In this section, a TDMA scheme is applied for traffic scheduling. One key benefit of using TDMA is that it guarantees collision free transmissions. In fact, various TDMA scheduling schemes are actually used in a few wide area wireless mesh network testbeds and network standards such as WiMAX. Based on TDMA scheduling, we provide a framework of non-asymptotic throughput derivation for WMNs.
The WMN model indicates that all wireless mesh routers contend for the same wireless channel of capacity
in backbone communications, and all mesh routers and mesh clients contend for capacity
in local communications. Therefore, the throughput of the
th mesh client when
gateways are deployed, denoted by
, is generally constrained by both
and
. Since
and
are orthogonal,
can be obtained by computing the throughput constrained by
and the throughput constrained by
separately, that is,
Here
is defined as the throughput of the
th mesh client in backbone communications when there are
gateways in the WMN and
is defined as the throughput of the
th mesh client in local communications. Note that
is independent of
in the WMN model.
indicates that a feasible per-client throughput can be achieved by taking the smaller one of
and
.
Since
and
should be split for uplink and downlink communications, respectively, it is assumed that
and
are assigned to downlink communications, and
and
are assigned to uplink communications, where
and c
2
are some constants between 0 and 1. Generally, throughput of a mesh client should be obtained as the sum of uplink throughput and downlink throughput. Choosing the value of
and
requires knowledge on actual applications running on clients, which is beyond the objectives of this paper. It is assumed in the following of this paper that downlink traffic is dominant in the WMN. Therefore, most of
and
will be assigned to downlink communications and throughput is decided by downlink throughput, which is constrained by
and
. This is not an uncommon case in today's applications of WMNs, for instance, in the application of Internet access. We shall note that the methodology proposed in this section can actually be used to obtain throughput of WMNs when both uplink traffic and downlink traffic are present. However, with the above simplified model, we can focus on the illustration of the key ideas without being distracted by trivial discussions.
4.1. Throughput in Backbone Communications
Time slots in backbone communications are first assigned to gateways so that no gateways interfere with each other. The TDMA scheduling scheme on gateways is assumed to satisfy the following two conditions:
time slots are assigned to each gateway with full fairness;
under the condition of
, each gateway should have as much as possible time slots for successful transmissions. In Section 3.2, an algorithm to obtain the optimal sharing efficiency on all the gateways, denoted by
, is provided and a traffic scheduling scheme satisfying the above two conditions is also constructed. In the scheme, the
th gateway can be guaranteed to have a number of time slots, which is equal to the total number of all time slots times
. Hence, the
th gateway is guaranteed to have an aggregate throughput of
in backbone communications. By the TDMA scheme, interfering gateways share the same wireless channel while noninterfering gateways can transmit simultaneously.
In the next step, time slots of a gateway will be further split into small time slots to have the following two properties:
each mesh client associated with the specific gateway should have separate small time slots for "interference free" transmissions;
each of such mesh clients should achieve a common throughput in backbone communications, that is,
, if mesh clients
and
are associated with the same gateway. It is assumed that a mesh router
has
-connected mesh clients and it located
hops away from its associated gateway. The second property requires that
be assigned
small time slots if there are no simultaneous transmissions along the way from the gateway to
. Figure 5. shows that simultaneous transmissions can be scheduled, if
is more than SRD-hops away from its gateway. SRD is defined as Slot Reuse Distance, for instance,
in Figure 5. Therefore, the actual time slot that an
-connected mesh client needs to meet the second property, denoted by
, has the following relationship with
:
With the first property all mesh clients associated with a specific gateway require total
small time slots for "interference free" transmissions in backbone communications. Hence, the k th gateway can guarantee the following per-client throughput for all its associated mesh clients in backbone communications:
With the consideration that a mesh router may have more than one potentially associated gateways and it will use all these gateways by round robin for fairness, the mesh router will assign all its time slots equally to its associated gateways. Therefore, the per-client throughput on the k th gateway can be modified to
where
denotes the number of the associated gateways with the mesh router
.
Assuming that the i th mesh client is connected with the mesh router
, finally, the per-client throughput of the i th mesh client in backbone communications is the averaged throughput over all its associated gateways:
An example is illustrated in Figure 6 for the throughput computation in backbone communications. In the example, there are 5 mesh routers, 2 of which are also gateways, denoted by
and
. It is assumed that both gateways have 50% sharing efficiency and all the mesh routers have 10 mesh clients. In the model, the mesh routers
and
are associated with
and
, respectively;
is associated with both
and
and it uses both the gateways by round robin. Thus, we have
By (9), we obtain
Finally, by (10), we obtain that each of the 30 mesh clients associated with
,
, and
, respectively, can achieve throughout of
sin the backbone communications.
The TDMA traffic scheduling scheme actually guarantees the full fairness among mesh clients for each gateway. Note that farther mesh clients from gateways are reserved more time slots for transmission so that their throughput is not throttled by closer ones.
The per-client throughput in backbone communications will be compared with the per-client throughput in local communications to decide the per-client throughput in the WMN. Note that if a mesh client is connected directly to a gateway, its throughput is decided only by the per-client throughput in local communications.
4.2. Throughput in Local Communications
Separate time slots are first assigned to different mesh routers so that simultaneous transmissions can only be carried out in cells that have enough distance in between; that is, simultaneous transmissions can only exist in cells that are
cells apart, where CRF is defined as Cell Reuse Factor. Hence, in downlink communications, each mesh router can only have one slot every CRF time-slots, as depicted in Figure 7, here
.
The above slot is further split into separate small-slots. Being assigned a different small-slot, each mesh client is guaranteed to obtain successful reception from its associated mesh router. Therefore,
With the above TDMA scheme, all the mesh clients associated with the same mesh router will have the same throughput in local communications, that is,
, if clients
and
are associated with the same mesh router.
4.3. Feasible Throughput in WMN
Combining (6)–(13), a feasible non-asymptotic throughput of the i th mesh client in the WMN can be obtained as follows:
and here i th mesh client is assumed to be connected with the mesh router
. It is important to note that this non-asymptotic throughput estimation is more realistic than the asymptotic throughput that is estimated when the number of nodes approaches infinity.
When all mesh routers are chosen as gateways, that is,
, throughput of the i th mesh client is only constrained by local communications, that is,
. Therefore, an upper bound is obtained for the aggregate throughput:
where
, if
has at least one connected client;
, if
has no connected client. And an upper bound is also obtained for the worst-case per-client throughput:
The above upper bounds are independent of
. Actually they are the maximal values that
and
can achieve for any number of gateways.
It should be noted that the throughput computation method is applicable to any gateway placement algorithm; that is, as long as a gateway placement is given, the results derived in this section can be used to calculate the throughput of WMNs.