An Efﬁcient Method for Proportional Differentiated Admission Control Implementation

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Introduction
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.

Problem Statement
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

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Here, (i) K: is a number of traffic classes. K ≥ 2; (ii) δ i : is the weight of class i, i = 1, . . . , K. This parameter reflects the traffic priority. By increasing the weight, we also increase the admittance priority of corresponding traffic class; (iii) ρ i : is the offered load of class i traffic; (iv) n i = C i /b i , C i is an allotted partition of the link capacity, b i is a bandwidth requirement of class i connections, and x is the largest integer not greater than x; (v) B(ρ i , n i ): is the Erlang loss function, that is, under the assumptions of exponential arrivals and general session holding times [5], it is the blocking probability for traffic of class i, i = 1, . . . , K.
It needs to find C 1 , C 2 , . . . , C K taking into account known δ i , ρ i , b i , i = 1, . . . , K and the restriction imposed by given link capacity, C: Let us remark that variations of C i imply a discrete changing of the function B(ρ i , n i ). 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 ρ > n, 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.

Approximate Erlang B Formula.
We assert that for the desired accuracy of the approximation (6) an offered load has to exceed a certain threshold. The concrete value of the threshold is given by the following theorem. Proof.
Here and below, we use the following designation: Assume that ρ > n. First, we rewrite the Erlang B formula Remark that Taking into account properties of geometrical progression, we have Hence To prove the second inequality of the theorem, we use the following upper bound of the Erlang loss function [6]: Transform this as follows: It implies We have n/ρ < 1. Hence, EURASIP Journal on Wireless Communications and Networking 3 Thus, for any such that it follows that From the inequality (20), we obtain the condition (9). The proof is completed.
Note that the approximate formula (6) can provide the required accuracy in the case of ρ < n + 1/ . Actually, if = 0.01, n = 200, then the required accuracy is reached for ρ = 270 < 300. Thus, the condition (9) is sufficient but not necessary. It guarantees the desired accuracy of the approximation for any small and n.
It is clear that for some values C, b i , ρ i , δ i , we can obtain C 1 > C in (24) or C i < 0 in (23). Therefore, the problem is unsolvable and PDAC implementation is impossible for the given parameters.
More precisely, if C 1 > C, then we have from (24) Using the following equality: we derive From the inequality C i < 0, we can write Therefore, By substituting the expressions (24) for the C 1 into (30), we get after some manipulations the following inequality: Note that the problem (22) has been formulated under the condition Actually, it implies Thus, the region of acceptability for PDAC problem (22) is defined by It follows from the theorem that the approximation (6) is applicable even for n = 1 and any small > 0 if ρ > 1/ − 1. In spite of this fact, the solution above cannot be useful for small values of the ratio C i /b i . In this case, the loss function B(ρ i , n i ) is sensitive to fractional part dropping under calculation n i = C i /b i . For example, if b i = 128 kb/s, ρ i = 2, and we obtain C i = 255 kb/s, then the approximate value of the blocking probability is about 0.004. But n i = C i /b i = 1 and B(1, 2) ≈ 0.67. Thus, the offered approximate formula is useful if the ratio C i /b i is relatively large.

Alternative Problem Statements.
Let n i be the number of channel assigned for class i traffic, i = 1, . . . , K. Each class i is characterized by a worst-case loss guarantee α i [7,8].

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Consider the following optimization problem: Assume that for all i ∈ {1, . . . , K} ∃n i ∈ N : B(ρ i , n i ) = α i . It is well known that the Erlang loss function B(ρ, n) is a decreasing function of n [9], that is, B(ρ, n 1 ) < B(ρ, n 2 ) if n 1 > n 2 . Therefore, the optimal solution (n * 1 , n * 2 , . . . , n * K ) of the problem (35) satisfies the mentioned condition If we designate δ i = 1/α i , then we get Thus, the optimization problem (35) is reduced to the problem (1).
Assume the approximation (6) is admissible. Therefore, the method from previous subsection is supposed to be used, but the optimal solution of the problem (35) can be computed by inverting the formula (36). Taking into account the approximation, we get Note that in practice the solution n * i is not usually integer; thus, it has to be as follows: We now consider the optimization of routing in a network through the maximization of the revenue generated by the network. The optimal routing problem is formulated as where n i is a fixed number of channels for class i traffic and r i is a revenue rate of class i traffic. Obviously, the Erlang loss function B(ρ, n) is an increasing function of ρ. Therefore, the optimal solution (ρ * 1 , ρ * 2 , . . . , ρ * K ) of the problem (40), (41) satisfies the following condition: Hence, the problem (40), (41) can be reduced to the problem (1) as well. Under the approximation, the optimal solution takes the form and the maximal total revenue is

Performance Evaluation
Let us illustrate the approximation quality. The difference Δ(ρ, n) = B(ρ, n) − β(ρ, n) is plotted as a function of offered load in Figure 1. If the number of channel n 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 B(n, ρ) decreases if the number of channels n increased relative accuracy . Let us designate ρ * = n + 1/ . If the approximation (2) is admissible for ρ * then it is also admissible for any ρ > ρ * . In Figure 2, the behavior of losses function B(n, ρ * ) 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 C = 640 Mb/s, K = 5, b i = 128 kb/s, ρ i = 1100, δ i = 1 − 0.1(i − 1), i = 1, . . . , 5. 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 5 i=1 C i = 640 Mb/s and three channels per 128 kb/s have not been used. We get It is easy to see that If K = 10, δ i = 1 − 0.05(i − 1), i = 1, . . . , 10, and other parameters are the same then max i, j δ i B ρ i , n i − δ j B ρ j , n j < 0.001.
If an obtained accuracy is not enough, then the formulas (23)-(25) provide efficient first approximation for numerical methods.

Conclusion
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.