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Joint spectrum load balancing and handoff management in cognitive radio networks: a non-cooperative game approach

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Abstract

We propose a non-cooperative game theory based algorithm for spectrum management problem in cognitive radio networks taking into account the spectrum handoff effects. The objective is to minimize the spectrum access time of Secondary Users (SUs) which are competing for spectrum opportunities in heterogeneous environment. In this paper, the preemptive resume priority (PRP) M/G/1 queuing model is used to characterize the multiple handoff and data delivery time of SUs. Also an explicit solution for channels selection probabilities of each SU is extracted for PRP M/M/1 model specifically. The effect of handoffs is considered as the interrupted packets which return to the SUs’ low priority queue when the high priority Primary User’s packets are arrived to take service. The queuing delay of SUs’ and the effect of these returned packets are considered in order to balance the load of SUs on channels so that the minimum spectrum access time is sensed by each SU. The non-cooperative spectrum load balancing with handoff management game is proposed to find a distributed solution for each SU. It is shown that this game has a unique Nash equilibrium point which can be achieved by SUs as decision makers. At this equilibrium, each SU incurs the minimum delay on all channels while the free spectrum holes of channels are utilized efficiently. Simulation results are provided to evaluate the performance of the proposed scheme in terms of spectrum access delay, fairness, and channels spectrum holes utilization.

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Authors and Affiliations

  1. Faculty of Computer Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Mina Fahimi & Abdorasoul Ghasemi

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  1. Mina Fahimi
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  2. Abdorasoul Ghasemi
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Correspondence to Abdorasoul Ghasemi.

Appendices

Appendix 1: Proof of proposition 1

If the strategy set of players is compact then a nonempty set of Nash equilibrium solution will exist [30]. In addition, the sufficient condition for the uniqueness of Nash equilibrium is that the cost function of each player is a strongly convex function in the strategy space. The domain of the problem which is given by (16) and (17) is a convex set. Using (28) and (29) we find that the derivation of the cost function in (10), exists and taking into account (18) its hessian is positive definite. Therefore, the cost function is strongly convex. In (29) and (30), we have:

$$\begin{aligned} I_{ji}&= {} E\left[ \left( X_{ji}^{(f)}\right) ^{2}\right] P_{ji}^{h}\lambda _j^{\left( SU \right) }+ E\left[ \left( X_{ji}^{(SU)}\right) ^{2}\right] \lambda _j^{\left( SU \right) } \\ B_{ji}&= {} E\left[ \left( X_i^{(PU)}\right) ^{2}\right] \lambda _i^{\left( PU \right) }+\sum _{k=1,k\ne j}^{N}\left( E\left[ \left( X_{ki}^{(f)}\right) ^{2}\right] P_{ki}^{h}s_{ki}\lambda _k^{\left( SU \right) }+ E\left[ \left( X_{ki}^{(SU)}\right) ^{2}\right] s_{ki}\lambda _k^{\left( SU \right) }\right) \\ C_{ji}&= {} E\left[ X_{ji}^{(f)}\right] P_{ji}^{h}\lambda _j^{\left( SU \right) }+ E\left[ X_{ji}^{(SU)}\right] \lambda _j^{\left( SU \right) } \\ D_{ji}&= {} E\left[ X_i^{(PU)}\right] \lambda _i^{\left( PU \right) }+\sum _{k=1,k\ne j}^{N}\left( E\left[ X_{ki}^{(f)}\right] P_{ki}^{h}s_{ki}\lambda _k^{\left( SU \right) }+ E\left[ X_{ki}^{(SU)}\right] s_{ki}\lambda _k^{\left( SU \right) }\right)\end{aligned}$$
$$\begin{aligned} \frac{\partial {c_j}}{\partial {s_{ji}}}&= {} \frac{1}{2\left( 1-E\left[ X_i^{\left( PU \right) }\right] \lambda _i^{\left( PU \right) }\right) }\left( \frac{I_{ji}s_{ji} +B_{ji}}{1-C_{ji}s_{ji} -D_{ji}} +\frac{\left( I_{ji}\left( 1-D_{ji}\right) + C_{ji}B_{ji}\right) s_{ji}}{\left( 1-C_{ji}s_{ji}-D_{ji}\right) ^2} \right) \\&\quad +\,E\left[ X_{ji}^{\left( SU \right) }\right] + \lambda _i^{\left( PU \right) }E\left[ X_{ji}^{\left( SU \right) }\right] \frac{E\left[ X_{i}^{\left( PU \right) }\right] }{1-\lambda _i^{\left( PU \right) }E\left[ X_{i}^{\left( PU \right) }\right] } \end{aligned}$$
(29)
$$\begin{aligned} \mathsf {H}\,&= {} \left( \begin{array}{llll} \frac{\partial ^2c_j}{\partial {s_{j1}^2}} &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ 0 &{}\quad \frac{\partial ^2c_j}{\partial {s_{j2}^2}} &{}\quad \ldots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad \frac{\partial ^2c_j}{\partial {s_{jM}^2}} \end{array}\right) \\&= {} \left( \begin{array}{llll} \frac{1}{1-E\left[ X_1^{\left( PU \right) }\right] \lambda _1^{\left( PU \right) }}\frac{\left( I_{j1} -I_{j1}D_{j1} +C_{j1}B_{j1}\right) \left( 1-D_{j1}\right) }{\left( 1-C_{j1}s_{j1} - D_{j1}\right) ^3} &{}\quad 0 &{}\quad \ldots &{}\quad 0 \\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad \frac{1}{1-E\left[ X_M^{\left( PU \right) }\right] \lambda _M^{\left( PU \right) }}\frac{\left( I_{jM} -I_{jM}D_{jM} +C_{jM}B_{jM}\right) \left( 1-D_{jM}\right) }{\left( 1-C_{jM}s_{jM} - D_{jM}\right) ^3} \end{array}\right) \end{aligned}$$
(30)

