2. Maximum-likelihood estimation in Mt/G/∞ queueing systems given interval censored data
3. Simulation study
4. Application example
5. Conclusions and discussion
Appendix A. Ranges
Appendix B. Formulations
Mt/G/Infinity queueing systems have been widely used to analyse complex systems, such as telephone call centres, software testing systems, and telecommunication systems. Statistical inferences of performance measures, such as the expected cumulative numbers of arrivals and departures, are indispensable for decision makers in analysing the current scenario, predicting future scenarios, and making cost-effective decisions. In most scenarios, we only obtain interval censored data, namely, counts in fixed time intervals, instead of complete data because we either do not want or are not able to monitor arrivals and departures. We provide a general framework for statistical inference in Mt/G/Infinity queueing systems given interval censored data. A maximum-likelihood estimation (MLE) method is proposed for inferring the arrival rate and service duration. This method is applicable to general forms of the arrival rate functions and general service duration distributions. More importantly, we propose a combination of the bootstrap method and the delta method for inferring the expected cumulative numbers of arrivals and departures. The results of the simulation study demonstrate that the point and interval estimates of the proposed MLE method are satisfactory overall. As the number of intervals increases, the estimates based on the proposed MLE approach the estimates based on MLE with complete data. Our procedure enables estimates to be obtained without the need to keep track of each item, thereby substantially reducing resource consumption for monitoring items and storing data. An application in a software testing system demonstrates that the goodness-of-fit performance of the proposed MLE method is satisfactory.
An Mt/G/∞ queueing system is a relatively simple queueing system that has a nonhomogeneous Poisson arrival process with a time-dependent deterministic arrival rate function λa ≡ λa(t), independent and identically distributed (i.i.d.) service durations that are independent of the arrival process, and infinitely many servers. Time-varying queueing models, including Mt/G/∞ models, are standard models for describing the dynamics of large-scale service systems, such as telecommunication systems, call centres, and healthcare systems, e.g., hospitals (Pender, 2016). Researchers have applied Mt/G/∞ models to service systems, such as telemarketing, police patrol, fire fighting, hospitals, copy machine repairs, and automatic teller machine operations. In these applications, an operating policy was to keep customer delays close to zero—a scenario that is consistent with the use of an infinite-server model (Green & Kolesar, 1998). Mt/G/∞ models have been applied to storage systems to assess the day-by-day adequacy of stock (Crawford, 1977), to analyse stock requirements (Hillestad & Carrillo, 1980), and to evaluate war-readiness spare requirements for aircraft (Crawford, 1981). They also have been applied to software testing programs (Vizarreta et al., 2018; Yang, 1996) and internet traffic systems (Fay, Roueff & Soulier, 2007). The Mt/G/∞ model is the offered load model for wireless and packet network systems, which describes the total packet carrying capacity of the channels or links in a packet network (Malhotra, Dey, van Doorn & Koonen, 2001; Palm, 1943; Singhai, Joshi & Bhatt, 2009).