The clusters in a blockchain computing system can be constructed to be elastic, thus supporting scalable computing and improving energy efficiency. To form an elastic cluster, the service nodes are dynamically divided into the working nodes and the reserved nodes. Specifically, the working nodes are active to meet the computing requirements of workloads, while the reserved nodes are switched to a low-power state for energy saving. Traditionally, workloads are distributed to working nodes in the mode of N-N service queues. But in this mode, the Quality of Service (QoS) of different working nodes may be diverse, because the requirements are various for the accumulated requests in different working nodes. As a result, the overall system capability is not sufficiently utilized, and the overall system QoS is dragged down. In this paper, we propose an N-1 queueing and on-demand resource provisioning method to process workloads in the mode of N-1 service queues. Different from N-N service queues, N-1 service queues prohibit the accumulation of requests in working nodes. Thereby, once there are idle working nodes, waiting requests can immediately be delivered to them. As a result, all the working nodes are sufficiently utilized, and the overall QoS is improved. Accordingly, after using the N-1 service queues, fewer working nodes are enough to meet the same Service Level Agreement (SLA) on same workloads. In addition, by using a resource demand monitor module, our method dynamically readjusts the number of working nodes to match workload demand. Finally, the energy efficiency of an elastic cluster can be measurably improved, due to that fewer working nodes are powered on while the same SLA can be met.