Appendix 2: Proof of proposition 2

In order to compute an explicit solution for (25), we use the logarithmic barrier method by applying KKT conditions [30, 31]. In optimization problem (25), there is one equality condition and three inequality conditions. Since \(\sum _{i =1}^{M}s_{ji}=1\) we can ignore the condition \(s_{ji} \le 1\). Let \(\alpha _j,\,\beta _{ji}\), and \(\eta _{ji}\) denote the Lagrange multipliers for constraints \(\sum _{i =1}^{M}s_{ji}=1,\,s_{ji} \ge 0\), and the last constraint in (25). The Lagrangian of (25) is given by:

$$\begin{aligned} L_j&= {} c_j^{m}\left( {\mathbf{s}}_j,\,{\mathbf{s}}_{-j}\right) + \alpha _j\sum _{i=1}^{M}\left( s_{ji} -1\right) + \sum _{i=1}^{M}\beta _{ji}\left( -s_{ji}\right) \\ &\quad +\,\sum _{i=1}^{M}\eta _{ji} \left( s_{ji}\lambda _{j}^{\left( SU \right) }\left( 1+P_{ji}^{h}\right) +\sum _{k=1,k\ne j}^{N}\left( s_{ki}\lambda _{k}^{\left( SU \right) }\left( 1+P_{ki}^{h}\right) \right) +\lambda _{i}^{\left( PU \right) }-\mu _i\right) \end{aligned}$$
(31)

In (37), \(\beta _{ji}(-s_{ji}) =0\) and \(\eta _{ji} \left( s_{ji}\lambda _{j}^{( SU )}(1+P_{ji}^{h})+\sum _{k=1,k\ne j}^{N}(s_{ki}\lambda _{k}^{( SU )}(1+P_{ki}^{h}) )+\lambda _{i}^{( PU )}-\mu _i\right) = 0, i=1,\ldots ,M.\)In order to find the optimum solution of this Lagrangian, we use Logarithmic barrier method which approximates the cost function without inequality conditions. In Logarithmic barrier method the cost function \(c_j^{m}({\mathbf{s}}_j,{\mathbf{s}}_{-j})\) can be written as:

$$\begin{aligned} c_j^{m}\left( {\mathbf{s}}_j,{\mathbf{s}}_{-j}\right) &\approx c_j^{m}\left( {\mathbf{s}}_j,{\mathbf{s}}_{-j}\right) + \sum _{i=1}^{M}\left( -\frac{1}{r}\right) \log (s_{ji}) \\&\quad +\,\sum _{i=1}^{M}\left( -\frac{1}{r}\right) \log \left( \mu _i-\left( \lambda _{i}^{( PU )} + s_{ji}\lambda _{j}^{( SU )}\left( 1+P_{ji}^{h}\right) +\sum _{k=1,k\ne j}^{N}\left( s_{ki}\lambda _{k}^{( SU )}\left( 1+P_{ki}^{h}\right) \right) \right) \right) \end{aligned}$$
(32)

and the Lagrangian is:

$$\begin{aligned} &L_j\approx c_j^{m}({\mathbf{s}}_j,{\mathbf{s}}_{-j})+ \sum _{i=1}^{M}\left( -\frac{1}{r}\right) \log (s_{ji}) \\&\quad +\,\sum _{i=1}^{M}\left( -\frac{1}{r}\right) \log \left( \mu _i-\left( \lambda _{i}^{( PU )} + s_{ji}\lambda _{j}^{( SU )}(1+P_{ji}^{h})+\sum _{k=1,k\ne j}^{N}\left( s_{ki}\lambda _{k}^{( SU )}(1+P_{ki}^{h})\right) \right) \right) \\&\quad +\,\alpha _j\sum _{i=1}^{M}(s_{ji}-1) \end{aligned}$$
(33)

