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

Abstract

We develop an analytical approach to the performance analysis and optimization of wireless cellular networks for which different types of calls are prioritized based on a channel reservation scheme. We assume that the channel occupancy time differs for new and handover calls. We obtain simple formulas for calculating quality of service (QoS) metrics and solve some problems related to finding the optimal values of guard channels as well as present the results of numerical experiments.

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.

Remark 2.1. The abovementioned channel occupancy times are defined not only in terms of required service times for different call types but also in terms of the subscriber mobility within cells. In all existing works, it has been assumed that the required service times of x-calls and the time spent by subscribers within the cell have exponential distributions with parameters 𝜏_𝑥 and 𝛾_𝑥, 𝑥∈{𝑜,ℎ}, respectively. Thus, the channel occupancy time of x-calls is defined as the minimum of two exponentially distributed stochastic variables; that is, for an x-call, it has an exponential form with parameter 𝜇_𝑥=𝜏_𝑥+𝛾_𝑥, 𝑥∈{𝑜,ℎ}. Based on practical considerations, the residence duration may differ greatly between different types of calls; hence, the channel occupancy times will also differ.

Cell functionality is described by a two-dimensional Markov chain (MC); that is, the state of the given system at an arbitrary moment in time is described by a two-dimensional vector 𝐤=(𝑘₀,𝑘_ℎ), where 𝑘_𝑜(𝑘_ℎ) indicates the number of n-calls (h-calls) in the cell. Thus, the state space of the corresponding MC is determined as follows:𝑆∶=𝐤∶𝑘_𝑜=0,1,...,𝑚;𝑘_ℎ=0,1,...,𝑁;𝑘_𝑜+𝑘_ℎ≤𝑁.(2.1)

Let us denote the stationary probability of state 𝐤∈𝑆 as 𝑝(𝐤). Stationary probabilities are found from the appropriate SGEE and then used to calculate all the required QoS metrics of the model. The primary QoS metrics are the loss probability of h-calls (𝑃_ℎ) and the blocking probability of o-calls (𝑃_𝑜). They are defined as follows when using the above-mentioned CGC admission control:𝑃𝑃_ℎ=_𝐤∈𝑆𝑝(𝐤)𝐼𝑘_𝑜+𝑘_ℎ=𝑁_𝑜,=_𝐤∈𝑆𝑝(𝐤)𝐼𝑘_𝑜+𝑘_ℎ≥𝑚.(2.2)

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

Step 1. Calculate the following parameters: 𝜌_𝑖(𝑗)=𝜈^𝑗ℎ/𝑗!𝜌_𝑖(0),𝑖=0,𝑚,𝑗=0,𝑁−𝑖,(2.3) where 𝜌_𝑖(0)=(∑^𝑚𝑗=0(𝜈^𝑗ℎ/𝑗!))⁻¹, 𝜋(⟨𝑖⟩)=𝜈^𝑖𝑜/𝑖!^𝑖𝑗=1Λ(𝑗)𝜋(⟨0⟩),𝑖=1,𝑚,(2.4) where (⟨0⟩)=(1+∑^𝑚𝑖=1(𝜈^𝑖𝑜/𝑖!)∏^𝑖𝑗=1Λ(𝑗)), Λ(𝑖+1)=𝜌_𝑖(0)∑^{𝑚−1−𝑖𝑗=0}(𝜈^𝑗ℎ/𝑗!), 𝑖=0,𝑚−1.

Step 2. Calculate the approximate values of the QoS metrics in (2.2): 𝑃_ℎ≈^𝑚_𝑖=0𝑃𝐸_𝐵𝜈_ℎ,𝑁−𝑖𝜋(⟨𝑖⟩),(2.5)_𝑜≈^𝑚_{𝑖=0𝑁−𝑖𝑗=𝑚−𝑖}𝜌_𝑖(𝑗)𝜋(⟨𝑖⟩),(2.6) where 𝐸_𝐵(𝜈,𝑠) is the Erlang B-formula, that is, 𝐸_𝐵(𝜈,𝑠)=(𝜈^s/𝑠!)(∑^𝑠𝑖=0(𝜈^𝑖/𝑖!))⁻¹.

