An efficient resource allocation for maximizing benefit of users and resource providers in ad hoc grid environment

Abstract

Ad hoc grids allow a group of individuals to accomplish a mission that involves computation and communication among the grid components, often without fixed structure. In an ad hoc grid, every node in the network can spontaneously arise as a resource consumer or a resource producer at any time when it needs a resource or it possesses an idle resource. At the same time, the node in ad hoc grid is often energy constrained. The paper proposes an efficient resource allocation scheme for grid computing marketplace where ad hoc grid users can buy usage of memory and CPU from grid resource providers. The ad hoc grid user agents purpose to obtain the optimized quality of service to accomplish their tasks on time with a given budget, and the goal of grid resource providers as profit-maximization. Combining perspectives of both ad hoc grid users and resource providers, the paper present ad hoc grid resource allocation algorithm to maximize the global utility of the ad hoc grid system which are beneficial for both grid users and grid resource providers. Simulations are conducted to compare the performance of the algorithms with related work.

Appendix

1.1 Proofs for theorem 1

We assume that each ad hoc grid user agent submits $\mathop{u}\nolimits_i^j$ to the CPU resource agent and $v_i^k$ to memory resource agent. Then, ${u_i} = [u_i^1......u_i^j]$ represents all payments of ad hoc grid user agents for jth CPU resource agent, ${v_i} = [v_i^1......v_i^k]$ represents all payments of ad hoc grid user agents for kth memory resource agent. Let ${m_i} = \sum\limits_j {u_i^j} + \sum\limits_k {v_i^k}$, m _i is the total payment of the ith ad hoc grid user agent. N ad hoc grid user agents compete for grid resources with finite capacity. The resource is allocated using a market mechanism, where the partitions depend on the relative payments sent by the ad hoc grid user agents. Let ${\text{p}}{{\text{x}}_{\text{j}}}$,${\text{p}}{{\text{y}}_{\text{k}}}$ denote the price of the resource unit of CPU resource agent j and memory resource agent k respectively. Let the pricing policy, $p{\text{x}} = (p{{\text{x}}_1},p{{\text{x}}_2}, \cdots, p{{\text{x}}_n})$, denote the set of CPU resource unit prices of all the CPU resource agents in the ad hoc grid, $p{\text{y}} = (p{{\text{y}}_1},p{{\text{y}}_2}, \cdots, p{{\text{y}}_k})$ is set of memory resource unit prices. The ith ad hoc grid user agent receives resources proportional to its payment relative to the sum of the grid resource agent’s revenue. Let $\mathop{x}\nolimits_i^j$, $\mathop{y}\nolimits_i^k$ be the fraction of resource units allocated to ad hoc grid user agent i by CPU resource agent j and memory resource agent k. The time taken by the ith ad hoc grid user agent to complete nth job is:

$$ \mathop{t}\nolimits_i^n = \frac{{{b_{{in}}}px{}_j}}{{{c_j}u_i^j}} + \frac{{{d_{{in}}}p{y_k}}}{{{s_k}v_i^k}} $$

Ad hoc grid user agent’s optimization can be reformulated as

$$ Max\left\{ {{U_{{user}}} = \phi_i^1\left( {{E_i} - \sum\limits_j {u_i^j} - \sum\limits_k {v_i^k} } \right) + \phi_i^2\left( {{T_i} - \sum\limits_{{n = 1}}^N {\frac{{{b_{{in}}}p{x_j}}}{{{c_j}u_i^j}} - } \sum\limits_{{n = 1}}^N {\frac{{{d_{{in}}}p{y_k}}}{{{s_k}v_i^k}}} - D} \right) + \phi_i^3\frac{g}{f}} \right\} $$

The Lagrangian for the ad hoc grid user agent’s utility is as follows.

$$ {L_{{user}}}\left( {u_i^j,v_i^k} \right) = \phi_i^1\left( {{E_i} - \sum\limits_j {u_i^j} - \sum\limits_k {v_i^k} } \right) + \phi_i^2\left( {{T_i} - \sum\limits_{{n = 1}}^N {\frac{{{b_{{in}}}p{x_j}}}{{{c_j}u_i^j}} - } \sum\limits_{{n = 1}}^N {\frac{{{d_{{in}}}p{y_k}}}{{{s_k}v_i^k}}} - D} \right) + \phi_i^3\frac{g}{f} + \lambda \left( {{T_i} - \sum\limits_{{i = 1}}^N {t_i^n} } \right) $$

