This paper develops estimators for market penetration level and arrival rate in finding queue lengths from probe vehicles at isolated traffic intersections. Closed-form analytical expressions for expectations and variances of these estimators are formulated. Derived estimators are compared based on squared error losses. Effect of number of cycles (i.e., short-term and long-term performances), estimation at low penetration rates, and impact of combinations of derived estimators on queue length problem are also addressed. Fully analytical formulas with unknown parameters are derived to evaluate how queue length estimation errors change with respect to percent of probe vehicles in the traffic stream. Developed models can be used for the real-time cycle-to-cycle estimation of the queue lengths by inputting some of the fundamental information that probe vehicles provide (e.g., location, time, and count). Models are evaluated using VISSIM microscopic simulations with different arrival patterns. Numerical experiments show that the developed estimators are able to point the true arrival rate values at 5% probe penetration level with 10 cycles of data. For low penetrations such as 0.1%, large number of cycles of data is required by arrival rate estimators which are essential for overflow queue and volume-to-capacity ratios. Queue length estimation with tested parameter estimators is able to provide cycle-to-cycle errors within ±5% of coefficient of variations with less than 5 cycles of probe data at 0.1% penetration for all arrival rates used. 
Queue length estimation from probe vehicles at isolated intersections: Estimators for primary parameters