Search

In this paper, we apply a supervised machine-learning approach to solve a fundamental problem in queueing theory: estimating the transient distribution of the number in the system for a G(t)/GI/1. We develop a neural network mechanism that provides a fast and accurate predictor of these distributions for moderate horizon lengths and practical settings. It is based on using a Recurrent Neural Network (RNN) architecture based on the first several moments of the time-dependant inter-arrival and the stationary service time distributions; we call it the Moment-Based Recurrent Neural Network (RNN) method (MBRNN ). Our empirical study suggests MBRNN requires only the first four inter-arrival and service time moments. We use simulation to generate a substantial training dataset and present a thorough performance evaluation to examine the accuracy of our method using two different test sets. We show that even under the configuration with the worst performance errors, the mean number of customers over the entire timeline has an error of less than 3%. While simulation modeling can achieve high accuracy, the advantage of the MBRNN over simulation is runtime, while the MBRNN analyzes hundreds of systems within a fraction of a second. This paper focuses on a G(t)/GI/1; however, the MBRNN approach demonstrated here can be extended to other queueing systems, as the training data labeling is based on simulations (which can be applied to more complex systems) and the training is based on deep learning, which can capture very complex time sequence tasks. In summary, the MBRNN can potentially revolutionize our ability to perform transient analyses of queueing systems.

In this paper, we analyze how well a machine can solve a general problem in queueing theory. To answer this question, we use a deep learning model to predict the stationary queue-length distribution of an $M/G/1$ queue (Poisson arrivals, general service times, one server). To the best of our knowledge, this is the first time a machine learning model is applied to a general queueing theory problem. We chose $M/G/1$ queue for this paper because it lies "on the cusp" of the analytical frontier: on the one hand exact solution for this model is available, which is both computationally and mathematically complex. On the other hand, the problem (specifically the service time distribution) is general. This allows us to compare the accuracy and efficiency of the deep learning approach to the analytical solutions. The two key challenges in applying machine learning to this problem are (1) generating a diverse set of training examples that provide a good representation of a "generic" positive-valued distribution, and (2) representations of the continuous distribution of service times as an input. We show how we overcome these challenges. Our results show that our model is indeed able to predict the stationary behavior of the $M/G/1$ queue extremely accurately: the average value of our metric over the entire test set is $0.0009$. Moreover, our machine learning model is very efficient, computing very accurate stationary distributions in a fraction of a second (an approach based on simulation modeling would take much longer to converge). We also present a case-study that mimics a real-life setting and shows that our approach is more robust and provides more accurate solutions compared to the existing methods. This shows the promise of extending our approach beyond the analytically solvable systems (e.g., $G/G/1$ or $G/G/c$).

We consider a queueing facility where customers decide when to arrive. All customers have the same desired arrival time (w.l.o.g.\ time zero). There is one server, and the service times are independent and exponentially distributed. The total number of customers that demand service is random, and follows the Poisson distribution. Each customer wishes to minimize the sum of three costs: earliness, tardiness and waiting. We assume that all three costs are linear with time and are defined as follows. Earliness is the time between arrival and time zero, if there is any. Tardiness is simply the time of entering service, if it is after time zero. Waiting time is the time from arrival until entering service. We focus on customers' rational behaviour, assuming that each customer wants to minimize his total cost, and in particular, we seek a symmetric Nash equilibrium strategy. We show that such a strategy is mixed, unless trivialities occur. We construct a set of equations that its solution provides the symmetric Nash equilibrium. The solution is a continuous distribution on the real line. We also compare the socially optimal solution (that is, the one that minimizes total cost across all customers) to the overall cost resulting from the Nash equilibrium.

We consider an M/G/1 queue in which the customers, while waiting in line, may renege from it. We study the Nash equilibrium profile among customers, and show that it is defined by two sequences of thresholds. For each customer, the decision is based on the observed past (which determines from what sequence the threshold is taken), and the observed queue length (which determines the appropriate element in the chosen sequence). We construct the a of equations that has the Nash equilibrium as its solution, and discuss the relationships between the properties of the service time distribution and the properties of the Nash equilibrium, such as uniqueness and finiteness.