Where λ is the Lagrangian constant. From Karush-Kuhn-Tucker Theorem we know that the optimal solution is given $\frac{{\partial {L_{\text{user}}}\left( {u,v} \right)}}{{\partial u}} = 0$ for λ>0.

$$ \frac{{\partial {L_{{user}}}\left( {\mathop{u}\nolimits_i^j, v_i^k} \right)}}{{\partial \mathop{u}\nolimits_i^j }} = - \phi_i^1 + \phi_i^2\frac{{{b_{{in}}}\mathop{{px}}\nolimits_j }}{{\mathop{{{c_j}\left( {\mathop{u}\nolimits_i^j } \right)}}\nolimits^2 }} + \lambda \frac{{{b_{{in}}}\mathop{{px}}\nolimits_j }}{{\mathop{{{c_j}\left( {\mathop{u}\nolimits_i^j } \right)}}\nolimits^2 }} $$

Let $\frac{{\partial {L_{\text{user}}}\left( {\mathop{u}\nolimits_i^j, v_i^k} \right)}}{{\partial \mathop{u}\nolimits_i^j }} = 0$ to obtain

$$ \mathop{u}\nolimits_i^j = \mathop{{\frac{{\left( {\phi_i^2 + \lambda } \right){b_{{in}}}\mathop{{p{\text{x}}}}\nolimits_j }}{{\phi_i^1{c_j}}}}}\nolimits^{{1/2}} $$

Using this result in the constraint equation, we can determine $\chi = \frac{{\phi_i^2 + \lambda }}{{\phi_i^1}}$ as

$$ \mathop{{\left( \chi \right)}}\nolimits^{{ - 1/2}} = \frac{{{T_i}}}{{\sum\limits_{{m = 1}}^N {\mathop{{\left( {\frac{{\mathop{{px}}\nolimits_m {b_{{im}}}}}{{{c_m}}}} \right)}}\nolimits^{{1/2}} } }} $$

We substitute χ to obtain $\mathop{{\mathop{u}\nolimits_i^j }}\nolimits^{ * }$

$$ \mathop{{\mathop{u}\nolimits_i^j }}\nolimits^{ * } = \mathop{{(\frac{{{b_{{in}}}\mathop{{p{\text{x}}}}\nolimits_j }}{{{c_j}}})}}\nolimits^{{1/2}} \frac{{\sum\limits_{{m = 1}}^N {\mathop{{(\frac{{{b_{{im}}}\mathop{{p{\text{x}}}}\nolimits_m }}{{{c_m}}})}}\nolimits^{{1/2}} } }}{{{T_i}}} $$

$\mathop{{\mathop{u}\nolimits_i^j }}\nolimits^{ * }$ is the unique optimal solution to the ad hoc grid user agent’s optimization problem. It means that ad hoc grid user agent want to pay$\mathop{{\mathop{u}\nolimits_i^j }}\nolimits^{ * }$ to CPU resource agent j for needed CPU resource under completion time constraint.

Using the similar method, let $\frac{{\partial L\left( {\mathop{u}\nolimits_i^j, v_i^k} \right)}}{{\partial \mathop{v}\nolimits_i^k }} = 0$

$$ \frac{{\partial L\left( {\mathop{u}\nolimits_i^j, v_i^k} \right)}}{{\partial \mathop{v}\nolimits_i^k }} = - \varphi_i^1 + \varphi_i^2\frac{{{d_{{in}}}\mathop{{py}}\nolimits_k }}{{\mathop{{{s_k}\left( {\mathop{v}\nolimits_i^k } \right)}}\nolimits^2 }} + \lambda \frac{{{d_{{in}}}\mathop{{py}}\nolimits_k }}{{\mathop{{{s_k}\left( {\mathop{v}\nolimits_i^k } \right)}}\nolimits^2 }} = 0 $$

We can get $\mathop{v}\nolimits_i^k = \mathop{{\left( {\frac{{\left( {\varphi_i^2 + \lambda } \right){d_{{in}}}\mathop{{py}}\nolimits_k }}{{\varphi_i^1{s_k}}}} \right)}}\nolimits^{{1/2}}$

Using this result in the constraint equation, we can determine $\rho = \frac{{\varphi_i^2 + \lambda }}{{\varphi_i^1}}$ as

$$ \mathop{{\left( \eta \right)}}\nolimits^{{ - 1/2}} = \frac{{{T_i}}}{{\sum\limits_{{m = 1}}^N {\mathop{{\left( {\frac{{\mathop{{py}}\nolimits_m {d_{{im}}}}}{{{s_m}}}} \right)}}\nolimits^{{1/2}} } }} $$

