Queue Inference Research Articles

Robust Queue Inference from Waiting Times

Modeling and decision making for queueing systems have been one of fundamental topics in operations research. For these problems, uncertainty models are established by estimation of key parameters such as expected interarrival and service times. In practice, however, their distributions are unknown, and decision makers are only given historical records of waiting times, which contain relevant but indirect information on the uncertainties. Their complex temporal dependence on the queueing dynamics and the absence of distributional information on the model primitives render estimation of queueing systems remarkably challenging. In the paper “Robust Queue Inference from Waiting Times” by Chaithanya Bandi, Eojin Han, and Alexej Proskynitopoulos, a new inference framework based on robust optimization is proposed to estimate unknown service times from waiting time observations. This new framework allows data-driven, distribution-free estimation on unknown model primitives by solving tractable optimization problems.

A survey of parameter and state estimation in queues

We present a broad literature survey of parameter and state estimation for queueing systems. Our approach is based on various inference activities, queueing models, observations schemes, and statistical methods. We categorize these into branches of research that we call estimation paradigms. These include: the classical sampling approach, inverse problems, inference for non-interacting systems, inference with discrete sampling, inference with queueing fundamentals, queue inference engine problems, Bayesian approaches, online prediction, implicit models, and control, design, and uncertainty quantification. For each of these estimation paradigms, we outline the principles and ideas, while surveying key references. We also present various simple numerical experiments. In addition to some key references mentioned here, a periodically updated comprehensive list of references dealing with parameter and state estimation of queues will be kept in an accompanying annotated bibliography.

An order-based method for robust queue inference with stochastic arrival and departure times

The internal structure and parameters of a queue are completely unobservable in some military and competitive commercial applications. Furthermore, arrival and departure times may be observable but subject to substantial uncertainty due to measurement error in an adversarial environment. This analysis estimates the number of servers in an internally unobservable, first-come, first-served GI/G/c queue using a novel, order-based approach. This new approach provides a lower bound and converges in probability to the correct value. Compared to the standard variance minimization method, the order-based approach has improved performance for small samples. The order-based method is robust to noise in arrival and departure time measurements, while the variance minimization approach exhibits poor performance with noisy data. An extension to last-come, first-served GI/G/c queues is also considered. The last-come, first-served order-based approach also provides a lower bound that converges in probability to the correct value. The empirical performance of the last-come, first-served order-based approach is statistically similar to the performance of the last-come, first-served variance minimization approach with respect to the number of observations required for approximate convergence.

Remarks on queue inference from departure data alone and the importance of the queue inference engine

Recently a generalization of Larson’s Queue Inference Engine (Q.I.E.) for computing conditional expected queue length was proposed in this journal. Although the conditional expectation calculation is correct and of independent interest, it does not represent the queuing inference quantities claimed. We demonstrate this with two examples with identical mean service times where the expectations claimed differ. If customer departure times are included, the examples demonstrate that queue inference cannot be made from departure data alone with unknown service distribution.

Queue inference from periodic reporting data

We introduce new algorithms for queue inference problems involving periodic reporting data. We assume that in each of several consecutive time periods, new customer arrivals follow a Poisson process with period-specific arrival rates, but only the number of departing customers during each period is observed. Rather than exploiting relations with order statistics, our algorithms rely on direct recursions involving Poisson probabilities. We illustrate with several examples, some of which are not typically viewed as queueing applications.

Inferring Balking Behavior From Transactional Data

Balking is the act of not joining a queue because the prospective arriving customer judges the queue to be too long. We analyze queues in the presence of balking, using only the service start and stop data utilized in Larson's Queue Inference Engine (Q.I.E.). Using an extension of Larson's congestion probability calculation to include balking we present new maximum likelihood, nonparametric, and Bayesian methods for inferring the arrival rate and balking functions. The methodology is applicable to businesses that wish to estimate lost sales because of balking arising from queuing-type congestion. The techniques are applied to a small transactional data set for illustrative purposes.

Order statistics applications to queueing and scheduling problems

We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We derive simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n, we show that the variance of the total penalty costs of the EDD is linear in n. The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)).

