Efficient QoS aware two-layer service allocation in hybrid mobile cloud

Abstract

The paper proposes QoS based service allocation optimization in hybrid mobile cloud. The optimization formulation of the hybrid mobile cloud utility adopts a network utility maximization framework. The proposed hybrid cloud service allocation framework leads to a decomposition of the overall hybrid mobile cloud system problem into a separate problem for mobile user and one for the hybrid cloud provider. The mathematic solution of proposed model is concerned with the variables of local and public cloud. When the equilibrium prices of public cloud services and local cloud services are obtained, the system performance can be optimized. The QoS based two-layer service allocation optimization problem leads to two sub-algorithms. Interactions between the two sub-problems are through optimal variables for capacities of local or public cloud resources and service need. When the convergence points of the two sub-problems of public and local cloud service allocation are obtained, the optimal solution for hybrid mobile cloud utility is acquired. The system design issues of QoS based two-layer service allocation model are also given. The experiments study how data size, request arrival rate, number of cloud nodes have the effect on proposed algorithm and other compared algorithms.

Appendix

1.1 Proofs for Theorem 1

The formulation of mobile user optimization is rewritten as follows:

$$ \begin{aligned} & MaxU_{mu} (\mathop u\nolimits_{ij}^{mu} ) = Max\left\{ {\left( {B_{i} - \sum\limits_{i} {\mathop u\nolimits_{ij}^{mu} } } \right) + \left( {T_{i} - \sum\limits_{n = 1}^{N} {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu} }}} } \right) + \left( {E_{i} - \sum\limits_{n} {\frac{{\mathop {\tau q}\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu} }}} } \right)} \right\} \\ & s.t\,T_{i} \ge \sum\limits_{n = 1}^{N} {t_{i}^{n} } , \\ & \sum\limits_{n} {\mathop {en}\nolimits_{i}^{n} } \le E_{i} \\ \end{aligned} $$

The Lagrangian expression of $MaxU_{mu} (\mathop u\nolimits_{ij}^{mu} )$ can be written as follows.

$$ \begin{aligned} L_{mu} (\mathop u\nolimits_{ij}^{mu} ) & = \left( {E_{i} - \sum\limits_{n} {\frac{{\mathop {\tau q}\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu} }}} } \right) + \left( {B_{i} - \sum\limits_{i} {\mathop u\nolimits_{ij}^{mu} } } \right) \\ & \quad + \left( {T_{i} - \sum\limits_{n = 1}^{N} {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu} }}} } \right) + \varepsilon \left( {E_{i} - \sum\limits_{n = 1}^{N} {en_{i}^{n} } } \right) + \eta \left( {T_{i} - \sum\limits_{n = 1}^{N} {t_{i}^{n} } } \right) \\ \end{aligned} $$

where $\varepsilon ,\eta$ are the multipliers of Lagrangian expression. By using Karush–Kuhn–Tucker Theorem, the optimal mobile user’s payment is gotten, if ${\raise0.7ex\hbox{${L_{mu} (\mathop u\nolimits_{ij}^{mu} )}$} \!\mathord{\left/ {\vphantom {{L_{mu} (\mathop u\nolimits_{ij}^{mu} )} {\partial \mathop u\nolimits_{ij}^{mu} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial \mathop u\nolimits_{ij}^{mu} }$}} = 0$ for $\varepsilon ,\eta > 0$.

$$ \partial L_{mu} (\mathop u\nolimits_{ij}^{mu} )/\partial \mathop u\nolimits_{ij}^{mu} = - 1 + (1 + \varepsilon )\sum\limits_{n = 1}^{N} {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu2} }}} + (1 + \eta )\sum\limits_{n} {\frac{{\mathop {\tau q}\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} \mathop u\nolimits_{ij}^{mu2} }}} $$

Let ${\raise0.7ex\hbox{${L_{mu} (\mathop u\nolimits_{ij}^{mu} )}$} \!\mathord{\left/ {\vphantom {{L_{mu} (\mathop u\nolimits_{ij}^{mu} )} {\partial \mathop u\nolimits_{ij}^{mu} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\partial \mathop u\nolimits_{ij}^{mu} }$}} = 0$

$$ \mathop u\nolimits_{ij}^{mu} = \mathop {\left( {\frac{{(1 + \varepsilon + \eta \tau )\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} }}} \right)}\nolimits^{1/2} $$

According to the constraint equation in mobile user optimization, let $\varsigma = 1 + \varepsilon + \eta \tau$

$$ \mathop {(\varsigma )}\nolimits^{ - 1/2} = \frac{{T_{i} }}{{\sum\nolimits_{i = 1}^{N} {\mathop {\left( {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} }}} \right)}\nolimits^{1/2} } }} $$

We replace $\varsigma$ to obtain $\mathop {\mathop u\nolimits_{ij}^{mu} }\nolimits^{ * }$, which is optimal payment given by the mobile user.

$$ \mathop {\mathop u\nolimits_{ij}^{mu} }\nolimits^{ * } = \mathop {\left( {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} }}} \right)}\nolimits^{1/2} \frac{{\sum\limits_{i = 1}^{N} {\left( {\frac{{\mathop q\nolimits_{i}^{n} p_{j}^{lc} }}{{C_{j}^{lc} }}} \right)^{1/2} } }}{{T_{i} }} $$

