### by Samira Shirzaei and Jeffery Smith (Auburn University)

##### As presented at the 2018 Winter Simulation Conference

We focus on a service system in which the customer arrivals are non-stationary and our goal is to determine a server staffing schedule that ensures that arriving customers do not experience long and/or unpredictable queue times. An airport ticket counter is an example of such a system. Passengers arrivals are nonstationary, yet arriving passengers do not wish to wait in long lines to check into their flights. Moreover, unpredictability is a significant issue in these environments as it often forces passengers to arrive earlier than necessary “just in case.” Unfortunately, we rarely know the precise form of the arrival process and must use observed samples to set the staffing policy. We show through a case study that simulation combined with a specialized input analysis tool can be used to determine good staffing policies in these environments.

## Introduction

Our goal is to optimize a service system operation like a check-in counter in an airport, by focusing on the staffing levels to best control the customers’ waiting times. We assume that passenger arrivals are nonstationary, but we do not know the precise form or parameters of the arrival process. It is clear that if the service rate is significantly larger than the maximum arrival rate, customers will generally experience a small amount of waiting time, but such overcapacity is expensive in terms of resource costs. While the expected queue time is important, we are more interested in the predictability/variance of the queue time as described by Smith and Nelson (2015) and our objective is to determine a resource schedule that can make the waiting times appear practically stationary when the arrivals are non-stationary

Input analysis is arguably one of the most expensive steps in most simulation studies and is essential to a successful simulation (Law 2009). An important step in input modeling is the assessment of data being independently and identically distributed (IID). While this is straightforward when modeling stationary stochastic processes, it becomes more challenging when the stochastic process follows a non-stationary pattern where the probability distribution or its parameters depend on time (Ansari et al. 2014). They proposed the Histogram and Rates for Input Analysis (HistoRIA) as a tool to facilitate input modeling. Smith and Nelson (2015) used a time bucket method for estimating virtual waiting time for the customers arriving to the queue of check-in section of the airport with non-stationary input arrivals. They showed that averages of waiting times within time buckets are more relevant to individual customers than the overall average. One factor that strongly affect this waiting time is number of assigned servers in each time bucket. If we can determine them properly we will be able to control customer’s satisfaction dramatically. It is obvious when we have infinite number of servers the waiting time is in the least amount but the cost of having infinite servers is too much and we are interested to reduce the cost of using them.

We focus on scheduling server levels in our case study with the goal of cost minimization under customer satisfaction constraints. In related work, Feldman et al. (2008) developed a method to determine appropriate staffing levels in call centers with the goal to achieve targeted time-stable performance and they assumed sinusoidal arrival-rate function 𝜆(𝑡). Jennings et al. (1996) considered a multi-server system with general nonstationary arrival and service time process. They developed an approximate procedure based on a time-dependent normal distribution where the mean and variance are determined by infinite-server approximation. Green et al. (2007) reviewed queueing-theory methods for setting staffing requirements in service systems with time-varying customer’s demand. They showed how to adapt stationary queueing models for use in nonstationary environments. Depending on level of targeted quality of service and service times, they discussed which method should be used and how must be modified. Whitt (2007) discussed methods to encounter with time-varying demand to set staffing levels in call centers, he showed when and why, and what to do when each of those methods fail. Izady et al. (2011) tried to set minimal medical staffing levels for reducing patient’s waiting time in the presence of complexities like time-varying demand, multiple types of patients, and resource sharing. Their proposed staffing algorithm relies on infinite server networks to compute the resources’ time dependent workload and highlights their ability in modeling complexities like multiple types of customers. These articles assumed arrival rate follows pre-determined distributions, sinusoidal arrival-rate function, but here we have only a set of sample data to work with and do not know the process characteristics. Furthermore, they used some approximations to obtain the appropriate resource schedule such as normal approximation and delay probability approximation and those estimations were affected by the size of parameters but in this paper we are interested to have less limitation in the process and system’s characteristics and perform staffing levels optimization with experimentation in simulation software with closer assumptions to make the model more realistic and observing practically stationary waiting time.