Convergence in the Calculation of the Handoff Arrival Rate: A Log-Time Iterative Algorithm

Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networking Volume 2006, Article ID 15876, Pages 1–11 DOI 10.1155/WCN/2006/15876 Convergence in the Calculation of the Handoff Arrival Rate: A Log-Time Iterative Algorithm Dilip Sarkar,1 Theodore Jewell,2 and S. Ramakrishnan3 1 Department of Computer Science, University of Miami, Coral Gables, FL 33124, USA 2 The Taft School, Watertown, CT 06795-2100, USA 3 Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA Received 23 March 2005; Revised 23 August 2005; Accepted 17 October 2005 Recommended for Publication by Vincent Lau Modeling to study the performance of wireless networks in recent years has produced sets of nonlinear equations with interrelated parameters. Because these nonlinear equations have no closed-form solution, the numerical values of the parameters are calculated by iterative algorithms. In a Markov chain model of a wireless cellular network, one commonly used expression for calculating the handoff arrival rate can lead to a sequence of oscillating iterative values that fail to converge. We present an algorithm that generates a monotonic sequence, and we prove that the monotonic sequence always converges. Lastly, we give a further algorithm that converges logarithmically, thereby permitting the handoff arrival rate to be calculated very quickly to any desired degree of accuracy. Copyright © 2006 Dilip Sarkar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION Wireless cellular networks provide service to mobile terminals, which can move from a given cell to any adjacent cell multiple times during the lifetime of a particular call. Therefore, a wireless network must take into account the rate at which ongoing calls arrive from neighboring cells, in addition to the arrival rate of new calls. When a user crosses the boundary from one cell to another, the network must react by handing off the call. However, there must be a channel available in the new cell for that call, or else the handoff fails and the service is abruptly terminated. One approach for improving the likelihood that a free channel is available when a handoff call arrives is the dedication of a certain number of channels in each cell purely for handoff calls. These dedicated channels are called guard channels, and earlier works have focused on the benefit of determining the number of guard channels dynamically. Models of cellular networks are very important for design as well as operation of the network. During the operation of a network, performance parameters can be estimated empirically by collecting data while the network is in operation. In fact, most of the current networks collect performance data and use it for decision making. However, if a network’s performance is outside the desired range, some of the control parameters will need adjustment. The amount of adjustment to be made is determined from a model. Simulation as well as analytical models are used for designing networks. Simulation models require a long computation time. However, in the absence of analytical models, simulation is the only available tool. Also, simulation models are necessary for the final evaluation of networks designed using approximate analytical models. For instance, even though call holding times and cell dwell times do not follow exponential distributions (see [1, 2]), analytical models assume that they are exponentially distributed. Therefore, a cellular network designed using such a model can be finetuned using simulation. On the other hand, analytical models are computationally efficient. One can estimate performance parameters very quickly. For instance, the (fuzzy associative memory) FAMbased call admission controller, reported in [3, 4], used a simulation model for development of the FAM. It took about two months of simulation time on a Pentium IV PC to develop the FAM. However, the FAMs for the call admission controllers reported in [5, 6] were developed using the algorithm presented in this paper requiring about a day on the same Pentium IV PC. Since the late eighties, the modeling of wireless cellular networks for analysis of their performance has produced sets 2 EURASIP Journal on Wireless Communications and Networking of nonlinear equations with interrelated parameters. These nonlinear equations have no closed-form solution, so the numerical values of the parameters are calculated by iterative algorithms. When these iterations fail to converge, however, the precise values of the parameters cannot be determined. The foregoing applies to wireless cellular networks, for which many studies have used Markov chains as models [7– 10]. Some of these models treat all calls identically, while others create a priority status for handoff calls. With respect to the prioritization of handoff calls, there are two basic approaches: (a) the early reservation of channels and (b) the use of guard channels that are dedicated exclusively to handoff calls [8–11]. The number of guard channels can be established in advance (statically) or as an ongoing process (dynamically) (see [12, Chapter 2]). In the former case, bandwidth may be underutilized or handoff call failure rate may be too high. In the latter case, there must be a continual computation of the optimal number of guard channels [7, 11]. This in turn creates a need for the computation of the handoff arrival rate. Note that although current handoff call arrival rate can be estimated from some “time averaging” of recent handoff call arrival records, the handoff call arrival rate that would result from the change of the number of guard channels must be determined from simulation or analytical model. Since simulations require a long time, analytical models are more desirable. The absence of a closed-form expression for the handoff arrival rate requires an alternate method, which commonly involves the use of iterative algorithms [7]. One standard formula for the calculation of the handoff arrival rate generates a sequence of approximations that may oscillate around the actual value. When this sequence converges, a result is obtained within any desired degree of accuracy. Convergence is not guaranteed, however, and in that instance the sequence develops a bifurcation and oscillates repeatedly between two values above and below the actual handoff rate value. In [13], fixed-point iteration for calculating the handoff arrival rate is proposed and used to overcome numerical overflow problems when a cell has a large number of channels. The paper also presents an algorithm for computing the optimal number of guard channels, but the optimization algorithm uses the proposed fixed-point iterative algorithm. The authors of that paper indicate that a proof of convergence of their algorithm is an open (...truncated)