Tracking mobile users in cellular wireless networks involves two basic functions: location update and paging. Location update refers to the process of tracking the location of mobile users that are not in conversation. Three basic algorithms have been proposed in the literature, namely the distance-based, time-based, and movement-based algorithms. The problem of minimizing the location update and paging costs has been solved in the literature by considering exponentially distributed Cell Residence Times (CRT) and Inter-Call Time (ICT), which is the time interval between two consecutive phone calls. In this paper we have selected the movement-based scheme since it is effective and easy to implement. Applying the theory for the delayed renewal process, we find the distribution of the number of cell crossings when the ICT is a mixtuer of exponentially distributed r.v.u27s and the CRT comes from any distribution with Laplace-Stieltjes transform. In particular, we consider the case in which the first CRT may have a different distribution from the remaining CRTu27s, which includes the case of circular cells. We aim at the total cost minimization in this case.
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