... can't recover from many years of lost probability theory knowledge (college so far out), while confronted with a real engineering issue, meant to lead to a particular choice of connectivity for a data center I'm building:
Given a set of "11" flows of data, of 10 Gbps each, coming simultaneously into a leaf switch, into its intake/downlink ports, what is the probability of 100% throughput at the other end (outgoing/uplink ports), if the outgoing port sizes are as follows:
1. 20 outgoing ports of 10Gbps/ea (i.e. same ports capacity as intake)
or
2. 5 outgoing ports of 40Gbps/ea (i.e. four times outgoing ports capacity, but reduced number of such)
or
3. 2 outgoing ports of 100Gbps/ea (i.e. ten times outgoing ports capacity, but further reduced number of such)
For those not aware of what a leaf switch and/or Gbps are, think of the problem above as a set of simultaneous incoming flows of water, each carrying a particular rate (the same/ea - in such case replacing Gigabits per second with gallons per min/GPM), assumed to enter a device capable of randomly distributing the incoming water out a specific number of channels, with the three types of outgoing flow capacity designated above.
I've seen vendor documentation advertising capabilities ref the above, but with no explanation on how they reached specific numbers, so I am trying to apply them to specific scenarios.
The answer to 1. appears to be easy: 20!/(20^11*9!), which is a "contraction" of (20/20)*(19/20)*...*(10/20), leading to 3% probability for none of the 11 flows going out through an exit port where another flow is going out, but the matter is complicated in the next two because the "size" of the incoming and outgoing "pipes" is different. The "pseudo" description of "2." would be something like: what is the probability that of the 11 simultaneous 10Gbps incoming data flows, none of the outgoing 5 40Gbps ports will see more than 4 simultaneous 10Gbps data attempting to go out ... if I can understand the formula accounting for the "pipe size" modification, then 2, or 3, or variations of such, from the categories above, will be easier (I am also contemplating 25Gbps ports, but that's another story).
TIA!
Given a set of "11" flows of data, of 10 Gbps each, coming simultaneously into a leaf switch, into its intake/downlink ports, what is the probability of 100% throughput at the other end (outgoing/uplink ports), if the outgoing port sizes are as follows:
1. 20 outgoing ports of 10Gbps/ea (i.e. same ports capacity as intake)
or
2. 5 outgoing ports of 40Gbps/ea (i.e. four times outgoing ports capacity, but reduced number of such)
or
3. 2 outgoing ports of 100Gbps/ea (i.e. ten times outgoing ports capacity, but further reduced number of such)
For those not aware of what a leaf switch and/or Gbps are, think of the problem above as a set of simultaneous incoming flows of water, each carrying a particular rate (the same/ea - in such case replacing Gigabits per second with gallons per min/GPM), assumed to enter a device capable of randomly distributing the incoming water out a specific number of channels, with the three types of outgoing flow capacity designated above.
I've seen vendor documentation advertising capabilities ref the above, but with no explanation on how they reached specific numbers, so I am trying to apply them to specific scenarios.
The answer to 1. appears to be easy: 20!/(20^11*9!), which is a "contraction" of (20/20)*(19/20)*...*(10/20), leading to 3% probability for none of the 11 flows going out through an exit port where another flow is going out, but the matter is complicated in the next two because the "size" of the incoming and outgoing "pipes" is different. The "pseudo" description of "2." would be something like: what is the probability that of the 11 simultaneous 10Gbps incoming data flows, none of the outgoing 5 40Gbps ports will see more than 4 simultaneous 10Gbps data attempting to go out ... if I can understand the formula accounting for the "pipe size" modification, then 2, or 3, or variations of such, from the categories above, will be easier (I am also contemplating 25Gbps ports, but that's another story).
TIA!