Many service systems use internet or smartphone app technology to notify customers about their expected waiting times or queue lengths via delay announcements, allowing customers to decide which queue to join. However, in many cases, either the information might be delayed or customers might require time to travel to the queue of their choice, thus causing a lag in information. We model multiple-queue systems through delay differential equations, and study how the delay in information affects the dynamics of the queues. When the delay is sufficiently large, the queues may oscillate indefinitely throughout time. We develop accurate approximations for the amplitude of these oscillations by implementing two numerical methods. The first technique is a classical analytic method that yields a closed-form approximation in terms of the model parameters. The second approximation method is novel, and it uses a statistical technique to deliver highly accurate approximations over a wider range of parameters. The oscillations in queue lengths are generally undesirable both for the service providers and the customers. This motivates us to explore how the delay announcement can be used to limit the oscillations. We show that, in some cases, using information about queue's velocity (the rate at which the queue length is changing) in the delay announcement can eliminate oscillations created by delays in information. We derive a fixed point equation for determining the optimal amount of velocity information that should be used and find closed form upper and lower bounds on its value. When the oscillations cannot be eliminated altogether, we identify the amount of velocity information that minimizes the amplitude of the oscillations. However, we also find that using too much velocity information can create oscillations in the queue lengths that would otherwise be stable. When the delay in information is caused by customers traveling to the queues, the delay may vary from customer to customer. We propose a queueing model that treats the individual's delay not as a constant, but as a random variable drawn from a fixed distribution. This generalized model allows us to identify the properties of dynamics that are independent of the delay distribution, such as the existence and uniqueness of the equilibrium state. However, the stability of the equilibrium and the presence of oscillations depend on the delay distribution. We therefore give an overview of the system's stability region for different common delay distributions, and finally offer a numerical method of approximating the stability region when the distribution is unknown.