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 two-layer 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: take-back upward and downward vertical handoff,
where the take-back 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 Markov-modulated 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 Strategy-1: 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 Erlang-B formula calculates the blocking probability of WLAN with the traffic
and the number of channels
as
This result can be easily extended to Erlang-C 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 Strategy-2: 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 Erlang-B 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 Strategy-3: Upward and Downward Vertical Handoffs
In this subsection, we describe the performance analysis of strategy-3. In strategy-3, 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 Erlang-B 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 Erlang-B 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 Strategy-4: Take-Back Vertical Handoff
In this subsection, we describe the performance analysis of strategy-4. In strategy-4, we consider take-back 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 take-back traffic rates to the CDMA system and WLAN by
and
, respectively. The notations
and
denote the take-back 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 take-back 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 Erlang-B 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 take-back 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 Erlang-B 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.