We investigate clusters of extremes defined as subsequent exceedances of high thresholds in a Lindley process. The latter is usually used to model the waiting time or the length of a queue in queuing systems. Distributions of the cluster and inter-cluster sizes of the Lindley process are obtained for a given value of the threshold assuming that the process begins from the zero value. An example of a M/M/1 queue and the impact of service and arrival rates on the cluster and inter-cluster distributions are shown.