A Comprehensive Analysis of Spectrum Handoff Under Different Distribution Models for Cognitive Radio Networks

Abstract

The static frequency allocation in wireless communication became a major concern for efficient spectrum utilization. Also due to spatio-temporal variation, some of the frequency channels are not utilized efficiently. The prologue of open spectrum and dynamic spectrum access (DSA) methodology provides the secondary (unlicensed) users, supported by cognitive radios (CRs) to opportunistically utilize the unused spectrum bands. When a primary user returns to the engaged spectrum band, secondary users should release it immediately through proper handing off to another spectrum band available to the network satisfying the quality of service (QoS) of both the network. Hence the focus of the new spectrum management policies is DSA technology based on CR. Spectrum mobility is considered to be the subsequent big challenge in CR technology. Spectrum mobility is associated with spectrum handoff which is directly associated with link maintenance and QoS. In this paper, we scrupulously investigate and analyze the probability of spectrum handoff under diverse realistic primary and secondary user traffic models. We have established a state of the art standard generalized form of probability of spectrum handoff without switching delay considering secondary user call duration and residual time of availability of spectrum holes as measurement metrics for diverse distribution functions designed for tele-traffic analysis. We thoroughly investigate different distribution models for the residual time under both zero switching delay and finite switching delay conditions. The switching delay (t _r ) between spectrum holes comprises of spectrum sensing time and transition delay for realizing all the related parameters. A comprehensive simulation results are presented to validate the generalized theory established in this paper.

Appendix: Derivation of Probability of ‘r’ Times Spectrum Handoff for Lognormal Distribution

The probability of zero spectrum handoff for lognormal distribution of residual time is calculated as:

$$ \begin{aligned} {\text{P}}_{0} & = {\text{Pr}}\{{\rm T}_{0} \le h\} =\mathop{\iint}\limits_{a < b} {\Phi (a)\Psi (b)dadb}=\int_{0}^{\infty } {\Psi (b)\int\limits_{0}^{b} {\Phi (a)dadb \,} } \\ & = \int_{0}^{\infty } {\Psi (b)\int\limits_{0}^{b} {\mu e^{ - \mu a} dadb} } =\int_{0}^{\infty } {\Psi (b)} (1 - e^{ - \mu b} )db \,\\ & = \int_{0}^{\infty } {\Psi (b)} db - \int\limits_{0}^{\infty } {\Psi (b)} \,e^{ - \mu b} db = 1- \int\limits_{0}^{\infty } {\frac{1}{{b\sqrt {2\pi \sigma^{2} } }}e^{{{\raise0.7ex\hbox{${ - (\log b - \beta )^{2} }$} \!\mathord{\left/ {\vphantom {{ - (\log b - \beta )^{2} } {2\sigma^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\sigma^{2} }$}}}} } e^{ - \mu b} db \\ \end{aligned} $$

Now

$$ \int\limits_{0}^{\infty } {\frac{1}{{b\sqrt {2\pi \sigma^{2} } }} \cdot e^{{{\raise0.7ex\hbox{${ - (\log b - \beta )^{2} }$} \!\mathord{\left/ {\vphantom {{ - (\log b - \beta )^{2} } {2\sigma^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\sigma^{2} }$}}}} } e^{ - \mu b} db $$

indicates the Laplace transform of lognormal distribution and which is approximated as [25],

$$ \begin{aligned} L = \int\limits_{0}^{\infty } {\frac{1}{{b\sqrt {2\pi \sigma^{2} } }} \cdot e^{{{\raise0.7ex\hbox{${ - (\log b - \beta )^{2} }$} \!\mathord{\left/ {\vphantom {{ - (\log b - \beta )^{2} } {2\sigma^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\sigma^{2} }$}}}} } e^{ - \mu b} db \hfill \\ =\left( {\frac{{\exp \left( { - \frac{{W^{2} \left( {\mu \sigma^{2} \exp \left( \beta \right)} \right){ + 2}W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)}}{{2\sigma^{2} }}} \right)}}{{\sqrt {1 + W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)} }}} \right) \hfill \\ \end{aligned} $$

where W(.) is the Lambert W function which is defined as the solution of the equation W(x) · exp(W(x)) = x, where exp(W(x)) is an exponential function and W(x) is any complex number.

Hence

$$ \begin{aligned} {\text{P}}_{0} & = 1- \int\limits_{0}^{\infty } {\frac{1}{{b\sqrt {2\pi \sigma^{2} } }} \cdot e^{{{\raise0.7ex\hbox{${ - (\log b - \beta )^{2} }$} \!\mathord{\left/ {\vphantom {{ - (\log b - \beta )^{2} } {2\sigma^{2} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${2\sigma^{2} }$}}}} } e^{ - \mu b} db \\ & = 1 - \left( {\frac{{\exp \left( { - \frac{{W^{2} \left( {\mu \sigma^{2} \exp \left( \beta \right)} \right){ + 2}W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)}}{{2\sigma^{2} }}} \right)}}{{\sqrt {1 + W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)} }}} \right) = 1 - L \\ \end{aligned} $$

Similarly,

$$ \begin{aligned} {\text{Pr}}\{{\rm H}_{1} < {\rm T}_{1}\} & = 1 - {\text{Pr}}\{h \,\ge {\text{ T}}_{1} \} = 1 - \mathop{\iint}\limits_{p < q} {\Phi (p)\Psi (q)dpdq} \\ & =\left( {\frac{{\exp \left( { - \frac{{W^{2} \left( {\mu \sigma^{2} \exp \left( \beta \right)} \right){ + 2}W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)}}{{2\sigma^{2} }}} \right)}}{{\sqrt {1 + W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)} }}} \right) = \,L \,\\ \end{aligned} $$

Also we can express,

$$ {\text{Pr}}\{{\rm H}_{r} \le {\rm T}_{r}\} = \left( {\frac{{\exp \left( { - \frac{{W^{2} \left( {\mu \sigma^{2} \exp \left( \beta \right)} \right){ + 2}W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)}}{{2\sigma^{2} }}} \right)}}{{\sqrt {1 + W\left( {\mu \sigma^{2} \exp \left( \beta \right)} \right)} }}} \right) = L \,$$

Therefore from Eq. (3), the generalized expression for probability of r handoff (r = 0, 1, 2,….) P _r is obtained as P _r  = (1 − 1 + L) (L)^r−1 (1 − L) = L ^r(1 − L) (Eq. 25).