Determining Back of Queue (Chapters 31 & 18)
We are working on designing an auxiliary southbound through lane (per the NCHRP report 707), and the procedure requires a reliable calculation of the back of queue to calculate. I am calculating the back of queue as described in Chapter 31, but many of the equations are listed in Chapter 18. On page 31-74 (last paragraph), the HCM states, “all flow rate variables are converted to common units of vehicles per second per lane.” Does this only apply to the calculation of Q1? The units for Q2 don’t make sense. Per Equation 31-136 on page 31-81:
The units of d2 are s/veh, Ca is in units of veh/hour, N is in units of lanes. The 3600 is apparently there to convert from hours to seconds, however if you do the unit conversion math, this leaves you with the units of (lane-1). However, the equation gives Q2 in veh/lane.
Furthermore, the d2 calculation (Equation 18-45) doesn’t seem to have units at all. If you complete the unit conversions on this equation, it leaves you with hour/sqrt(veh), and not sec/veh as given in Equation 31-136.
For my application, I am trying to convert everything into veh/sec/lane, as instructed on pg 31-74. I would like to account for a lane utilization of 75%-25% due to the auxiliary through lane. When calculating Q1 I calculate q for the lane with the highest demand as follows:
I am able to follow the units through the Q1 calculation, and am confident in the result. However, Q2 is calculated using lane groups, and mixes hours and seconds in several places. Ca can be in veh/hour, or veh/sec, which should just require removing the 3600 (sec/hour) conversion factor. Equation 18-45 introduces the analysis period in hours. I’m trying to follow my units and make sure this is being calculated accurately, but I’m not sure. As I adjust for units my results are varying widely between 05 vehicles, and >500 vehicles per lane. I can provide my spreadsheet on request.
Kittelson & Associates, Inc.
The comment that "all flow rate variables are converted to common units of vehicles per second per lane" specifically applies to the flow rate variables q and q_g in the Q1 calculation. There are no flow rate variables in Q2. The variables Q1 and Q2 have units of vehicles.
Regarding the observation "However, the equation gives Q2 in veh/lane." The equation in question is based on queuing theory. In queuing theory, the variable d_2 has units of "seconds" and represents the average waiting time. (see http://en.wikibooks.org/wiki/Fundamentals_of_Transportation/Queueing) However, d_2 is shown as having units of "seconds/vehicle" for consistency with its use in other equations throughout Chapters 18 and 31..
Regarding the observation "the d2 calculation (Equation 18-45) doesn’t seem to have units at all". Again, the equation in question is based on queuing theory. The term in brackets has units of "1/veh". The term outside the brackets has units of "seconds".
Regarding the comment "I’m trying to follow my units and make sure this is being calculated accurately, but I’m not sure." The variable definitions (and units) associated with Equations 31-136 and 18-45 and related equations are correct as shown in the HCM 2010.
As for the application of NCHRP Report 707, take a look at the spreadsheet posted on this page: http://www.trb.org/Main/Blurbs/166520.aspx.
That page is only available to registered users of the HCM2010.
Registration is free! Please create an account or log in to access Methodological Details, Interpretations & Errata, the Technical Reference Library, and the Applications Guide.