Optimal dynamic service rate control in time dependent M/M/S/N queues

In this paper, deterministic optimal control theory is used to find the time dependent optimal service rate in an 5-server, finite capacity (N), markovian queue (M/M/S/A). Chapman-Kolmogorov differential equations are used as the state equations of the control problem with N +1 state variables and one control variable. The objective to be minimized is the cost of waiting customers plus the cost of service over a specified time interval. A final time penalty cost of deviations from a desired expected queue length is also included in the objective. Optimal dynamic service rate is found by using Pontryagin's minimum principle which gives rise to a two point boundary value problem and is solved numerically by applying the Newton-Raphson boundary iteration. An example illustrates the results.