If r is large enough, \(\beta _{ji}(-s_{ji}) = \frac{1}{r}\) and \(\eta _{ji} \left( s_{ji}\lambda _{j}^{( SU )}(1+P_{ji}^{h})\,+\sum _{k=1,k\ne j}^{N}(s_{ki}\lambda _{k}^{( SU )}(1+P_{ki}^{h}) )+\lambda _{i}^{( PU )}-\mu _i\right) =\frac{1}{r}.\) The derivation and hessian of this Lagrangian can be computed by (33) and (34):

$$\begin{aligned} G_{ji}&= {} \frac{\partial {c_j^{m}({\mathbf{s}}_j,{\mathbf{s}}_{-j})}}{\partial {s_{ji}}} -\frac{1}{r\,s_{ji}} \\&\quad -\,\frac{1}{r(\mu _i-(\lambda _{i}^{( PU )} + s_{ji}\lambda _{j}^{( SU )}(1+P_{ji}^{h})+\sum _{k=1,k\ne j}^{N}(s_{ki}\lambda _{k}^{( SU )}(1+P_{ki}^{h}) )))} \end{aligned}$$
(34)
$$\begin{aligned} H_{ji} &= {} \frac{\partial ^{2}{c_j^{m}({\mathbf{s}}_j,{\mathbf{s}}_{-j})}}{\partial {s_{ji}^2}} +\frac{1}{r\,(s_{ji})^2} \\ &\quad +\,\frac{(\lambda _{j}^{( SU )}(1+P_{ji}^{h}))^2}{r(\mu _i-(\lambda _{i}^{( PU )} + s_{ji}\lambda _{j}^{( SU )}(1+P_{ji}^{h})+\sum _{k=1,k\ne j}^{N}(s_{ki}\lambda _{k}^{( SU )}(1+P_{ki}^{h}) )))^2} \end{aligned}$$
(35)

By applying Newton step in logarithmic barrier method, we can compute the explicit solution for \(\alpha _j\) and \(s_{ji}\). The Newton step in logarithmic barrier method is given by [31]:

$$rH_{ji}\varDelta {s_{ji}} +\alpha _j= -rG_{ji}$$
(36)

By considering the condition (17), \(\sum _{i =1}^{M}\varDelta {s_{ji}} =0\) and by this condition the value of \(\alpha _j\) in each step can be computed by:

$$\sum _{i =1}^{M}\varDelta {s_{ji}} +\sum _{i =1}^{M}\frac{\alpha _j}{rH_{ji}} = \sum _{i =1}^{M}\frac{-rG_{ji}}{H_{ji}}$$
(37)
$$\alpha _j = r\,\frac{\sum _{i =1}^{M}\frac{-G_{ji}}{H_{ji}}}{\sum _{i =1}^{M}\frac{1}{H_{ji}}}$$
(38)

By computing the derivation, hessian and Lagrange multiplier we can reach to optimum solution of (25) iteratively by:

$$s_{ji}^{l+1} = s_{ji}^{l} + \kappa \left( \frac{-G_{ji}}{H_{ji}} - \frac{\alpha _j}{H_{ji}}\right)$$
(39)

In optimum solution, which is denoted by \(s_{ji}^{*}\), we have \(-\alpha _j = G_{ji}\). We can compute the explicit solution for \(s_{ji}^{*}\) in M/M/1 queuing model. For M/M/1, the value of \(G_{ji}\) is:

$$G_{ji}=\frac{2s_{ji}\mu _{ji}\lambda _j^{( SU )}(1+P_{ji}^{h}) -\left( s_{ji}\lambda _j^{( SU )}(1+P_{ji}^{h})\right) ^2+R_{ji}\mu _{ji}}{\left( \mu _i-\lambda _i^{( PU )}\right) (\mu _{ji} - s_{ji}\lambda _j^{( SU )}(1+P_{ji}^{h}))^2} +\, \frac{1}{\mu _i} +\, \frac{\lambda _i^{( PU )}}{\mu _i}\left( \frac{1}{\mu _i-\lambda _i^{( PU )}}\right)$$
(40)

By solving the \(-\alpha _j = G_{ji}\), we have:

$$\left( s_{ji}\lambda _j^{( SU )}\left(1+P_{ji}^{h}\right)\right) ^2(1+Q)-2\left( s_{ji} \mu _{ji}\lambda _j^{( SU )}\left(1+P_{ji}^{h}\right)\right) (1+Q)+\left(Q \mu _{ji}^2 - R_{ji}\mu _{ji}\right)=0$$
(41)

where \(Q = \left( -\alpha _j -\frac{1}{\mu _i}- \frac{\lambda _i^{( PU )}}{\mu _i} \left(\frac{1}{\mu _i-\lambda _i^{( PU )}}\right)\right) \left(\mu _i-\lambda _i^{( PU )}\right)\).

Therefore if there exists a feasible solution for the system the best strategy of \(SU _j\) for selecting channel \(F_i\) at the Nash equilibrium, \(s_{ji}^*\), is given by:

$$s_{ji}^* = \frac{\mu _{ji}(1+Q)+ \sqrt{\mu _{ji}^2 (1+Q)^2-(1+Q)\left( Q\mu _{ji}-R_{ji}\mu _{ji}\right) }}{1+Q}$$
(42)

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Fahimi, M., Ghasemi, A. Joint spectrum load balancing and handoff management in cognitive radio networks: a non-cooperative game approach. Wireless Netw 22, 1161–1180 (2016). https://doi.org/10.1007/s11276-015-1023-z

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