Abstract
<title>Abstract</title> <p>We consider the dynamic scheduling problem in an M/G/$N$ queue with impatient customers and an objective function that takes into account both holding costs and abandonment penalties. Many papers consider this scheduling problem assuming that both the service times and the abandonment times have exponential distributions. Even with these additional assumptions, the exact solution is known only in very few special cases. Near-optimal policies have been produced, e.g., by fluid-scaling methods and utilizing the Whittle index approach. Our aim in this paper is to get rid of these restrictive additional assumptions by allowing general service time distributions and IHR abandonment times. We apply the Whittle index approach to find a reasonable heuristic solution for this tricky problem. A special challenge in this approach is the resulting two-dimensional state space. Our main theoretical achievements are proving that the closed version of the corresponding discrete-time problem is indexable and deriving an explicit expression for the Whittle index. The discrete-time results are utilized to develop the Whittle index policy for the original continuous-time scheduling problem. By numerical simulations, we demonstrate that the developed Whittle index policy systematically outperforms the well-known $c\mu/\theta$-rule.</p>