M/PH/C queue under a congestion-based staffing policy with applications in steel industry operations

To avoid either idle servers or an over-congested situation, we analyse a queueing system with a variable number of servers. Specifically, if the queue length exceeds an upper threshold, all of the servers are serving customers, and if the number of idle servers reaches a threshold (or the number of customers is below a lower threshold), these idle servers are turned off. We call this policy Congestion-Based Staffing (CBS) with two thresholds. Optimising these thresholds under a certain cost structure is the focus of this paper. The key factor in modelling a manufacturing or service system with random service requests by a queueing model is to realistically model the random service times. Although the exponential distribution has been used successfully to model the service times of a call centre, it is not appropriate for manufacturing or service systems. We propose to use a Phase-type (PH) distribution for modelling the service times in a manufacturing system, as it is more flexible and can fit any shape of distribution in theory. Therefore, we build an M/PH/C model with the CBS policy and develop a solution procedure for computing the queue length stationary distribution. Based on this stationary distribution, we investigate a real-world system in the Shanghai Baoshan Iron and Steel Complex. Using the real data and a realistic cost structure, we determine the optimal CBS policy in terms of minimising the operating cost. This policy yields an operating cost that is considerably smaller than the operating cost under a practical policy.