Performance analysis of time-dependent queueing systems: survey and classification

Many queueing systems are subject to time-dependent changes in system parameters, such as the arrival rate or number of servers. Examples include time-dependent call volumes and agents at inbound call centers, time-varying air traffic at airports, time-dependent truck arrival rates at seaports, and cyclic message volumes in computer systems.There are several approaches for the performance analysis of queueing systems with deterministic parameter changes over time. In this survey, we develop a classification scheme that groups these approaches according to their underlying key ideas into (i) numerical and analytical solutions,(ii)approaches based on models with piecewise constant parameters, and (iii) approaches based on mod-ified system characteristics. Additionally, we identify links between the different approaches and provide a survey of applications that are categorized into service, road and air traffic, and IT systems

On the Timing of the Peak Mean and Variance for the Number of Customers in an M(t)/M(t)/1 Queueing System

Revised October 1994This paper examines the time lag between the peak in the arrival rate and the peaks in the mean and variance for the number of customers in an M(t)/M(t)/1l system. We establish a necessary condition for the time at which the peak in the mean is achieved. In cases in which system utilization exceeds one during some period, we show that the peak in the mean occurs after the end of this period

Strong approximations for time-varying infinite-server queues with non-renewal arrival and service processes

In real stochastic systems, the arrival and service processes may not be renewal processes. For example, in many telecommunication systems such as internet traffic where data traffic is bursty, the sequence of inter-arrival times and service times are often correlated and dependent. One way to model this non-renewal behavior is to use Markovian Arrival Processes (MAPs) and Markovian Service Processes (MSPs). MAPs and MSPs allow for inter-arrival and service times to be dependent, while providing the analytical tractability of simple Markov processes. To this end, we prove fluid and diffusion limits for MAP(t)/MSPt/ queues by constructing a new Poisson process representation for the queueing dynamics and leveraging strong approximations for Poisson processes. As a result, the fluid and diffusion limit theorems illuminate how the dependence structure of the arrival or service processes can affect the sample path behavior of the queueing process. Finally, our Poisson representation for MAPs and MSPs is useful for simulation purposes and may be of independent interest.111sciescopu

Approximate solutions for multi-server queuing systems with Erlangian service times and an application to air traffic management

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 209-213).This thesis is concerned with approximations of certain M(t)/G(t)/n(t)/n(t) + q queueing systems. More specifically, we are interested in such systems under very general conditions such as time-varying demand and capacity, and high utilization, including occasional oversaturation. Conditions such as these cannot be addressed with existing methodologies. We focus on M(t)/G(t)/n(t)/n(t) + q systems that can be approximated fairly well by M(t)/E&(t)/n(t)/n(t) + q systems. The latter have a large number of system states, that increase with the system parameters k, n, q and the utilization ratio, and involve complicated state transition probabilities. We propose numerical methods to solve the corresponding Chapman-Kolmogorov equations, exactly and approximately We first describe the exact solution technique of M(t)/Ek(t)/n(t)/n(t) + q queueing systems. Then, we develop two heuristic solution techniques of M(t)/E&(t)/ndt)/n(t) + q queueing systems, and provide the corresponding complete state descriptions. We compare the exact and approximate results to validate our heuristics and to select the heuristic that best approximates the exact results in steady-state and under stationary conditions. We also propose two algorithms to vary the number of servers in the system, since many real-life problems involve such changes in response to variations in demand. Further results using our ELC heuristic show that our practical approach behaves well under nonstationary conditions, including varying capacity, and during the transient period to steady-state. We conclude that our heuristic approach is an excellent alternative for studying and analyzing M(t)/E&(t)/n(t)/n(t)+q models and, as a by-product, many M(t)/G(t)/n(t)/n(t) +q systems that arise in practice. Finally, we present an application of the M(t)/E&(t)/n(t)/n(t) + q queueing model in the context of Air Traffic Management. This model appears to be a reasonable approach to estimating delays and congestion in an en-route sector in the air traffic system and can be used as an important building block in developing an analytical model of the entire Air Traffic Management system.by Marcos Escobar Fernández de la Vega.Ph.D