Comments on HOV computation (by Alex Kurzhanskiy on 9 Oct 2008)

[Andy]

1. I am not sure about constraint (4):

My understand of it is to 'ensure the ratio of HOV to SOV be constant
before and after leaving the node'.

In my opinion, a good thing of HOV lane to HOV drivers is that they
can get out from a bottleneck sooner than SOVs because the HOV lane is
reserved for those HOVs even though all other lanes are jammed ... but
constraint(4) seems to prohibit that ...


2. The multiple solutions are inevitable unless the system is
completely supply-driven (i.e. over-saturated). If the system is
demand-driven (i.e. the demand dominates the supply/capacity), drivers
(especially those HOV drivers) can always choose to drive on any link
at any time as soon as they are not constrainted by the traffic
condition. I don't think you can model such 'arbitrary' behavior and
you need to specify a 'default split ratio' (which could be from
measurement or other kinds of estimation) for under-saturated
condition.


3. Following pt 2 above, I think the split ratio should simply be
equal to the default value, it will only be affected when the supply
term kicks in (i.e. when the system or parts of the system get over-
saturated).


4. I could not quite follow procedure for 'partially known A and B':
is the 'j' in the bracket a index for iteration instead for link?
Which mean the two 'j' (one with a bracket, one without) have two
different meanings?


5. I don't think you need to implement any LP solver or procedure for
modeling HOV. Note that CTM is a solution of a LP (or more precisely,
an approximated solution of a set of partial differential equations),
while you don't need to implement any solver at all for CTM.

The LP formulation should simply be used for deriving an analytical
expression for calculating outflow from a node to each of the
downstream links, I believe that you can always come up with an
analytical solution for all cases:

y(i,j) = min (A,B);

in which y represents flow from link i to link j; 'A' is kind of
'demand flow' on upstream link i; and 'B' is kind of 'supply or
capacity' on downstream link j.

Or, at least, you would come up with some sorts of 'if-then' rules to
what values of 'y' that you should take under certain conditions (I
have derived some of those rules for a simpler formulation in the
slides which I circulated last week ... )


6. I don't quite understand HOV why lane would have a different
fundamental diagram (FD) than SOV lanes (on page 4). The lanes should
be physically identical, and the amount of flow should not affect the
FD at all ..


7. In my opinion, we could confine to cases not more complicated than
' 3-by-3' (HOV + SOV + ramp) to (HOV + SOV +ramp).


8. Nie, Y have derived a general expression (with some
approximations..) for n-by-m case in his PhD thesis. Nie's result is
similar to what you have been doing. Nie doesn't require any LP solver
or solution procedure to get the eventual node inflows and outflows.
Jeff Ban and I have studied Nie's work for some time. A note was
posted on:

http://path.berkeley.edu/topl/reports/080125_JBan_AChow__NodeModel.pdf