Maintenance of infinite-server service systems subjected to random shocks

• We introduce the repairable infinite-server service model with deteriorating service and controllable repair rate.
• We prove that the expected number of customers in the queue is a convex function of the repair rate.
• We prove that the total expected cost of repair and delay is a unimodal function.
• We give conditions under which the cost function is one of three simple forms.
• We present numerical examples.

Motivated by the need to study traffic incident management, we consider a Markovian infinite server queue that is subjected to randomly occurring shocks. These shocks affect the service of all servers to deteriorate, i.e., increase the service time of all servers, and might also cause other shocks, thus causing further service deterioration. There are a finite number of service levels, zero corresponding to the normal service with the highest service rate and the last level corresponding to the slowest service rate which could even be equal to zero, implying the complete service breakdown. The repair process is performed only at the last level. These types of queues also represent an approximation of multi-server call-centers with deteriorating service. We derive the mean and variance of the stationary number in the system, and show that the mean is convex with respect to the repair rate. Furthermore, we study the optimal repair rate that minimizes the expected long-run average cost incurred due to delay and repairs. We show that the expected total cost per unit time as a function of repair rate is unimodal. We derive conditions under which the cost function is in one of three simple forms, so that the optimum repair rate can easily be obtained. Numerical examples are also provided.