Architecture of Computing Systems (Lecture Notes in Computer ...

106 K. Kloch et al.
constant. This is a very surprising result. Finally, in the third regime, the infection
ratio starts to grow very rapidly with R. The three regimes are analyzed
and described analytically in the following sections.
3.1 First Scaling Regime
In the first regime, we consider the case of very small radio ranges. Thus, agents
travel long distances between coming into contact with each other and the infected
and uninfected agents are typically spread uniformly across the board
(see Fig. 3(a)). In order to derive the R dependence of K, wenotethatthe
unimpeded infection rate K specifies the expected number of new infections per
time unit if no reset is involved. Hence, each collision of an infected agent with
another agent has a high probability to cause a new infection, since the other
agent is usually uninfected. By collision we mean that the distance between the
twoagentsbecomesassmallasR. To calculate the expected number of collisions
within one time step, we adapt the mean free path approach well-known from
the kinetic theory of gases for a two-dimensional setting, agents of different sizes,
and non-negligible radius of infected agents.
Consider an infected agent moving along a straight path with speed ¯v. During
the next time unit the agent’s radio range R will reach a previously untouched
area of size ¯v · 2R. All previously uninfected agents in this area will then be
infected. The mean number of agents in the area is given by its size times the
mean density of agents,
K =(¯v · 2R) N/L 2 . (3)
Note that ¯v is not equal to the infected agent’s velocity v in general, since both,
infected and uninfected agents, are moving. Since in the presented scenario also
the uninfected agents move randomly, ¯v has to be calculated as the expected
relative velocity of the uninfected agents with respect to the infected agent.
Therefore, we consider a reference system in which the infected agent is at
rest. Without loss of generality, we can assume that it is moving to the right
with velocity v1 =(v, 0) (the two components of the vector indicating the speeds
in x and y direction, respectively). In the considered moving reference system,
the speed of the uninfected agents is thus given by v(ϕ) =v2 − v1 = v (cos ϕ −
1, sin ϕ) where0� ϕ