In this section, we describe handoff strategies and metrics that we use to quantify the performance. We consider a large geographical area covered by contiguous WLANs. WLAN constitutes the lower layer of the twolayer hierarchy. All the WLANs are overlaid by a large CDMA system. The overlaying CDMA system forms the upper cell layer. Each CDMA system is allocated traffic channels, and the number of channels allocated to the WLAN cell is . In the case of speech calls, the number of WLAN channels is the maximum number of users who can communicate with the access point (AP) while satisfying both the QoS and delay jitter conditions at the same time. All channels are shared among new calls and handoff calls. In our system, mobile stations (MSs) are traversing randomly the coverage area of WLAN and CDMA systems. We distinguish two classes of MSs: fast and slow MSs, respectively. We further assume that an MS does not change its speed during a call.
Figure 5 shows the traffic flows between different wireless networks with related parameters. In our system, we have classified them into four handoff strategies as follows:

(i)
strategy 1: no vertical handoff;

(ii)
strategy 2: only upward vertical handoff;

(iii)
strategy 3: upward and downward vertical handoff;

(iv)
strategy 4: takeback upward and downward vertical handoff,
where the takeback vertical handoff means that the vertical handoff traffics, which have been connected to the CDMA (or WLAN) as overflow, are taken back to a WLAN (or CDMA) of the appropriate layer as soon as the traffic channels become available. This capability has the effect that the number of MSs with different speeds is minimized in the considered cell layer. In general, the slow MS is connected to the WLAN according to the network selection algorithm. If no other AP is available, the slow MS first is connected to the CDMA cell. Next, if an AP becomes available, the slow MS is back to the WLAN. The four strategies enable the network to clear the handoff target cell depending on the user's mobility. The four strategies can be used to estimate the velocity threshold () for various handoff admission controls.
In this paper, all WLANs of the lower layer are treated equally to simplify the overflow. We present analytical results for the proposed system. As stated, our objective is to focus on simple and tractable mechanisms for which analytical results can give an insight into the handoff mechanism between different networks. According to the velocity threshold, all the mobile users are divided into two groups: slower moving users () and fast moving users (). In order to determine the optimal threshold velocity, which is one of the main goals of this study, a few assumptions related to mobility characteristics are made in the system model.
The assumptions we employ in the mobility models are taken from [22] as cells are circular with radius , mobiles are uniformly distributed in the system, mobiles making new calls in WLAN move along a straight line with a direction uniformly distributed between , and mobiles crossing cell boundary enter a neighbor cell with the incident angle which assumes the distribution: .
WLAN cells assume two types of new call traffics, represented by the call arrival rates and , respectively, and modeled by the Markovmodulated Poisson process (M/M/k/k, in voice traffic model) [23]. Let random variables and denote the straight mobile paths for new calls and handoff calls, respectively. With the assumption of unique WLAN cell size and the same speed for the MSs, WLAN cell boundary crossing rate per call (), provided that no handoff failure occurs [22], is . New calls are assumed to finish within the average call duration time, , or the call handoffs to an adjacent cell. The proportion of the channels returned by the handoff is [22]. In other words, the rate of channel release and that of the call completion due to handoff are and , respectively.
4.1. Handoff Strategy1: No Vertical Handoff
In this strategy, we consider the reference system in which each layer in the overlaid WLAN/CDMA network is kept completely independent. Slow mobile users are traversing only in the WLAN and fast mobile users are traversing in the CDMA system. Horizontal handoff is allowed but vertical handoff is not allowed in this strategy.
We denote the blocking probability of calls from the CDMA system and WLAN by and , respectively. The handoff traffic from slow and fast mobiles is denoted as follows. and are the rates of fast and slow mobile handoff traffic in a CDMA system, respectively. and are the rates of fast and slow mobile handoff traffic in a WLAN, respectively.
4.1.1. The New Call Blocking Probability
The Call Blocking Probability in WLAN
The total traffic rate into the WLAN due to a slow MS is computed as follows:
where the superscript denotes the slow MS. The subscript 1 is for WLAN. The subscripts and denote the new call and the handoff call, respectively.
