![](bg5.png)
delivered first in any h
x
. So there is a shorter path from S(h
y
) to h
x
than m
x
has taken.
That is a contradiction to the tree-structure and FIFO message exchange. ut
Proof (safety property, total order). Suppose messages m
x
∈ g
x
and m
y
∈ g
x
and
m
x
is delivered before m
y
in any MH h
x
. Suppose m
y
is delivered before m
x
in any
MH h
y
∈ H \ {h
x
}. That means there is a shorter path from the sequencer to h
y
in
case of delivering m
y
then in case of delivering m
x
. So we have a contradiction to the
tree-structure used and the FIFO message exchange. ut
Proof
(liveness property). Links between all network nodes support reliable message
exchange. There is only a bounded number of messages to be handled at each MSS
(unmarked multicast messages are spread away from sender and the unique sequencer
spreads these messages further away and back to sender). Thus, assuming each handling
takes a bounded time, each message is delivered after a finite amount of time. ut
4 Performance Evaluation
We have evaluated the performance of the main algorithm (total order using the de-
scribed approach) using a simulation on OMNeT++ [?]. The used network parameters
are shown in Table 1. We want to evaluate the performance of the network of MSSs, so
propagation delay and bandwidth of wireless links are neglected. We also use only one
MG. Our results describe the hypothetical maximum number of processable multicasts.
Using a real mobile network protocol, such as GPRS [?], there are technical limits:
Assuming 56 kbit/s downstream and a message size of 100 byte the maximum number
of multicasts per second without buffering is approx. 560. But with our hypothetical
number we obtain information about the behavior of the connecting network. We also
recognize limitations and can derive the behavior in complex cases, such as more MGs.
In the experiments we evaluate the influence of several parameters on the average
network traffic per multicast and processing time per multicast (time per multicast =
reciprocal value of the average number of processable multicasts). Here we plot time
per multicast. We assume an equal distribution of multicasts between MHs. Also an
equal distribution of MHs to MSSs and a symmetric network topology is assumed,
which is shown in Fig. 2 with the density parameter σ = 3 (number of subordinate
MSSs) and 4 levels of MSSs. In the following subsections we describe the evaluation of
the influence of the number of MSSs, number of MHs, and the location of sequencers.
Number of MSS.
When the number of MSSs is increased the average time to deliver a
multicast should decrease significantly. The network traffic per multicast should differ
only to a small extend because the main load is produced by the delivery of multicasts
to MHs. The influence on time and network load per multicast is shown in Fig. 3(a), (b).
In the experiments we choose 3 different settings. The sequencer is the central network
node, the density parameter is chosen with σ = 3. We vary the number of MSSs from 1
to 150. As expected the time per multicast decreases significantly (logarithmic scale in
Fig. 3(a)) if the number of MSSs grows. Using 150 MSSs the delivery time is approx.
67 to 136 times faster than using 1 MSS. Adding more MSSs means only a very low
increment of network traffic. The graph shows almost constant network load, because
delivery to the MHs is really more crucial than communication among the MSSs.
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