A Probabilistic Protocol for Multihop Routing in VANETs

Algorithm 2

A practical variant of SteadyTraffic.

Algorithm SteadyTraffic2

Inputs:

 𝑇, 𝐻, 𝑀, 𝑅, 𝑃(𝑢₁,𝑢₂), 𝑠,𝑑∈𝑀: same as SteadyTraffic,

  𝜉(𝑥₁,𝑥₂): 1 hop packet error ratio from 𝑥₁∈𝑀 to 𝑥₂∈𝑀,

 𝑒(𝑥): expected car density at 𝑥∈𝑀,

 𝛽,𝜖∈(0,1): numeric parameters,

 𝑑𝑖𝑟∈{(0,−1),(0,1),(1,0),(−1,0)}: vector representing the moving

  direction.

Output:

  Approximate communication probability 𝜌(𝑠,𝑑,𝑇)

from 𝑠 to 𝑑 with time limit 𝑇.

begin

  (1). For every space time 𝑢=(𝑥,𝑡) do Answer(𝑥,𝑡,0)=0;

  (2). Answer(𝑠,0,0)=1;

  (3). for 𝑘=1 to 𝐻 do

  3-1. for every space time 𝑢=(𝑥,𝑡) do

   3-1-1. Answer(𝑥,𝑡,𝑘)=𝑃((𝑠,0),𝑢);

   3-1-2. for every other space time 𝑢₂=(𝑥₂,𝑡₂) and 𝑢₁=(𝑥₁,𝑡₂)

     such that

     i) 𝑡₂≤𝑡, ii) 𝑥₁∈𝐷(𝑥₂), iii) 𝑃(𝑢₁,𝑢)≥𝜖,

     iv) (𝑥₁−𝑥₂)⋅𝑑𝑖𝑟≥0 and iv) (𝑥−𝑥₁)⋅𝑑𝑖𝑟≥0

     /^∗ 𝐷(𝑥) is the set of 𝑅-neighbors of 𝑥∈𝑀 ^∗/

     /^∗ ⋅ represents the inner product of two vectors ^∗/

     do

    3-1-2-1. 𝑝₁=𝑃(𝑢₁,𝑢)(𝑒(𝑥₁)𝑒(𝑥))^𝛽;

    3-1-2-2. 𝑝₂=Answer(𝑥₂,𝑡₂,𝑘−1)(1−𝜉(𝑥₁,𝑥))𝑝₁;

    3-1-2-3. Answer(𝑥,𝑡,𝑘)=Answer(𝑥,𝑡,𝑘)(1−𝑝₂)+𝑝₂;

   3-1-3. end for

  3-2. end for

   (4). end for

   (5). return 1−∏_{𝑥∈𝐷(𝑑)𝑡≤𝑇}(1−Answer(𝑥,𝑡,𝐻)(1−𝜉(𝑥,𝑑)));

 end