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 in [6] also introduced an approximation algorithm for solving this problem, which was based on the following heuristic considerations. Because the QoS metrics 𝑃_ℎ and 𝑃_𝑜 increase in proportion to the incoming traffic load 𝜈_ℎ and 𝜈_𝑜, a one-dimensional birth and death process is used as a research model. The different channel occupation times are replaced with their averages, and the traffic intensities are replaced by their loads; that is, the following approximations are made: 𝜇_𝑜=𝜇_ℎ=1, 𝜆_𝑜=𝜈_𝑜, 𝜆_ℎ=𝜈_ℎ. The authors in [6] proposed the following formulas for approximating the QoS metrics in (2.2): 𝑃𝑃_ℎ𝑜≈𝜌_𝑁,≈^𝑁_𝑖=𝑚𝜌_𝑖,(2.7) where 𝜌_𝑖=⎧⎪⎨⎪⎩𝜈𝑖!^𝑖⋅𝜌⁰𝜈,if𝑖≤𝑚,_𝑚⋅𝜈^{ℎ𝑖−𝑚}𝑖!⋅𝜌⁰𝜌,if𝑚+1≤𝑖≤𝑁,₀=^𝑚_𝑖=0𝜈^𝑖𝑖!+^𝑁_𝑖=𝑚+1𝜈^𝑚⋅𝜈^{ℎ𝑖−𝑚}𝑖!⁻¹,𝜈=𝜈_𝑜+𝜈_ℎ.(2.8)

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).

Table 1 

Comparison of different algorithms for the case 𝑁=10, 𝜆_𝑜=1, 𝜆_ℎ=5, 𝜇_𝑜=1, 𝜇_ℎ=5.

Table 2 

Comparison of different algorithms for the case 𝑁=10, 𝜆_𝑜=5, 𝜆_ℎ=1, 𝜇_𝑜=5, 𝜇_ℎ=1.

Table 3 

Comparison of different algorithms for the case 𝑁=10, 𝜆_𝑜=𝜆_ℎ=5, 𝜇_𝑜=𝜇_ℎ=5.

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].

The stationary distribution of state probabilities is defined by solving the appropriate SGEE. The solution of an SGEE again presents problems with exact calculation for high-dimensional state spaces (2.1). To overcome this problem, the authors in [6] applied the above-mentioned heuristic assumption and proposed the following algorithm for the approximate calculation of QoS metrics:𝑃𝑃_ℎ0≈𝜌_𝑁,≈^𝑁_𝑖=01−𝛽_𝑖𝜌_𝑖,𝛽_𝑁=0,(3.1) where𝜌_𝑖=1𝑖!⋅^𝑖−1_𝑗=0𝛽_𝑗𝜌₀𝜈_𝑜+𝜈_ℎ⋅𝜌₀,=^𝑁_𝑖=01𝑖!⋅^𝑖−1_𝑗=0𝛽_𝑗𝜈_𝑜+𝜈_ℎ⁻¹.(3.2)

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.

In the FGC scheme, as in the CGC scheme, the cell state at any time is described by a bidimensional vector 𝐤=(𝑘_𝑜,𝑘_ℎ), where 𝑘_𝑜(𝑘_ℎ) denotes the number of n-calls (h-calls) occupying channels. Additionally, the state space of the model is defined by (2.1). The intensities 𝑞(𝐤,𝐤^), 𝐤, 𝐤′∈𝑆 of transitions between states of this two-dimensional MC are defined as follows:𝑞(𝐤,𝐤′)=⎧⎪⎪⎨⎪⎪⎩𝜆^𝑜𝜆_ℎ𝛽_𝑖,if𝑘_𝑜+𝑘_ℎ=𝑖,𝐤^=𝐤+𝐞₁,𝑘_𝑜,if𝐤^=𝐤+𝐞_𝟐,𝑘_ℎ𝜇_ℎ,if𝐤^=𝐤−𝐞₁,0𝜇_ℎ,oifth𝐤e^r=wi𝐤se−,𝐞₂,(3.3) where 𝐞₁=(1,0), 𝐞_𝟐=(0,1).

