**1.1 Previously related work**

In (Fang, 2001; Zeng et al. 2002), it is observed that, depending on the variance of CDT, the mean channel holding time for new calls (CHTn) can be greater than the mean channel holding time for handoff calls (CHTh). However, in these works, it is neither explained nor discussed the physical reasons for this observed behavior. This phenomenon (which is addressed in Section 3.1) and the lack of related published numerical results have motivated the present chapter.

Most of the previously published papers that have developed mathematical models for the performance analysis of mobile cellular systems considering general probability distribution for cell dwell time have either only presented numerical results for the Erlang (Wang & Fan, 2007; Fang et al., 1999; Rahman & Alfa, 2009; Kim & Choi, 2009) or Gamma distributions with shape parameter greater than one1 (Yeo & Jun, 2002; Fang, 2005), and/or only for the CHTh2 (Fang, 2001; Fang & Chlamtac, 1999), or have not presented numerical results at all (Fang, 2005; Alfa & Li, 2002; Soong & Barria, 2000). Thus, numerical results both for values of the coefficient of variation (CoV) of CDT greater than one and/or for the CHTn have been largely ignored. Exceptions of this are the papers (Orlik & Rappaport, 1998; Fang et al., b, 1997; Pattaramalai, et al., 2009).

On the other hand, probability distribution of CHT has been determined under the assumption of the staged distributions sum of hyper-exponentials, Erlang, and hyper-Erlang for the CDT (Orlik & Rappaport, 1998; Soong & Barria, 2000). However, to the best of the authors' knowledge, probability distribution of CHT in mobile cellular networks with neither hyper-exponential nor Coxian distributed CDT has been previously reported in the literature.

In this Chapter, the statistical relationships among residual cell dwell time (CDTr), CDT, and CHT for new and handoff calls are revisited and discussed. In particular, under the assumption that UST is exponentially distributed and CDT is phase-type distributed, a novel algebraic set of general equations that examine the relationships both between CDT and CDTr and between CDT and channel holding times are obtained. Also, the condition upon which the mean CHTn is greater than the mean CHTh is derived. Additionally, novel mathematical expressions for determining the parameters of the resulting CHT distribution as functions of the parameters of the CDT distribution are derived for hyper-exponentially or Coxian distributed CDT.

<sup>1</sup> For the Erlang distribution and for the Gamma distribution with shape parameter greater than one, the coefficient of variation of its associated random variable is smaller than one.

<sup>2</sup> Also referred as handoff call channel occupancy time.
