The paper proposes a new method for modelling multiservice cellular networks with traffic overflow. The proposed method employs a model of Erlang’s Ideal Grading (EIG) with multiservice traffic and differentiated availability. The fundamental advantage of the proposed method, as compared to other relevant methods, is a major simplification in modelling systems with traffic overflow that results from the elimination of the necessity of a determination of the parameters of overflow traffic, that is, the average value and the variance. According to the proposed method, calculations in the overflow system can be reduced to calculations in a system composed of one grading only. The paper presents the method for determining availability in such a grading that models a system with traffic overflow. The results of analytical calculations were compared with the results of simulation experiments. The results of the research study confirm high accuracy of the proposed method.
New technologies used in building mobile networks evolve very quickly. Not much long ago, UMTS networks (Universal Mobile Telecommunications System) [
In order to ensure optimum usage of resources of mobile networks that belong to a given operator, it is necessary to take advantage of various traffic management mechanisms [
In 2G networks, the traffic overflow mechanism has been introduced to achieve optimum usage of GSM900 and GSM1800 network resources [
The overflow mechanism is also used with the coexistence of 2G and 3G networks in a given area. In this case, calls are transferred from the 3G network to the 2G network based on the IRAT HO (Inter Radio Access Technology HandOver) connection transfer procedure [
The phenomenon of traffic overflow in telecommunications is well-known and has been studied since the mid-20th century. The mechanism was used for the first time in hierarchical networks in which resources (historically called link groups (defining network resources as the link group resulted from the structure of the first telecommunications networks in which the basic resource was the link, while the link corresponded to the physical pair of wires connecting, e.g., telephone exchanges. In the paper, the term “network resources” will be used to define resources of modern telecommunications networks, whereas, in the descriptions of analytical models of traffic overflow systems, the notion of the group, viewed as system resources expressed in allocation units, e.g., links, channels, basic bandwidth units [
All the methods for modelling traffic overflow systems that have been developed so far require first a determination of characteristics of overflow traffic, that is, the average value of overflow traffic intensity and the related variance. The present paper proposes a simpler analytical method for overflow systems calculation that is based on the Erlang’s Ideal Grading model [
The paper is organized as follows. Section
The first models of networks with traffic overflow were proposed in the 1950s. The literature of the subject includes a number of models of full-availability groups (full-availability group is the model of a resource with complete sharing policy [
In both the ERT method and Fredericks-Hayward method, the parameters
The most common form of overflow in telecommunications networks is when call streams from a number of direct resources overflow to one alternative resource. If we assume that PCT1 (pure chance traffic of Type 1 (PCT1) [
In the ERT method all direct resources are replaced by one equivalent resource with the capacity
Graphic presentation of the ERT method: (a) overflow scheme for real network and (b) overflow scheme for equivalent system.
It is worthwhile to mention at this point that in this method the so-called overflow scheme was used for the graphical representation of a network with traffic overflow. The overflow scheme shows in a simplified way and independently of the network technology the dependencies between direct and alternative resources. Such an exemplary overflow scheme, used in the ERT method, is presented in Figure
The other method for modelling of single-service systems with traffic overflow, the so-called Fredericks-Hayward method [ capacity of a single component resource: traffic offered to a single component resource: peakedness coefficient
The value of the coefficient
Equation (
Fredericks-Hayward method and the ERT method that both make it possible to determine traffic properties of systems that are characterized by the exponential service time and service calls generated according to the Poisson distribution triggered further work on single-service systems with overflow traffic. The author of [
First studies in modelling systems with multiservice traffic began in the 1980s [
Overflow scheme for multiservice overflow traffic networks.
The variance of each of call streams is determined approximatively by carrying out a decomposition of each real resource the average value of the intensity of traffic of class the variance the peakedness coefficient for class
An alternative method for a determination of the capacity of fictitious groups is proposed in [
To determine the blocking probability of calls in the resource servicing multiservice overflow traffic, the Hayward method described in Section
Another general approach to the analysis of multiservice, hierarchical networks with overflow traffic were proposed in [
All the methods presented in earlier sections are based in their operation on the full-availability group model. It is necessary in these models to determine values of additional parameters
The proposed method relates to the remark given by the authors of the book on the life and lifetime achievements of Agner Krarup Erlang [
Erlang’s Ideal Grading (EIG) for single-service traffic was defined by A. K. Erlang [
Erlang’s Ideal Grading (a) offered traffic distribution and (b) concept of availability [
Figure
In 1993, in [
Erlang’s Ideal Grading with different availability (a) offered traffic distribution, (b) concept of availability [
According to [
Probability
The blocking probability for calls of class
Let us consider now a new approach to modelling systems with multiservice traffic overflows. The presented multiservice traffic overflow scheme shown in Figure
Presentation of the method for modelling systems with traffic overflow with the application of Erlang’s Ideal Grading (a) overflow scheme for real network and (b) equivalent grading.
