This paper develops methods to determine appropriate staffing levels in call centers and other many-server queueing systems with time-varying arrival rates. The goal is to achieve targeted time-stable performance, even in the presence of significant time variation in the arrival rates. The main contribution is a flexible simulation-based iterative-staffing algorithm (ISA) for the M t/G/s t + G model--with nonhomogeneous Poisson arrival process (the M t) and customer abandonment (the + G). For Markovian M t/M/s t + M special cases, the ISA is shown to converge. For that M t/M/s t + M model, simulation experiments show that the ISA yields time-stable delay probabilities across a wide range of target delay probabilities. With ISA, other performance measures--such as agent utilizations, abandonment probabilities, and average waiting times--are stable as well. The ISA staffing and performance agree closely with the modified-offered-load approximation, which was previously shown to be an effective staffing algorithm without customer abandonment. Although the ISA algorithm so far has only been extensively tested for M t/M/s t + M models, it can be applied much more generally--to M t/G/s t + G models and beyond.contact centers, call centers, staffing, nonstationary queues, queues with time-dependent arrival rates, many-server queues, capacity planning, queues with abandonment, time-varying Erlang models

We consider a multiserver service system with general nonstationary arrival and service-time processes in which s(t), the number of servers as a function of time, needs to be selected to meet projected loads. We try to choose s(t) so that the probability of a delay (before beginning service) hits or falls just below a target probability at all times. We develop an approximate procedure based on a time-dependent normal distribution, where the mean and variance are determined by infinite-server approximations. We demonstrate that this approximation is effective by making comparisons with the exact numerical solution of the Markovian M t/M/s t model.operator staffing, queues, nonstationary queues, queues with time-dependent arrival rates, multiserver queues, infinite-server queues, capacity planning

We establish some general structural results and derive some simple formulas describing the time-dependent performance of the Mt/G/∞ queue (with a nonhomogeneous Poisson arrival process). We know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. Our results show how the time-dependent mean function m depends on the time-dependent arrival-rate function λ and the service-time distribution. For example, when λ is quadratic, the mean m(t) coincides with the pointwise stationary approximation λ(t)E[S], where S is a service time, except for a time lag and a space shift. It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/∞ model; the time-dependent mean function m depends on the service-time distribution beyond its mean. The service-time stationary-excess distribution plays an important role. When λ is decreasing before time t, m(t) is increasing in the service-time variability, but when λ is increasing before time t, m(t) is decreasing in service-time variability. We suggest using these infinite-server results to approximately describe the time-dependent behavior of multiserver systems in which some arrivals are lost or delayed.

The purpose of this paper is to develop a unified theory of stochastic ordering for Markov processes on countable partially ordered state spaces. When such a space is not totally ordered, it can induce a wide range of stochastic orderings, none of which are equivalent to sample path comparisons. Similar comparison theorems are also developed for non-Markov processes that are functions of Markov processes and for time-inhomogeneous Markov processes. Such alternative orderings can be quite useful when analyzing multi-dimensional stochastic models such as queueing networks.

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