The generation rate of the handoff traffic of a slow mobile station in a WLAN is given by
The offered load in a WLAN is . The ErlangB formula calculates the blocking probability of WLAN with the traffic and the number of channels as
This result can be easily extended to ErlangC or M/M/k/k queue models.
The Call Blocking Probability in CDMA System
The total traffic rate into the CDMA cellular system due to a fast MS is computed as follows
The generation rate of the handoff traffic of a fast mobile station in a CDMA system is given by
The offered load to a CDMA system is calculated as . Similar to the new call blocking probability of WLAN, the CDMA system's blocking probability can be expressed as
4.1.2. The Handoff Call Dropping Probability
The Handoff Call Dropping Probability in WLAN
Slow MS users are supposed to use WLAN channels. The probability of handoff call drop in WLAN can be calculated as follows. is defined in such a way that the th handoff request is successful but the th request is dropped:
where and . The variable describes the probability that the handoff fails due to the channel shortage, and is the probability of successful handoff.
The Handoff Call Dropping Probability in the CDMA System
Similar to the call dropping probability of WLAN, the probability of call dropping in CDMA systems can be calculated as follows:
The overall probability of either dropping or handoff failure can be expressed as follows:
where and are the fractions of slow and fast MSs, respectively.
4.2. Handoff Strategy2: Upward Vertical Handoff
The system in this strategy allows upward vertical handoff from the WLAN to the CDMA system. Only upward vertical handoff of new MS and handoff traffic for a slow MS to the CDMA system is allowed.
4.2.1. The New Call Blocking Probability
The New Call Blocking Probability in WLAN
The total traffic rate in WLAN due to a slow MS is the same as (4), where is the new call generation rate in WLAN due to a slow MS, and is the rate of handoff call in a WLAN of a slow MS. Notice also that the generation rate of the handoff traffic of a slow mobile station in a WLAN is the same as (5).
The offered load in a WLAN is . The ErlangB formula (6) calculates the blocking probability of WLAN with the traffic and the number of channels .
The New Call Blocking Probability in the CDMA System
The total traffic rate in the CDMA cellular system due to a fast MS assumes the same expression as in (7). The total traffic rate into a CDMA system due to a slow MS is given by
where denotes the number of WLANs in an overlay CDMA cellular system. The generation rate of the handoff traffic of a fast mobile station in a CDMA system assumes the same expression as in (8). The generation rate of the handoff traffic of a slow mobile station in a CDMA system is given by
The offered load to a CDMA system is calculated as . Finally, the blocking probability of the CDMA system can be expressed as in (9).
4.2.2. The Handoff Call Dropping Probability
The Handoff Call Dropping Probability in the WLAN
The probability of handoff call drop in the WLAN can be calculated as follows:
The notation denotes the probability that a slow MS fails to be handed over to a near WLAN, and to be handed over to the overlaying CDMA system. The notation denotes the probability that a slow MS fails to be handed over to the CDMA system during a call.
The notation is defined in such a way that the th handoff request is successful but the th request is dropped:
is calculated as follows:
The Handoff Call Dropping Probability in the CDMA System
The probability of call dropping of a fast mobile station in the CDMA system is the same as (11). The overall probability of dropping is the same as (12).
4.3. Handoff Strategy3: Upward and Downward Vertical Handoffs
In this subsection, we describe the performance analysis of strategy3. In strategy3, we consider upward and downward vertical handoffs between WLAN and the CDMA system.
4.3.1. The New Call Blocking Probability
The New Call Blocking Probability in the WLAN
The total traffic rate into the WLAN due to a slow MS is the same as (4). The total traffic rate into the WLAN due to a fast MS is expressed as
The generation rate of the handoff traffic of a slow MS in a WLAN is the same as (5). The generation rate of the handoff traffic of a fast moving MS in a WLAN is characterized by
The parameter is the actual offered load to a WLAN from the new call arrival and the handoff call arrival. Invoking this important property, we can use as the offered load to WLAN. The ErlangB formula (6) can be used then to calculate the blocking probability with the traffic and the number of channels [22].
