  We study the problem of strategic choice of arrival time to a single-server
queue with opening and closing times when there is uncertainty regarding
service speed. A Poisson population of customers choose their arrival time with
the goal of minimizing their expected waiting times and are served on a
first-come first-served basis. There are two types of customers that differ in
their beliefs regarding the service time distribution. The inconsistent beliefs
may arise from randomness in the server state along with noisy signals that
customers observe. Customers are aware of the two types of populations with
differing beliefs. We characterize the Nash equilibrium dynamics for
exponentially distributed service times and show how they substantially differ
from the model with homogeneous customers. We further provide an explicit
solution for a fluid approximation of the game. For general service time
distributions we provide an algorithm for computing the equilibrium in a
discrete time setting. We find that in equilibrium customers with different
beliefs arrive during different (and often disjoint) time intervals. Numerical
analysis further shows that the mean waiting time increases with the
coefficient of variation of the service time. Furthermore, we present a
learning agent based model (ABM) in which customers make joining decisions
based solely on their signals and past experience. We numerically compare the
long-term average outcome of the ABM with that of the equilibrium and find that
the arrival distributions are quite close if we assume (for the equilibrium
solution) that customers are fully rational and have knowledge of the system
parameters, while they may greatly differ if customers have limited information
or computing abilities.
