Stochastic Programming of Multiple Channel Service Systems with Deterministic Inflow and Stochastic Service Times

The problem of allocating customers of different types to various channels of a service system is considered. Service times are assumed to be independent identically distributed random variables whose distribution functions depend on the type of customers as well as the service channel. The total loading time of each channel consists of the sum of service times of all customers which were allocated to it and is thus a random variable also. If the loading time of a given channel exceeds (falls short) its nominal capacity, an overtime (idle time) penalty is incurred. Penalties are assumed to be proportional to the time lapse involved. There is also a revenue gain which is proportional to the number of customers served. The objective is to find the optimal allocation of customers to channels, x_ij, such that the expected net gain, revenue minus losses, is maximized.

It is shown that the distribution function of a loading time depends on the choice of the x_ij and hence that, in general, no claims can be made with respect to desirable convexity properties of the objective function. It is further shown that if the service times are assumed to be normally distributed, then the objective function depends also on the means and the variances of the loading times. The mathematical properties of the program are utilized to ascertain that the solution obtained via a suggested algorithm is global. The nonlinear program is reduced to a (possibly iterative) solution of a linear program by using previous results obtained by the first author.