An Analytical Approach to the Analysis of Guard-Channel-Based Call Admission Control in Wireless Cellular Networks

1. Introduction

In wireless cellular networks, when a subscriber crosses the boundary of a cell (while on a call), the subscriber releases this cell’s channel and requests an empty channel in a neighboring cell. This process is called a handover. If a neighboring cell has at least one empty channel, then such a handover call (h-call) is delivered continuously and almost transparently to the subscriber; otherwise, the call is dropped. Usually, dropping an ongoing call from a different cell is less desirable than blocking new attempts originating within the cell (o-calls). Thus, h-calls are considered to be more important (have a higher priority) than o-calls.

There are various prioritization schemes for h-calls in wireless networks. Efficient methods for minimizing the rate of dropped h-calls include reserving a number of channels (guard channels) specifically for h-calls, imposing a threshold for o-calls, and queuing h-calls. A detailed review of papers investigating these methods can be found in [1–3].

In this paper, we consider a model with call admission control (CAC) based on guard channels schemes (GC schemes) and without queuing of h-calls. In almost all existing papers, it has been assumed that h-calls and o-calls have identical channel occupancy times. Due to this assumption, the function of a cell is described by a one-dimensional Markov chain, and the authors managed to find simple recursive formulas for calculating the probabilities of dropping h-calls and of blocking o-calls. The classical guard channels (CGC) reservation scheme [4], as well as a generalized reservation scheme called fractional guard channels (FGC) and its own special case, that is, uniform fractional guard channels (UFGC) [5], was considered. However, the identity channel occupancy times assumption for calls of different types is rather limiting and unrealistic [6–8]. Therefore, one-dimensional Markov chains do not adequately describe such networks, and further research on multidimensional models is required. However, when using multidimensional models, we face the famous “curse of dimensionality”. To address this problem, a few papers have suggested various approximation schemes. A detailed review of approximate methods can be found in [9].

In this paper, we develop an analytical approach to the calculation of models without h-call queues, in which o- and h-calls are not assumed to have identical channel occupancy times. First, we briefly review the results of well-known papers that study such models. One of the first works to investigate such models was [6]. This paper examined models featuring various reservation strategies for h-calls or establishing thresholds for o-calls. For the latter type of model, simple formulas are easily developed for calculating the loss probability of h-calls as well as the blocking probability of o-calls. The development of simple formulas is possible because there is an explicit solution to an appropriate set of global equilibrium equations (SGEE) for the state probabilities. However, there is no closed form solution to the corresponding equations for models with CAC based on the GC scheme; therefore, the authors suggested approximation formulas for QoS calculations based on some heuristic considerations. The accuracy of the suggested formulas was determined by simulation. The authors also suggested another approximation scheme, which they dubbed traditional and in which different occupancy times are replaced by a single time by following a defined procedure. It was shown that the traditional scheme gives a rough approximation, especially as the true occupancy times of different calls differ substantially. The common idea underlying both approximation methods is the unification of the channel occupancy times of polytypic calls with appropriate schemes. A similar approximation approach based on this idea is proposed in [7], and another approach is suggested in paper [8], which offers high accuracy in models for which the parameters for different call types differ substantially. For models with identical parameters, however, its results are less accurate.

In this paper, we develop an analytical approach for calculating models of a wireless network with nonidentical channel occupancy times for h- and o-calls in which the CAC are based on CGC, FGC, and UFGC schemes. We prove that the formulas suggested in [6, 7] are not merely approximations but are exact if the model satisfies a local balance condition (for more information about this condition, see [10]); otherwise, these formulas are approximations. We use these formulas to find the optimal number of guard channels for CAC based on CGC scheme.

Even though we consider a monoservice network for the sake of simplifying both our models and any intermediate calculations, the results that we achieve can be easily adapted for multiservice networks.

This paper is organized as follows. In Section 2, well-known QoS calculation algorithms are given for the CGC scheme. In Section 3, a new analytical approach is developed for the study of models with the FGC scheme, and, as a particular case, we obtain results for the CGC scheme. Section 4 contains the results of selecting the optimal number of guard channels for the CAC based on CGC scheme. In Section 5, we provide some concluding remarks. Finally, in an appendix, we provide proofs of all required facts.

2. Approximate Methods for CAC Based on the CGC Scheme

First, we will analyze the model of a cell belonging to a homogeneous wireless network with a CAC based on the CGC scheme. Henceforth, a homogeneous network is defined as one for which the traffic parameters of all cells within the network are statistically identical; that is, we can study the function of a representative cell in isolation. This assumption is true for practically all networks with small cells (e.g., networks with microcells).

The representative cell contains N channels, 1 < N < ∞, which are used by Poisson flows of o- and h-calls. The intensity of an x-call is denoted by λ_x, where x ∈ {o, h}. If at least one free channel exists upon the arrival of an h-call, then the call seizes one of the free channels; otherwise, the call is dropped. A new arriving call is accepted only when the number of busy channels is less than m, for some fixed m satisfying 1 ≤ m ≤ N (in the case where m = N, there are no restrictions on new calls); otherwise, the new call is blocked.

The distribution functions corresponding to the channel occupancy times of both types of calls are exponential, but their parameters differ by the intensity of handling of n- and h-calls, which are given by μ_o and μ_h, respectively. Generally speaking, μ_o ≠ μ_h.

Henceforth, let I(A) denote the indicator function of event A. From Formulas (2.2), we obtain P_o = P_h  when m = N.

We should note that for practical networks, this SGEE has large dimension, which explains why its exact solution is computationally difficult. In [8], for the case λ_o ≪ λ_h, μ_o ≪ μ_h the following algorithm for approximating the values of the QoS metrics from (2.2) is developed based on strict mathematical justifications.

