The paper proposes a formal derivation of recurrent equations describing the occupancy distribution in the full-availability group with multirate Binomial-Poisson-Pascal (BPP) traffic. The paper presents an effective algorithm for determining the occupancy distribution on the basis of derived recurrent equations and for the determination of the blocking probability as well as the loss probability of calls of particular classes of traffic offered to the system. A proof of the convergence of the iterative process of estimating the average number of busy traffic sources of particular classes is also given in the paper.

Dimensioning and optimization of integrated networks, that is, Integrated Services Digital Networks (ISDN) and Broadband ISDN (B-ISDN) as well as wireless multiservice networks (e.g., UMTS), have recently developed an interest in multirate models [

Multirate systems can be analysed on the basis of statistical equilibrium equations resulting from the multidimensional Markov process that describe the service process in the considered systems [

Nowadays, in the analysis and optimization of multirate systems, the recurrent algorithms are usually used. This group of algorithms is based on the approximation of the multidimensional service process in the considered system by the one-dimensional Markov chain. Such approach leads to a determination of the occupancy distribution in systems with state-independent admission process and state-independent arrival process (in teletraffic engineering such system is called the full-availability group with Erlang traffic streams) on the basis of simple Kaufman-Roberts recurrence [

Because of the simplicity of the Kaufman-Roberts equation, in many works the attempts of its modification in order to analyse the systems with BPP traffic were undertaken. In [

The aim of this paper is to formally prove that the MIM-BPP algorithm [

The paper is organized as follows. Section

Let us consider a model of the full-availability group with the capacity of

Full-availability group with the Erlang, Engset, and Pascal traffic stream.

The call arrival rate for Erlang traffic of class

The total intensity of Erlang traffic of class

Let us consider now a fragment of the multidimensional Markov process in the full-availability group with the capacity of

Fragment of a diagram of Markov process in the full-availability group with BPP traffic.

The multidimensional service process in the Erlang-Engset-Pascal model is a reversible process. In concordance with Kolmogorov reversibility test considering any cycle for the microstates shown in Figure

It is convenient to consider the multidimensional process occurring in the considered system at the level of the so-called macrostates. Each macrostate

The macrostate probability

In (

In (

In order to determine the average number of calls serviced in particular states of the system, let us consider a fragment of the one-dimensional Markov chain presented in Figure

A fragment of the one-dimensional Markov process in the full-availability group with multirate traffic, servicing two call streams (

Let us notice that each state of the Markov process in the full-availability group (Figure

Let us notice that, in order to determine the parameter

On the basis of the reasoning presented above, in [

Consider the following steps.

Determination of the value of Erlang traffic

Setting the iteration step:

Determination of initial values of the number

Increase in each iteration step:

Determination of the value of Engset traffic

Determination of the state probabilities

Determination of the average number of serviced calls

Repetition of steps (3)–(6) until predefined accuracy

Determination of the blocking probability

In this section we prove that the process for a determination of the average number of serviced traffic sources proposed in the MIM-BPP method is, in the case of multiservice Engset sources, a convergent process. Thus, the following theorem needs to be proved.

The sequence

In order to prove Theorem

Regardless of the iteration step, for every

Let us demonstrate now that the process of a determination of the average number of serviced traffic sources proposed in the MIM-BPP method is a convergent process also in the case of multiservice Pascal sources. The following theorem will be then proved.

The sequence

Proceeding in the analogical way as we did in the case of sequence (

Therefore, in order to show that sequence

Consider the elements of series (

The presented iterative algorithm for systems with state-independent admission process (i.e., the full-availability group) makes it possible to determine exactly the occupancy distribution and the blocking and loss probabilities in systems that service Erlang (Poisson distribution of call streams), Engset (binomial distribution of call streams), and Pascal traffic streams (negative binomial distribution of call stream). The call stream of the types investigated in the paper are typical streams to be considered in traffic theory. They are used for modelling at the call level, where any occupancy of resources of the system, for example, effected by a telephone conversation or by a packet stream with characteristics defined at the packet level, can be treated as a call [

The application of the notion of the basic bandwidth unit (BBU) used in the notation of the presented method makes it possible to obtain high universality for the method. BBU is determined as the highest common divisor of all demands that are offered to the system. Depending on a system under consideration, the basic bandwidth unit can be expressed in bits per second or as the percentage of the occupancy of the radio interface (the so-called interference load) [

The algorithm worked out for modelling systems with BPP traffic can be treated as an extension to the Kaufman-Roberts model [

The paper introduces a formula that makes it possible to determine exactly the occupancy distribution in systems with state-independent call admission process. It is then demonstrated that the algorithm for a determination of the average number of serviced traffic sources of particular classes used in the MIM-BPP method is convergent.

In order to present the convergence of the MIM-BPP method (the number of required iterations), in Table

Relative errors of the number of busy class 3 sources in relation to the number of iterations.

The step of iteration | ||||

2 | 7 | |||

0.3 | ||||

0.4 | ||||

0.5 | ||||

0.6 | ||||

0.7 | ||||

0.8 | ||||

0.9 | ||||

1.0 | ||||

1.1 | ||||

1.2 |

In this section we limit ourselves to just presenting the results of the convergence of the presented algorithm for one selected system. A comparison of the analytical results for the blocking/loss probability with the results of the simulation is presented in earlier works, for example, [

In the paper recurrent equations describing—at the macrostate level—the service process in the full-availability group with multirate BPP traffic were derived. The derived equations made it possible to formulate an exact iterative algorithm for determining the occupancy distribution, blocking probability, and loss probability of calls of particular classes offered to the system. The convergence of the proposed process of estimating the average number of busy sources of Engset and Pascal traffic was proved.