5.2.2 Modeling Mobility
In wireless networks, one needs to deal with the time variation of the topology due to mobility and the time variation of the channel due to interference, noise, environment, and other variables. The time variation of the channel imposes conditions on the establishment or breakage of wireless links, so that a physical layer issue has a relevant effect on the topology of networks such as those for reconfigurable devices. Diversity can help to overcome some of the problems presented at the physical layer due to the time variation of the channel, especially new diversity forms such as the cooperative kind where at some point there is always a device with a better channel response than other devices.
In the end, performance experienced by users is the relevant objective that needs to be achieved by any operation or management function in the network. As discussed in the previous section, mobility imposes some restrictions on performance, where from the point of view of traffic we see that blocking is increased as mobility increases since handoffs increase.
The model introduced in the previous section provides a simple way to evaluate mobility in a cellular network by varying the proportion of handoff calls being offered through the use of routing probabilities in Equation (5.2). The model has been extended to the case of several classes of traffic by considering the solution of Markov chains for each of the cells and capturing mobility effects through the traffic equations. This has been shown in Vargas-Rosales et al. , in which a net revenue equation is evaluated as
where vjk is calculated with Equation (5.2), λk is the new calls offered to cell k, wk is the revenue generated by accepting a new call in cell k, ck is the cost of dropping a handoff call in cell k, and Bk and Bhk are the new call and handoff blocking probabilities, respectively. In Figure 5.14, we can see the effects of mobility and reservation in net revenue. The reservation helps give priority to handoff calls, and as this increases, net revenue decreases.
FIGURE 5.14 Network net revenue for low mobility and reservation.
The model has also been considered in evaluating vertical handoffs in a wireless ATM network in Garca-Berumen and Vargas , where mobility also is an important aspect of capacity. An extension of the model to consider outage effects together with mobility is presented in Gallegos and Vargas . Among the advantages of the model using the Jackson network approach, we find the low computational complexity to be implemented, the easy way to modify mobility by varying the routing probabilities, and the mobility effects captured through the traffic equations in Equation (5.2). Disadvantages include the lack of clear sensibility to holding times, which turn out to be a fundamental part of the definition of offered traffic in erlangs, and the difference between cell residence times and call holding times, which are dependent upon the routing probabilities.
Examples of works that discuss at these times in cellular networks with mobility are Orlik and Rappaport  and Vargas-Rosales et al. . Both use a multidimensional Markov chain. The main difference is that in  call holding times are given by the sum of several exponential random variables for calls undergoing handoffs so that in general erlang-k distribution is obtained. In contrast, in  general distributions are used.
Vertical handoffs among several wireless providers have also been treated as an application of the simple model. See Mora-Zamorano and Vargas  for a model where resources are shared via random and sequential strategies, and Vargas-Rosales and Stevents  for the case in which resources are shared in an adaptive way by considering a state-dependent Markov chain, which helps to obtain better results than those in Miranda-Guardiola andVargas-Rosales  due to its adaptability to traffic overload. In this case, the least-loaded resource is the one chosen to accommodate users offered as handoffs to other networks or technologies.
The use of the least-loaded algorithm guarantees that idle infrastructure will be used and will generate revenue. In the aforementioned papers, a rate of return function is formulated that considers the carried traffic in the network, and a nonlinear optimization problem is solved where the maximum offered rates are obtained in terms of several degrees of mobility. Reservation is also used for handoff calls.
The model can be applied to amulticarrier system with the same mobility and offered traffic conditions. This could be the initial step toward modeling of the vertical handoffs that need to be performed in a 4G network. The main goal of the multicarrier system is understood as the organized integration of carriers to benefit from all idle resources. The use of reservation needs to be carefully evaluated to obtain the trade-off required. In Figure 5.15, we can see the network net revenue when vertical handoffs are allowed even for incoming new calls that find no free channel.
FIGURE 5.15 Network net revenue for (a) low mobility and (b) reservation.
We can see that the use of reservation, in both cases of mobility, benefits net revenue since it keeps increasing as offered traffic increases. In contrast, we can see how net revenue decreases as offered traffic increases when no reservation is used. We can also see that reservation needs to vary according to mobility levels; that is, it needs to increase as mobility increases.
When we want to change the scenario toward the reconfigurable networks, we need to consider different issues regarding mobility. Because we lack an infrastructure, the use of location areas and cells with paging and control messages has no application. A general scenario will need mobile nodes to act as beacons or references to the remaining nodes in the network. These reference nodes can be fixed or mobile.
The issues of modeling mobility are important since they play important roles in performance when introduced in more general models to predict or evaluate link establishment and maintenance that affect topology management. Some of these models can be simple, such as a square area where nodes move like billiard balls in the area, and when an edge is reached, two options are used - one where a complementary angle defines the new direction of movement and another where the node appears suddenly on the other side of the square region. In both cases, the nodes might be moving at constant speed with a direction determined randomly in [0, 2π]. Also, it can be more complicated by considering a speed varying randomly as well within an interval [0, vMAX]. Another way would be to have a discrete simulation where at each discrete event, nodes randomly select velocity, direction, and time in discrete slots through which they will travel at those values.
Simulations are important since it is easier to achieve better modeling of node movements with more realistic behavior as discrete events are executed. In general, one of the mobility models widely used is the random waypoint , which includes as a special case the one described as the billiard model in the previous paragraph.
The existence of a mean trip duration implies that the distribution of node mobility converges to a stationary distribution (see Le Boudec and Vojnovic  and Navidi and Camp ). This model is simple and tractable for implementation in simulations together with other aspects of networking. One of the disadvantages is that it does not capture realistic behavior of nodes, especially in contrasting scenarios such as indoor and outdoor areas, since mobility will have different degrees of freedom as scales change .
A generic framework that intends to present a mobility model that integrates heterogeneous wireless networks with vertical and horizontal handoffs is introduced in Zahran and Liang . The introduction of distributions of the phase type is also an important aspect of this framework. The model tries to integrate technologies such as 3G, WiMAX, and WiFi by carrying out vertical handoffs that are related to cell residence times for cellular technologies and zone residence times for other technologies. The use of phase (PH)-type distributions for residence times fitted to mobility traces is also discussed. The advantage in terms of analysis is that PH-type distributions have simple results when superpositioned, giving another PH-type distribution. Examples of PH-type distributions are the hypoexponential and hyperexponential distributions, and the Coxian random variable, which gives rise to the Coxian model .
Simulations for a network integrating 3G and WLAN technologies are presented in . Evaluation is in terms of cellular session utilization and the rate of vertical handoffs. These measures are also evaluated analytically to find good agreement with the simulations. The models used include the one based on the Coxian model  and a zone residence time determined by PH distributions. The mobility models just discussed where zone and cell residence times are considered can be classified as macroscopic mobility models since they describe large-scale user mobility. When one wants a better representation of mobility at small scales such as that encountered in reconfigurable networks, microscopic models are the best choice .
In , mobility modeling is carried out via an approach known as behavioral mobility (BM), where modeling concentrates on the realistic representation of mobility as a function of behavioral patterns. A discussion of models, such as the Markovian, which forms part of the macroscopic classification, is introduced, where applications of cellular networks are straightforward and the level of detail is usually low. Also discussed are models comprising a mixture of micro and macro that have applications to the representation of mobility in reconfigurable networks with medium granularity. These models include the random waypoint mobility model. An enhancement of that model is also discussed as an example of a microscopic mobility model with high accuracy and applications in reconfigurable networks. Pedestrian-level and group-mobility models are also introduced in Legendre et al. .