservice server with the capacity

The occupancy distribution in the system with the SD-FIFO discipline is determined on the basis of the following recurrence [

It is possible to determine on the basis of the distribution

Thus, Formula (

On the basis of distribution (

Let us consider the multiservice queueing system presented in Figure

Queueing system with cFIFO queueing discipline.

Let us consider now the operation of a queueing system with the system with the parameters

Queueing system with cFIFO discipline (

The model of a queueing system proposed in the article has been developed at the macrostate level. This means that the occupancy state is described by just one parameter, namely, the total number of occupied AU in the system (i.e., together in the server and buffer). Each macrostate

Multiservice telecommunications systems can be generally modelled by algorithms that analyze either the dependencies between microstates or macrostates. The one approach is characterized by a large computational complexity [

Let us consider now the operation of a convolution algorithm with a multiservice system that is composed of a server with the capacity

The input data for the convolution algorithm are the occupancy distributions for single classes

In the case of Engset and Pascal traffic, the distributions of single classes can be determined on the basis of the following formulas:

The occupancy distribution

In the case of other traffic streams (of the type

By having the single distributions for all classes offered to the system

Let us note that, for example, as a result of the convolution operation of two normalized distributions with the length

In (

The convolution operation (

Then, on the basis of the distributions

Let us consider now the queueing system with the cFIFO service discipline the operation of which is presented in Section

Paper [

Assuming, then, that the method for a determination of the occupancy distribution in the queueing system and in the system with zero buffer (when the system is offered Erlang call classes) is known, it is possible to determine the value of the parameter

The Markov process in the server with zero buffer to which a mixture of Erlang traffic classes is offered is a reversible process [

The convolution operation makes a determination of the occupancy distribution

In Section

Occupancy distributions in SD-FIFO and cFIFO queueing systems (

Occupancy distributions in SD-FIFO and cFIFO queueing systems (

The simulation results are shown with 95% confidence interval determined on the basis of the Student distribution for 5 series, 100,000 calls each. The results for the presented comparison indicate a very good convergence of occupancy distributions in the SD-FIFO and cFIFO queueing systems. The simulation experiments for other systems differentiated by their capacity, number, and demands of offered traffic classes conducted by the authors earlier confirm strong convergence of occupancy distributions of both queueing systems.

It should be stressed that queueing characteristics (such as the average number of calls of particular classes in the queue in corresponding occupancy states of the system) are not characterized by just as good convergence as the occupancy distributions. This would be the first reason why the SD-FIFO model cannot be directly applied to determine queueing characteristics for the cFIFO system. The other reason is the fact that to determine the parameter

By having the knowledge of the coefficients

Let us consider now a method for a determination of the average number

In states

To determine the occupancy distribution in a queueing system with the cFIFO queueing discipline and offered BPP traffic (or traffic with any distribution of the call stream) we will use the general recurrence dependencies derived for a system with Erlang traffic (Section

It should be emphasized that the application of the convolution algorithm in the proposed model makes it possible to determine characteristics of queueing systems for call streams other than BPP streams. Distributions of single classes that are the input data for the convolution algorithm can be then determined empirically on the basis of measurements or by simulation experiments.

On the basis of the occupancy distribution

Let us consider now the method for a determination of the average queue length for calls of a single class

In the case of Engset traffic, the parameter

Independently of the considered call streams, the proposed model can be written in the form of the following method, henceforth called the cFIFO method.

Determination of occupancy distributions for individual classes:

Determination of nonnormalized aggregated occupancy distributions of all classes, except a class

Determination of the average number

Determination, by convolution algorithm, of the occupancy distribution

Determination of transformation coefficients

Determination of the occupancy distribution

Determination of the average queue length

Determination of the average number

Determination of the average number

In order to verify the proposed analytical method the results of the calculations were compared with the results provided by the simulation experiments. For this purpose, a dedicated simulation program for a cFIFO queueing system evaluation was developed. The simulator uses an experiment with steady system time in which the process interaction method is used. The classes with finite number of sources were implemented according to the principles described in [

Each simulation experiment to determine the characteristics of the system under investigation for particular values

Figures

The blocking probability and the average queue length (

The blocking probability and the average queue length (

The blocking probability and the average queue length (

The next stage in the process of verification of the proposed analytical model was to test its accuracy for systems to which a mixture of Erlang, Engset, and Pascal traffic was offered (Figure

The blocking probability and the average queue length (

In the first case of considered types of non-Poissonian offered traffic (denoted by symbol NE), each call stream is described by normal distribution

The blocking probability and the average queue length (

Figure

The blocking probability and the average queue length (

The results presented on Figure

The blocking probability and the average queue length (

The results of the study showed in Figures

This article proposes an approximate analytical model of a queueing system that services multiservice traffic with the cFIFO queueing discipline. The model is based on a convolution algorithm in which to determine certain characteristics, such as the average number of serviced calls, the formulas derived on the basis of accurate Markovian models for multiservice systems with zero and nonzero buffer (with the SD-FIFO service discipline for the queue) are used. The introduction of these assumptions makes it possible to construct a general model of a multiservice queueing system with the cFIFO discipline. The proposed model is characterized by high accuracy, which makes an analysis of cFIFO systems for call streams with any, mutually independent, probabilistic distributions possible. This high accuracy stems from the fact that any errors that may possibly result from the adopted approach will be minimized by the convolution operation.

The application of multirate analytical models to analyze present-day broadband telecommunications systems is possible with a proper bandwidth discretization. Discretization allows the capacity of a system and bit rates demanded by calls to be expressed in the so-called allocation unit (AU).

The discretization process itself consists in replacing a variable bit rate (VBR) of call streams with a specific constant bit rate (CBR) called the equivalent bandwidth. There are many methods and algorithms specific methods of determining the equivalent bandwidth for given network types and services in the literature of the subject, for example, [

The procedure of discretization process for system services

Determination of the equivalent bandwidth for all offered call classes:

Determination of the allocation unit (AU). The value of this parameter can be calculated as the greatest common divisor (GCD) of all equivalent bandwidths in a considered system:

Determination of the number of allocation units demanded by calls of class

Determination of the capacity of the system (expressed in allocation units):

where

The authors declare that there is no conflict of interests regarding the publication of this paper.

The present work is financed with the Ministry of Science and Higher Education resources for academic purposes in the year 2016 within the frame of own research project entitled “The Structure, Analysis and Designing of Modern Switching Systems and Telecommunications Networks.”