In this paper, we investigate the effect of bursty traffic and random availability of caching helpers in a wireless caching system. More explicitly, we consider a general system consisting of a caching helper with its dedicated user in proximity and another non-dedicated user requesting for content. Both the non-dedicated user and the helper have limited storage capabilities. When the user is not able to locate the requested content in its own cache, then its request shall be served either by the caching helper or by a large data center. Assuming bursty request arrivals at the caching helper from its dedicated destination, its availability to serve other users is affected by the request rate, which will further affect the system throughput and the delay experienced by the non-dedicated user. We characterize the maximum weighted throughput and the average delay per packet of the considered system, taking into account the request arrival rate of the caching helper, the request probability of the user and the availability of the data center. Our results provide fundamental insights in the throughput and delay behavior of such systems, which are essential for further investigation in larger topologies.