We obtain $\mathop{{\mathop{v}\nolimits_i^k }}\nolimits^{ * }$

$$ \mathop{{\mathop{v}\nolimits_i^k }}\nolimits^{ * } = \mathop{{\left( {\frac{{{d_{{im}}}\mathop{{py}}\nolimits_k }}{{{s_k}}}} \right)}}\nolimits^{{1/2}} \frac{{\sum\limits_{{m = 1}}^N {\mathop{{\left( {\frac{{{d_{{im}}}\mathop{{py}}\nolimits_m }}{{{s_m}}}} \right)}}\nolimits^{{1/2}} } }}{{{T_i}}} $$

It means that ad hoc grid user agent want to pay $\mathop{{\mathop{v}\nolimits_i^k }}\nolimits^{ * }$ to memory resource agent k for needed memory resource under completion time constraint.

1.2 Proofs for theorem 2

We take derivative and second derivative with respect to $\mathop{x}\nolimits_i$:

$$ {U_{{provider}}}\prime \left( {x_i^j} \right) = \frac{{u_i^j}}{{x_i^j}} + {w_j}\quad \,{U_{{provider}}}\prime \prime \left( {x_i^j} \right) = - \frac{{u_i^j}}{{x_i^{{j2}}}} $$

${U_{\text{provider}}}\prime \prime \left( {x_i^j} \right) < 0$ is negative due to $0 < {\text{x}}_i^{\text{j}}$. The extreme point is the unique value maximizing the utility of CPU resource agents. The Lagrangian for U _provider is ${L_{\text{provider}}}(x_i^j,y_i^k)$ .

$$ \begin{gathered} {L_{{provider}}}\left( {x_i^j,y_i^k} \right) = \sum {{w_j}\left( {{c_j} - \sum {x_i^j} } \right) + \sum {u_i^j\left( {\log x_i^j + 1} \right)} } + \sum {{\sigma_k}\left( {{s_k} - \sum {y_i^k} } \right) + \sum {v_i^k\left( {\log y_i^k + 1} \right)} } \\ + \xi \left( {{c_j} - \sum\limits_i {x_i^j} } \right) + \zeta \left( {{s_k} - \sum\limits_i {y_i^k} } \right) \\ \end{gathered} $$

Where ξ, ζ are the Lagrangian constants. From Karush-Kuhn-Tucker Theorem we know that the optimal solution is given $\frac{{\partial {L_{\text{provider}}}\left( {x_i^j,y_i^k} \right)}}{{\partial x_i^j}} = 0$ for ξ, ζ > 0.

Let $\frac{{\partial {L_{\text{provider}}}\left( {x_i^j,y_i^k} \right)}}{{\partial x_i^j}} = 0$ to obtain $x_i^j = \frac{{u_i^j}}{{\left( {{w_j} + \xi } \right)}}$

Using this result in the constraint Eq. ${{\text{c}}_{\text{j}}} \geqslant \sum\limits {{\text{x}}_i^{\text{j}}}$, we can determine θ = w_j + ξ as

$$ \theta = \frac{{\sum\limits_{{d = 1}}^n {u_i^d} }}{{{c_j}}} $$

We substitute θ into ${\text{x}}_i^{\text{j}}$ to obtain

$$ \mathop{{x_i^j}}\nolimits^{*} = \frac{{u_i^j{c_j}}}{{\sum\limits_{{d = 1}}^n {u_i^d} }} $$

$x_i^{{j*}}$ is the unique CPU allocation for maximizing the utility of CPU resource provider j.

Using the similar method, we can solve memory allocation optimization problem.

Let $\frac{{\partial {L_{{provider}}}\left( {x_i^j,y_i^k} \right)}}{{\partial y_i^k}} = 0$ to obtain $y_i^k = \frac{{v_i^k}}{{\left( {{\sigma_k} + \zeta } \right)}}$

Using this result in the constraint Eq. ${{\text{s}}_k} \geqslant \sum\limits {y_i^k}$, we can determine $\tau = {\sigma_{\text{k}}} + \zeta$ as

$$ \tau = \frac{{\sum\limits_{{k = 1}}^K {v_i^k} }}{{{s_k}}} $$

We substitute τ into $y_i^k$ to obtain $\mathop{{y_i^k}}\nolimits^{*} = \frac{{v_i^k{s_k}}}{{\sum\limits_{{k = 1}}^K {v_i^k} }}$

$y{_i^{k*}}$ is the unique optimal memory resource allocation for maximizing the utility of memory resource provider k.