The New Call Blocking Probability in the CDMA System
The total traffic rate into the CDMA system due to a fast MS is the same as (7). The total traffic rate into the CDMA due to slow MS is expressed as (13). The total traffic rate into the CDMA system due to a fast MS is the same as (8). The generation rate of the handoff traffic of a fast MS in the CDMA system is calculated as
The generation rate of the handoff traffic of a slow MS in the CDMA system is computed as (14). The probability of call blocking is given by the ErlangB formula because it does not depend on the distribution of the session time. Invoking this important property, we can use as the offered load to the CDMA system, and the blocking probability can be expressed as in (9).
4.3.2. The Handoff Call Dropping Probability
The Handoff Call Dropping Probability in WLAN
Slow MSs are supposed to use WLAN channels. However, since the handoff to the CDMA system is also allowed, the probability of handoff call drop in WLAN can be calculated as follows. Let denote the probability that a slow MS fails to be handed over to a near WLAN. The probability of calls in a WLAN, , denotes the probability of failed upward vertical handoffs to the overlaying CDMA system due to channel shortages. Then the handoff call dropping probability can be expressed as (15).
The Handoff Call Dropping Probability in the CDMA System
The probability of call droppings of a fast mobile station in the CDMA system can be approximated by
The overall probability of dropping is the same as (12).
4.4. Handoff Strategy4: TakeBack Vertical Handoff
In this subsection, we describe the performance analysis of strategy4. In strategy4, we consider takeback vertical handoff between the WLAN and the CDMA system.
4.4.1. New Call Blocking Probability
New Call Blocking Probability in the WLAN
We denote the takeback traffic rates to the CDMA system and WLAN by and , respectively. The notations and denote the takeback probabilities from the CDMA system and the WLAN, respectively.
The total traffic rate into the WLAN due to a slow MS is computed as follows:
where the takeback traffic rate component is given by
The total traffic rate into the WLAN due to a fast MS is expressed as
The generation rate of the handoff traffic of a slow MS in a WLAN is given by
The generation rate of the handoff traffic of a fast moving MS in a WLAN is characterized by
The parameter is the actual offered load to a WLAN from the new call arrival and the handoff call arrival. Invoking this important property, we can use as the offered load to the WLAN. Notice that the ErlangB formula (6) calculates the blocking probability with the traffic and the number of channels .
The New Call Blocking Probability in the CDMA System
The total traffic rate into the CDMA system due to a fast MS is computed as follows:
Here, the takeback traffic rate component takes the expression
Thus the total traffic rate into the CDMA system due to a slow MS is given by
The generation rate of the handoff traffic of a fast MS in the CDMA system is
The generation rate of the handoff traffic of a slow MS in the CDMA system is computed as
The probability of call blocking is given by the ErlangB formula because it does not depend on the distribution of the session time. Invoking this important property, we can use as the offered load to the CDMA system, and the blocking probability can be expressed as in (9).
4.4.2. The Handoff Call Dropping Probability
The handoff call dropping probability in WLAN
Slow MSs are supposed to use WLAN channels. However, since handoff to the CDMA system is also allowed, the probability of handoff call drop in WLAN can be calculated as follows. The handoff call dropping probability is the same as (15).
The handoff call dropping probability in the CDMA system
The probability of call dropping probability of a fast mobile station in the CDMA system can be calculated as follows:
The overall probability of either dropping or handoff failure is given by (12).
4.5. The Number of Handoffs and Grade of Service
We will use the term handoff rate to refer to the mean number of handoffs per call. We use geometric models to predict handoff rates per call as the cell shapes and sizes are varied. Approximating the cell as a circle with radius and the speed of the mobile station with , the expected mean sojourn time in the call initiated cell and in an arbitrary cell can be found [22], and are given, respectively, by
A user will experience a handoff if he moves out of the radio coverage of the base station with which he/she currently communicates. The faster the user travel, probably the more handoffs he/she will experience. Using a result from renewal theory, the expected number of handoffs given the speed of the user can be found [22]:
Among many system performance measures, GoS is the most widely used. In fact, users complain much more for call droppings than for call blockings. GoS is evaluated using the prespecified weights and [22]:
where and represent the blocking and dropping probabilities of the involved systems, respectively. The weight emphasizes the dropping effect with its value in general larger than one half. In this paper, we use due to the fact that the dropping effect is more critical for calling users.