Macroparticle tracking is a prominent method for studying the collective beam instabilities in accelerators. However, the heavy computationalload often limits the capabilities of tracking codes. One widely used macroparticle tracking code for simulating collective instabilities in storage rings is mbtrack. The Message Passing Interface (MPI) has already been implemented in mbtrack to accelerate the simulations. However, many Central Processing Unit (CPU) threads are requested in mbtrack for the analyses of the coupled-bunch instabilities. Therefore, computer clusters or desktops with many CPU cores are needed. Since these are not always available, we employ a Graphics Processing Unit (GPU) with a CUDA programming interface as an alternative to run such simulations in a stand-alone workstation. All the heavy computations have been moved to the GPU. The benchmarks confirm that mbtrack-cuda can be used to analyze coupled-bunch instabilities of 484 bunches. Compared to mbtrack on an 8-core CPU, a 36-core CPU and a cluster, mbtrack-cuda is faster for simulations of up to 3 bunches. For 363 bunches, mbtrack-cuda requires approximately six times the execution time of the cluster and twice that of the 36-core CPU. The multibunch instability analysis demonstrates that the length of the ion-cleaning gap does not have a substantial influence, at least at ¾ filling.