Because the loss probability of an h-call and the blocking probability of an o-call are determined by the number of busy channels (see Formulas (2.2)), consider the following partition of the state space (2.1):𝑆=^𝑁_𝑖=0𝑆_𝑖,𝑆_𝑖𝑆^𝑗=∅,𝑖≠𝑗,(3.4) where 𝑆={𝐤∈𝑆∶𝑘_𝑜+𝑘_ℎ=𝑖}.

Let us denote the probability of the merged state 𝑆_𝑖 by 𝜋(𝑖), that is, 𝜋(𝑖)∶=_{𝐤∈𝑆𝑖}𝑝(𝐤).(3.5)

Thus,^𝑁_𝑖=0𝜋(𝑖)=1.(3.6)

Then, the initial QoS metrics from (2.2) are defined via the stationary probabilities of the merged states as follows:𝑃𝑃_ℎ=𝜋(𝑁),_𝑜=^𝑁_𝑖=01−𝛽_𝑖𝜋(𝑖).(3.7)

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.

Proposition 3.1. If the system satisfies the local balance condition, then the merged state probabilities are defined as follows: 𝜋(𝑗)=1𝑗!^𝑗−1_𝑖=0𝜈_𝑜𝛽_𝑖+𝜈_ℎ𝜋(0),𝑗=1,…,𝑁,(3.8) where 𝜋(0)=(∑^𝑁𝑗=0(1/𝑗!)∏^{𝑗−1𝑖=0}(𝜈_𝑜𝛽_𝑖+𝜈_ℎ))⁻¹.

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.

By applying these formulas to the special cases, we obtain exact formulas for the QoS metrics (2.2) when using the CAC based on both the CGC scheme and the UFGC scheme. Thus, for the CGC scheme we have that 𝑃_ℎ=𝜋(0)⋅𝜈𝜈_ℎ^𝑚⋅𝜈^𝑁ℎ𝑃𝑁!_𝑜,=𝜋(0)⋅𝜈𝜈_ℎ^𝑚⋅^𝑁_𝑖=_𝑚+1𝜈^𝑖ℎ𝑖!,(3.9) where 𝜋(0)=^𝑚_𝑖=0𝜈^𝑖𝑖!+𝜈𝜈_ℎ^𝑚⋅^𝑁_𝑖=_𝑚+1𝜈^𝑖ℎ𝑖!⁻¹.(3.10) For the UFGC scheme, the following formulas are obtained: 𝑃_ℎ=𝐴^𝑁𝑃𝑁!⋅𝜋(0),_𝑜=1−𝛼,(3.11) where 𝐴∶=𝛼𝜈_𝑜+𝜈_ℎ, 𝜋(0)=(∑^𝑁𝑗=0(𝐴^𝑗/𝑗!))⁻¹.

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.

However, in real life, absolutely fair handling is not required, and so we may choose to introduce the concept of 𝜀-fair handling, which denotes handling at which 𝑃_𝑜−𝑃_ℎ<𝜀, for a given number 𝜀>0. Thus, the problem of finding 𝜀-fair handling can be defined mathematically as follows:𝑚^∗=argmin_𝑚𝑃_𝑜(𝑚)−𝑃_ℎ(𝑚)<𝜀.(4.1)

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.

Table 4 

Solutions for the problem (4.1).

