Hbj Model

• ${\displaystyle p}$ is number of processors.
• ${\displaystyle n}$ is the problem size.
• ${\displaystyle m}$ is number of machine words in a packet sent over the network.
• ${\displaystyle \tau }$ is the latency, or time at which a processor takes to initiate a communication on a network.
• ${\displaystyle \sigma }$ is the bandwidth, or time per machine word at which a processor can inject or receive ${\displaystyle m}$ machine words from the network.
• ${\displaystyle T_{comp}}$ is the largest computation time expended on a processor.
• ${\displaystyle T_{comm}}$ is the time spent in communication on the network.

This model assumes that for any subset of ${\displaystyle q}$ processors, a block permutation among the ${\displaystyle q}$ processors takes ${\displaystyle (\tau +\sigma m)}$ time, where ${\displaystyle m}$ is the size of the largest block.

Complexities of common parallel algorithms contained in the MPI libraries:

• Point to point communication: ${\displaystyle O(\tau +\sigma m)}$ 
• Reduction :${\displaystyle O(log(p)(\tau +\sigma m))}$ 
• Broadcast: ${\displaystyle O(log(p)(\tau +\sigma m))}$ 
• Parallel prefix: ${\displaystyle O(log(p){n \over p}(\tau +\sigma m))}$ 
• All to all: ${\displaystyle O(p(\tau +\sigma m)))}$