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Efficient Structured Policies for Admission Control in Heterogeneous Wireless Networks

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Abstract

In the near future, demand for heterogeneous wireless networking (HWN) is expected to increase. QoS provisioning in these networks is a challenging issue considering the diversity in wireless networking technologies and the existence of mobile users with different communication requirements. In HWNs with their increased complexity, “the curse of dimensionality” problem makes it impractical to directly apply the decision theoretic optimal control methods that are previously used in homogeneous wireless networks to achieve desired QoS levels. In this paper, optimal call admission control policies for HWNs are considered. A decision theoretic framework for the problem is derived by a dynamic programming formulation. We prove that for a two-tier wireless network architecture, the optimal policy has a two-dimensional threshold structure. Further, this structural result is used to design two computationally efficient algorithms, Structured Value Iteration and Structured Update Value Iteration. These algorithms can be used to determine the optimal policy in terms of thresholds. Although the first one is closer in its operation to the conventional Value Iteration algorithm, the second one has a significantly lower complexity. Extensive numerical observations suggest that, for all practical parameter sets, the algorithms always converge to the overall optimal policy. Further, the numerical results show that the proposed algorithms are efficient in terms of time-complexity and in achieving the optimal performance.

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Correspondence to Amin Farbod.

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This work was supported in part by a grant from LG Electronics and by Bell Canada through its Bell University Laboratories R&D program.

Appendix

Appendix

Proof of Lemma 1 V k + 1(i,j ) as shown in Eq. 2 consists of nine terms, T 1 ...T 9. Here for clarity, we factor out the common 1/v max term and label the costs by their class number. We define

$$ D_{m}(i,\!j\,)\kern16pt = T_{m}(i,\!j\,)-T_{m}(i-1,\!j\,) $$
(16)
$$ V_{k+1}(i,\!j\,)\kern11pt = \frac{1}{v_{\max}} \sum\limits_{m=1}^{9} T_{m}(i,\!j\,) \quad $$
(17)
$$ \Delta^i V_{k+1}(i,\!j\,) = \frac{1}{v_{\max}} \sum\limits_{m=1}^{9} D_{m}(i,\!j\,). $$
(18)

We use induction on k to prove V k + 1(i,j ) is convex and monotonically non-decreasing in i for every fixed j (proof for convexity in j is similar). This is equal to showing \(\Delta^i V_{k+1}(i+1,\!j\,) \geq \Delta^i V_{k+1}(i,\!j\,)\) and \(\Delta^i V_{k+1}(i,j\,) \geq 0\). The induction hypothesis is the following: V k (i,j ) is (I) monotonically non-decreasing in i and j and (II) convex in i (or j ) for every fixed j (or i ).□

Proof of (I) We show \(\Delta^i V_{k+1}(i,\!j\,) \geq 0\) by evaluating each D m term individually. The basis step is trivial since ∀ (i,j ) ≤ (C c ,C w ): V 0(i,j ) = 0 and ∀ j: V 0(C + 1,j ) = ∞. Consider D 1(i,j ), when λ c is factored out. Let

$$ \begin{array}{lll} &&{\kern-10pt} D'_{1}(i,\!j\,)\\ && = [(T_{1}(i,\!j\,) -T_{1}(i-1,\!j\,)]/\lambda_{c} \\ && = \min(V_{k}(i,\!j\,)+C_1,V_{k}(i+1,\!j\,)) \\ && \quad - \min(V_{k}(i-1,\!j\,)+C_1,V_{k}(i,\!j\,)) \\ && = \underbrace{\min(C_1,V_{k}(i+1,\!j\,)-V_{k}(i,\!j\,))}_{\geq 0 \quad \texttt{by hypo.\textit{(I)} for $V_{k}$}} + V_{k}(i,\!j\,) \\ && \quad - \min(C_1,V_{k}(i,\!j\,)-V_{k}(i-1,\!j\,))- V_{k}(i-1,\!j\,) \\ && \geq V_{k}(i,\!j\,)-V_{k}(i-1,\!j\,)\\ && \quad - \min(C_1,V_{k}(i,\!j\,)-V_{k}(i-1,\!j\,)) \\ && \geq 0 \quad \texttt{(by hypo.\textit{(I)} for $V_{k}$)}. \end{array} $$
(19)

