Performance Modeling and Benchmarking of Event-Based ... - DVS

4.1. MODELING METHODOLOGY FOR EBS 39
where l = HC L(q) and r = HR C (q). Assuming that connections use dedicated network links,
the utilization Uq
NET of the network link corresponding to connection c q is:
|E|
∑
Uq NET = τqS t t,q NET
(4.9)
t=1
An approximation for the mean response time Rt,q
NET
can be computed according to:
Rt,q NET = SNET t,q
1 − U NET
q
of events of type e t at connection c q
(4.10)
If multiple connections are sharing a network link, the utilization of the network link due
to each of these connections must be taken into account when computing the mean response
times at the connections. Assume for example that connections c q1 , c q2 , ..., c qm all share a single
physical network link. The relative utilization of the link due to connection c qi is given by:
|E|
∑
Uq NET
i
= τq t i
St,q NET
i
(4.11)
t=1
The total utilization of the network link can be computed by summing up the relative
utilizations due to the connections that share it:
U NET
q 1,...,q m
=
m∑
i=1
U NET
q i
(4.12)
An approximation for the mean response time Rt,q NET
i
of events of type e t at connection c qi
can then be obtained by substituting Uq NET
1,...,q m
for Uq
NET in Eq. 4.10. Now that we have approximations
for the mean response times of events at the system nodes and network links, we
can use this information to derive an approximation for the mean event delivery latency. In
order to do that we need to capture the paths that events follow on their way from publishers
to subscribers.
Definition 6 (Delivery Path) A delivery path of an event is every ordered sequence of nodes
(n i1 , n i2 , ..., n im ) without repetitions that is followed by the event upon its delivery to a subscriber
(the event is published at node n i1 and delivered to a subscriber at node n im ).
Event delivery paths can be determined by monitoring the system during the experiments
conducted to measure the routing probabilities ν t,k
i,j (Section 4.1.2). Every delivery path can be
seen as a vector ⃗w = (n i1 , n i2 , ..., n im ) whose elements are system nodes.
Definition 7 (Dissemination Tree) The set W of all delivery paths of an event will be referred
to as the dissemination tree of the event.
Let W t,k be the union of the dissemination trees of all events of type e t published by publisher
p k . By definition, W t,k = ∅, if publisher p k does not publish any events of type e t . Let
W t be the union of the dissemination trees of all events of type e t irrespective of the publisher,
i.e., W t = ⋃ |P |
k=1 W t,k is the set of all delivery paths of events of type e t . Let Q(i, j) for
1 ≤ i < j ≤ |N| be the id of the connection between nodes n i and n j assuming that such a
connection exists.
Definition 8 (Mean Delivery Latency) If ˜W ⊆ W t , the mean delivery latency L t ( ˜W ) of
an event of type e t over the set of delivery paths ˜W is defined as the average time it takes to
deliver an event of type e t over a randomly chosen path from ˜W .