On Continuous-space Embedding of Discrete-parameter Queuing Systems

In simulation-based optimization of queuing systems, the presence of discrete-valued parameters (such as buffer sizes and the number of servers) makes the optimization difficult. We propose a novel technique for embedding such discrete parameters into a continuous space, so that optimization can be performed efficiently using continuous-space methods. Unlike spatial interpolation, our embedding technique is based on a randomization of the simulation model. The interpolated value can be computed using a single simulation of this randomized model, irrespective of the number of parameters. We first study the theoretical properties of such a randomization scheme applied to M/M/1 and Geom/Geom/1 queues. We prove that the randomization produces valid interpolations of the steady-state performance functions with respect to an integer service-time parameter. We then show that such an embedding is possible for more complex queueing networks whose parameters can include deterministic service times, queue capacities, and the number of servers, and empirically demonstrate that the interpolation is well behaved. To demonstrate the utility of the embedding technique, we solve a 6-parameter queuing network optimization problem by embedding the discrete parameters into a continuous space. The technique produces smooth interpolations of the objective function, and a continuous optimizer applied directly over this embedding converges rapidly, producing good solutions. Keywords— Queuing, Simulation, Discrete Optimization, Interpolation 1 ar X iv :1 60 6. 02 90 0v 2 [ cs .P F] 8 J an 2 01 7