Abstract
In this paper, we propose an adaptive call admission algorithm based on learning automata. The proposed algorithm uses a learning automaton to specify the acceptance/rejection of incoming new calls. It is shown that the given adaptive algorithm converges to an equilibrium point which is also optimal for uniform fractional channel policy. To study the performance of the proposed call admission policy, the computer simulations are conducted. The simulation results show that the level of QoS is satisfied by the proposed algorithm and the performance of given algorithm is very close to the performance of uniform fractional guard channel policy which needs to know all parameters of input traffic. The simulation results also confirm the analysis of the steady-state behaviour.
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Appendix: Proof of Theorems and Lemmas
Appendix: Proof of Theorems and Lemmas
In this appendix, we give the proof of some lemmas and theorems given in this paper.
1.1 Proof of Lemma 1
Before we begin to prove the lemma, we introduce some definitions and notations. To count how many calls are arrived, we introduce concept of local time for each type of calls. The local time for each type of calls starts with 0 and incremented by 1 when a call of given type is arrived. Let us to define \(n^n\) and \(n^h\) as the local times for new and hand-off calls, respectively. Then, we define two sequences of random variables \(n^n_m\) (\(n^n_1 < n^n_2<\cdots \)) and \(n^h_m\) (\(n^h_1 < n^h_2 < \cdots \)), where \(n^n_m(n^h_m)\) is the global time when the \(m^{th}\) new (hand-off) call is arrived.
The proof for penalty probability of \(c_1(p)\) is trivial, because action \(\mathrm{ACCEPT}\) is penalized when all allocated channels are busy. Since the probability of all channels being busy is equal to \(P_C\), then \(c_1(p)\) is equal to \(P_C\). To find expression for \(c_2(p)\), we define \(X_n\) as the indicator of dropping of a hand-off call at the hand-off local time \(n\), where \(X_n=1\) if a hand-off call arrives at hand-off local time \(n^h=n\) and dropped and \(X_n=0\) if a hand-off call arrives at hand-off local time \(n^h=n\) and accepted. Since in interval \([n,n+1]\), it is possible that \(M \ge 0\) new calls to be accepted or \(N \ge 0\) calls to be completed, then the state of the Markov chain describing cell at hand-off local time \(n+1\) is independent of its state at the hand-off local time \(n\) when \(N+M > 0\). Although there is an exception \(N+M=0\), which we ignore in our analysis due to the violation of Markov chain properties. Therefore, \(X_1,X_2,\ldots ,X_n\) are independent identically distributed (i.i.d) random variables with the following first- and second-order statistics.
Using the central limit theorem \(\bar{X}_n=\hat{B}_h=\frac{1}{n}\sum _{k=0}^nX_k\) is a random variable with normal distribution \((\hat{B}_h \sim N(\mu _b,\sigma _b))\) with the following mean and variance [30].
Thus, the value of penalty probability of \(c_2(p)\) is equal to
which completes the proof of this lemma. \(\square \)
1.2 Proof of Lemma 2
The proofs for items one through three are trivial using Eq. (7) and we only give the proof of item 4. From Eq. (7) and when \(\rho < C\), we have
Using Eq. (7), we obtain
Increasing \(p_2\), decreases the probability of accepting new calls and hence the number of busy channels decreased. Therefore, the dropping probability of hand-off calls is decreased or \(c_2(p)=\mathrm{Prob} \left[ \hat{B}_h < p_h\right] \) is increased. Thus, we have
However, by choosing the proper value for parameters, condition \(\frac{\partial c_2(p)}{\partial p_2} > 0\) is also satisfied. From Eqs. (22) and (24), Eq. (9) is concluded, from Eqs. (22) and (25), Eq. (10) is concluded and from Eqs. (23) and (24), Eq. (11) is concluded. This completes the proof of this lemma. \(\square \)
1.3 Proof of Lemma 3
Consider \(f(p)\) at its two end points
Since \(f(p)\) is a continuous function of \(p_1\) and \(p_2\), there exists at least a \(p^*\) such that \(f(p^*)=0\). For proving the uniqueness of \(p^*\), the derivative of \(f(p)\) with respect to \(p_1\) is computed and then using Lemma 2, we obtain