The same method can be applied directly to T {2,3,5,7}. However, the rest of the terms have to be considered altogether. We first extend D 9(i,j ) as follows,

$$ \begin{array}{lll} &&{\kern-10pt} D_{9}(i,\!j\,)\\ && = T_{9}(i,\!j\,) -T_{9}(i-1,\!j\,) \\ && = (v_{\max}-{v_{\rm out}(i,\!j\,)})V_{k}(i,\!j\,) \\ && \quad {-}\: (v_{\max}-{v_{\rm out}(i-1,\!j\,)})V_{k}(i-1,\!j\,) \\ && = {(C_c- i)\eta_{\rm hcc}}V_{k}(i,\!j\,) - (C_c - i +1 )\eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ && \quad+ \: {(C_c- i)\eta_{\rm hcw}}V_{k}(i,\!j\,) - (C_c - i +1 )\eta_{\rm hcw}V_{k}(i-1,\!j\,) \\ && \quad+ \: {(C_w- j\,)\eta_{\rm hwc}}V_{k}(i,\!j\,) - (C_w - j\,)\eta_{\rm hwc}V_{k}(i-1,\!j\,) \\ && \quad+ \: (C_c- i)\mu_c V_{k}(i,\!j\,) - (C_c - i +1 )\mu_c V_{k}(i-1,\!j\,) \\ && \quad+ \: (C_w- j\,)\mu_w V_{k}(i,\!j\,) - (C_w - j )\mu_w V_{k}(i-1,\!j\,).\\ \end{array} $$
(20)

Although it is straight forward to show that the third and fifth terms above are non-negative, for other terms it is not trivial. We show that every of these terms in D 9(i,j ) if combined with other terms in \(\Delta^i V_{k}(i,\!j\,)\) can be proved to be non-negative. As an example, consider the first term which we can rewrite as

$$ \begin{array}{lll} D_9^{\eta_{\rm hcc}}&=&\! {(C_c- i)\eta_{\rm hcc}}V_{k}(i,\!j\,)\! - (C_c\! - i +1 )\eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ & =&\! {(C_c- i)\eta_{\rm hcc}}\underbrace{\left\{V_{k}(i,\!j\,)- V_{k}(i-1,\!j\,) \right\}}_{\Delta^i V_{k}(i,\!j\,) \geq 0}\\&& - \eta_{\rm hcc}V_{k}(i-1,\!j\,). \end{array} $$
(21)

Equation 21 contains η hcc, and from Eq. 2 we see that term T 8 has η hcc as well, so we will use D 8(i,j ). We consider the sum of them:

$$ \begin{array}{lll} &&{\kern-10pt} D_9^{\eta_{\rm hcc}} + D_{8}(i,\!j\,) \\ &&= {(C_c- i)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,)- V_{k}(i-1,\!j\,) \right\} - \eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ && \quad + \: i\eta_{\rm hcc}V_{k}(i-1,\!j\,) - (i-1)\eta_{\rm hcc}V_{k}(i-2,\!j\,) \\ && = {(C_c- i)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,)- V_{k}(i-1,\!j\,) \right\} \\ && \quad + \: {(i-1)\eta_{\rm hcc}}\left\{V_{k}(i-1,\!j\,)- V_{k}(i-2,\!j\,) \right\} \\ && = {(C_c- i)\eta_{\rm hcc}} \Delta^i V_{k}(i,\!j\,) + {(i-1)\eta_{\rm hcc}} \Delta^i V_{k}(i-1,\!j\,) \\ && \geq 0. \end{array} $$
(22)

The other two terms in D 9(i,j ) can be matched with T 4 and T 6 similarly. Hence, \(\Delta^i V_{k+1}(i,\!j\,)= \sum_{m=1}^9 D_m \geq 0\). □