Now let the QoS for different call types be measured by the limit values of the loss and blocking probabilities; that is, the limitations for their upper bounds are set as follows:𝑃𝑃_𝑜ℎ(𝑚)≤𝜀_𝑜,(𝑚)≤𝜀_ℎ,(4.2) where 𝜀_𝑜>0 and 𝜀_ℎ>0 are given parameters.

Then, the optimization problem is formulated as follows: find the extreme values of the parameter 𝑚 satisfying (4.2). Let us first consider the following problem:𝑚=argmin_𝑚𝑃_𝑜(𝑚)≤𝜀_𝑜,𝑃_ℎ(𝑚)≤𝜀_ℎ.(4.3)

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

Step 1. If 𝜀_𝑜<𝑃_𝑜(𝑁) or 𝜀_ℎ<𝑃_ℎ(1), then there is no solution.

Step 2. If 𝜀_𝑜>𝑃_𝑜(1) and 𝜀_ℎ>𝑃_ℎ(1), then 𝑚=1.

Step 3. If 𝑃_𝑜(𝑁)<𝜀_𝑜<𝑃_𝑜(1) and 𝑃_ℎ(1)<𝜀_ℎ<𝑃_𝑜(𝑁), then solve the following problem: 𝑚₁=argmin_𝑚𝑃_𝑜(𝑚)≤𝜀_𝑜(4.4).

Step 4. If 𝑃_ℎ(𝑚₁)>𝜀_ℎ, then there is no solution; otherwise the solution is given by 𝑚=𝑚₁.

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

A similar method is developed for solving the following problem:𝑚=argmax_𝑚𝑃_𝑜(𝑚)≤𝜀_𝑜,𝑃_ℎ(𝑚)≤𝜀_ℎ.(4.5)

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.

Table 5 

Solutions for the problem (4.3), (4.5) in the case of 𝑁=10.

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.

Appendix

To prove the above-mentioned proposition, we must first prove the following lemmas.

Lemma A.1. If the system satisfies the local balance condition, then the following equations hold: 𝜈_𝑜𝛽𝜈_𝑖−1𝜋(𝑖−1)=𝐸𝑘_𝑜∣𝑖𝜋(𝑖),(A.1)_ℎ𝜋(𝑖−1)=𝐸𝑘_ℎ∣𝑖𝜋(𝑖),𝑖=1,…,𝑁,(A.2) where E(⋅∣⋅) denotes conditional mathematical expectation.

Lemma A.2. If the system satisfies the local balance condition, then the following equations hold: 𝜈_𝑜𝛽_𝑖−1+𝜈_ℎ𝜋(𝑖−1)=𝑖𝜋(𝑖),𝑖=1,…,𝑁.(A.3)

Proof of Lemma A.1. We use a technique proposed in [11], and we begin with proving the first equation. From (3.3), we conclude that the set of local equilibrium equations (SLEE) for the states 𝐤∈𝑆_𝑖, 𝑖=1,…,𝑁 has the form 𝜈_𝑜𝛽_𝑖−1𝑝𝑘_𝑜−1,𝑘_ℎ𝐼𝑘_𝑜≥1=𝑘_𝑜𝑝𝑘_𝑜,𝑘_ℎ.(A.4) Summing both parts of (A.4) for all possible 𝐤∈𝑆_𝑖, we obtain 𝜈_𝑜𝛽_𝑖−1𝐤_∈𝑆𝑖𝑝𝑘_𝑜−1,𝑘_ℎ𝐼𝑘_𝑜≥1=_{𝐤∈𝑆𝑖}𝑘_𝑜𝑝𝑘_𝑜,𝑘_ℎ.(A.5) From (3.5), we conclude that _𝐤_∈𝑆𝑖𝑝𝑘_𝑜−1,𝑘_ℎ𝐼𝑘_𝑜≥1=𝜋(𝑖−1).(A.6) Let us rearrange the right hand side of (A.5) as follows: _𝐤_∈𝑆𝑖𝑘_𝑜𝑝(𝐤)=_{𝐤∈𝑆𝑖}𝑘_𝑜⋅𝑝(𝐤)𝜋(𝑖)⋅𝜋(𝑖).(A.7) From the definition of conditional probability, we obtain that 𝑃(𝐤∣𝑖)=𝑃𝐤∣𝑘_𝑜+𝑘_ℎ=𝑖=⎧⎪⎨⎪⎩𝑝𝜋((𝐤𝑖)),if𝐤∈𝑆^𝑖0,otherwise,.(A.8) Then, in light of (A.8), we deduce the following from (A.7): _𝐤_∈𝑆𝑖𝑘_𝑜𝑝(𝐤)=⎛⎜⎜⎝_{𝐤∈𝑆𝑖}𝑘_𝑜𝑃(𝐤∣𝑖)⎞⎟⎟⎠𝜋(𝑖)=𝐸𝑘_𝑜∣𝑖𝜋(𝑖).(A.9) Finally, using (A.6) and (A.9), we obtain 𝜈_𝑜𝛽_𝑖−1𝜋(𝑖−1)=𝐸𝑘_𝑜∣𝑖𝜋(𝑖).(A.10) The second equation is proved similarly.

