An Efficient Method for Proportional Differentiated Admission Control Implementation

Efficient implementation of admission control mechanisms is a key point for next-generation wireless network development. Actually, over the last few years an interrelation between pricing and admission control in QoS-enabled networks has been intensively investigated. Call admission control can be utilized to derive optimal pricing for multiple service classes in wireless cellular networks [1]. Admission control policy inspired in the framework of proportional differentiated services [2] has been investigated in [3]. The proportional differentiated admission control (PDAC) provides a predictable and controllable network service for real-time traffic in terms of blocking probability. To define the mentioned service, proportional differentiated service equality has been considered and the PDAC problem has been formulated. The PDAC solution is defined by the inverse Erlang loss function. It requires complicated calculations. To reduce the complexity of the problem, an asymptotic approximation of the Erlang B formula [4] has been applied. However, even in this case, the simplified PDAC problem remains unsolved.

In this paper, we improve the previous results in [3] and withdraw the asymptotic assumptions of the used approximation. It means that for the desired accuracy of the approximate formula an offered load has to exceed a certain threshold. The concrete value of the threshold has been derived. Moreover, an explicit solution for the considered problem has been provided. Thus, we propose a method for practical implementation of the PDAC mechanism.

The rest of the paper is organized as follows. In the next section, we give the problem statement. In Section 3, we first present a nonasymptotic approximation of the Erlang B formula. We then use it for a proportional differentiated admission control implementation and consider some alternative problem statements for an admission control policy. In Section 4, we present the results of numerous experiments with the proposed method. Section 5 is a brief conclusion.

Let us consider the concept of admission control inspired in the framework of proportional differentiated services. In the above paper [3], whose notation we follow, PDAC problem is defined as

Here,

It needs to find taking into account known and the restriction imposed by given link capacity, :

Let us remark that variations of imply a discrete changing of the function . Hence, it is practicably impossible to provide the strict equality in (1). It is reasonable to replace (1) by an approximate equality as follows:

But, even in this case, the above problem is difficult and complex combinatorial problem. For its simplification, the following asymptotic approximation has been used [3]. If the capacity of link and the offered loads are increased together:

and , then the Erlang loss function

can be approximated by

Taking into account the PDAC problem, the authors of [3] consider the limiting regime when

and . Under these conditions, the asymptotic approximation of the Erlang B formula has been used and (1) has been replaced by simplified equations as follows:

In practice, the limited regime (7) is not appropriate. But the simplification (8) can be used without the conditions (7). Actually, the approximation (6) can be applied without the condition (4). We prove it below.

Let us illustrate the approximation quality. The difference is plotted as a function of offered load in Figure 1. If the number of channel is relatively small then high accuracy of approximation is reached for heavy offered load. Let us remark that heavy offered load corresponds to high blocking probability. Generally, this situation is abnormal for general communication systems, but the blocking probability decreases if the number of channels increased relative accuracy . Let us designate . If the approximation (2) is admissible for then it is also admissible for any . In Figure 2, the behavior of losses function according to different is shown. Thus, the provided approximation is attractive for a performance measure of queuing systems with a large number of devices.

Next, we consider a numerical example to evaluate the quality of a PDAC implementation based on the proposed method. Assume that  Mb/s,  kb/s, . In average, there are 1000 channels per traffic class. Following the theorem above, we conclude that the blocking probability can be replaced by the approximation (6) with accuracy about 0.01. Using (23)–(25), find a solution of the simplified PDAC problem and calculate the blocking probability for the obtained values. The results are shown in the Table 1.

Note that  Mb/s and three channels per 128 kb/s have not been used. We get

It is easy to see that

If , and other parameters are the same then

If an obtained accuracy is not enough, then the formulas (23)–(25) provide efficient first approximation for numerical methods.

In this paper, a simple nonasymptotic approximation for the Erlang B formula is considered. We find the sufficient condition when the approximation is relevant. The proposed result allows rejecting the previously used limited regime and considers the proportional differentiated admission control under finite network resources. Following this way, we get explicit formulas for PDAC problem. The proposed formulas deliver high-performance computing of network resources assignment under PDAC requirements. Thus, an efficient method for proportional differentiated admission control implementation has been provided.