Staffing under Taylor's Law: A Unifying Framework for Bridging Square-root and Linear Safety Rules

Staffing rules serve as an essential management tool in service industries to attain target service levels. Traditionally, the square-root safety rule, based on the Poisson arrival assumption, has been commonly used. However, empirical findings suggest that arrival processes often exhibit an ``over-dispersion'' phenomenon, in which the variance of the arrival exceeds the mean. In this paper, we develop a new doubly stochastic Poisson process model to capture a significant dispersion scaling law, known as Taylor's law, showing that the variance is a power function of the mean. We further examine how over-dispersion affects staffing, providing a closed-form staffing formula to ensure a desired service level. Interestingly, the additional staffing level beyond the nominal load is a power function of the nominal load, with the power exponent lying between $1/2$ (the square-root safety rule) and $1$ (the linear safety rule), depending on the degree of over-dispersion. Simulation studies and a large-scale call center case study indicate that our staffing rule outperforms classical alternatives.Comment: 55 page

Product-form solutions for integrated services packet networks and cloud computing systems

We iteratively derive the product-form solutions of stationary distributions of priority multiclass queueing networks with multi-sever stations. The networks are Markovian with exponential interarrival and service time distributions. These solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies of large scale networks with multi-processor shared-memory switches and cloud computing systems with parallel-server stations. Numerical comparisons with existing Brownian approximating model are provided to indicate the effectiveness of our algorithm.Comment: 26 pages, 3 figures, short conference version is reported at MICAI 200

Optimal Rate Scheduling via Utility-Maximization for J-User MIMO Markov Fading Wireless Channels with Cooperation

We design a dynamic rate scheduling policy of Markov type via the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a class of generalized processor-sharing queues living in a random environment, whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain (FS-CTMC). Whereas the scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state and since the closed-form solution for the performance of such a queueing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching (RDRS) model for its measures of performance and justify its asymptotic optimality through deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic and identifying a cost function related to the utility function, which is minimized through minimizing the workload process in the diffusion limit. More importantly, our queueing model includes both J-user multi-input multi-output (MIMO) multiple access channel (MAC) and broadcast channel (BC) with cooperation and admission control as special cases. In these wireless systems, data from the J users in the MAC or data to the J users in the BC is transmitted over a common channel that is fading according to the FS-CTMC. The J-user capacity region for the MAC or the BC is a set-valued stochastic process that switches with the FS-CTMC fading. In any particular channel state, we show that each of the J-user capacity regions is a convex set bounded by a number of linear or smooth curved facets. Therefore our queueing model can perfectly match the dynamics of these wireless systems.Comment: 53 pages, Originally submitted on June 17, 2010; Revised version submitted on December 24, 201