In the above algorithm, the following notation is used: ν_x = λ_x/μ_x, x ∈ {o, h}. The results achieved with this algorithm nearly coincide with those from exact calculations under the indicated above conditions: λ_o ≪ λ_h, μ_o ≪ μ_h. Under these conditions, the exact values of the QoS metrics for a model of moderate dimensions are calculated with SGEE.

The authors noted that it is impossible to measure the accuracy of their proposed formulas analytically; therefore, they demonstrated the high accuracy of these formulas with simulations.

Another way of transforming a two-dimensional model to an approximate one-dimensional model proceeds by replacing different average channel occupation times with the weighted average μ = ν/λ, where λ = λ_o + λ_h. Such an approach to the problem solution is called traditional [6]. In this case, QoS parameter calculations require the use of formulas (2.7); however, in formulas (2.8), parameters ν_h and ν_o are defined as follows: ν_h = λ_h/μ, ν_o = λ_o/μ.

The authors in [6] also proved the low accuracy of this traditional approach, especially when μ_o and μ_h vary greatly. As was expected, the results of the last two approaches coincide at μ_o = μ_h.

Tables 1, 2, and 3 contain the results of numerical experiments for the above three methods. Comparison of these methods shows that the results contained in [6], do not differ significantly, even when the condition λ_o ≪ λ_h, μ_o ≪ μ_h does not hold (see Tables 2 and 3).

3. An Analytical Method for CAC Based on FGC and UFGC Schemes

In this method, as in the CGC scheme, an h-call is accepted if there is at least one free channel upon its arrival; otherwise, the call is dropped. An arriving o-call is accepted with probability β_i if upon its arrival the number of busy channels equals i, i = 0,1, …, N − 1; thus, the call is blocked with probability 1 − β_i. Obviously, the CGC scheme is a special case of this scheme. Indeed, by setting β_i = 1 for all i = 0,1, …, m − 1 and β_i = 0 for all i = m, m + 1, …, N − 1 in the FGC scheme, we obtain the CAC based on the CGC scheme; by setting β_i = α for all i = 0,1, …, N − 1, the FGC scheme reduces to the UFGC scheme with admission probability α [5].

Here, we offer strict analysis of these formulas; moreover, we will prove that the formulas are exact if the system satisfies the local balance condition.

Hence, to calculate the QoS metrics in (2.2), it is sufficient to find the merged state probabilities π(i) rather than calculating the probabilities of all states k ∈ S.  

The proof is provided in the appendix.

Thus, taking (3.7) into account, we deduce that the proposed algorithm (8)–(11) in [6] is accurate and not merely an approximation as long as the system satisfies the local balance condition.

Note that closed-form solutions (3.9)–(22) might be considered refined approximations for the similar model that features identical channel occupancy times for heterogeneous calls [5], that is, when μ_o = μ_h. Indeed, formulas (3.11) completely coincide with the exact results given in [5] in the case that the admission probability α is equal to 1; small deviations arise when α is close to 1.

4. Selection of Optimal Values for the Parameters of CAC Based on the CGC Scheme

The problem of how to provide a given level of handling quality for different call types is scientifically and practically very interesting. The solution of such problems requires some regulated parameters. Thus, in some networks, for which the distribution of channels between cells is fixed, only call admission control parameters can be regulated because controlling loads present a difficult task and one that is sometimes practically impossible.

It is extremely difficult to develop a method for finding the optimal parameter values for the FGC scheme, as this strategy contains a large number (exactly N) of controlled parameters (i.e., β_i, i = 0,1, …, N − 1). However, even if it was possible to calculate the optimal values of β_i, i = 0,1, …, N − 1, the practical realization of the resulting randomized call admission strategies would still include methodological difficulties. Thus, we address several problems arising from trying to find optimal parameter values for the CGC scheme to meet the required QoS level. The following scheme has only one controlled parameter (m).

First, we will study the problem of organizing the fair handling of different call types. As stated above, as m increases, the probability of h-call loss increases, whereas the probability of o-call blocking decreases; thus, at m = N, we obtain an equilibrium of P_o = P_h. In other words, at m = N, we have absolutely fair handling.

Considering that P_o  (N) = P_h  (N) and assuming the monotonicity of P_o(m) and P_h(m), we can apply the dichotomy method to solve the equation in (4.1). We now describe the general algorithm. The initial uncertainty interval [1, N] is split in half; that is, we set b = int (N + 1)/2, where int (x) is the integer part of x. If condition (4.1) is satisfied at this point, then we next consider the interval [1, b]; otherwise, we consider the interval [b,   N]. The algorithm correctly terminates by locating an interval of unitary length whose right limit satisfies condition (4.1) but whose left limit does not. As a result, this right limit will be the required value of m^*. This algorithm has a finite number of steps because by the condition that P_o  (N) = P_h  (N), condition (4.1) is satisfied in the worst case when m = N. The results of solving (4.1) are given in Table 4.

Given the monotonicity of P_o(m) and P_h(m), we propose the following algorithm for solving problem (4.3).

Note that the dichotomy method can be used to solve the problem in Step 3 due to the monotonicity of P_o(m).

Combining the solutions of problems (4.3) and (4.5), we obtain a range of values for the parameter m meeting conditions (4.2). The results of the solutions of these problems are given in Table 5, where the symbol ∅ indicates no solution.

It is possible to pose other similar QoS optimization problems; however, due to space limitations, they are not considered here.

5. Conclusion

We have provided exact and simple formulas for the calculation and optimization of QoS metrics for traditional wireless network models, assuming that the two fundamentally different types of calls (new and handover) do not have identical channel occupancy times. It is important to note that the adaptation of the proposed approach for next generation networks is straightforward.