Since the derivative of \(f(p)\) with respect to \(p_1\) is negative, \(f(p)\) is a strictly decreasing function of \(p_1\). Thus there exists one and only one point \(p^*\) for which function \(f(p)\) crosses the horizontal line and hence the lemma. \(\square \)
1.4 Proof of Lemma 4
Define \(p^2=p^Tp\) for vector \(p\). Let \(p=p_1\) and
Since \(w(p) < 0\) when \(p > p^*\) and \(w(p) > 0\) when \(p<p^*\), \(g(p)\) is positive and continuous in interval \([0,1]\). Hence, there exists a \(R > 0\) such that \(g(p) \ge R\). Thus, we have
for all probability \(p\), then computing
and taking expectation on both sides, cancelling \(\mathrm{E}\left[ p(n)-p^*\right] ^2\) and dividing by \(2a\), we obtain
or
Since, we have only bounded variables, \(\tilde{S} \left( p(n)\right) \) is also bounded; thus, there exists a \(K > 0\) such that \(\mathrm{E}\left[ \tilde{S} \left( p(n)\right) \right] \le K\). Hence, we obtain
Using this Eq. (28), we obtain
and hence the lemma.\(\square \)
1.5 Proof of Lemma 5
To prove Eq. (15), let us to define
Since \(w(.)\) is Lipshitz with bound \(\beta \), we have \(|w\left( p(n)\right) -w\left( p^*\right) | \le K |p(n) - p^*|,\) where \(K>0\) is a constant. Using this Eq. (29), we obtain
Let
where \(\lambda \in [0,1]\). It follows that
Since \(h'(.)\) is continuous, we have
Subtracting \(w'(x)(y-x)\) from both sides of the above equation, we obtain
Since \(w(.)\) is Lipschitz with bound \(\beta \), we obtain
Substituting \(y\) with \(p(n)\) and \(x\) with \(p^*\) in the above equation, we obtain
Using this Eqs. (29) and (30) and Lemma 4, we obtain
Multiplying both sides of the above equation by \(\sqrt{a}\), we obtain
or
which implies Eq. (15). To derive Eq. (16), let us to define
By subtracting \(\tilde{S}(p(n))\) from both sides of the above equation, we obtain
Since \(\tilde{S}(.)\) is Lipschitz, we have
Substituting this Eq. (30) into Eq. (35), we obtain
Using Lemma 4, we have \(\mathrm{E}\left[ p(n)-p^*\right] ^2 \le Ka \) and \(\mathrm{E}\left[ p(n)-p^*\right] \le K\sqrt{a} \). Thus, we obtain \(|\eta - \tilde{S}(p^*)| = o(a)\). Hence, as a consequence, we have \(\mathrm{E}|\eta - \tilde{S}(p^*)| \rightarrow 0\) as \(a \rightarrow 0\), which confirms Eq. (16). Equation (17) follows by observing that
Substituting Eq. (17) into the above equation, we obtain
where \(\xi a ^{3/2} \rightarrow 0\) as \(a \rightarrow 0\). This completes the proof of this lemma. \(\square \)
1.6 Proof of Theorem 2
Let \(h(u)=\mathrm{E}\left[ e^{iuz(n)}\right] \) be the characteristic function of \(z(n)\). Then using the third-order taylor’s expansion of \(e^{iu}\) for real \(u\), we obtain
where \(k \le 1/6\); thus
Cancelling \(h(u)\) and dividing by \(u\), results
Thus, using estimates of Lemma 5, we have
From Eqs. (15) and (17), it is evident that \(\mathrm{E}[|z(n)|] < \infty \) when \(a\) is small or
Dividing the above equation by \(aw'(p^*)\) and using fact \(w'(p^*)<0\), we obtain
where
as \(a \rightarrow 0\). Since \(h(0)=1\), it follows that
where \(\sigma ^2=\frac{\tilde{S}(p^*)}{2 \left| w'(p^*)\right| }\). But we have
as \(a \rightarrow 0\); thus
Then using the facts that each characteristic function determines the distribution uniquely and \(h(u)\) is characteristic function of \(N(0,\sigma ^2)\), thus we obtain
and hence the theorem. \(\square \)
1.7 Proof of Theorem 3
In the equilibrium state, the average penalty rates for both actions are equal or \(f_1(p^*)=f_2(p^*),\) which results \(c_1\pi ^*=c_2(1-\pi ^*)\). Thus we have
where \(\delta =\mathrm{Prob} \left[ \hat{B}_h < p_h\right] \). Thus average number of blocked new calls, \(\bar{N}_n\), is equal to
Computing derivative of \(\bar{N}_n\) with respect to \(\delta \) results
Thus \(\bar{N}_n\) is a strictly decreasing function of \(\delta \). Since the adaptive UFC algorithm gives the higher priority to the hand-off calls, it attempts to minimize the dropping probability of hand-off calls. Using this fact and Eq. (40), it is evident that \(\bar{N}_n\) is minimized which results in minimization of the blocking probability of new calls and hence the theorem.\(\square \)
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Beigy, H., Meybodi, M.R. A learning automata-based adaptive uniform fractional guard channel algorithm. J Supercomput 71, 871–893 (2015). https://doi.org/10.1007/s11227-014-1330-7
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DOI: https://doi.org/10.1007/s11227-014-1330-7