Proof of Lemma A.2. Summing over the parts of the equation in Lemma A.1, we obtain 𝜈_𝑜𝛽_𝑖−1+𝜈_ℎ𝜋(𝑖−1)=𝐸𝑘_𝑜∣𝑖+𝐸𝑘_𝑘∣𝑖𝜋(𝑖)=𝐸𝑘_𝑜+𝑘_ℎ∣𝑖𝜋(𝑖)=𝑖𝜋(𝑖).(A.11) Thus, Lemma A.2 also follows.
Consequently, from Lemma A.2, taking into account the normalization condition in (3.6), we conclude that the proposition holds.
Now let us suppose that the system does not satisfy the local balance condition; one can use the following approach. Taking into account the relations in (3.3), we conclude that the SGEE for the states 𝐤∈𝑆_𝑖 has the following form:𝜆_𝑜𝛽_𝑖−1+𝜆_ℎ+𝑘_𝑜𝜇_𝑜+𝑘_ℎ𝜇_ℎ𝑝(𝐤)=+𝜆_𝑜𝛽𝑘_𝑖−2ℎ𝑝𝐤−𝐞₁+𝜆_ℎ𝑝𝐤−𝐞₂+𝑘_𝑜+1𝜇_𝑜𝑝𝐤+𝐞₁+1𝜇_ℎ𝑝𝐤+𝐞₂.(A.12)
For notational simplicity, we assume that the states 𝐤, 𝐤+𝐞_𝑖, 𝐤−𝐞_𝑖, 𝑖=1,2 appearing in (A.12) are in the state space (2.1); otherwise, the corresponding terms are zeroed. Summing both sides of (A.12) over all possible 𝐤∈𝑆_𝑖, after collecting similar terms and accounting for (3.5), one obtains𝜆_𝑜𝛽_𝑖−1+𝜆_ℎ𝜋(𝑖−1)=_{𝐤∈𝑆𝑖}𝑘_𝑜𝜇_𝑜+𝑘_ℎ𝜇_ℎ𝑝(𝐤).(A.13)
Now, various schemes may be used for the unification of channel occupancy times. If we use the heuristic from [6], that is, we set 𝜇_𝑜=𝜇_ℎ=1 and 𝜆_𝑜=𝜈_𝑜, 𝜆_ℎ=𝜈_ℎ then from (A.13), we obtain𝜈_𝑜𝛽_𝑖−1+𝜈_ℎ𝜋(𝑖−1)=_{𝐤∈𝑆𝑖}𝑘_𝑜+𝑘_ℎ𝑝(𝐤).(A.14)
By using the proof schemes from our lemmas, we conclude that right side of (A.14) is equal to 𝑖𝜋(𝑖); that is, we obtain equation from Lemma A.2.

Acknowledgment

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant no. 2010-0003269).