 |
The LogGP model is an extension of the LogP model and adds a parameter
to effectively capture long-message bandwidth [Alexandrov95].
|
|
According to this model, in a simple communication operations, sending a message from one processor to another, requires a time of L+2o. In the case of request/response, the time required to for a message to travel from one processor to another is 2L + 4o. This model and others like it (BSP, C3, etc.) may be used to measure the throughput of "thought" architectures from the human brain to supercomputers in space. For further details about these models, the papers listed in the References should be reviewed.
Variables used in the models L = The latency to communicate a small message from its source component to its target. o = The processor time overhead required to transmit or receive a message, during which the processor cannot perform other operations. g = The gap, or the minimum time interval between consecutive message transmission or consecutive message receptions at a processor. 1/g is the per processor communication bandwidth. P = The number of processors (processor/memory elements) in the network. G = The gap per byte for long messages
To provide the highest capacity, computer architecture engineers must maximize message bandwidth (for problems involving many small messages) or message capacity (for problems involving many large messages) or some balance between these two goals. The LogP and LogGP models provide insights into where the limits are in these architectures and what needs to be improved to increase capacity.
| Lat | = | Latency of the axon transmission of the electrical signal |
| Lnr | = | Latency of the signal transduction for neurotransmitter release (electrical to chemical) |
| Lnd | = | Latency of the neurotransmitter diffusion across the synaptic cleft |
| Lnu | = | Latency of the neurotransmitter uptake and signal transduction (chemical to electrical) |
| Ldt | = | Latency of the dendrite transmission of the electrical signal |
If one assumes that there is effective computation occuring at the synaptic cleft, then the LogP model would need to be modified to a greater degree to account for two types of processors, the synaptic chemical transmission processor and the neuronal electrical signal processor.
Current models for memory and neural network training suggest that the synaptic processors may play a role in the overall capabilities of the network on long time scales (minutes to days) but perform little computation on short time scales (seconds or less). They effectively function as capacitors in the electrical network, delaying the signal transmission from the axon to the dendrite. Interestingly, synapses of high strength (rapid or greater neurotransmitter release) would correspond to small capacitance capacitors (rapid charge times) in electrical networks, while synapses of low strength (slow or little neurotransmitter release) would correspond to large capacitance capacitors (slow charge times) in electrical networks.
In brains the dominant factor in signal propagation is the transmission time for the electrical signal in unmyelinated axons and is of the order of milliseconds. It is worth noting the myelination is used by nature to provide rapid signal propagation in sensory neurons. The same effect is achieved in some species by making the axons physically larger. Since neither of these approaches is used in the brain, one may suggest that the decreased signal propagation times would be offset by a physically larger brain that would be subject to one or more of: (a) diminished thought "density"; (b) heat removal constrants; (c) energy supply constraints. It seems doubtful that nature would not have tried brain neuron myelination at some point in the course of evolution (since it has the genes available to perform the function). Since it is not observed we may assume it was not a successful strategy for greater survivability. Further work exploring why this may be true would be useful.
In nanocomputers (Diamondoid, ~1 cm3, as proposed by Drexler in Nanosystems), the signals are most likely to be acoustic signals limited by the speed of sound in diamond. While light-speed signals are possible, the wavelengths of light are large and thus require large transmitters, receivers and physical carriers compared with the atomic-scale signal propagation envisioned in nanocomputers. Any decrease in signal propagation times provided by photonic signals may be offset by the increases in the physical size of the computer and the delays involved in converting between mechanical and optical energies (Os & Or in Figure 1). In actual practice, a combination of acoustic and optical networks might be used, where acoustic signals are used between relatively nearby processors (within microns to millimeters), while optical fibers surrounding the nanocomputer (or evacuated tunnels within the nanocomputer) could be used for communication with "distant" processors (on opposite cube faces or corners).
In Dyson Shell supercomputers of the Matrioska Brain type (MBrain) the adjacent nodes communicate optically using arrays of lasers (VCSELs) and CCD or CID array receivers. In space, this provides light-speed transmission times. The inter-node distances are determined by the power collection and heat radiator sizes and may range from less than a meter to 100's of km for nancomputers operating at the ~105 W limit proposed by Drexler. While adjacent nodes may have relatively short communication delays (see Table 1), because of the large number of nodes, the radii of MBrain layers is that of planetary orbits, so communication from one side of the MBrain to an opposite side may take minutes to hours. This may be decreased somewhat by providing routing nodes, that have shorter path (straight line) communications directly across empty space, but this will be limited for two reasons.
First, even laser communications are subject to beam dispersal across large distances, to provide the necessary signal at the receiver, the transmitter power must be increased (so there is a limit to how many other nodes one may communicate with). One could offset this by providing optical fiber "carriers" between distant nodes, but even if the problems of maintaining such fibers in networks of orbiting MBrain nodes could be solved, the fibers require mass that will be in short supply (since the computers and especially the radiators would require the material that would otherwise be used to construct fibers). In practice, if the communication needs are great enough, one might use high power routing nodes and/or limited material resources for fiber "trunk" lines. These would minimize the propagation delays and costs between distant nodes but not eliminate them.
Second, in a classical MBrain architecture, the nodes are nested, with inner shells having higher operating temperatures and the innermost "shell" being the sun itself. Since these shells are opaque, one cannot transmit "line-of-sight" directly across a shell. One can only transmit to those nodes that are unobstructed by the next inner shell. So the communications must "hop" around the circumference of each layer along "line-of-sight" paths, or be routed through inner layers. Which of these is faster will depend largely on communications priorities and routing traffic. An optimal architecture would probably have high priority messages that go through inner nodes at priorities above normal inner node traffic, while lower priority messages would go point-to-point through the routers of the outer layer.
In nanocomputers and MBrains, the processor overhead (o) will be low (or effectively "non-existent", if dedicated communications sub-processors are used) and the latency parameter (L) dominates communications costs. In MBrain architectures that are thermodynamically constrained by heat radiation requirements, the coolest, outermost nodes must have greater inter-node distances, and so the latency in communications must increase. As previously discussed, this will require increased amounts of energy or mass to effectively execute a transmission that is easily accomplished by nodes that are closer together in inner layers. Greater energy may be made available but that runs into heat dissipation problems that require increasing the radii of the MBrain layers that increases the inter-node latency. More material may be made available, for say inter-node fibers, but fibers are subject to losses that currently require rare elements (e.g. erbium) for optical power amplifiers. Such materials may be harvested from remote solar systems or stars or are manufactured locally in breeder reactors from more abundant elements. Both remote harvesting and local breeding are subject to energy costs and time delays. The element requirements may be bypased by the use of electro-optical amplifier nodes, but this places one back in the realm of increased power and heat dissipation requirements.
Vitanyi [1988], has studied the problem of inter-node communications costs at the theoretical level in a variety of architectures (n-cubes, cube-connected cycles, edge-symmetric graphs and binary trees) and summarizes the findings with the statement, "realistic models for non-sequential computation should charge extra for communications, in terms of time and space". He suggests that even if we had greater than 3 dimensions in which to construct inter-node interconnects, they would be increasingly less efficient.
As MBrains become larger, their communications costs increase and their message throughput and/or bandwidth decreases. At some point the marginal benefit (e.g. an additional 10-20 in thought capacity) does not justify the expense of design time, energy or materials required to add that additional thought capacity.