1.2 Proofs for Theorem 2

The Lagrangian expression of $MaxU_{pc}$ is $L_{pc} (s_{jl}^{pc} ,\mathop s\nolimits_{jk}^{b} )$.

$$ \begin{aligned} L_{pc} (s_{jl}^{pc} ,\mathop s\nolimits_{jk}^{b} ) & = \sum\limits_{i = 1}^{N} {(v_{jk}^{b} \log \mathop s\nolimits_{jk}^{b} + v_{jl}^{lc} \log s_{jl}^{pc} } ) \\ & \quad + \lambda \left( {C_{l}^{pc} - \sum\limits_{j} {s_{jl}^{pc} } } \right) + \beta \left( {C_{k}^{b} - \sum\limits_{j} {\mathop s\nolimits_{jk}^{b} } } \right) \\ \end{aligned} $$

In above equation, $\lambda ,\beta$ are the Lagrangian constants.

Let $L_{pc} (s_{jl}^{pc} ,\mathop s\nolimits_{jk}^{b} )/s_{jl}^{pc} = 0$

$$ s_{jl}^{pc} = \frac{{v_{jl}^{lc} }}{\lambda } $$

According to the constraint equation $C_{l}^{pc} \ge \sum\limits_{j} {s_{jl}^{pc} }$, $\lambda$ is written as follows:

$$ \lambda = \frac{{\sum\nolimits_{j = 1}^{n} {v_{jl}^{lc} } }}{{C_{l}^{pc} }} $$

We replace $\lambda$ to obtain $s_{jl}^{pc*}$

$$ s_{jl}^{pc*} = \frac{{v_{jl}^{lc} C_{l}^{pc} }}{{\sum\nolimits_{j = 1}^{n} {v_{jl}^{lc} } }} $$

$s_{jl}^{pc*}$ is the public cloud service allocated to local cloud agent for maximizing the revenue.

The bandwidth allocation can be solved by the similar approach. $\mathop s\nolimits_{jk}^{b*}$ is optimal bandwidth allocated to local cloud agent.

$$ \mathop s\nolimits_{jk}^{b*} = \frac{{v_{jk}^{b} C_{k}^{b} }}{{\sum\nolimits_{j = 1}^{n} {v_{jk}^{b} } }} $$

1.3 Proofs for Theorem 3

$$ s_{jl}^{pc} = C_{l}^{pc} \frac{{v_{jl}^{lc} }}{{p_{l}^{pc} }} $$

$$ \mathop s\nolimits_{jk}^{b} = C_{k}^{b} \frac{{v_{jk}^{b} }}{{p_{k}^{b} }} $$

The utility of local cloud agent is rewritten as follows.

$$ \begin{aligned} U_{lc} \left( {v_{jl}^{lc} ,v_{jk}^{b} } \right) & = - \sum\limits_{j = 1}^{J} {(v_{jl}^{lc} + v_{jk}^{b} } ) \\ & \quad - K\left( {\sum\limits_{j} {\frac{{p_{l}^{pc} }}{{v_{jl}^{lc} C_{l}^{pc} }}} + \sum\limits_{j} {\frac{{p_{k}^{b} }}{{v_{jk}^{b} C_{k}^{b} }}} } \right) \\ \end{aligned} $$

The Lagrangian function for local cloud agent can be formulated as follows.

$$ \begin{aligned} L_{lc} \left( {v_{jl}^{lc} ,v_{jk}^{b} } \right) & = - \sum\limits_{j = 1}^{J} {(v_{jl}^{lc} + v_{jk}^{b} } ) \\ & \quad - K\left( {\sum\limits_{j} {\frac{{p_{l}^{pc} }}{{v_{jl}^{lc} C_{l}^{pc} }}} + \sum\limits_{j} {\frac{{p_{k}^{b} }}{{v_{jk}^{b} C_{k}^{b} }}} } \right) \quad + \vartheta \left( {D_{j} - \sum\limits_{l = 1}^{L} {(v_{jl}^{lc} + v_{jk}^{b} } )} \right) \\ \end{aligned} $$

where $\vartheta$ is the Lagrangian constant.

Let $\partial L_{lc} (v_{jl}^{lc} ,v_{jk}^{b} )/\partial v_{jl}^{lc} = 0$

$v_{jl}^{lc*}$ can be obtained

$$ v_{jl}^{lc*} = \mathop {\left( {\frac{{p_{l}^{pc} }}{{C_{l}^{pc} }}} \right)}\nolimits^{1/2} \frac{{\sum\limits_{l = 1}^{N} {\mathop {\left( {\frac{{p_{l}^{pc} }}{{C_{l}^{pc} }}} \right)}\nolimits^{1/2} } }}{{D_{i} }} $$

$v_{jl}^{lc*}$ is the optimal payment of the local cloud agent given to public cloud supplier.

By using the same approach, $v_{jk}^{b*}$ is local cloud agent’s payment for bandwidth provider.

$$ v_{jk}^{b*} = \mathop {\left( {\frac{{p_{k}^{b} }}{{C_{k}^{b} }}} \right)}\nolimits^{1/2} \frac{{\sum\limits_{k = 1}^{N} {\left( {\frac{{p_{k}^{b} }}{{C_{k}^{b} }}} \right)^{1/2} } }}{{D_{i} }} $$