Proof of (II) We need to show \(\Delta^i V_{k+1}(i+1,\!j\,) \geq \Delta^i V_{k+1}(i,\!j\,)\). This is equal to \(\sum_{m=1}^{9} [ D_{m}(i+1,\!j\,)- D_{m}(i,\!j\,)] \geq 0\). Similar Similar to the last part, the basis step is trivial since ∀ (i,j ) ≤ (C c ,C w ): V 0(i,j ) = 0 and ∀ j: V 0(C + 1,j ) = ∞. Let us define Y m (i,j ) = D m (i,j ) − D m (i − 1,j ). Again, we prove terms are non-negative either individually or when combined with other terms. Let us start with Y 1, having factored out λ c :

$$ \begin{array}{lll} &&{\kern-6pt} Y'_{1}(i+1,\!j\,)\\ && = [D_{1}(i+1,\!j\,)- D_{1}(i,\!j\,)]/\lambda_{c} \\ && = \min(C_1,V_{k}(i+2,\!j\,)-V_{k}(i+1,\!j\,)) + V_{k}(i+1,\!j\,) \\ && \quad - \: \min(C_1,V_{k}(i+1,\!j\,)-V_{k}(i,\!j\,)) - V_{k}(i,\!j\,) \\ && \quad - \: \min(C_1,V_{k}(i+1,\!j\,)-V_{k}(i,\!j\,)) - V_{k}(i,\!j\,) \\ && \quad + \: \min(C_1,V_{k}(i,\!j\,)-V_{k}(i-1,\!j\,))+ V_{k}(i-1,\!j\,) \\ && = \underbrace{\min(C_1,\Delta^i V_{k}(i+2,\!j\,)) - \min(C_1,\Delta^i V_{k}(i+1,\!j\,))}_{\geq 0 \quad \texttt{by hypo.\textit{(II)} for $V_{k}$}} \\ && \quad + \: \Delta^i V_{k}(i+1,\!j\,)-\Delta^i V_{k}(i,\!j\,) \\ && \quad - \: \left [ \min(C_1,\Delta^i V_{k}(i+1,\!j\,)) - \min(C_1,\Delta^i V_{k}(i,\!j\,))\right ] \\ && \geq \Delta^i V_{k}(i+1,\!j\,)-\Delta^i V_{k}(i,\!j\,) \\ && \quad - \: \left [ \min(C_1,\Delta^i V_{k}(i+1,\!j\,)) - \min(C_1,\Delta^i V_{k}(i,\!j\,))\right ] \\ && \geq 0 \texttt{ (by hypo.\textit{(II)} for $V_{k}$)}. \end{array} $$
(23)

Again, the proof for Y {2,3,5,7} is the same as for Y 1. For other terms, we have to take an approach similar to the last part. Let us extend Y 9, focusing on the terms containing η hcc:

$$ \begin{array}{lll} &&{\kern-6pt} Y_{9}(i+1,\!j\,)\\ && = D_{9}(i+1,\!j\,) -D_{9}(i,\!j\,)\\ && = \big( v_{\max}-{v_{\rm out}(i+1,\!j\,)}\big) V_{k}(i+1,\!j\,)\\ && \quad {-}\: \big(v_{\max}-{v_{\rm out}(i,\!j\,)}\big) V_{k}(i,\!j\,)\\ && \quad{-}\: \big\{ \big(v_{\max}-{v_{\rm out}(i,\!j\,)}\big) V_{k}(i,\!j\,)\\ && \quad{\kern15pt} {-}\: \big(v_{\max}-{v_{\rm out}(i-1,\!j\,)}\big) V_{k}(i-1,\!j\,) \big\}\\ && = {(C_c- i-1)\eta_{\rm hcc}}V_{k}(i+1,\!j\,) - (C_c - i )\eta_{\rm hcc}V_{k}(i,\!j\,)\\ && \quad- \: {(C_c- i)\eta_{\rm hcc}}V_{k}(i,\!j\,) + (C_c - i +1 )\eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ && \quad+ \ldots. \end{array} $$
(24)