Efficient Computation of Probabilities of Events Described by Order Statistics and Applications to Queue Inference

This paper derives recursive algorithms for efficiently computing event probabilities related to order statistics and applies the results in a queue inferencing setting. Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. Both a forward and a backward recursive O(N3) algorithm are developed for computing the probability that the order statistics vector lies in a given N-rectangle. The new algorithms have applicability in inferring the statistical behavior of Poisson arrival queues, given only the start and stop times of service of all N customers served in a period of continuous congestion. The queue inference results extend the theory of the “Queue Inference Engine” (QIE), originally developed by Larson in 1990 (Larson, R. C. 1990. The queue inference engine: Deducing queue statistics from transactional data. Management Sci. 36(5) 586–601, and 1991. The queue inference engine: Addendum. Management Sci. 37(6) 1062.). The methodology is extended to a third O(N3) algorithm, employing both forward and backward recursion, that computes the conditional average probability (averaged over all N customers in a congestion period) that the in-queue wait is less than t minutes, given the departure time data and assuming first come, first served service. To our knowledge, this result is the first O(N3) exact algorithm for computing points on the in-queue waiting time distribution function, conditioned on the start and stop time data. The paper concludes with an extension to the computation of certain correlations of in-queue waiting times. Illustrative computational results are included throughout. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.

Random Walk and the Area Below its Path

A simple random walk with reflected origin is considered. The walk starts at the origin and it must return to the origin at time 2n. We show that the expected area below the path of this walk is n22n−1/Cn2n. If, however, the walk is required to return to the origin for the first time at time 2n, then the expected area below the path of this Bernoulli excursion is (2n − 1)22n−1/Cn2n. We also show that if V1 < ⋯ < Vn is the order statistics based on a sample of size n from a uniform distribution over (0, 1), and that if U1 < ⋯ < Un is another independent set of order statistics from the same distribution, then E [∑i=1n |Vi − Ui|] = n22n−1/[(2n + 1)Cn2n] We use this result to find an average-case performance of the Earliest Due Date (EDD) heuristic for one machine scheduling problem with earliness and tardiness penalties. We also apply some of the results to Larson's Queue Inference Engine (1990).

Some Pitfalls of Black Box Queue Inference: The Case of State-Dependent Server Queues

In several queueing systems the service rate of a server is affected by the work load present in the system. For example, a teller at a bank or a checker at a check-out counter in a supermarket may change the service rate depending on the number of customers present in the system. But the service rate as a function of the number in the system can rarely be measured. Consequently, in a typical model of such a system it is assumed that the service rate is constant. Hence, such systems with a single stage are often modeled by GI/GI/c queueing systems with mutually independent arrival and service processes. Then the observed service times are used to find a sample distribution that will represent the distribution of the assumed i.i.d. service times. The purpose of this paper is to explore the effect of this black box queue inference (BBQI) in its ability to predict the performance of the actual system. In this regard, we have shown that when the arrival process is Poisson, if the servers react favorably [unfavorably] to higher work loads (i.e., if the server increases [decreases] the service rate as the number of customers in the system increases) then the BBQI predictions will be pessimistic [optimistic]. This result can be used to identify the server's attitude toward higher work load.

Deducing Queueing from Transactional Data: The Queue Inference Engine, Revisited

R. Larson proposed a method to statistically infer the expected transient queue length during a busy period in O(n5) solely from the n starting and stopping times of each customer's service during the busy period and assuming the arrival distribution is Poisson. We develop a new O(n3) algorithm which uses these data to deduce transient queue lengths as well as the waiting times of each customer in the busy period. We also develop an O(n) on-line algorithm to dynamically update the current estimates for queue lengths after each departure. Moreover, we generalize our algorithms for the case of a time-varying Poisson process and also for the case of i.i.d. interarrival times with an arbitrary distribution. We report computational results that exhibit the speed and accuracy of our algorithms.

The Queue Inference Engine: Addendum

A simple modification of one equation in Larson (1990) reduces the computational complexity of the algorithm for the Queue Inference Engine (QIE) from O(N5) to O(N3), where N is the number of customers queued during a congestion period.

The Queue Inference Engine: Deducing Queue Statistics from Transactional Data

The transactional data of a queueing system are the recorded times of service commencement and service completion for each customer served. With increasing use of computers to aid or even perform service one often has machine readable transactional data, but virtually no information about the queue itself. In this paper we propose a way to deduce the queueing behavior of Poisson arrival queueing systems from only the transactional data and the Poisson assumption. For each congestion period in which queues may form (in front of a single or multiple servers), the key quantities obtained are mean wait in queue, time-dependent mean number in queue, and probability distribution of the number in queue observed by a randomly arriving customer. The methodology builds on arguments of order statistics and usually requires a computer to evaluate a recursive function. The results are exact for a homogeneous Poisson arrival process (with unknown parameter) and approximately correct for a slowly time varying Poisson process.