The considered network with traffic overflow services calls from the set
The intensity of PCT1 traffic of class
In the proposed method, calculations of the blocking probability in the overflow system composed of many full-availability groups (both direct and alternative resources) can be reduced to calculations of this probability in a system that is composed of one group only: Erlang’s Ideal Grading (Figure
At the first stage of defining the function
In (
Probability
The value of the parameter
The proposed model assumes that the capacity of Erlang’s Ideal Grading used for the calculations is determined as the sum of the capacities of the resources in the system under consideration, that is, the capacity of all direct resources and the alternative resource:
The values of the availability parameters for individual call classes are determined in the following way: a call of class
Taking into consideration the availability for class
The sequence of calculations in the proposed method can be written in the form of the Overflow-EIG method:
Define capacity Determine availabilities Determine availabilities Determine the occupancy distribution in Erlang’s Ideal Grading (formula ( Determine the blocking probability in Erlang’s Ideal Grading (formula (
Observe that the analytical model of the network with multiservice traffic overflow (each of direct resources services many traffic classes) proposed in the paper is simplified to the model considered in [
The method for modelling networks with multiservice overflow traffic proposed in the paper is presented for the case of systems in which direct resources are offered Poisson call streams. It should be stressed, however, that the method can be easily adopted for modelling systems with traffic overflow in which direct resources are offered multiservice BPP (Bernoulli-Poisson-Pascal) call streams [
The proposed method can be easily adopted for modelling of multitier overflow systems. An example of a scheme for traffic overflow in three-tier systems is shown in Figure
Scheme of three-tier overflow.
In the considered case, the availability for traffic
The presented method for modelling multiservice systems with overflow traffic is an approximate method. To evaluate the accuracy of the proposed solution, the results of the analytical calculations of the blocking probability in cellular networks with traffic overflow have been compared with simulation data. The study was conducted for many scenarios of traffic overflow that occur in currently used mobile 2G, 2.5G, 3G, and 4G networks. The studies on systems with traffic overflow, for the scenarios under consideration, were carried out at the call level. In the simulation model, each cell was represented as an object with a given capacity and offered traffic. Depending on the current load of cells to which calls generated by subscribers were directly offered, these calls were either serviced by the cells or directed to the alternative cell. The analysis of the system at the call level makes it possible to consider traffic overflow regardless of the technology employed to construct a mobile network. A detailed description of the methods for modelling mobile systems at the call level in wireless networks of different technologies is presented in, for example, [
The diagram for the case of traffic overflow that was considered first is presented in Figure
Traffic overflow between cell of the GSM900 system: (a) traffic overflow between cells and (b) overflow scheme (
Blocking probability in mobile system GSM900 with overflow traffic, only voice traffic.
Blocking probability in mobile system GSM900 with overflow traffic, two traffic classes: voice and data.
Blocking probability in mobile system GSM900 with overflow traffic, various capacities of direct resources.
Overflow between cells of systems GSM1800 and GSM900: (a) traffic overflow between cells and (b) overflow scheme (
Blocking probability in mobile system with overflow traffic between cells of GSM1800 and GSM900 systems.
Traffic overflow between cells of the UMTS system and GSM1800 and GSM900: (a) traffic overflow between cells and (b) overflow scheme (
Blocking probability in mobile system with overflow traffic: overflow between cells of the UMTS system and GSM1800 and GSM900.
Overflow between cells of the systems: LTE and UMTS, (a) traffic overflow between cells and (b) overflow scheme (
Blocking probability in mobile system with overflow traffic: overflow between cells of the LTE system and UMTS.
The results for the blocking probability obtained for the considered case of traffic overflow (Figure
Figure
Figure
Figure
The results of the blocking probability for the considered network with traffic overflow between cells of the GSM1800 and GSM900 systems (according to the overflow scheme presented in Figure
Blocking probability in mobile system with overflow traffic: overflow between cells of the LTE system (different capacities of LTE cells) and UMTS;
The next considered case was a network with traffic overflow between the systems: UMTS and GSM (Figure
The last considered case of traffic overflow involves traffic overflow between the systems: LTE and UMTS. We assume that these two systems are comparable as far as data transmission possibilities are concerned. Traffic overflow in this particular case relates to all types of connections (Figure
In the last part of our research we have considered the network with traffic overflow between the LTE and the UMTS systems (Figure
The analysis of the presented results allows us to state that for each of the considered traffic overflow scheme the presented method ensures high accuracy of calculations, which is sufficient for practical purposes in engineering applications at the stage of analysis and dimensioning of present-day cellular networks.
This paper proposes a simple analytical method for a determination of traffic characteristics of mobile systems with traffic overflow, both single service and multiservice. The proposed method is based on the Erlang’s Ideal Grading model with different availabilities and makes it possible to determine the blocking probability in overflow systems based only on the information on the capacity of individual network resources (cells) and the value of offered traffic. The proposed method is characterised by the complexity of order
The paper considers a number of cases that involved traffic overflow in mobile networks, that is, both between cells (sectors of cells) of different systems (GSM, UMTS, and LTE), and between cells (sectors of cells) of one system. For each of the considered overflow scenarios, blocking probabilities obtained on the basis of the proposed analytical method are evaluated. These results are then compared with the data obtained in the simulations, which confirms very high accuracy of the proposed model. The model, far more simpler than any earlier developed methods, can be thus used for solving practical problems in design and optimization of modern cellular networks.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The presented work has been funded by the Polish Ministry of Science and Higher Education within the status activity task “Struktura, analiza i projektowanie nowoczesnych systemów komutacyjnych i sieci telekomunikacyjnych” in 2015.