248 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 1, JANUARY 2004 Performance Analysis of Hierarchical Cellular Networks With Generally Distributed Call Holding Times and Dwell Times Marco Ajmone Marsan, Fellow, IEEE, Gabriele Ginella, Roberta Maglione, and Michela Meo, Member, IEEE Abstract—In this paper, we propose a new technique for the performance analysis of cellular mobile communication networks based on a frequency division multiple access/time division mul- tiple access scheme (such as global system for mobile communi- cations), in which the utilization of two separate frequency bands leads to a complex cellular structure with overlapping microcells and macrocells. Call durations and dwell times are described by random variables with general distributions; the call duration dis- tribution is then approximated by a two-phase hyper-exponential distribution with the same first and second moments as the orig- inal general distribution. The analysis technique is based on Mar- kovian assumptions as regards the traffic flows entering both mi- crocells and macrocells, as well as an assumption of flow balance between handovers into and out of any cell. The analytical model is validated against results of detailed simulation experiments for various system configurations, and shown to provide quite accu- rate predictions. Index Terms—Cellular radio networks, Markovian processes, performance evaluation, phase-type distributions. I. INTRODUCTION T HE global system for mobile communications (GSM) standard is based on the use of two separate frequency bands, around 900 MHz and 1.8 GHz, respectively. Cells served by frequencies in the 900-MHz band are rather large (up to 35 km around the base station), whereas cells served by frequencies in the 1.8-GHz band are much smaller (typically less than 1 km), due to the much worse propagation characteristics of microwaves in the latter frequency range through the atmosphere. For this reason, cells served by frequencies in the 900-MHz band are normally called “macrocells,” whereas cells served by frequen- cies in the 1.8 GHz band are often called “microcells.” The main advantage related to the use of microcells in a GSM network lies in an improved spatial reuse of frequencies, hence in substantial capacity increases, with the consequent possibility of offering, in addition to telephony, data services at medium-high rates (up to hundreds of kb/s), and even multimedia services through the integration of voice and data traffic flows. The design and planning of GSM networks require accurate models for the computation of the number of frequencies to be activated in cells (each frequency can support up to eight traffic Manuscript received April 10, 2001; revised July 17, 2002; accepted January 8, 2003. The editor coordinating the review of this paper and approving it for publication is K. K. Leung. This work was supported in part by the Italian Na- tional Research Council. The authors are with the Dipartimento di Elettronica, Politecnico di Torino, Italy (e-mail: ajmone@polito.it; michela@polito.it). Digital Object Identifier 10.1109/TWC.2003.821154 channels), so as to obtain acceptable performance. In particular, an accurate model must allow the realistic description of the call duration and of the time spent by a user in each microcell or macrocell. In addition to being accurate, a tool for the design and planning of GSM networks must also be efficient and flexible, so that different system configurations, traffic conditions, and user mobility patterns can be modeled. In this paper, we present an analytical approach which has the required characteristics of flexibility and simplicity while being extremely accurate in predicting performance measures. Markovian models have been traditionally used for the de- sign and planning of mobile cellular telephony networks, con- sidering one cell at a time (see, for example, [1]–[3]). While this approach proved adequate for networks comprising only macrocells, it cannot be directly transferred to the dual-band environment, where the minimum network element that has to be considered consists of one macrocell and all the microcells comprised within the macrocell. This network element will be called a “cell cluster.” This subsystem is rather complex for the direct development of Markovian models. For this reason, approximate models were proposed in the literature. Approxi- mations in many cases concentrate on the interaction between microcells and macrocells. In [4], the overflow traffic of users entering the macrocell from underlying microcells is modeled as an Interrupted Poisson Process (IPP). Instead, in [5] an IPP is used to model the overflow from each microcell; the aggre- gate overflow traffic from all microcells into the macrocell is described by the Markov Modulated Poisson Process (MMPP) which derives from the superposition of all IPPs. A similar ap- proach is adopted also in [6]; the aggregate overflow traffic is modeled by a MMPP which approximates the superposition of IPPs; the approximation is such that the two processes have the same values of the first three moments of the distribution of interarrival times. A different approach is proposed in [7] in which the overflow process is modeled by computing the av- erage residual call service time; the call duration is assumed to have negative exponential distribution while the distribution of the dwell time is taken to be general. From the point of view of the interaction between microcells and macrocells, the approach we adopt in this paper is similar to that in [8]–[10]; i.e., the overflow process is simply approximated by a Poisson process. The novelty of the approach proposed in this paper is the combination of this simple approximation (which is however discussed in detail) with the use of phase-type distribu- tions for call durations and general distributions for dwell times. The phase-type distributions of call durations are combined with 1536-1276/04$20.00 © 2004 IEEE