Separating the terms with η hcc, we obtain

$$ \begin{array}{lll} && {\kern-6pt} {Y_{9}^{\eta_{\rm hcc}}(i+1,\!j\,)} \\ && ={(C_c- i-1)\eta_{\rm hcc}}V_{k}(i+1,\!j\,) - (C_c - i )\eta_{\rm hcc}V_{k}(i,\!j\,) \\ && \quad- \: {(C_c- i)\eta_{\rm hcc}}V_{k}(i,\!j\,) + (C_c - i +1 )\eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ && = {(C_c- i-1)\eta_{\rm hcc}}\left\{V_{k}(i+1,\!j\,) - V_{k}(i,\!j\,)\right\} \\ && \quad - \: {(C_c- i-1)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,) - V_{k}(i-1,\!j\,)\right\} \\ && \quad - \: 2\:\eta_{\rm hcc}V_{k}(i,\!j\,) + 2\:\eta_{\rm hcc}V_{k}(i-1,\!j\,). \end{array} $$
(25)

This term has to be evaluated along with Y 8(i + 1,j ) in order to show that the sum of both terms is non-negative. For Y 8(i + 1,j ) we have

$$ \begin{array}{lll} &&{\kern-6pt} {Y_{8}(i+1,\!j\,)} \\ && = (i+1)\eta_{\rm hcc}V_{k}(i,\!j\,) - i\eta_{\rm hcc}V_{k}(i-1,\!j\,) \\ && \quad - \: i\eta_{\rm hcc}V_{k}(i-1,\!j\,) + (i-1)\eta_{\rm hcc}V_{k}(i-2,\!j\,) \\ && = {(i-1)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,) - V_{k}(i-1,\!j\,)\right\} \\ && \quad - \: {(i-1)\eta_{\rm hcc}}\left\{V_{k}(i-1,\!j\,) - V_{k}(i-2,\!j\,)\right\} \\ && \quad + \: 2\:\eta_{\rm hcc}V_{k}(i,\!j\,) - 2\:\eta_{\rm hcc}V_{k}(i-1,\!j\,), \end{array} $$
(26)

and the sum of these two terms is

$$ \begin{array}{lll} &&{\kern-6pt} {Y_{9}^{\eta_{\rm hcc}}(i+1,\!j\,)+ Y_{8}(i+1,\!j\,)} \\ && = {(C_c- i-1)\eta_{\rm hcc}}\left\{V_{k}(i+1,\!j\,) - V_{k}(i,\!j\,)\right\} \\ && \quad - \: {(C_c- i-1)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,) - V_{k}(i-1,\!j\,)\right\} \\ && \quad + \: {(i-1)\eta_{\rm hcc}}\left\{V_{k}(i,\!j\,) - V_{k}(i-1,\!j\,)\right\} \\ && \quad - \: {(i-1)\eta_{\rm hcc}}\left\{V_{k}(i-1,\!j\,) - V_{k}(i-2,\!j\,)\right\} \\ && = {(C_c- i-1)\eta_{\rm hcc}}\underbrace{\left\{ \Delta^i V_{k}(i+1,\!j\,) - \Delta^i V_{k+1} (i,\!j\,) \right\}}_{\geq 0 \quad \texttt{by hypo.\textit{(II)} for $V_{k}$}} \\ && \quad + \: {(i-1)\eta_{\rm hcc}} \underbrace{ \left\{ \Delta^iV_{k} (i,\!j\,) - \Delta^i V_{k-1}(i,\!j\,) \right\}}_{\geq 0 \quad \texttt{by hypo.\textit{(II)} for $V_{k}$}} \\ && \geq 0. \end{array} $$
(27)

It is similar to show non-negativity for the other terms of Y m . Since we have ∀ m: Y m  ≥ 0, the inequality \(\Delta^i V_{k+1}(i+1,\!j\,) \geq \Delta^i V_{k+1}(i,\!j\,)\) holds, and hence, the cost function is convex. □

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Farbod, A., Liang, B. Efficient Structured Policies for Admission Control in Heterogeneous Wireless Networks. Mobile Netw Appl 12, 309–323 (2007). https://doi.org/10.1007/s11036-008-0045-5

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